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1.19.1: Perfect and Real Gases

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.19%3A_Perfect_and_Real_Gases/1.19.1%3A_Perfect_and_Real_Gases

In a description of the properties of gases, the term ‘perfect’ means that there are no intermolecular forces, either attractive or repulsive. The equation of state for \(\mathrm{n}_{\mathrm{j}}\) moles of perfect gas \(\mathrm{j}\) takes the following form where \(\mathrm{R}\) is the Gas Constant, \(8.314 \mathrm{J K}^{-1} \mathrm{~mol}^{-1}\). \[\mathrm{p} \, \mathrm{V}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} \, \mathrm{R} \, \mathrm{T}\]The chemical potential of a perfect gas \(\mu_{j}^{\text {id }}\) at temperature \(\mathrm{T}\) is related to pressure \(\mathrm{p}_{j}\) using equation (b). \[\mu_{\mathrm{j}}^{\mathrm{id}}\left(\mathrm{T}, \mathrm{p}_{\mathrm{j}}\right)=\mu_{\mathrm{j}}^{\mathrm{id}}\left(\mathrm{T}, \mathrm{p}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{p}_{\mathrm{j}} / \mathrm{p}^{0}\right)\]Thus \(\mu_{j}^{\mathrm{id}}\left(\mathrm{T}, \mathrm{p}_{\mathrm{j}}\right)\) is the chemical potential of gas \(j\) at pressure \(\mathrm{p}_{j}\) whereas \(\mu_{\mathrm{j}}^{\mathrm{id}}\left(\mathrm{T}, \mathrm{p}^{0}\right)\) is the corresponding chemical potential at the standard pressure \(\mathrm{p}^{0}\).The ratio \(\left(\mathrm{V}_{\mathrm{j}} / \mathrm{n}_{\mathrm{j}}\right)\) is the molar volume of gas \(j\), \(\mathrm{V}_{\mathrm{mj}}\). Equation (a) describing a perfect gas can be written as follows. \[\mathrm{p}_{\mathrm{j}}^{\mathrm{id}} \, \mathrm{V}_{\mathrm{mj}}=\mathrm{R} \, \mathrm{T}\]No real gas is perfect at all temperatures and pressures although at high temperatures and low pressures the product \(\mathrm{p}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{mj}}\) is arithmetically almost equal to the product, \(\mathrm{R} \, \mathrm{T}\). Generally however equation (c) does not describe real gases. The properties of real gases are described in several ways.In one approach \(\mu_{j}\left(T, p_{j}\right)\) is related to \(\mu_{\mathrm{j}}^{\mathrm{id}}\left(\mathrm{T}, \mathrm{p}^{0}\right)\) using equation (d) where \(\mathrm{f}_{j}\) is the fugacity. \[\mu_{\mathrm{j}}\left(\mathrm{T}, \mathrm{p}_{\mathrm{j}}\right)=\mu_{\mathrm{j}}\left(\mathrm{T}, \mathrm{p}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{f}_{\mathrm{j}} / \mathrm{p}^{0}\right)\]Thus \[\operatorname{limit}\left(\mathrm{p}_{\mathrm{j}} \rightarrow 0\right) \mathrm{f}_{\mathrm{j}}=\mathrm{p}_{\mathrm{j}}\]Another approach uses virial coefficients. Thus pressure \(\mathrm{p}_{j}\) is related to molar volume \(\mathrm{V}_{\mathrm{mj}\) using a power series in the term \(\mathrm{V}_{\mathrm{mj}}\). Thus, \[\mathrm{p}_{\mathrm{j}}=\frac{\mathrm{R} \, \mathrm{T}}{\mathrm{V}_{\mathrm{mj}}}\left[1+\frac{\mathrm{B}}{\mathrm{V}_{\mathrm{mj}}}+\frac{\mathrm{C}}{\mathrm{V}_{\mathrm{mj}}^{2}}+\ldots \ldots\right]\]In the event that a given gas is only slightly imperfect the terms C, D,…. are negligibly small. Then, \[\mathrm{p}_{\mathrm{j}}=\frac{\mathrm{R} \, \mathrm{T}}{\mathrm{V}_{\mathrm{mj}}}\left[1+\frac{\mathrm{B}}{\mathrm{V}_{\mathrm{mj}}}\right]\]At low temperatures \(\mathrm{B}\) tends to be negative but at high temperatures \(\mathrm{B}\) is positive. For equation (a), \[\left[\mathrm{N} \mathrm{m}^{-2}\right] \,\left[\mathrm{m}^{3}\right]=[\mathrm{mol}] \,\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}]\]where \([\mathrm{J}]=[\mathrm{Nm}]\) \[\mu_{\mathrm{j}}^{\mathrm{id}}\left(\mathrm{T}, \mathrm{p}_{\mathrm{j}}\right)=\left[\mathrm{J} \mathrm{mol}^{-1}\right] \quad \mathrm{R} \, \mathrm{T}=\left[\mathrm{J} \mathrm{mol}^{-1} \mathrm{~K}^{-1}\right] \,[\mathrm{K}]=\left[\mathrm{J} \mathrm{mol}^{-1}\right]\] I. Prigogine and R. Defay, Chemical Thermodynamics, transl. D. H. Everett, Longmans Green, London, 1953, chapter 11.This page titled 1.19.1: Perfect and Real Gases is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.19.2: Perfect Gas: The Gas Constant

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.19%3A_Perfect_and_Real_Gases/1.19.2%3A_Perfect_Gas%3A_The_Gas_Constant

Throughout these Topics, the Gas Constant, symbol \(\mathrm{R}\), plays an important role. Here we examine how such an important quantity emerges.An important concept in chemical thermodynamics is the perfect gas. In practice the properties of real gases differ from those of the perfect gas but the concept provides a useful basis for understanding the properties of real gases and by extension the properties of liquid mixtures and solutions. After all, nothing is perfect.The starting point for the analysis is the following equation (see Topic 2500) for the change in thermodynamic energy of a closed system \(\mathrm{dU}\) at temperature \(\mathrm{T}\), pressure \(\mathrm{p}\) and affinity for spontaneous change \(\mathrm{A}\). \[\mathrm{dU}=\mathrm{T} \, \mathrm{dS}-\mathrm{p} \, \mathrm{dV}-\mathrm{A} \, \mathrm{d} \xi\]Then for processes at equilibrium where \(\mathrm{A}\) is zero, \[\mathrm{dU}=\mathrm{T} \, \mathrm{dS}-\mathrm{p} \, \mathrm{dV}\]For one mole of chemical substance \(\mathrm{j}\), equation (b) can be written in the following form. \[\mathrm{dU}_{\mathrm{j}}=\mathrm{T} \, \mathrm{dS}_{\mathrm{j}}-\mathrm{p} \, \mathrm{dV_{ \textrm {j } }}\]Then, \[\mathrm{dS}_{\mathrm{j}}=\frac{\mathrm{dU}_{\mathrm{j}}+\mathrm{p} \, \mathrm{dV} \mathrm{V}_{\mathrm{j}}}{\mathrm{T}}\]The molar isochoric heat capacity \(\mathrm{C}_{\mathrm{Vj}}\) describes the differential dependence of molar thermodynamic energy \(\mathrm{U}_{j}\) on temperature at fixed volume. Thus \[\mathrm{C}_{\mathrm{Vj}}=\left(\partial \mathrm{U}_{\mathrm{j}} / \partial \mathrm{T}\right)_{\mathrm{V}(\mathrm{j})}\]Using equation (d), \[\mathrm{dS}_{\mathrm{j}}=\frac{\mathrm{C}_{\mathrm{v}_{\mathrm{j}}}}{\mathrm{T}} \, \mathrm{dT}+\frac{\mathrm{p}}{\mathrm{T}} \, \mathrm{dV}_{\mathrm{j}}\]The latter equation emerges from an equation expressing the molar entropy of an ideal gas \(j\) as a function of the independent variables \(\mathrm{T}\) and \(\mathrm{V}_{j}\). Thus, \[\mathrm{S}_{\mathrm{j}}=\mathrm{S}_{\mathrm{j}}\left[\mathrm{T}, \mathrm{V}_{\mathrm{j}}\right]\]According to Joules Law. The molar thermodynamic energy of a perfect gas depends only on temperature. Hence from equation (e) the molar isochoric heat capacity \(\mathrm{C}_{\mathrm{Vj}}\) is solely a function of temperature. Therefore equation (f) yields the following two important equations. \[\left(\frac{\partial S_{j}}{\partial T}\right)_{v}=\frac{C_{v_{j}}}{T}\]\[\left(\frac{\partial S_{j}}{\partial V}\right)_{T}=\frac{p}{T}\]According to Boyles Law, the molar volume of gas \(j\) is inversely proportional to the pressure at fixed temperature. Thus \[V_{j}=f(T) / p\]Alternatively \[\mathrm{p}=\mathrm{f}(\mathrm{T}) / \mathrm{V}_{\mathrm{j}}\]Hence using equation (i), \[\left(\frac{\partial S_{j}}{\partial V}\right)_{T}=\frac{f(T)}{T \, V_{j}}\]A calculus condition requires that \[\frac{\partial}{\partial V}\left(\frac{\partial S}{\partial T}\right)=\frac{\partial}{\partial T}\left(\frac{\partial S}{\partial V}\right)\]In other words, \[\frac{\partial\left(\mathrm{C}_{\mathrm{vj}} / \mathrm{T}\right)}{\partial \mathrm{V}}=\frac{\partial(\mathrm{p} / \mathrm{T})}{\partial \mathrm{T}}\]Or, using equation (k), \[\frac{\partial\left(\mathrm{C}_{\mathrm{v}_{\mathrm{j}}} / \mathrm{T}\right)}{\partial \mathrm{V}}=\frac{\partial\left[\mathrm{f}(\mathrm{T}) / \mathrm{T} \, \mathrm{V}_{\mathrm{j}}\right]}{\partial \mathrm{T}}\]But the isochoric heat capacity \(\mathrm{C}_{\mathrm{Vj}}\) is independent of volume. Hence \[\frac{\partial\left(\mathrm{C}_{\mathrm{v}_{\mathrm{j}}} / \mathrm{T}\right)}{\partial \mathrm{V}}=0\]Then, \[\frac{\partial\left[\mathrm{f}(\mathrm{T}) / \mathrm{T} \, \mathrm{V}_{\mathrm{j}}\right]}{\partial \mathrm{T}}=0\]In other words \([\mathrm{f}(\mathrm{T}) / \mathrm{T}]\) must be a constant, conventionally called the Gas Constant with symbol \(\mathrm{R}\). As the name implies \(\mathrm{R}\) is a constant used to describe the properties of all gases. We can therefore rewrite equation (k) as follows (recalling that \(\mathrm{V}_{\mathrm{j}}\) is the molar volume of a perfect gas). \[\mathrm{p} \, \mathrm{V}_{\mathrm{j}}=\mathrm{R} \, \mathrm{T}\]The perfect gas is an artificial chemical substance having defined properties. The link with reality stems from the idea that the properties of real gases approach those of an ideal gas as the pressure is reduced.In addition to the definition given by equation (q), the ideal gas is defined by the following equation which requires that the thermodynamic energy of an ideal gas is independent of volume, being nevertheless a function of temperature. \[\left(\partial \mathrm{U}_{\mathrm{j}} / \partial \mathrm{V}_{\mathrm{j}}\right)_{\mathrm{T}}=0\] I. Prigogine and R Defay, Chemical Thermodynamics, transl. D. H. Everett, Longmans Green, London, 1954, chapter X. Reference 1, page 116. \[\begin{aligned} &\mathrm{R}=8.31450 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1} \\ &\mathrm{R}=\mathrm{N}_{\mathrm{A}} \, \mathrm{k} \end{aligned}\]where \(\mathrm{N}_{\mathrm{A}} =\) Avogadro’s constant and k = Boltzmann’s constant \[\begin{aligned} &\mathrm{k}=1.380658 \times 10^{-23} \mathrm{~J} \mathrm{~K}^{-1} \\ &\mathrm{~N}_{\mathrm{A}}=6.0221367 \times 10^{23} \mathrm{~mol}^{-1} \end{aligned}\] G. N. Lewis and M. Randall, Thermodynamics, McGraw-Hill, 1923, page 63. P. W. Atkins, Concepts in Physical Chemistry, Oxford University Press, Oxford,1995.This page titled 1.19.2: Perfect Gas: The Gas Constant is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.19.3: Real Gases: Liquefaction of Gases

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.19%3A_Perfect_and_Real_Gases/1.19.3%3A_Real_Gases%3A_Liquefaction_of_Gases

In the nineteenth century a major theme in science concerned the properties of gases and their liquefaction. The challenge offered by the liquefaction of gases also prompted the development of thermodynamics and the production of low temperatures. Michael Faraday is noteworthy in this context. In 1823 Faraday had liquefied chlorine, ammonia and sulfur dioxide using a combination of pressure and low temperatures.Based on the observation that the densities of liquids are higher than gases the expectation was that liquefaction of a given gas would follow application of high pressure. However this turned out not to be the case. For example, Newton showed that application of 2790 atmospheres (\(\equiv 2.8 \times 10^{8} \mathrm{~N} \mathrm{~m}^{-2}\)) did not liquefy air. In 1877 Cailletet and Pictet working independently obtained a mist of oxygen by sudden expansion of gas compressed at 300 atmospheres (\(\equiv 3 \times 10^{7} \mathrm{~N} \mathrm{~m}^{-2}\)) and cooled by \(\mathrm{CO}_{2}(\mathrm{s})\). However in developing the background to this subject we turn attention to the work of Joule.A gas, chemical substance \(j\), is held in a closed system. The molar thermodynamic energy \(\mathrm{U}_{j}\) is defined by equation (a) where \(\mathrm{V}_{j}\) is the molar volume and \(\mathrm{T}\), the temperature. \[\mathrm{U}_{\mathrm{j}}=\mathrm{U}_{\mathrm{j}}\left[\mathrm{T}, \mathrm{V}_{\mathrm{j}}\right]\]The complete differential of equation (a) describes the change in \(\mathrm{U}_{\mathrm{j}}, \mathrm{dU}_{\mathrm{j}}\), as a function of temperature and volume. \[\mathrm{dU}_{\mathrm{j}}=\left(\frac{\partial \mathrm{U}_{\mathrm{j}}}{\partial \mathrm{T}}\right)_{\mathrm{v}} \, \mathrm{dT}+\left(\frac{\partial \mathrm{U}_{\mathrm{j}}}{\partial \mathrm{V}_{\mathrm{j}}}\right)_{\mathrm{T}} \, \mathrm{dV}_{\mathrm{j}}\]The molar isochoric heat capacity \(\mathrm{C}_{\mathrm{Vj}}\) is defined by equation (c). \[\mathrm{C}_{\mathrm{Vj}}=\left(\frac{\partial \mathrm{U}_{\mathrm{j}}}{\partial \mathrm{T}}\right)_{\mathrm{V}}\]The first law of thermodynamics relates the change in \(\mathrm{U}_{j}\) to the work done on the system \(\mathrm{w}\) and heat \(\mathrm{q}\) passing from the surroundings into the system. Thus, \[\mathrm{dU}_{\mathrm{j}}=\mathrm{q}+\mathrm{w}\]Hence, \[\mathrm{q}=\left(\frac{\partial \mathrm{U}_{\mathrm{j}}}{\partial \mathrm{T}}\right)_{\mathrm{V}} \, \mathrm{dT}+\left(\frac{\partial \mathrm{U}_{\mathrm{j}}}{\partial \mathrm{V}_{\mathrm{j}}}\right)_{\mathrm{T}} \, \mathrm{dV} \mathrm{V}_{\mathrm{j}}-\mathrm{w}\]The apparatus used by Joule comprised two linked vessels having equal volumes. A tube joining the two vessels included a tap. In an experiment, one vessel was filled with gas \(j\) at a known pressure whereas the second vessel was evacuated. When the tap was opened gas flowed into the second vessel, equalizing the pressure in the two vessels. By flowing into an evacuated vessel the gas did no work because there was no confining pressure; i.e. \(\mathrm{w} =\) zero. The temperature of the gas in the containing vessel fell and that in the originally empty vessel rose by an equal amount. In other words \(\mathrm{dT}\) for the two vessel system is zero. Hence \[\left(\frac{\partial \mathrm{U}_{j}}{\partial \mathrm{T}}\right)_{\mathrm{V}}=0\]The clear hope was that the temperature would fall dramatically leading to liquefaction of the gas. In fact and with the benefit of hindsight the overall change in temperature \(\mathrm{dT}\) was too small to be measured. More sophisticated apparatus would show that \(\left(\frac{\partial \mathrm{U}_{\mathrm{j}}}{\partial \mathrm{T}}\right)_{\mathrm{V}} \neq 0\) because as the gas expands work is done against cohesive intermolecular interaction. It would only be zero for a perfect gas.In a series of famous experiments carried out in an English brewery, Joule and Thomson used an apparatus in which the gas under study passed through a porous plug from high to low pressures. The plug impeded the flow of the gas such that the pressure of the gas on the high pressure side and the pressure of gas on the low pressure side remained constant. It was observed that the temperature of the gas decreased as a consequence of the work done by the gas against intermolecular cohesion.A technological breakthrough was now made. A portion of the cooled gas was re-cycled to cool the gas on the input side. On passing through the plug the temperature of the gas fell to a lower temperature. As this process continues a stage was reached where a fraction of the gas is liquefied.As noted above, the cooling emerges because work is done on expansion of the gases against intermolecular interaction. This is a quite general observation. When the pressure drops the mean intermolecular distance increases with the result that the temperature decreases. However there are exceptions to this generalisation. If the pressure is high the dominant intermolecular force is repulsion. Consequently when the pressure drops, work is done by the repulsive forces increasing the intermolecular distances thereby raising the temperature. The account given here is based on that given by N. K. Adam, Physical Chemistry, Oxford, The Clarendon Press, 1956, chapter III. Thomson ≡ Lord Kelvin J. P. Joule and W. Thomson, Proc. Roy. Soc.,1853,143,3457. G.N. Lewis and M. Randall, Thermodynamics, revised by K. S. Pitzer and L Brewer, McGraw-Hill, 2nd. edn.,1961, New York, pages 47-49.This page titled 1.19.3: Real Gases: Liquefaction of Gases is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.2.1: Affinity for Spontaneous Chemical Reaction

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.02%3A_Affinity_for_Spontaneous_Chemical_Reaction/1.2.01%3A_Affinity_for_Spontaneous_Chemical_Reaction

A given closed system is prepared using ethyl ethanoate(aq) in an alkaline solution. The composition of the system changes spontaneously as a consequence of chemical reaction. The latter is described by the following chemical equation. \[\mathrm{CH}_{3} \mathrm{COOC}_{2} \mathrm{H}_{5}(\mathrm{aq})+\mathrm{OH}^{-}(\mathrm{aq}) \rightarrow \mathrm{CH}_{3} \mathrm{COO}^{-}(\mathrm{aq})+\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(\mathrm{aq})\]At each stage, the extent of chemical reaction is represented by the symbol \(\xi\). The composition of the system varies with time as the reaction proceeds. At any given instant we characterise the rate of chemical reaction by \(\mathrm{d} \xi / \mathrm{dt}\). We also ask ‘why did chemical reaction proceed in this direction?’ The answer is − the chemical reaction is driven in that direction by the affinity for spontaneous change, symbol \(\mathrm{A}\). The affinity \(\mathrm{A}\) for spontaneous chemical reaction is defined by the second law of thermodynamics which states that, \[\mathrm{T} \, \mathrm{dS}=\mathrm{q}+\mathrm{A} \, \mathrm{d} \xi\]\[\text { where } A \, d \xi \geq 0\]This page titled 1.2.1: Affinity for Spontaneous Chemical Reaction is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.2.2: Affinity for Spontaneous Reaction- Chemical Potentials

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.02%3A_Affinity_for_Spontaneous_Chemical_Reaction/1.2.02%3A_Affinity_for_Spontaneous_Reaction-_Chemical_Potentials

For a closed system containing \(\mathrm{k}\) − chemical substances, the differential dependence of Gibbs energy on temperature, pressure and chemical composition, is given by the following equation. \[\mathrm{dG}=-\mathrm{S} \, \mathrm{dT}+\mathrm{V} \, \mathrm{dp}+\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{k}}\left(\frac{\partial \mathrm{G}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{i} \neq \mathrm{j})} \, \mathrm{dn} \mathrm{j}_{\mathrm{j}}\]The condition at constant \(n(i \neq j)\) indicates that the amounts of each \(i\) chemical substance except chemical substance \(j\) is constant. The Gibbs energy of a closed system is a thermodynamic potential function; equation (b). \[\mathrm{dG}=-\mathrm{S} \, \mathrm{dT}+\mathrm{V} \, \mathrm{dp}-\mathrm{A} \, \mathrm{d} \xi\]Here \(\mathrm{A}\) is the affinity for spontaneous chemical reaction producing a change in extent of reaction, \(\mathrm{d} \xi\), in this case a change in composition. Further the chemical potential of chemical substance \(j\), \[\mu_{\mathrm{j}}=\left(\frac{\partial \mathrm{G}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{i} \neq \mathrm{j})}\]Comparison of equations (a) and (b) yields equation(d). \[-A \, d \xi=\sum_{j=1}^{j=k} \mu_{j} \, d n_{j}\]The stoichiometry in a chemical reaction for chemical substance \(j\), \(ν_{j}\) is defined such that \(ν_{j}\) is positive for products and negative for reactants; a mnemonic is ‘P for P’. \[v_{j}=d n_{j} / d \xi\]Hence the affinity for spontaneous change, \[A=-\sum_{j=1}^{j=k} v_{j} \, \mu_{j}\]But at equilibrium, the affinity for spontaneous change \(\mathrm{A}\) is zero. \[\text { Then, } \sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{k}} \mathrm{v}_{\mathrm{j}} \, \mu_{\mathrm{j}}^{\mathrm{eq}}=0\]Equation (g) in terms of its simplicity is misleading. Chemists are experts at assaying a system at equilibrium in order to determine the chemical substances present and their amounts. For example, an assay of a given system yields (for defined temperature and pressure) the amounts of un-dissociated acid \(\mathrm{CH}_{3}\mathrm{COOH}(\mathrm{aq})\), and the conjugate base \(\mathrm{CH}_{3}\mathrm{COO}^{−} (\mathrm{aq})\) and hydrogen ions at equilibrium. We write equation (g) as follows. \[-\mu^{\mathrm{eq}}\left(\mathrm{CH}_{3} \mathrm{COOH} ; \mathrm{aq}\right)+\mu^{\mathrm{eq}}\left(\mathrm{CH}_{3} \mathrm{COO}^{-} ; \mathrm{aq}\right)+\mu^{\mathrm{eq}}\left(\mathrm{H}^{+} ; \mathrm{aq}\right)=0\]Or, representing a balance of chemical potentials, (a useful approach) \[\mu^{\mathrm{eq}}\left(\mathrm{CH}_{3} \mathrm{COOH} ; \mathrm{aq}\right)=\mu^{\mathrm{eq}}\left(\mathrm{CH}_{3} \mathrm{COO}^{-} ; \mathrm{aq}\right)+\mu^{\mathrm{eq}}\left(\mathrm{H}^{+} ; \mathrm{aq}\right)\]The concept of a balance of equilibrium chemical potentials at thermodynamic equilibrium is often the starting point for a description of the properties of closed systems.This page titled 1.2.2: Affinity for Spontaneous Reaction- Chemical Potentials is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.2.3: Affinity for Spontaneous Chemical Reaction- Phase Equilibria

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.02%3A_Affinity_for_Spontaneous_Chemical_Reaction/1.2.03%3A_Affinity_for_Spontaneous_Chemical_Reaction-_Phase_Equilibria

A given system comprises two phases, I and II, both phases comprising i-chemical substances. We consider the transfer of one mole of chemical substance \(j\) from phase I to phase II. The affinity for the transfer is given by equation (a). \[\mathrm{A}_{\mathrm{j}}=\mu_{\mathrm{j}}(\mathrm{I})-\mu_{\mathrm{j}}(\mathrm{II})\]If \(\mu_{j}(\mathrm{II})<\mu_{\mathrm{j}}(\mathrm{I})\), \(\mathrm{A}_{j}\) is positive and the process is spontaneous. If the system is at fixed \(\mathrm{T}\) and pressure, the gradient of Gibbs energy is negative. \[\mathrm{A}_{\mathrm{j}}=-(\partial \mathrm{G} / \partial \xi)_{\mathrm{T}, \mathrm{p}}=\mu_{j}(\mathrm{I})-\mu_{\mathrm{j}}(\mathrm{II})\]We suppose the mole fraction of substance \(–j\) in phases I and II are \(x_{j}(\mathrm{I})\) and \(x_{j}(\mathrm{II})\). We express the chemical potentials as functions of the mole fraction compositions of the two phases. \[\begin{aligned} A_{j}=\mu_{j}^{*}(I)+& R \, T \, \ln \left[x_{j}(I) \, f_{j}(I)\right] \\ &-\mu_{j}^{*}(I I)-R \, T \, \ln \left[x_{j}(I I) \, f_{j}(I I)\right] \end{aligned}\]Here \(\mathrm{f}_{j}(\mathrm{I})\) and \(\mathrm{f}_{j}(\mathrm{II})\) are rational activity coefficients of substance \(j\) in phases \(\mathrm{I}\) and \(\mathrm{II}\) respectively. \[\text { At all } T \text { and p, both } \operatorname{limit}\left(x_{j}(I) \rightarrow 1\right) f_{j}(I)=1\]\[\text { and } \operatorname{limit}\left(\mathrm{x}_{\mathrm{j}} \text { (II) } \rightarrow 1\right) \mathrm{f}_{\mathrm{j}}(\text { II })=1\]\[\text { By definition } \mu_{j}^{*}(\mathrm{II})-\mu_{\mathrm{j}}^{*}(\mathrm{I})=-\mathrm{R} \, \mathrm{T} \, \ln [\mathrm{K}(\mathrm{T}, \mathrm{p})]\]\(\mathrm{K}_{j}(\mathrm{T}, p)\) is a measure of the difference in reference chemical potentials of substance \(j\) in phases \(\mathrm{I}\) and \(\mathrm{II}\). If the two phases are in equilibrium, there is no affinity for substance \(j\) to pass spontaneously between the two phases. At equilibrium, \(\mathrm{A}_{j}\) is zero. Hence from equations (c) and (f), for the non-equilibrium state, \[A_{j}=R \, T \, \ln \left[K_{j}(T, p)\right]+R \, T \, \ln \left[\frac{x_{j}(I) \, f_{j}(I)}{x_{j}(I I) \, f_{j}(I I)}\right]\]\[\frac{A_{j}}{T}=R \, \ln \left[K_{j}(T, p)\right]+R \, \ln \left[\frac{x_{j}(I) \, f_{j}(I)}{x_{j}(I I) \, f_{j}(I I)}\right]\]Equation (g) yields the affinity for chemical substance \(j\) to pass between the phases in a non-equilibrium state. In applications of equation (h), we describe the dependence of \(\left(\mathrm{A}_{j} / \mathrm{T}\right)\) on temperature, pressure and composition of the two phases. In other words we require the general differential of equation (h) which is written in the following form.\[\begin{aligned} d\left(\frac{A_{j}}{T}\right)=R \,\left(\frac{\partial \ln K_{j}(T, p)}{\partial T}\right) \, d T \\ &+R \,\left(\frac{\partial \ln K_{j}(T, p)}{\partial p}\right) \, d p+R \, d \ln \left[\frac{x_{j}(I) \, f_{j}(I)}{x_{j}(I I) \, f_{j}(I I)}\right] \end{aligned}\]For the transfer process described by \(\mathrm{K}_{j}(\mathrm{T}, p)\) we obtain equation (j) where \(\Delta_{\text {trans }} \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{~T}, \mathrm{p})\) and \(\Delta_{\text {trans }} V_{j}^{0}(T, p)\) are the standard enthalpy and volume for transfer for chemical substance \(j\). \[\begin{aligned} \mathrm{d}\left(\frac{\mathrm{A}_{\mathrm{j}}}{\mathrm{T}}\right)=\left(\frac{\Delta_{\text {trans }} \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{~T}, \mathrm{p})}{\mathrm{T}^{2}}\right) \, \mathrm{dT} \\ &-\left(\frac{\Delta_{\text {trans }} \mathrm{V}_{\mathrm{j}}^{0}(\mathrm{~T}, \mathrm{p})}{\mathrm{T}}\right) \, \mathrm{dp}+\mathrm{R} \, \mathrm{d} \ln \left[\frac{\mathrm{x}_{\mathrm{j}}(\mathrm{I}) \, \mathrm{f}_{\mathrm{j}}(\mathrm{I})}{\mathrm{x}_{\mathrm{j}}(\mathrm{II}) \, \mathrm{f}_{\mathrm{j}}(\mathrm{II})}\right] \end{aligned}\]\(\Delta_{\text {trans }} \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{~T}, \mathrm{p})\) and \(\Delta_{\text {trans }} \mathrm{V}_{\mathrm{j}}^{0}(\mathrm{~T}, \mathrm{p})\) are properties of pure chemical substance \(j\); i.e. are not dependent on the composition of phases \(\mathrm{I}\) and \(\mathrm{II}\).Footnotes Consider the freezing of water; \[\text { water }(\lambda) \rightarrow \text { water }(\mathrm{s})\]For this process, \(v\left(\mathrm{H}_{2} \mathrm{O} ; \lambda\right)=-1 ; v\left(\mathrm{H}_{2} \mathrm{O} ; \mathrm{s}\right)=1\)The general rule is — Positive for Products. The affinity for spontaneous change, \[A=-\left(\frac{\partial G}{\partial \xi}\right)_{T, p}=-\sum_{j=1}^{j=i} v_{j} \, \mu_{j}=\mu^{*}\left(H_{2} \mathrm{O} ; \lambda\right)-\mu^{*}\left(H_{2} \mathrm{O} ; \mathrm{s}\right)\]At equilibrium (at fixed \(\mathrm{T}\) and \(p\)), \(\mathrm{A} = 0\). \[\text { Then, } \mu^{*}\left(\mathrm{H}_{2} \mathrm{O} ; \lambda\right)=\mu^{*}\left(\mathrm{H}_{2} \mathrm{O} ; \mathrm{s}\right)\]This page titled 1.2.3: Affinity for Spontaneous Chemical Reaction- Phase Equilibria is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.2.4: Affinity for Spontaneous Reaction- General Differential

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A given closed system is prepared using \(\mathrm{n}_{\mathrm{i}}^{0}\) moles of each chemical substance \(i\). At extent of chemical reaction \(\xi\) the ratio \((\mathrm{A}/\mathrm{T})\) where \(\mathrm{A}\) is the affinity for spontaneous chemical reaction is defined by independent variables, \(\mathrm{T}\), \(p\) and \(\xi\). \[(\mathrm{A} / \mathrm{T})=(\mathrm{A} / \mathrm{T})[\mathrm{T}, \mathrm{p}, \xi]\]The general differential of this equation has the following form. \[\mathrm{d}(\mathrm{A} / \mathrm{T})=\left[\frac{\partial(\mathrm{A} / \mathrm{T})}{\partial \mathrm{T}}\right]_{\mathrm{p}, \xi} \, \mathrm{dT}+\frac{1}{\mathrm{~T}} \,\left[\frac{\partial \mathrm{A}}{\partial \mathrm{p}}\right]_{\mathrm{T}, \xi} \, \mathrm{dp}+\frac{1}{\mathrm{~T}} \,\left[\frac{\partial \mathrm{A}}{\partial \xi}\right]_{\mathrm{T}, \mathrm{p}} \, \mathrm{d} \xi\]Footnote Equation (b) forms the basis of equations describing the dependence of A on T at fixed p and on p at fixed T.This page titled 1.2.4: Affinity for Spontaneous Reaction- General Differential is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.2.5: Affinity for Spontaneous Reaction- Dependence on Temperature

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Using the definition of the Gibbs energy \(\mathrm{G} [=\mathrm{U}+\mathrm{p} \, \mathrm{V}-\mathrm{T} \, \mathrm{S}=\mathrm{H}-\mathrm{T} \, \mathrm{S}]\), we form an equation for the entropy of a closed system. Thus \(\mathrm{T} \, \mathrm{S}=-\mathrm{G}+\mathrm{H}\). The entropy of the closed system is perturbed by a change in composition/organisation, \(\xi\) at fixed \(\mathrm{T}\) and \(p\). \[\text { Then, } T \,\left(\frac{\partial S}{\partial \xi}\right)_{T, p}=-\left(\frac{\partial G}{\partial \xi}\right)_{T, p}+\left(\frac{\partial H}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}\]\[\text { But the affinity for spontaneous reaction, } A=-\left(\frac{\partial G}{\partial \xi}\right)_{T, p}\]A Maxwell equation requires that, \(\left(\frac{\partial \mathrm{S}}{\partial \xi}\right)_{\mathrm{T}_{, \mathrm{p}}}=\left(\frac{\partial \mathrm{A}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \xi}\). \[\text { Hence, } \mathrm{T} \,\left(\frac{\partial \mathrm{A}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \xi}=\mathrm{A}+\left(\frac{\partial \mathrm{H}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}\]Equation (c) is rearranged to yield the following interesting equation. \[\mathrm{A}-\mathrm{T} \,\left(\frac{\partial \mathrm{A}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \xi}=-\left(\frac{\partial \mathrm{H}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}]The affinity for spontaneous change and its dependence on temperature are simply related to the enthalpy of reaction at fixed \(\mathrm{T}\) and \(p\). We exploit this link by considering the derivative \(\mathrm{d}(\mathrm{A} / \mathrm{T}) / \mathrm{dT}\) (at fixed \(p\) and fixed \(\xi\)). \[\mathrm{d}(\mathrm{A} / \mathrm{T}) / \mathrm{dT}=(1 / \mathrm{T}) \,(\mathrm{dA} / \mathrm{dT})-\mathrm{A} / \mathrm{T}^{2}\]\[\text { Hence }\left[\frac{\partial(\mathrm{A} / \mathrm{T})}{\partial \mathrm{T}}\right]_{\mathrm{p}, \xi}=-\frac{1}{\mathrm{~T}^{2}} \,\left[\mathrm{A}-\mathrm{T} \,\left(\frac{\partial \mathrm{A}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \xi}\right]\]\[\text { Using equation }(\mathrm{d}),\left[\frac{\partial(\mathrm{A} / \mathrm{T})}{\partial \mathrm{T}}\right]_{\mathrm{p}, \xi}=\frac{1}{\mathrm{~T}^{2}} \,\left(\frac{\partial \mathrm{H}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}\]The latter equation is an analogue of the Gibbs-Helmholtz Equation relating the change in Gibbs energy to the enthalpy of reaction, \(\left(\frac{\partial H}{\partial \xi}\right)_{T, \mathrm{p}}\). The background to equation (g) is the definition of the dependent variable (\(\mathrm{A} / \mathrm{T}\)) in terms of independent variables, \(\mathrm{T}\), \(p\) and \(\xi\). \[\text { Thus } \quad(\mathrm{A} / \mathrm{T})=(\mathrm{A} / \mathrm{T})[\mathrm{T}, \mathrm{p}, \xi]\]The general differential of the latter equation has the following form. \[\mathrm{d}(\mathrm{A} / \mathrm{T})=\left[\frac{\partial(\mathrm{A} / \mathrm{T})}{\partial \mathrm{T}}\right]_{\mathrm{p}, \xi} \, \mathrm{dT}+\frac{1}{\mathrm{~T}} \,\left[\frac{\partial \mathrm{A}}{\partial \mathrm{p}}\right]_{\mathrm{T}, \xi} \, \mathrm{dp}+\frac{1}{\mathrm{~T}} \,\left[\frac{\partial \mathrm{A}}{\partial \xi}\right]_{\mathrm{T}, \mathrm{p}} \, \mathrm{d} \xi\]But, from equation (e) \[\mathrm{d}(\mathrm{A} / \mathrm{T})=-\left(\mathrm{A} / \mathrm{T}^{2}\right) \, \mathrm{dT}+(1 / \mathrm{T}) \, \mathrm{dA}\]\[\text { Or, } \mathrm{dA}=\mathrm{T} \, \mathrm{d}(\mathrm{A} / \mathrm{T})+(\mathrm{A} / \mathrm{T}) \, \mathrm{dT}\]We incorporate equation (i) for the term (\(\mathrm{A} / \mathrm{T}\)). Thus \[\mathrm{dA}=\left[\frac{1}{\mathrm{~T}} \,\left(\frac{\partial(\mathrm{A} / \mathrm{T})}{\partial \mathrm{T}}\right)_{\mathrm{p}, \xi}+\frac{\mathrm{A}}{\mathrm{T}}\right] \, \mathrm{dT}+\left[\frac{\partial \mathrm{A}}{\partial \mathrm{p}}\right]_{\mathrm{T}, \xi} \, \mathrm{dp}+\left[\frac{\partial \mathrm{A}}{\partial \xi}\right]_{\mathrm{T}, \mathrm{p}} \, \mathrm{d} \xi\]Then using equation (g), \[\mathrm{dA}=\left[\frac{1}{\mathrm{~T}} \,\left(\frac{\partial \mathrm{H}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}+\frac{\mathrm{A}}{\mathrm{T}}\right] \, \mathrm{dT}-\left[\frac{\partial \mathrm{V}}{\partial \xi}\right]_{\mathrm{T}, \mathrm{p}} \, \mathrm{dp}+\left[\frac{\partial \mathrm{A}}{\partial \xi}\right]_{\mathrm{T}, \mathrm{p}} \, \mathrm{d} \xi\]The latter is a general equation for the change in affinity. We rearrange this equation as an equation for a change in extent of reaction. \[\begin{aligned} \mathrm{d} \xi=-\left(\frac{\partial \xi}{\partial \mathrm{A}}\right)_{\mathrm{T}, \mathrm{p}} \,\left[\frac{1}{\mathrm{~T}}\right.&\left.\,\left(\frac{\partial \mathrm{H}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}+\frac{\mathrm{A}}{\mathrm{T}}\right] \, \mathrm{dT}+\left[\frac{\partial \mathrm{V}}{\partial \xi}\right]_{\mathrm{T}, \mathrm{p}} \,\left(\frac{\partial \xi}{\partial \mathrm{A}}\right)_{\mathrm{T}, \mathrm{p}} \, \mathrm{dp} \\ &+\left(\frac{\partial \xi}{\partial \mathrm{A}}\right)_{\mathrm{T}, \mathrm{p}} \, \mathrm{dA} \end{aligned}\]The latter equation has the form of a general differential for the extent of reaction written as, \[\xi=\xi[T, \mathrm{p}, \mathrm{A}]\]\[\text { Or, } \mathrm{d} \xi=\left(\frac{\partial \xi}{\partial \mathrm{T}}\right)_{\mathrm{p}, \mathrm{A}} \, \mathrm{dT}+\left(\frac{\partial \xi}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{A}} \, \mathrm{dp}+\left(\frac{\partial \xi}{\partial \mathrm{A}}\right)_{\mathrm{T}, \mathrm{p}} \, \mathrm{dA}\]Hence from equation (n), \[\left(\frac{\partial \xi}{\partial T}\right)_{\mathrm{p}, \mathrm{A}}=-\left(\frac{\partial \xi}{\partial \mathrm{A}}\right)_{\mathrm{T}, \mathrm{p}} \,\left[\frac{1}{\mathrm{~T}} \,\left(\frac{\partial \mathrm{H}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}+\frac{\mathrm{A}}{\mathrm{T}}\right]\]Equation (q) describes the dependence of extent of reaction on temperature at fixed pressure and affinity for spontaneous reaction. Then from equation (n), \[\left(\frac{\partial \xi}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{A}}=+\left[\frac{\partial \mathrm{V}}{\partial \xi}\right]_{\mathrm{T}, \mathrm{p}} \,\left(\frac{\partial \xi}{\partial \mathrm{A}}\right)_{\mathrm{T}, \mathrm{p}}\]Equation (r) describes the dependence of extent of reaction at fixed temperature and fixed affinity for spontaneous change.This page titled 1.2.5: Affinity for Spontaneous Reaction- Dependence on Temperature is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.2.6: Affinity for Spontaneous Reaction - Dependence on Pressure

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The Gibbs energy of a given closed system is defined by equation (a) where \(\xi\) describes the chemical composition. \[\mathrm{G}=\mathrm{G}[\mathrm{T}, \mathrm{p}, \xi]\]We consider the dependence of Gibbs energy on pressure and extent of reaction at fixed temperature \(\mathrm{T}\). \[\frac{\partial}{\partial p}\left(\frac{\partial \mathrm{G}}{\partial \xi}\right)=\frac{\partial}{\partial \xi}\left(\frac{\partial \mathrm{G}}{\partial \mathrm{p}}\right)\]But volume \(\mathrm{V}=\left(\frac{\partial \mathrm{G}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \xi}\) and affinity \(A=-\left(\frac{\partial G}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}\).Volume \(\mathrm{V}\) and affinity \(\mathrm{A}\) are given by first differentials of the Gibbs energy, \(\mathrm{G}\). \[\text { Then }-\left(\frac{\partial \mathrm{A}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \xi}=\left(\frac{\partial \mathrm{V}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}\]Here \(\left(\frac{\partial \mathrm{V}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}\) is the volume of reaction, being the increase volume accompanying unit increase in extent of reaction, \(\xi\).This page titled 1.2.6: Affinity for Spontaneous Reaction - Dependence on Pressure is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.2.7: Affinity for Spontaneous Reaction - Stability

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.02%3A_Affinity_for_Spontaneous_Chemical_Reaction/1.2.07%3A_Affinity_for_Spontaneous_Reaction_-_Stability

A given closed system at temperature \(T\) and pressure \(p\) undergoes a spontaneous change in chemical composition. Chemical reaction is driven by the affinity for spontaneous change such that the Gibbs energy decreases.Thus\[\mathrm{A}=-\left(\frac{\partial \mathrm{G}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}} \label{a}\]The plot of Gibbs energy \(G\) against composition \(\xi\) shows a gradual decrease until \(\mathrm{G}\) reaches a minimum where the affinity \(\mathrm{A}\) is zero at chemical equilibrium. An imagined plot beyond equilibrium would show an increase in Gibbs energy. In other words spontaneous chemical reaction stops at the point where \(\mathrm{G}\) is a minimum (at fixed \(\mathrm{T}\) and \(p\)). If the chemical reaction stops, the rate of chemical reaction is zero. We link the thermodynamic definition of chemical equilibrium and the definition of chemical equilibrium which emerges from the Law of Mass Action with reference to the kinetics of chemical reaction.An accompanying plot shows a gradually decreasing affinity when plotted against \(\xi\), passing zero at \(\xi_{\mathrm{eq}}\). The gradient of the plot in the neighbourhood of equilibrium is negative;\[\text { i.e. }\left(\frac{\partial \mathrm{A}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}<0 \label{b}\]Equation \ref{b} is the thermodynamic condition for a stable chemical equilibrium. The composition does not change no matter how long we wait. Indeed that is the experience of chemists and Equation \ref{b} expresses quantitatively this observation.One might ask--- how does the system ‘know’ it is at a minimum in Gibbs energy?Within the system, any fluctuation in composition leads to an increase in Gibbs energy. This tendency is opposed spontaneously; i.e. these fluctuations are opposed.This page titled 1.2.7: Affinity for Spontaneous Reaction - Stability is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.2.8: Affinity for Spontaneous Chemical Reaction - Law of Mass Action

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.02%3A_Affinity_for_Spontaneous_Chemical_Reaction/1.2.08%3A_Affinity_for_Spontaneous_Chemical_Reaction_-_Law_of_Mass_Action

The differential change in Gibbs energy of a closed system \(\mathrm{dG}\) is related to the change in chemical composition – organisation using equation (a) where \(\mathrm{A}\) is the affinity for spontaneous chemical reaction such that \(\mathrm{A} \, \mathrm{d} \xi \geq 0\).\[\mathrm{dG}=-\mathrm{S} \, \mathrm{dT}+\mathrm{V} \, \mathrm{dp}-\mathrm{A} \, \mathrm{d} \xi \label{a}\]Spontaneous chemical reaction is driven by the affinity for spontaneous change, \(\mathrm{A}\). Eventually the system reaches a minimum in Gibbs energy \(\mathrm{G}\) where the affinity for spontaneous change is zero. \[\text { In general terms, } \mathrm{A}^{\mathrm{eq}}=-\sum \mathrm{v}_{\mathrm{j}} \, \mu_{\mathrm{j}}^{\mathrm{eq}}=0 \label{b}\]We identify chemical equilibrium as the state where the chemical potentials driving the chemical flow from reactants to products are balanced by the chemical potentials driving chemical flow from products to reactants. The condition given in Equation \ref{b} is based on the first and second laws of thermodynamics. These two laws do not lead to quantitative statements concerning the rate of change of chemical composition; i.e. the dependence of the concentration of reactants and products on time.At a given time \(\mathrm{t}\), the rate of change of composition \(v\) is defined by equation (c).\[\mathrm{v}=\mathrm{d} \xi / \mathrm{dt}\]\[\text { Hence, with } A \, d \xi>0, A \, V>0\]Therefore for chemical reaction in a closed system, the signs of \(\mathrm{A}\) and \(\mathrm{v}\) are identical. Moreover if the system is at thermodynamic equilibrium such that \(\mathrm{A}\) is zero, then \(\mathrm{v}\) is zero. The latter sentence establishes a crucial link between chemical kinetics and chemical thermodynamics. However away from equilibrium we have no information concerning the rate of change of composition. The required property is the ratio \(\mathrm{d} \xi / \mathrm{dt}\) at time \(\mathrm{t}\) characterising the rate of change of the extent of chemical reaction. Intuitively one might argue that the rate depends on the affinity for spontaneous reaction—the greater the affinity for reaction the faster the reaction. The key equation might take the following form.\[\mathrm{d} \xi / \mathrm{dt}=\mathrm{L} \, \mathrm{A}\]Here \(\mathrm{L}\) is a phenomenological parameter, describing the phenomenon of chemical reaction. Unfortunately no further progress can be made because we have no way of measuring the affinity \(\mathrm{A}\) for chemical reaction; no affinity meter is available which we can plunge into the reacting system and ‘read off’ the affinity. In these terms the analysis comes to a halt.In the context of chemical kinetics, the rate of chemical reaction is defined by equation (f) where \(n_{j}\) refers to the amount of chemical substance \(j\) as either product or reactant; positive for product\[\mathrm{v}=\pm \mathrm{V}^{-1} \, \mathrm{dn} \mathrm{n}_{\mathrm{j}} / \mathrm{dt}\]For chemical reaction involving solutes in dilute solution the volume \(\mathrm{V}\) of the system (at fixed \(\mathrm{T}\) and \(p\)) is effectively independent of time.\[\text { Then, } v=\pm \mathrm{dc}_{\mathrm{j}} / \mathrm{dt}\]The kinetics of chemical reactions in solution are simpler than those for reactions in the gas phase and we confine comment to the former.The relationship between velocity \(v\) and the chemical composition of a solution is described by the Law of Mass Action as proposed by Guldberg and Waage in 1867. The developments reported by Harcourt and Essen, Bredig and Stern and by Lapworth in 1904 and by Goldschmidt in 1930 were important. Hammett comments that by the 1930s the subject had emerged from the ‘dark ages’. Hammett draws attention to the contributions made by Bartlett, Ingold and Pedersen in the decades of 1920 and 1930. Effectively these authors showed that the rate of chemical reaction at time \(t\) is a function of the concentrations of substances in the systems at that time, \(t\). In the textbook case the spontaneous chemical reaction between two chemical substances \(\mathrm{X}\) and \(\mathrm{Y}\) at fixed \(\mathrm{T}\) and \(p\) in aqueous solution, has the following form.\[\mathrm{x} . \mathrm{X}(\mathrm{aq})+\mathrm{y} . \mathrm{Y}(\mathrm{aq})->\text { products }\]\[\text { Rate of reaction }=\mathrm{k} \,[\mathrm{X}]^{\alpha} \,[\mathrm{Y}]^{\beta}\]Here \(\alpha\) and \(\beta\) are orders of reaction with respect to substances \(\mathrm{X}\) and \(\mathrm{Y}\). These orders have to be determined from the experimental kinetic data because they do not necessarily correspond to the stoichiometric coefficients in the chemical equation.We develop the argument by considering a chemical reaction in solution. An aqueous solution is prepared containing n1 moles of water(l) and \(\mathrm{n}_{\mathrm{X}}^{0}\) moles of chemical substance \(\mathrm{X}\) at time, \(t = 0\) where \(n_{X}^{0}>>n_{1}\). Spontaneous chemical reaction leads to the formation of product \(\mathrm{Y}\), where at ‘\(t=0\)’, \(\mathrm{n}_{\mathrm{Y}}^{0}\) is zero. Chemical reaction is described using equation (j).\[\begin{array}{llll} & \mathrm{X}(\mathrm{aq}) & \rightarrow & \mathrm{Y}(\mathrm{aq}) \\ \mathrm{At} \mathrm{t}=0 & \mathrm{n}_{\mathrm{X}}^{0} & \mathrm{n}_{\mathrm{y}}^{0}=0 & \mathrm{~mol} \end{array}\]The convention is for the chemical reaction to be written in the form ‘reactants → products’, such that the affinity for reaction \(\mathrm{A}\) is positive and hence \(\mathrm{d}\xi\) and \(\mathrm{d} \xi / \mathrm{dt}\) are positive. Many chemical reactions of the form shown in equation (j) go to completion.\[\text { Thus } \lim \mathrm{it}(\mathrm{t} \rightarrow \infty) \mathrm{n}_{\mathrm{X}}=0 ; \mathrm{n}_{\mathrm{Y}}=\mathrm{n}_{\mathrm{X}}^{0}\]The minimum in Gibbs energy (where \(\mathrm{A} = 0\)) is attained when all reactant has been consumed.A given closed system contains \(\mathrm{n}_{\mathrm{X}}^{0}\) moles of chemical substance \(\mathrm{X}\) which \(\mathrm{X}\) decomposes to form chemical substance \(\mathrm{Z}\). At time \(t\), \(\xi\) moles of reactant \(\mathrm{X}\) have formed product \(\mathrm{Z}\).\[\begin{array}{lcc} \multicolumn{1}{c}{\text { Chemical Reaction }} & \mathrm{X} & \mathrm{Z} \\ \text { Amounts at } \mathrm{t}=0 ; & \mathrm{n}_{\mathrm{X}}^{0} & 0 \mathrm{~mol} \\ \text { Concentrations }(\mathrm{t}=0) & \mathrm{n}_{\mathrm{X}}^{0} / \mathrm{V} & 0 \mathrm{~mol} \mathrm{~m}^{-3} \\ \text { At time } \mathrm{t} & \mathrm{n}_{\mathrm{X}}^{0}-\xi & \mathrm{mol} \\ \text { Amounts } & \left(\mathrm{n}_{\mathrm{X}}^{0}-\xi\right) / \mathrm{V} & \xi / \mathrm{V} \mathrm{mol} \mathrm{m} \end{array}\]The law of mass action is the extra-thermodynamic assumption which relates the rate of change of concentration t o the composition of the system. \[\text {Then, } -\frac{\mathrm{d}\left[\left(\mathrm{n}_{\mathrm{X}}^{0}-\xi\right) / \mathrm{V}\right]}{\mathrm{dt}}=\frac{\mathrm{d}[\xi / \mathrm{V}]}{\mathrm{dt}}=\mathrm{k} \, \frac{\left[\mathrm{n}_{\mathrm{X}}^{0}-\xi\right]}{\mathrm{V}}\]The constant of proportionality, rate constant \(\mathrm{k}\), in this case has units of \(\mathrm{s}^{-1}\).\[\text { Then in terms of reactant } \mathrm{X}, \frac{\mathrm{d} \xi}{\mathrm{dt}}=\mathrm{k} \,\left[\mathrm{n}_{\mathrm{X}}^{0}-\xi\right]\]Chemical reaction proceeds leading to a decrease in the Gibbs energy of the system until \(\mathrm{n}_{\mathrm{X}}^{0}=\xi\) such that all reactant has been consumed. At this point \((\mathrm{d} \xi / \mathrm{dt})\) is zero, the system is at chemical equilibrium and the Gibbs energy is a minimum. Further from equation (m),\[\int_{\xi=0}^{\xi} \frac{d \xi}{\left(n_{X}^{0}-\xi\right)}=\int_{t=0}^{t} k \, d t\]The Law of Mass Action is the most important extra-thermodynamic equation in chemistry.A given closed system contains \(\mathrm{n}_{\mathrm{X}}^{0}\) and \(\mathrm{n}_{\mathrm{Y}}^{0}\) moles of chemical substances \(\mathrm{X}\) and \(\mathrm{Y}\) respectively at fixed \(\mathrm{T}\) and \(p\). Spontaneous chemical reaction produces chemical substance \(\mathrm{Z}\). At time \(t\), \(\xi\) moles of product \(\mathrm{Z}\) are formed from chemical substances \(\mathrm{X}\) and \(\mathrm{Y}\).\[\begin{array}{lccc} {c}{\text { Chemical Reaction }} & \mathrm{X} & +\mathrm{Y} & \mathrm{Z} \\ \text { Amounts at } \mathrm{t}=0 ; & \mathrm{n}_{\mathrm{X}}^{0} & \mathrm{n}_{\mathrm{Y}}^{0} & 0 \mathrm{~mol} \\ \text { Concentrations }(\mathrm{t}=0) & \mathrm{n}_{\mathrm{X}}^{0} / \mathrm{V} & \mathrm{n}_{\mathrm{Y}}^{0} / \mathrm{V} & 0 \mathrm{~mol} \mathrm{~m} \\ \text { At time t } & & & \\ \text { Amounts } & \mathrm{n}_{\mathrm{X}}^{0}-\xi & \mathrm{n}_{\mathrm{Y}}^{0}-\xi & \xi \mathrm{mol} \\ \text { Concentrations } & \left(\mathrm{n}_{\mathrm{X}}^{0}-\xi\right) / \mathrm{V} & \left(\mathrm{n}_{\mathrm{Y}}^{0}-\xi\right) / \mathrm{V} & \xi / \mathrm{V} \mathrm{mol} \mathrm{m} \end{array}\]The Law of Mass Action is the extra-thermodynamic assumption, relating the rate of change of concentration to the composition of the system. Then,\[-\frac{\mathrm{d}\left[\left(\mathrm{n}_{\mathrm{X}}^{0}-\xi\right) / \mathrm{V}\right]}{\mathrm{dt}}=-\frac{\mathrm{d}\left[\left(\mathrm{n}_{\mathrm{Y}}^{0}-\xi\right) / \mathrm{V}\right]}{\mathrm{dt}}=\frac{\mathrm{d}[\xi / \mathrm{V}]}{\mathrm{dt}}=\mathrm{k} \, \frac{\left[\mathrm{n}_{\mathrm{X}}^{0}-\xi\right]}{\mathrm{V}} \, \frac{\left[\mathrm{n}_{\mathrm{Y}}^{0}-\xi\right]}{\mathrm{V}}\]The unit of rate constant \(\mathrm{k}\) is ‘\(\mathrm{m}^{3} \mathrm{~mol}^{-1} \mathrm{~s}^{-1}\)’.\[\text { Then, } \frac{\mathrm{d} \xi}{\mathrm{dt}}=\mathrm{k} \, \mathrm{V}^{-1} \,\left[\mathrm{n}_{\mathrm{X}}^{0}-\xi\right] \,\left[\mathrm{n}_{\mathrm{Y}}^{0}-\xi\right]\]\[\text { Or, } \quad \int_{\xi=0}^{\xi} \frac{\mathrm{d} \xi}{\left[n_{X}^{0}-\xi\right] \,\left[n_{Y}^{0}-\xi\right]}=\int_{t=0}^{t} k \, V^{-1} \, d t\]In applications of equation (q), rate constant \(\mathrm{k}\) and volume \(\mathrm{V}\) are usually treated as independent of time.The foregoing analysis of kinetics of chemical reactions illustrates the application of the variable \(\xi\) in describing the composition of a closed system. Most accounts of chemical kinetics start out with a consideration of concentrations of chemical substances in a given system.Nevertheless for each and every chemical reaction, the form of the relevant ‘Law of Mass Action’ has to be determined from the observed dependence of composition on time. The latter sentence does not do justice to the skills of chemists in this context.In the previous section we considered those cases where chemical reaction goes to completion in that one or more of the reactants are consumed. For many cases this is not the case. Here we imagine that a dilute solution has been prepared using \(\mathrm{n}_{\mathrm{X}}^{0}\) and \(\mathrm{n}_{\mathrm{Y}}^{0}\) moles of solute reactants \(\mathrm{X}\) and \(\mathrm{Y}\). Chemical reaction at fixed \(\mathrm{T}\) and \(p\) proceeds spontaneously. The Gibbs energy of the system decreases reaching a minimum where the affinity for spontaneous reaction is zero. Chemical analysis shows that the resulting system contains product \(\mathrm{Z}\) together with reactants \(\mathrm{X}\) and \(\mathrm{Y}\), and that the chemical composition is independent of time; i.e. chemical kinetic equilibrium. Thus,\[\begin{gathered} \mathrm{X}(\mathrm{aq}) \quad+\quad \mathrm{Y}(\mathrm{aq}) \quad \Leftrightarrow \quad \mathrm{Z}(\mathrm{aq}) \\ \mathrm{At} \mathrm{} \mathrm{t}=0 \quad \mathrm{n}_{\mathrm{X}}^{0} \quad \mathrm{n}_{\mathrm{Y}}^{0} \quad \mathrm{n}_{\mathrm{Y}}^{\mathrm{e}}=\mathrm{n}_{\mathrm{Y}}^{0}-\xi^{\mathrm{eq}} \quad \mathrm{n}_{\mathrm{eq}}^{\mathrm{eq}}=\xi^{\mathrm{eq}} \quad \mathrm{mol} \\ \mathrm{At} \mathrm{} \mathrm{t}=\infty \quad \mathrm{n}_{\mathrm{x}}^{\mathrm{eq}}=\mathrm{n}_{\mathrm{X}}^{0}-\xi^{\mathrm{eq}} \quad \mathrm{n}_{\mathrm{eq}} \\ \text { or, } \quad \mathrm{c}_{\mathrm{X}}^{\mathrm{eq}}=\left(\frac{\mathrm{n}_{\mathrm{X}}^{0}-\xi^{\mathrm{eq}}}{\mathrm{V}}\right) \quad \mathrm{c}_{\mathrm{eq}}^{\mathrm{eq}}=\left(\frac{\mathrm{n}_{\mathrm{Y}}^{0}-\xi^{\mathrm{eq}}}{\mathrm{V}}\right) \quad \mathrm{c}_{\mathrm{Z}}^{\mathrm{eq}}=\left(\frac{\xi^{\mathrm{eq}}}{\mathrm{V}}\right) \end{gathered}\]A number of assumptions are based on the Law of Mass Action. \(\text { At time } t \text {, rate of forward reaction }=k_{f} \, c_{X}(t) \, c_{Y}(t) \text { and rate of the reverse reaction }=\mathrm{k}_{\mathrm{r}} \, \mathrm{c}_{\mathrm{Z}}(\mathrm{t})\)Rate constants \(\mathrm{k}_{\mathrm{f}}\) and \(\mathrm{k}_{\mathrm{r}}\) are initially assumed to be independent of the extent of reaction. A key conclusion is now drawn. Because at ‘\(t \rightarrow\) infinity’, the properties of the system are independent of time, the system is ‘at chemical equilibrium where the rates of forward and reverse reactions are balanced.\[\text { Then, } \mathrm{k}_{\mathrm{f}}^{\mathrm{eq}} \, \mathrm{c}_{\mathrm{X}}^{\mathrm{eq}} \, \mathrm{c}_{\mathrm{Y}}^{\mathrm{eq}}=\mathrm{k}_{\mathrm{r}}^{\mathrm{eq}} \, \mathrm{c}_{\mathrm{z}}^{\mathrm{eq}}\]\[\text { Hence, } \quad \mathrm{d} \xi^{\mathrm{eq}} / \mathrm{dt}=0\]\[\text { and } \mathrm{k}_{\mathrm{eq}}^{\mathrm{eq}} \,\left(\frac{\mathrm{n}_{\mathrm{X}}^{0}-\xi^{\mathrm{eq}}}{\mathrm{V}}\right) \,\left(\frac{\mathrm{n}_{\mathrm{Y}}^{0}-\xi^{\mathrm{eq}}}{\mathrm{V}}\right)=\mathrm{k}_{\mathrm{r}}^{\mathrm{eq}} \, \frac{\xi^{\mathrm{eq}}}{\mathrm{V}}\]At this point we encounter a key problem -- we cannot determine \(\mathrm{k}_{\mathrm{f}}^{\mathrm{eq}\) and \(\mathrm{k}_{\mathrm{r}}^{\mathrm{eq}\) because at equilibrium the composition of the system is independent of time. Nevertheless we can express the ratio of rate constants as a function of the composition at equilibrium.\[\left(\mathrm{k}_{\mathrm{f}}^{\mathrm{eq}} / \mathrm{k}_{\mathrm{r}}^{\mathrm{eq}}\right) \,\left(\frac{\mathrm{n}_{\mathrm{X}}^{0}-\xi^{\mathrm{eq}}}{\mathrm{V}}\right) \,\left(\frac{\mathrm{n}_{\mathrm{Y}}^{0}-\xi^{\mathrm{eq}}}{\mathrm{V}}\right)=\frac{\xi^{\mathrm{eq}}}{\mathrm{V}}\]The ratio \(\mathbf{k}_{\mathrm{f}}^{\mathrm{eq}} / \mathbf{k}_{\mathrm{r}}^{\mathrm{eq}}\) is characteristic of the system (at defined \(\mathrm{T}\) and \(p\)), defining what we might call a ‘Law of Mass Action equilibrium constant’, \(\mathrm{K}(\operatorname{lma})\).\[\text { Thus } \mathrm{K}(\operatorname{lm} a)=\mathrm{k}_{\mathrm{f}}^{\mathrm{eq}} / \mathrm{k}_{\mathrm{r}}^{\mathrm{eq}}\]In other words, the property \(\mathrm{K}(\operatorname{lma})\) is based on a balance of reaction rates whereas the thermodynamic equilibrium constant is based on a balance of chemical potentials. With reference to equation (v), at fixed \(\mathrm{T}\) and \(p\), the thermodynamic condition is given in equation (w).\[\mu_{\mathrm{X}}^{\mathrm{eq}}(\mathrm{aq})+\mu_{\mathrm{Y}}^{\mathrm{eq}}(\mathrm{aq})=\mu_{\mathrm{Z}}^{\mathrm{eq}}(\mathrm{aq})\]\[\text { For } \mathrm{p} \cong \mathrm{p}^{0}, \mathrm{~K}^{0}(\mathrm{~T})=\frac{\left(\mathrm{m}_{\mathrm{Z}} \, \gamma_{\mathrm{Z}} / \mathrm{m}^{0}\right)^{\mathrm{eq}}}{\left(\mathrm{m}_{\mathrm{X}} \, \gamma_{\mathrm{X}} / \mathrm{m}^{0}\right)^{\mathrm{eq}} \,\left(\mathrm{m}_{\mathrm{Y}} \, \gamma_{\mathrm{Y}} / \mathrm{m}^{0}\right)^{\mathrm{eq}}}\]Therefore the question is raised--- how is \(\mathrm{K}(\operatorname{lma})\) related to \(\mathrm{K}^{0}(\mathrm{T})\)? In the absence of further information, a leap of faith by chemists sets \(\mathrm{K}(\operatorname{lma})\) equal to \(\mathrm{K}^{0}(\mathrm{T})\). We avoid debating the meaning of the phrase ‘rate constants at equilibrium’.Spontaneous chemical reaction involving a single solute \(\mathrm{X}(\mathrm{aq}\)) can be described using a rate constant \(\mathrm{k}\) for a solution at fixed \(\mathrm{T}\) and \(p\) where the concentration of solute \(\mathrm{X}\) at time \(t\) is \(\mathrm{c}_{\mathrm{X}}(\mathrm{aq})\).If we can assume that in solution there are no solute-solute interactions, rate constant \(\mathrm{k}\) is not dependent on the composition of the solution. In thermodynamic terms, the thermodynamic properties of solute \(\mathrm{X}\) are ideal; for such a system rate constant \(\mathrm{k}\) is independent of time and initial concentration of solute \(\mathrm{X}\). In other words experiment yields the property \(\mathrm{k}(\mathrm{T}, p)\) for a given solution indicating that the rate constant is a function of temperature and pressure. For nearly all chemical reactions in solution rate constants increase with increase in temperature, a dependence described by the Arrhenius equation.\[\mathrm{k}=\mathrm{A} \, \exp \left(-\mathrm{E}_{\mathrm{A}} / \mathrm{R} \, \mathrm{T}\right)\]Then rate constant, \(\mathrm{k}\) increases with increase in temperature. The idea emerges that spontaneous conversion of reactant to products is inhibited by an ‘energy’ barrier. Further in the \(\operatorname{limit}(\mathrm{T} \rightarrow \infty) \ln (\mathrm{k})=\mathrm{A}\), the pre-exponential factor which has the same units as rate constant \(\mathrm{k}\).The assumption in the foregoing comments is that \(\mathrm{E}_{\mathrm{A}}\) is independent of temperature such that for example a first order rate constant \(\ln \left(\mathrm{k} / \mathrm{s}^{-1}\right)\) is a linear function of \(\mathrm{T}^{-1}\). \[\text { Thus, } \quad \ln \left(\mathrm{k} / \mathrm{s}^{-1}\right)=\ln \left(\mathrm{A} / \mathrm{s}^{-1}\right)-\left(\mathrm{E}_{\mathrm{A}} / \mathrm{R} \, \mathrm{T}\right)\]The latter pattern is generally observed but there are many well-documented cases where the plot is not linear. In other words it is incorrect to conclude that equation (y) somehow predicts how rate constants depend on temperature. There is no substitute for actually measuring this dependence.The law of mass action and the concept of an activation energy for a given chemical reaction are extrathermodynamic. This conclusion is unfortunate, implying that the treatment of kinetic data for reactions in solution is completely divorced from the thermodynamic treatment of the properties of solution. One can understand therefore why Transition Sate Theory (TST) attracts so much interest. At the very least, this theory offers analysis of kinetic data a patina of thermodynamic respectability. We describe TST with respect to a chemical reaction where the dependence of composition on time is described using a first order rate constant.For chemical reactions in the gas phase, statistical thermodynamics offers a reasonably straightforward approach to the description of both reactants and a transition state in which one vibrational mode for the transition state is transposed into translation along the reaction co-ordinate. The theory was re-expressed in terms of equations which could be directly related to the thermodynamics of the process of reaction in solutions.Chemical reaction proceeds from reactant \(\mathrm{X}(\mathrm{aq})\) to products through a transition state \(\mathrm{X}^{\neq}(\mathrm{aq})\). As the reaction proceeds the amount of solute \(\mathrm{X}\), \(n_{\mathrm{X}}(\mathrm{aq})\) decreases but at all times reactant \(\mathrm{X}(\mathrm{aq})\) and transition state \(\mathrm{X}^{\neq}(\mathrm{aq})\) are in chemical equilibrium.\[\text { Thus } \mathrm{X}(\mathrm{aq}) \Leftrightarrow=\mathrm{X}^{7} \rightarrow \text { products }\]The condition ‘chemical equilibrium’ is quantitatively expressed in terms of chemical potentials.\[\mu_{\mathrm{X}}^{\mathrm{eq}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{\neq}^{\mathrm{eq}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\]Conventionally where kinetics of reactions in solution are addressed, the composition of solutions is expressed in terms of concentrations using the unit, \(\mathrm{mol dm}^{-3}\). Then equation (zc) is formed assuming that ambient pressure is close to the standard pressure, \(p^{0}\); \(\mathrm{c}_{\mathrm{r}} =1 \mathrm{~mol dm}^{-3}\). Hence,\[\mu_{\mathrm{X}}^{0}(\mathrm{aq} ; \mathrm{T})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{c}_{\mathrm{X}} \, \mathrm{y}_{\mathrm{X}} / \mathrm{c}_{\mathrm{r}}\right)^{\mathrm{eq}}=\mu_{\neq}^{0}(\mathrm{aq} ; \mathrm{T})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{c}_{\neq} \, \mathrm{y}_{\neq} / \mathrm{c}_{\mathrm{r}}\right)^{\mathrm{eq}}\]The standard Gibbs energy of activation \(\Delta^{\neq} \mathrm{G}^{0}(\mathrm{aq} ; \mathrm{T})\) is given by equation (zd) leading to the definition of an equilibrium constant \({ }^{*} \mathrm{~K}^{0}(\mathrm{aq} ; \mathrm{T}) ; \text { with } \mathrm{p} \approx \mathrm{p}^{0}\).\[\Delta^{\neq} \mathrm{G}^{0}(\mathrm{aq} ; \mathrm{T})=\mu_{\neq}^{0}(\mathrm{aq} ; \mathrm{T})-\mu_{\mathrm{X}}^{0}(\mathrm{aq} ; \mathrm{T})\]\[\text { Then } \quad \Delta^{\neq} \mathrm{G}^{0}(\mathrm{aq} ; \mathrm{T})=-\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{c}_{\neq} \, \mathrm{y}_{\neq} / \mathrm{c}_{\mathrm{X}} \, \mathrm{y}_{\mathrm{X}}\right)^{\mathrm{eq}}\]\[\text { By definition, } \Delta^{\neq} \mathrm{G}^{0}(\mathrm{aq} ; \mathrm{T})=-\mathrm{R} \, \mathrm{T} \, \ln \left[{ }^{\neq} \mathrm{K}^{0}(\mathrm{aq} ; \mathrm{T})\right]\]\[\text { Therefore, } \mathrm{c}_{\neq}={ }^{\neq} \mathrm{K}^{0}(\mathrm{aq} ; \mathrm{T}) \, \mathrm{c}_{\mathrm{X}} \, \mathrm{y}_{\mathrm{X}} / \mathrm{y}_{\neq}\]Through the course of chemical reaction, as the concentration of reactant \(\mathrm{X}(\mathrm{aq})\) decreases, the condition given in equation (zb) holds. Chemical reaction is not instantaneous because \(\mu_{z}^{0}(\mathrm{aq} ; \mathrm{T})>\mu_{\mathrm{X}}^{0}(\mathrm{aq} ; \mathrm{T})\); a barrier exists to chemical reaction. Consequently the concentration \(\mathrm{c}_{\neq}\) is small. The analysis up to equation (zf) is based on a thermodynamic description of equilibrium between reactant and transition states. In the limit that the solution is very dilute, \(y_{X}=y_{\neq}=1\) at all time \(t\), \(\mathrm{T}\) and \(p\).\[\text { Thus at given } T \text { and } p, \quad k=\frac{k_{B} \, T}{h} \, \exp \left(\frac{-\Delta^{\neq} G^{0}}{R \, T}\right)\]\[\text { Then, } \mathrm{k}=\frac{\mathrm{k}_{\mathrm{B}} \, \mathrm{T}}{\mathrm{h}} \, \exp \left(-\frac{\Delta^{\neq} \mathrm{H}^{0}}{\mathrm{R} \, \mathrm{T}}+\frac{\Delta^{\neq} \mathrm{S}^{0}}{\mathrm{R}}\right)\]\(\Delta^{\neq} \mathrm{G}^{0}\) is the standard Gibbs energy of activation defined in terms of reference chemical potentials of transition state and reactants. Here \(\mathrm{k}_{\mathrm{B}}\) is the Boltzmann constant and \(\mathrm{h}\) is Planck’s constant.\[\Delta^{\neq} \mathrm{G}^{0}=-\mathrm{R} \, \mathrm{T} \, \ln \left(\frac{\mathrm{h} \, \mathrm{k}(\mathrm{T})}{\mathrm{k}_{\mathrm{B}} \, \mathrm{T}}\right)\]At temperature \(\mathrm{T}\), \(\Delta^{\neq} \mathrm{G}^{0}(\mathrm{aq} ; \mathrm{T})\) is re-expressed in terms of standard enthalpy and standard entropy of activation.\[\text { Then, } \Delta^{\neq} \mathrm{G}^{0}(\mathrm{aq} ; \mathrm{T})=\Delta^{\neq} \mathrm{H}^{0}(\mathrm{aq} ; \mathrm{T})-\mathrm{T} \, \Delta^{\neq} \mathrm{S}^{0}(\mathrm{aq} ; \mathrm{T})\]The standard isobaric heat capacity of activation,\[\Delta^{\neq} \mathrm{C}_{\mathrm{p}}^{0}(\mathrm{aq})=\left(\frac{\partial \Delta^{\neq} \mathrm{H}^{0}(\mathrm{aq})}{\partial \mathrm{T}}\right)_{\mathrm{p}}\]In summary transition state theory allows kinetic data to be analysed using the protocols and equations of thermodynamics.Analysis of the dependence of rate constants at fixed temperature as a function of pressure yields standard volumes of activation \(\Delta^{\neq} \mathrm{V}^{0}(\mathrm{aq} ; \mathrm{T})\). In further exercises this volumetric parameter is measured as functions of pressure at temperature \(\mathrm{T}\) and of temperature at fixed pressure.Nevertheless a word of caution is in order. Johnston points out that diagrams describing the progress of chemical reaction through several intermediates are often misleading. Such diagrams should be based on reference chemical potentials otherwise misleading conclusions can be drawn; see also [22 - 28]. I. Prigogine and R. Defay, Chemical Thermodynamics, trans. D. H. Everett, Longmans Green, London , 1953. In fact in the treatment of data obtained using fast reaction techniques (e.g. temperature jump and pressure-jump) where the displacement from equilibrium is small, it is assumed that the rate of response is linearly related to the affinity for chemical reaction. (a) E. F. Caldin, Fast Reactions in Solution, Blackwell, Oxford, 1964. (b) M. J. Blandamer, Introduction to Chemical Ultrasonics, Academic Press, London, 1973. L. P. Hammett, Physical Organic Chemistry, McGraw-Hill, New York, 1970, 2nd. Edition. K. J. ‘Neither did they ( i.e. Guldberg and Waage] make any contributions to kinetics since they worked in terms of forces and not of rates, although they did tentatively suggest that rates might be proportional to forces’. The condition ‘spontaneous change‘ signals a ‘natural’ direction but that does mean that the process is instantaneous. The properties of the system are dependent on time. We make this point to counter such statements as ‘Thermodynamics … deals exclusively with system showing no temporal change; reacting systems are outside its province’; cf. E. A. Moelwyn-Hughes, Kinetics of Reactions in Solution, Clarendon Press, Oxford, 1947, page 162. Thermodynamics in the form of the first and second laws offers no way forward. Intuitively one might argue that the rate of change of composition would be directly related to the affinity for spontaneous reaction, \(\mathrm{A}\) where \(\pi\) is a proportionality factor characteristic of the system, temperature and pressure. \[\text { Thus, } \mathrm{d} \xi / \mathrm{dt}=\pi \, \mathrm{A} \quad \text { where } \quad \lim \mathrm{it}(\mathrm{A} \rightarrow 0) \mathrm{d} \xi / \mathrm{dt}=0\]In fact we might draw an analogy with Ohm’s law whereby electric current \(\mathrm{i}\) (= rate of flow of charge) is proportional to the electric potential gradient, \(\Delta \phi\), the constant of proportionality being the electrical conductivity, \(\mathrm{L}\); \(\mathrm{i}=\mathrm{L} \, \Delta \phi\) where \(\mathrm{L}\) is characteristic of the system, temperature and pressure. Indeed such a kinetic force-flow link might be envisaged. Indeed an link emerges with the phenomenon of electric potential driving an electric current through an electrical circuit.Electric current = rate of flow of charge, unit = Ampere.Electric potential has the unit, volt.The product, \(\mathrm{I}. \(\mathrm{V}=[\mathrm{A}] \,\left[\mathrm{J} \mathrm{A}^{-1} \mathrm{~s}^{-1}\right]=[\mathrm{W}]\) Here \(\mathrm{W}\) is the SI symbol for the unit of power, watt. Rate of chemical reaction \(\mathrm{d} \xi / \mathrm{dt}=\left[\mathrm{mol} \mathrm{s} \mathrm{s}^{-1}\right]\) Affinity \(\mathrm{A}=\left[\mathrm{J} \mathrm{mol}^{-1}\right]\) Then \(\mathrm{A} \, \mathrm{d} \xi / \mathrm{dt}=\left[\mathrm{J} \mathrm{mol} \mathrm{m}^{-1}\right] \,\left[\mathrm{mol} \mathrm{s}^{-1}\right]=[\mathrm{W}]\) Interestingly chemists rarely refer to the 'wattage' of a chemical reaction. One cannot help be concerned with accounts which describe chemical equilibrium in terms of rates of chemical reaction. As we understand the argument runs along the following lines. For a chemical equilibrium having the following stochiometry, \(\mathrm{X}+\mathrm{Y} \Leftrightarrow\mathrm{Z}\) at equilibrium the rates of forward and reverse reactions are balanced.\[\mathrm{k}_{\mathrm{f}} \,[\mathrm{X}] \,[\mathrm{Y}]=\mathrm{k}_{\mathrm{r}} \,[\mathrm{Z}]\]Here \(\mathrm{k}_{\mathrm{f}}\) and \(\mathrm{k}_{\mathrm{f}}\) are the forward and reverse rate constants.\[\text { Then } \mathrm{K}=\mathrm{k}_{\mathrm{f}} / \mathrm{k}_{\mathrm{r}}=[\mathrm{Z}] /[\mathrm{X}] \,[\mathrm{Y}]\]The argument loses some of its force if one turns to accounts dealing with chemical kinetics when questions of order and molecularity are raised. In any case one cannot measure rates of chemical reactions 'at equilibrium' because at equilibrium 'nothing is happening'. Even in those cases where the rates of chemical reactions 'at equilibrium ' are apparently measured the techniques rely on following the return to equilibrium when the system is perturbed. In nearly all applications of the law of mass action to chemical reactions in solution a derived rate constant is based on on a description of the composition on time at fixed T and p. Therefore the calculated rate constant is an isobaric-isothermal property of the system. Nevertheless the volume is usually taken as independent of time. Certainly in most applications in solution chemistry the solutions are quite dilute and so throughout the course of the reaction the volume is effectively constant. An interesting point now emerges in that a given rate constant is an isothermal-isobaric-isochoric property. K. J. Laidler and N. Kallay, Kem. Ind.,1988,37,183. M. J. Blandamer, Educ. Chemistry, 1999,36,78. \(\mathrm{E}_{\mathrm{A}}=\left[\mathrm{J} \mathrm{mol}^{-1}\right] \text { such that } \mathrm{E}_{\mathrm{A}} / \mathrm{R} \, \mathrm{T}=\left[\mathrm{J} \mathrm{mol}^{-1}\right] /\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}]=\) The account given here stresses the link with classical thermodynamics. The key equations should be developed using statistical thermodynamics; see H. Eyring, J. Chem. Phys., 1935, 3,107. S. Glasstone, K. J. Laidler and H. Eyring , The Theory of Rate Processes, McGraw-Hill, New York, 1941. \(\frac{\mathrm{K}_{\mathrm{B}} \, \mathrm{T}}{\mathrm{h}}=\frac{\left[\mathrm{J} \mathrm{K}^{-1}\right] \,[\mathrm{K}]}{[\mathrm{J} \mathrm{s}]}=\left[\mathrm{s}^{-1}\right]\) \(\frac{\mathrm{k}_{\mathrm{B}} \, \mathrm{T}}{\mathrm{h}} \, \mathrm{c}_{\neq}=\left[\mathrm{s}^{-1}\right] \,\left[\mathrm{mol} \mathrm{dm}{ }^{-3}\right]=\left[\mathrm{mol} \mathrm{dm}^{-3} \mathrm{~s}^{-1}\right]\) R. E. Robertson, Prog. Phys. Org. Chem.,1967, 4, 213. M. J. Blandamer, J. M. W. Scott and R. E. Robertson, Prog. Phys. Org. Chem., 1985,15,149. M. J. Blandamer, Chemical Equilibria in Solution, Ellis Horwood, PTR Prentice Hall, New York,, 1992. H. S. Johnston, Gas Phase Reaction Rate Theory, Ronald Press, New York, 1966. R. B. Snadden, J.Chem.Educ.,1985,62,653. M. I. Page, Educ. Chem.,1981,18,52. R. D. Levine, J.Phys.Chem.,1979,83,159. I. H. Williams, Chem. Soc. Rev.,1993, 22,277. A. Williams, Chem. Soc. Rev.,1994,23,93. A. Drljaca, C. D. Hubbard, R. van Eldik, T. Asano, M. V. Basilevsky and W. J. Le Noble, Chem. Rev.,1998,98,2167. R. K. Boyd, Chem. Rev.,1977,77,93.This page titled 1.2.8: Affinity for Spontaneous Chemical Reaction - Law of Mass Action is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.2.9: Affinity for Spontaneous Chemical Reaction - Isochoric Condition and Controversy

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.02%3A_Affinity_for_Spontaneous_Chemical_Reaction/1.2.09%3A_Affinity_for_Spontaneous_Chemical_Reaction_-_Isochoric_Condition_and_Controversy

According to the First and Second Laws of thermodynamics, the change in Helmholtz energy \(\mathrm{dF}\) accompanying chemical reaction, change in volume and change in temperature is given by Equation \ref{a}.\[\mathrm{dF}=-\mathrm{S} \, \mathrm{dT}-\mathrm{p} \, \mathrm{dV}-\mathrm{A} \, \mathrm{d} \xi \label{a}\]where,\[\mathrm{A} \, \mathrm{d} \xi \geq 0\]At constant \(\mathrm{T}\) and \(\mathrm{V}\):\[\mathrm{dF}=-\mathrm{A} \, \mathrm{d} \xi ; \mathrm{A} \, \mathrm{d} \xi \geq 0\]If we monitor the change in composition of a closed system held at constant temperature and constant volume, equilibrium corresponds to a minimum in Helmholtz energy. In practice chemists concerned with spontaneous chemical reaction in solutions held at constant temperature, do not constrain the system to a constant volume. Rather they constrain the system to constant temperature and pressure. Therefore the following equation is the key. \[\mathrm{dG}=-\mathrm{S} \, \mathrm{dT}+\mathrm{V} \, \mathrm{dp}-\mathrm{A} \, \mathrm{d} \xi\]\[\text { At constant } \mathrm{T} \text { and } \mathrm{p}, \mathrm{dG}=-\mathrm{A} \, \mathrm{d} \xi ; \quad \mathrm{A} \, \mathrm{d} \xi \geq 0\]The rate at which chemical reaction drives the system to a minimum in Gibbs energy is described by the Law of Mass Action. Presumably a similar law would describe the approach to equilibrium of a system held at constant \(\mathrm{T}\) and \(\mathrm{V}\), the system moving spontaneously towards a lower Helmholtz energy. We do not explore this point. Nevertheless the isochoric condition is often invoked and we examine how this comes about.A key problem revolves around the role of the solvent in determining activation parameters; e.g. standard Gibbs energy of activation and standard enthalpy of activation. An extensive literature examines the role of solvents. But densities of solvents are a function of temperature and pressure. Then in attempting to understand the factors controlling kinetic activation parameters there is the problem that intermolecular distances (e.g. solvent-solvent and solvent-solute) are a function of temperature. In 1935 Evans and Polanyi suggested that isochoric activation parameters for chemical reaction in aqueous solution might be more mechanistically informative than conventional isobaric activation parameters; i.e. \([\partial \ln (\mathrm{k}) / \partial \mathrm{T}]_{\mathrm{V}}\) rather than \([\partial \ln (\mathrm{k}) / \partial \mathrm{T}]_{\mathrm{p}}\) where \(\mathrm{k}\) is the rate constant for spontaneous chemical reaction.With reference to chemical reactions in dilute aqueous solution the isochoric standard internal energy of activation \(\Delta^{\neq} \mathrm{U}_{\mathrm{V}}^{0}\) is related to the isobaric standard enthalpy of activation \(\Delta^{\neq} \mathrm{H}_{\mathrm{p}}^{0}\) at temperature \(\mathrm{T}\) and the standard volume of activation \(\Delta^{\neq} V^{0}\) using equation (f) where \(\alpha_{p 1}^{*}\) and \(\kappa_{\mathrm{T} 1}^{*}\) are respectively the isobaric expansibilities and isothermal compressibilities of water. \[\Delta^{\neq} \mathrm{U}_{\mathrm{V}}^{0}=\Delta^{\neq} \mathrm{H}_{\mathrm{p}}^{0}-\mathrm{T} \,\left[\alpha_{\mathrm{p} 1}^{*} / \kappa_{\mathrm{T} 1}^{*}\right] \, \Delta^{\neq} \mathrm{V}^{0}\]Baliga and Whalley noted that the dependence on solvent mixture composition of \(\Delta^{z} \mathrm{U}_{\mathrm{V}}^{0}\) is less complicated than that for \(\Delta^{\neq} \mathrm{H}_{\mathrm{p}}^{0}\) for solvolysis of benzyl chloride in ethanol + water mixtures at \(298.15 \mathrm{~K}\). A similar pattern was reported by Baliga and Whalley for the hydrolysis of 2-chloro-2-methyl propane in the same mixture at \(273.15 \mathrm{~K}\).The proposal concerning isochoric activation parameters sparked enormous interest and debate. Much of the debate centred on the isochoric condition and the answer to the question ‘what volume is held constant?’ With reference to equation (f), \(\alpha_{p l}^{*}\) and \(\kappa_{\mathrm{T1}}^{*}\) depend on temperature. Then the volume identified by subscript \(\mathrm{V}\) on \(\Delta^{\neq} \mathrm{U}_{\mathrm{V}}^{0}\) is dependent on temperature. Further the molar volumes of binary liquid mixtures depend on \(\mathrm{T}\) and \(p\). In other words the isochoric condition is not global across the given data set. Haak et al. noted that either side of the TMD of water there are pairs of temperatures where the molar volumes of water at ambient pressure are equal. Hence rates of reaction for chemical reactions in dilute aqueous solution at such temperatures would yield pairs of isochoric rate constants. Kinetic data for spontaneous hydrolysis of 1–benzoyl–1,2,4 triazole in aqueous solution at closely spaced temperatures close to the TMD of water reveal no unique features associated with the isochoric condition.The isochoric condition nevertheless remains interesting. There is however an important point to note. For the most part kinetic experiments investigate the rates of chemical reactions in solution at constant \(\mathrm{T}\) and \(p\). In other words spontaneous change is driven by the decrease in Gibbs energy. The latter is the operating thermodynamic potential function, both \(\mathrm{T}\) and \(p\) being held constant; the rate constant is an isobaric-isothermal property. Further the dependence of rate constant is often monitored on temperature at constant pressure, recognising that the volume of the system changes as the temperature is altered.In principle spontaneous chemical reaction could be monitored, holding the system at constant \(\mathrm{T}\) and volume \(\mathrm{V}\). Here the direction of spontaneous chemical reaction would be towards a minimum in Helmholtz energy, \(\mathrm{F}\). The dependence of the isochoric – isothermal rate constant on temperature could again in principle be measured. The technological challenge would be immense because one might expect enormous pressures to be required to hold the volume constant when the temperature was increased.Similar concerns over the definition of isochoric emerge in the context of the dependence on \(\mathrm{T}\) and \(p\) of acid dissociation constants in aqueous solution of ethanoic acid.The migration of ions through a salt solution under the influence of an applied electric potential can be envisaged as a rate process, analogous to the rate of chemical reaction. In most studies the applied electric field is weak such that the ions are only marginally displaced from their ‘equilibrium’ positions. The derived property is the molar conductivity characterising a given salt in solution at defined \(\mathrm{T}\) and \(p\). \[\text { Then, } \quad \ln (\Lambda)=\ln (\Lambda[\mathrm{T}, \mathrm{p}])\]The isobaric dependence of \(\ln (\Lambda)\) on temperature yields the isobaric energy of activation \(\mathrm{E}_{\mathrm{p}} \quad\left[=\mathrm{R} \,\left\{\partial \ln \left(\Lambda^{0}\right) / \partial(1 / \mathrm{T})\right\}_{\mathrm{p}}\right]\). An extensive literature describes the isochoric energies of activation defined by equation (h). \[\mathrm{E}_{\mathrm{V}}=-\mathrm{R} \, \mathrm{T}^{2}\left\{\partial \ln \left(\Lambda^{0}\right) / \partial\right\}_{\mathrm{V}}\]The property \(\mathrm{E}_{\mathrm{V}}\) has attracted considerable attention and debate [18-,21]. Nevertheless the same question (i.e. which volume is held constant?) can be asked. The debate emerges partly form the observation that \(\mathrm{T}\) and \(p\) are intensive variables whereas \(\mathrm{V}\) is an extensive variable.Footnotes M. G. Evans and M. Polanyi, Trans. Faraday Soc., 1935, 31, 875. M. G. Evans and M. Polanyi, Trans. Faraday Soc.,1937,33,448. D. M. Hewitt and A. Wasserman, J. Chem. Soc.,1940,735. B. T. Baliga and E. Whalley, J. Phys. Chem., 1967, 71, 1166. B. T. Baliga and E. Whalley, Can. J. Chem., 1970, 48, 528. See also E. Whalley, J. Chem. Soc. Faraday Trans.,1987,83,2901. L. M. P. C. Albuquerque and J. C. R. Reis, J. Chem. Soc., Faraday Trans. 1, 1989, 85, 207; 1991, 87,1553. H. A. J. Holterman and J. B. F. N. Engberts, J. Am. Chem. Soc., 1982, 104, 6382. P. G. Wright, J. Chem. Soc., Faraday Trans. 1, 1986, 82,2557. M. J. Blandamer, J. Burgess, B. Clarke, R. E. Robertson and J. M. W. Scott. J. Chem. Soc., Faraday Trans. 1, 1985, 81, 11. M. J. Blandamer, J. Burgess, B. Clarke, H. J. Cowles, I. M. Horn, J. F. B. N. Engberts, S. A. Galema and C. D. Hubbard, J. Chem. Soc. Faraday Trans. 1, 1989, 85, 3733. M. J. Blandamer, J. Burgess, B. Clarke and J. M. W. Scott, J. Chem. Soc., Faraday Trans. 1, 1984, 80, 3359. J. R. Haak, J. B. F. N. Engberts and M. J. Blandamer, J. Am. Chem. Soc., 1985, 107, 6031. M. J. Blandamer, N. J. Buurma, J. B. F. N. Engberts and J. C. R. Reis, Org. Biomol. Chem.,2003, 1,720. D. A. Lown, H. R. Thirsk and Lord Wynne-Jones, Trans. Faraday Soc., 1970, 66, 51. \(\Lambda=\left[\mathrm{S} \mathrm{} \mathrm{m}^{2} \mathrm{~mol}^{-1}\right]\) S. B. Brummer and G. J. Hills, Trans. Faraday Soc.,1961,57,1816,1823. A. F. M. Barton, Rev. Pure Appl. Chem.,1971,21,49. S. B. Brummer, J. Chem. Phys.,1965,42,1636.This page titled 1.2.9: Affinity for Spontaneous Chemical Reaction - Isochoric Condition and Controversy is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.20.1: Surface Phase: Gibbs Adsorption Isotherm

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A given plane surface phase contains i-chemical substances having amounts \(\mathrm{n}_{j}\) for each \(j\)-chemical substance. The plane surface phase is perturbed leading to a change in thermodynamic energy \(\mathrm{dU}^{\sigma}\) where the symbol \(\sigma\) identifies the surface phase. Using the Master Equation as a guide we set down the corresponding fundamental equation for the plane surface phase. \[\mathrm{dU}^{\sigma}=\mathrm{T} \, \mathrm{dS}^{\sigma}-\mathrm{p} \, \mathrm{dV}^{\sigma}+\gamma \, \mathrm{dA}+\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mu_{\mathrm{j}} \, \mathrm{dn}_{\mathrm{j}}^{\sigma}\]With reference to equation (a), \(\mathrm{T}, \mathrm{~p}, \gamma \text { and } \mu_{j}\) are intensive variables whereas \(\mathrm{~U}, \mathrm{~S}, \mathrm{~V}, \mathrm{~A} \text { and } \mathrm{n}_{j}\) are extensive variables. We integrate equation (a) to yield equation (b). \[\mathrm{U}^{\sigma}=\mathrm{T} \, \mathrm{S}^{\sigma}-\mathrm{p} \, \mathrm{V}^{\sigma}+\gamma \, \mathrm{A}+\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mu_{\mathrm{j}} \, \mathrm{n}_{\mathrm{j}}^{\sigma}\]The general differential of equation (b) is equation (c). \[\begin{gathered} \mathrm{dU}^{\sigma}=\mathrm{T} \, \mathrm{dS}{ }^{\sigma}+\mathrm{S}^{\sigma} \, \mathrm{dT}-\mathrm{p} \, \mathrm{dV}{ }^{\sigma}-\mathrm{V}^{\sigma} \, \mathrm{dp}+\gamma \, \mathrm{dA}+\mathrm{A} \, \mathrm{d} \gamma \\ +\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mu_{\mathrm{j}} \, \mathrm{dn}{ }_{\mathrm{j}}^{\sigma}+\sum_{\mathrm{j}=1}^{\mathrm{o}} \mathrm{n}_{\mathrm{j}}^{\sigma} \, \mathrm{d} \mu_{\mathrm{j}} \end{gathered}\]Using equations (a) and (c). \[0=\mathrm{S}^{\sigma} \, \mathrm{dT}-\mathrm{V}^{\sigma} \, \mathrm{dp}+\mathrm{A} \, \mathrm{d} \gamma+\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{n}_{\mathrm{j}}^{\sigma} \, \mathrm{d} \mu_{\mathrm{j}}\]Hence for a surface phase \(\sigma\) at fixed \(\mathrm{T}\) and \(\mathrm{p}\), \[0=\mathrm{A} \, \mathrm{d} \gamma+\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{n}_{\mathrm{j}}^{\sigma} \, \mathrm{d} \mu_{\mathrm{j}}\]We restrict attention to two component systems, comprising components labelled 1 and 2 wherein there are two phases \(\alpha\) and \(\beta\). In practice the boundary between phases \(\alpha\) and \(\beta\) comprises a region across which the compositions of small sample volumes change from pure \(\mathrm{i}\) to pure \(\mathrm{j}\). We imagine that the boundary layer can be replaced by a surface. For component \(j\), \(\Gamma_{j}\) is the amount of chemical substance \(j\) adsorbed per unit area; i.e. the surface concentration expressed in \(\mathrm{mol m}^{-2}\). Then, \[0=\mathrm{d} \gamma+\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \Gamma_{\mathrm{j}} \, \mathrm{d} \mu_{\mathrm{j}}\]Consider the case of a system prepared using two chemical substances, 1 and 2. We set the interphase between the two phases by a mathematical plane where \(\Gamma_{1}\) is zero. Then \[\Gamma_{2}=-\mathrm{d} \gamma / \mathrm{d} \mu_{2}\]The definition based on \(\Gamma_{1}\) defines the surface excess per unit area of the surface separating the two phases.Chemical substance 1 is the solvent (e.g. water) and chemical substance 2 is the solute. The surface divides the liquid and vapor phases. Equation (g) describes the surface excess of the solute. We assume that the surface and bulk aqueous phases are in (thermodynamic) equilibrium. Moreover we assume that the thermodynamic properties of the solutions are ideal. Then in terms of the concentration scale (where \(\mathrm{c}_{r} = 1 \mathrm{~mol dm}^{-3}\)), \[\mu_{2}(a q)=\mu_{2}^{0}(a q)+R \, T \, \ln \left(c_{2} / c_{r}\right)\]Then, \[\mathrm{d} \mu_{2}(\mathrm{aq})=\mathrm{R} \, \mathrm{T} \, \mathrm{d} \ln \left(\mathrm{c}_{2} / \mathrm{c}_{\mathrm{r}}\right)\]Hence, \[\Gamma_{2}=-\frac{1}{\mathrm{R} \, \mathrm{T}} \, \frac{\mathrm{d} \gamma}{\mathrm{d} \ln \left(\mathrm{c}_{2} / \mathrm{c}_{\mathrm{r}}\right)}\]Equation (j) is the ‘Gibbs adsorption equation’ for a two-component system using the Gibbs definition of surface excess. The validity of the Gibbs treatment was confirmed in 1932 by McBain who used an automated fast knife to remove a layer between 5 and 1 mm thick from the surface of a solution. The compositions of this layer and the solution were than analyzed. As N. K. Adam comments ‘in every case so far examined’, the measured adsorption agreed with that predicted by Equation (j).If \(\gamma\) decreases with increase in \(\mathrm{c}_{2}\), \(\Gamma_{2}\) is positive as is the case for organic solutes then these solutes are positively adsorbed at the air-water interface. The reverse pattern is observed for salt solutions. N. K. Adam, The Physics and Chemistry of Surfaces, Dover, New York, 1968; a corrected version of the third edition was published in 1941 by Oxford University Press. N. K. Adam, Physical Chemistry, Oxford, 1956, chapter XVII. S. E. Glasstone, Physical Chemistry, MacMillan, London, 1948, chapter XIV. As forcefully expressed to undergraduates, N. K. Adam did not like the symbols used in this reference. G. N. Lewis and M. Randall, Thermodynamics, revised by K. S. Pitzer and L. Brewer, McGraw-Hill, New York, 2nd edn., 1961, chapter 29. J. W. McBain and C. W. Humphreys, J. Phys.Chem.,1932,36,300.This page titled 1.20.1: Surface Phase: Gibbs Adsorption Isotherm is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.20.2: Surfactants and Micelles

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‘Nonsense, McBain’. The story is told that with these words an eminent scientist, chairman of a meeting of the Royal Society in London in the early 20th Century, reacted to the proposal by J. W. McBain that surfactants [ = surface active agents] might aggregate in aqueous solution. Subsequent events confirmed that the nub of McBain’s model is correct.The enormous industry based on ‘soaps’ and detergents prompts intensive studies of these complicated systems, supported by monographs and detailed reviews.In the context of aqueous solutions, surfactant molecules are amphipathic meaning that the solutes have dual characteristics : amphi ≡ dual and pathi ≡ sympathy. These dual characteristics emerge because a solute molecule contains both hydrophobic and hydrophilic parts. The subject is complicated because there is no general agreement concerning the nature-structure of these solute aggregates in aqueous solutions. Amphipathic molecules are broadly classified as either ionic or non-polar. Ionic surfactants are typified by salts such as sodium dodecylsulfate (\(\mathrm{C}_{12}\mathrm{H}_{25}\mathrm{OSO}_{3}^{-}\mathrm{~Na}^{+}\)) and hexyltrimethylammonium bromide (\(\mathrm{C}_{16}\mathrm{H}_{33}\mathrm{N}^{+} \mathrm{~Me}_{3}\mathrm{Br}^{-}\); CTAB). Non-ionic surfactants are typified by those based on ethylene oxide; e.g. hexaethylene glycol dodecylether, (\(\mathrm{C}_{16}\mathrm{H}_{25}\left(\mathrm{OCH}_{2}\mathrm{CH}\right)_{6}\mathrm{OH}\)).When small amounts of a given surfactant are gradually added to a given volume of water (\(\ell\)), the properties of the aqueous solutions are unexceptional until the concentration of the surfactant exceeds a characteristic concentration of surfactant (at defined \(\mathrm{T}\) and \(\mathrm{p}\)) called the critical micellar concentration, cmc. At this point further added surfactant exists in solution as aggregates of generally 20 to 100 monomers, which are called micelles. The formation of micelles is often signalled by a change in the pattern of the dependence of a given property \(\mathrm{P}\) of a solution on surfactant concentration. The property can be surface tension, molar conductance of an ionic surfactant, uv-visible absorption spectra of water soluble dyes (or, an iron complex). When more surfactant is added, the micelles cluster to form more complicated aggregates.Micelles are not formed by the gradual association of monomers, forming dimers, trimers….. . Rather micelles are examples of organized structures spontaneously formed by simple molecules. A quoted aggregation number is not a stoichiometric number. The quoted number (e.g. approx. 90 for CTAB at \(298 \mathrm{~K}\)) is taken as an ‘average’ over the micelles in a given system.In terms of the structure of micelles in aqueous solutions, key questions centre onThe two questions are linked by the question – to what extent does the terminal group in, for example, the hexadecyl chains of CTAB come into contact with the aqueous solution? If the answer is ‘never’, micelles have a structure in which there is a well organized hydrophobic core. If the answer is ‘frequently’ the micelles are very dynamic assemblies with continuous changes in organization/structure. We do not become involved in this debate. However we note that there is such a debate over what precisely the thermodynamic analysis is asked to describe.Description of micellar-surfactant systems emerges from the Phase Rule. The number of phases = 3; vapor, aqueous solution and micelle. The number of components = 2; water and surfactant. Hence having defined the temperature, the remaining intensive variables are defined; i.e. vapor pressure, mole fraction of surfactant monomer in aqueous solution, and mole fraction of surfactant in micellar phase.The starting point for a thermodynamic analysis is the assumption of an equilibrium between surfactant monomers and micelles in solution (at defined \(\mathrm{T}\) and \(\mathrm{p}\)). A key feature of these systems is that above the cmc when more surfactant is added the concentration of monomers remains essentially constant, the added surfactant existing in micellar form. Two models for micelle formation are discussed;Here we assume that the closed system containing solvent and surfactant is at equilibrium at defined \(\mathrm{T}\) and \(\mathrm{p}\). Each system is at a unique minimum in Gibbs energy. The system is at ambient pressure, which for our purposes is effectively the standard pressure \(\mathrm{p}^{0}\). We characterize micelle formation using thermodynamic variables describing monomers and micelles. Then \(\Delta_{\text {mic }} G^{0}\) is the standard Gibbs energy of micelle formation which is dependent on temperature. The latter dependence is characterized by the standard enthalpy of micelle formation, \(\Delta_{\text {mic }} H^{0}\), where, \[\Delta_{\text {mic }} \mathrm{G}^{0}=\Delta_{\text {mic }} \mathrm{H}^{0}-\mathrm{T} \, \Delta_{\text {mic }} \mathrm{S}^{0}\]Clearly the definition of \(\Delta_{\text {mic }} \mathrm{G}^{0}\) is directly associated with the definition of standard states for both the simple salt in solution and the micelles in the aqueous system. These thermodynamic variables together with aggregation numbers are extensively documented. F. M. Menger, Acc. Chem.Res.,1979,12,111; and references therein. E. M. Kirshner, Chem. Eng. News, 1998,76,39 J. H. Clint, Surfactant Aggregation, Blackie, Glasgow, 1992. Y. Mori, Micelles, Plenum Press, New York, 1992. D. F. Evans and H. Wennerstrom, The Colloidal Domain, VCH, New York, 1994. B. Lindman and H. Wennerstrom, Top. Curr. Chem., 1980, 87,1. G. C. Kresheck in, Water: A Comprehensive Treatise, ed. F. Franks, Plenum Press, New York, 1973, volume 4,chapter 2. J. E. Desnoyers, Polymer-surfactant interactions; Solubilisation by micelles; C. Treiner, Chem. Soc. Rev.,1994,23,349. Alcohol-micelle interactions; R. E. Verrall, Chem. Soc. Rev.,1995,24,79. S. Backlund, H. Hoiland, O. J. Krammen and E. Ljosland, Acta Chem. Scand,,Ser. A, 1982, 87,1169. But see D. Schuhman, Prog. Colloid Polym. Sci.,1989,71,338. See for example, K. Shinoda, Langmuir, 1991,7,2877. F. Menger and D. W. Doll , J. Am. Chem.Soc.,1984,106,1109. S. Puvvada and D. Blankschtein, J.Phys.Chem.,1992,96, 5567. N. M. van Os, J. R. Haak and L A. M. Rupert, Physico-Chemical Properties of Selected Anionic, Cationic and Non-Ionic Surfactants, Elsevier, Amsterdam,1993.This page titled 1.20.2: Surfactants and Micelles is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.20.3: Surfactants and Micelles: Non-Ionics

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The properties of aqueous solutions containing non-ionic surfactants can be described using two models.We envisage a non-ionic surfactant \(\mathrm{X}\). When chemical substance \(\mathrm{X}\) is added to \(\mathrm{n}_{1}\) moles of water (at fixed \(\mathrm{T}\) and \(\mathrm{p}\)), solute \(\mathrm{X}\) exists as a simple solute \(\mathrm{X}(\mathrm{aq})\) until the concentration of solute \(\mathrm{X}\), \(\mathrm{c}_{\mathrm{X}}\) reaches a characteristic concentration \(\mathrm{cmc}_{\mathrm{X}}\) when a trace amount of the micellar phase appears. Each micelle comprises \(\mathrm{n}\) molecules of surfactant \(\mathrm{X}\). The equilibrium between monomer surfactant \(\mathrm{X}(\aq) and surfactant in the micelles is described by the following equation. \[\mu_{\mathrm{X}}^{\mathrm{eq}}(\mathrm{aq})=\mu_{\mathrm{X}}^{*}(\mathrm{mic})\]If \(\mathrm{X}(\mathrm{aq})\) is a typical neutral solute in aqueous solution \(\mu_{\mathrm{x}}^{\mathrm{eq}}(\mathrm{aq})\) is related to the cmc of \(\mathrm{X}(\mathrm{aq})\) in solution at the point where only a trace amount of micellar phase exists. Hence, \[\mu_{\mathrm{X}}^{0}(\mathrm{aq} ; \mathrm{c}-\mathrm{scale})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{cmc} \, \mathrm{y}_{\mathrm{X}} / \mathrm{c}_{\mathrm{r}}\right)=\mu_{\mathrm{X}}^{\mathrm{*}}(\mathrm{mic})\]Here \(\mathrm{y}_{\mathrm{X}}\) is the solute activity coefficient for \(\mathrm{X}(\mathrm{aq})\) taking account of solute-solute interactions in the aqueous solution. Therefore, by definition, \[\Delta_{\text {mic }} \mathrm{G}^{0}(\mathrm{aq})=-\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{K}_{\text {mic }}^{0}\right)=\mu_{\mathrm{X}}^{*}(\mathrm{mic})-\mu_{\mathrm{x}}^{0}(\mathrm{aq} ; \mathrm{c}-\text { scale })\]Equilibrium constant \(\mathrm{K}_{\text {mic }}^{0}\) describes the phase equilibrium involving surfactant X in aqueous solution and micellar phase. \[\Delta_{\text {mic }} \mathrm{G}^{0}(\mathrm{aq})=-\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{K}_{\mathrm{mic}}^{0}\right)=\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{cmc} \, \mathrm{y}_{\mathrm{X}} / \mathrm{c}_{\mathrm{r}}\right)\]If \(\mathrm{X}(\mathrm{aq})\) is a neutral solute and the cmc is low, a useful approximation sets \(\mathrm{y}_{\mathrm{X}}\) at unity. Therefore \(\Delta_{\text {mic }} \mathrm{G}^{0}(\mathrm{aq})\) is the standard increase in Gibbs energy when one mole of surfactant \(\mathrm{X}(\mathrm{aq})\) forms one mole of \(\mathrm{X}\) in the micellar phase. Combination of equations (c) and (d) yields equation (e). \[\mathrm{K}_{\text {mic }}^{0}=\left(\mathrm{cmc} / \mathrm{c}_{\mathrm{r}}\right)^{-1}\]Here \(\mathrm{K}_{\text {mic }}^{0}\) describes the equilibrium between surfactant in the micellar phase and the aqueous solution. A famous equation suggested by Harkin relates the cmc to the number of carbon atoms in the alkyl chain, \(\mathrm{n}_{\mathrm{C}}\); equation (f) \[\log \left(\mathrm{cmc} / \mathrm{c}_{\mathrm{r}}\right)=\mathrm{A}-\mathrm{B} \, \mathrm{n}_{\mathrm{C}}\]With equation (e), \[\log \left(\mathrm{K}_{\text {mic }}^{0}\right)=A-\left(B \, n_{C}\right)\]The above analysis is also used for ionic surfactants if it can be assumed the degree of counter ion binding by the micelles is small, the thermodynamic properties of the solution are ideal and the aggregation number is high.The aqueous phase comprises an aqueous solution of solute \(\mathrm{X}\), \(\mathrm{X}(\mathrm{aq})\). The micellar phase comprises both water and surfactant \(\mathrm{X}\) such that the mole fraction of surfactant in the micellar phase equals \(\mathbf{X}_{\mathrm{X}}^{\mathrm{eq}}\). We treat the micellar phase using the procedures used to describe the properties of a binary liquid mixture. For the micellar phase the chemical potential of \(\mathrm{X}\) is given by the following equation. \[\mu_{\mathrm{x}}(\mathrm{mic})=\mu_{\mathrm{x}}^{*}(\text { mic })+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{\mathrm{x}} \, \mathrm{f}_{\mathrm{x}}\right)^{\mathrm{eq}}\]where \[\operatorname{limit}\left(x_{x} \rightarrow 1\right) f_{x}=1 \text { at all } \mathrm{T} \text { and } \mathrm{p}\]But at equilibrium for a system containing a trace of the micellar phase, \[\mu_{\mathrm{x}}^{\mathrm{eq}}(\mathrm{mic})=\mu_{\mathrm{x}}^{\mathrm{eq}}(\mathrm{aq})\]Then, \[\mu_{\mathrm{X}}^{\circ}(\text { mic })+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{\mathrm{X}} \, \mathrm{f}_{\mathrm{X}}\right)^{\mathrm{cq}}=\mu_{\mathrm{X}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{cmc}_{\mathrm{x}} \, \mathrm{y}_{\mathrm{X}} / \mathrm{c}_{\mathrm{r}}\right)\]By definition \[\Delta_{\text {mic }} \mathrm{G}^{0}=-\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{K}_{\text {mic }}^{0}\right)=\mu_{\mathrm{x}}^{*}(\mathrm{mic})-\mu_{\mathrm{x}}^{0}(\mathrm{aq})\]Then, \[\mathrm{K}_{\text {mic }}^{0}=\left[\mathrm{x}_{\mathrm{X}} \, \mathrm{f}_{\mathrm{X}}\right]^{\mathrm{eq}} /\left[\mathrm{cmc} / \mathrm{c}_{\mathrm{r}}\right]\]If the micelle is only ‘damp’ rather than wet , a reasonable assumption sets \(\mathrm{f}_{\mathrm{X}}\) equal to unity although it is not obvious how \(\mathrm{x}_{\mathrm{X}}\) might be determined.Micelle formation is described as an equilibrium between \(\mathrm{X}(\mathrm{aq})\) as a solute in aqueous solution and a micellar aggregate in aqueous solution formed by n molecules of the monomer \(\mathrm{X}(\mathrm{aq})\). Then at the point where micelles are first formed, the following equilibrium is established. \[\mathrm{nX}(\mathrm{aq}) \Leftrightarrow \mathrm{X}_{\mathrm{n}}(\mathrm{aq})\]The total amount of surfactant in the system equals \(\mathrm{N}(\mathrm{X} ; \mathrm{aq})+\mathrm{n} \, \mathrm{N}\left(\mathrm{X}_{\mathrm{n}} ; \mathrm{mic}\right)\) where \(\mathrm{N}(\mathrm{X} ; \mathrm{aq})\) is the amount of monomer surfactant and where \(\mathrm{N}\left(\mathrm{X}_{\mathrm{n}} ; \mathrm{mic}\right)\) is the total amount of micelles , each micelle containing n surfactant molecules. But for the micellar aggregate \(\mathrm{X}_{\mathrm{n}}(\mathrm{aq})\) treated as a single solute, \[\mu\left(\mathrm{X}_{\mathrm{n}} ; \mathrm{aq}\right)=\mu^{0}\left(\mathrm{X}_{\mathrm{n}} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{c}\left(\mathrm{X}_{\mathrm{n}}\right) \, \mathrm{y}\left(\mathrm{X}_{\mathrm{n}}\right) / \mathrm{c}_{\mathrm{r}}\right]\]Here \(c\left(X_{n}\right)\left[=N\left(X_{n} ; a q\right) / V\right.\) where \(\mathrm{V}\) is the volume of the system] is the concentration of micelles in the system, activity coefficient \(\mathrm{y}\left(\mathrm{X}_{\mathrm{n}}\right)\).The latter can be assumed to be unity if there are no micelle-micelle interactions and no micelle-monomer interactions in the aqueous system. Although this approach seems similar to that used to describe chemical equilibria, the procedure has problems in the context of determining \(\mathrm{c}\left(\mathrm{X}_{\mathrm{n}}\right)\). J. E. Desnoyers, G. Caron, R. DeLisi, D. Roberts, A. Roux and G. Perron, J. Phys. Chem.,1983,87, 1397. G. Olofsson, J.Phys.Chem.,1985, 89,1473. J. E. Desnoyers, Pure Appl.Chem.,1982, 54,1469. M. J. Blandamer, J. M. Permann, J. Kevelam, H. A. van Doren, R. M. Kellogg and J. B. F. N. Engberts, Langmuir, 1999,15,2009. M. J. Blandamer, K. Bijma and J. B. F. N. Engberts, Langmuir, 1998,14,79. M. J. Blandamer, B. Briggs, P. M. Cullis, J. B. F. N. Engberts and J. Kevelam, Phys.Chem.Chem.Phys.,2000,2,4369.This page titled 1.20.3: Surfactants and Micelles: Non-Ionics is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.20.4: Surfactants and Miceles: Ionics

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An intense debate concerns the structure of micelles, particularly those formed by ionic surfactants such as SDS and CTAB. It seems generally agreed that micelles are essentially spherical in shape. The polar head groups ( e.g. \(– \mathrm{N}^{+} \mathrm{Me}_{3}\)) are at the surface of each micelle, having strong interactions with the surrounding solvent. In close proximity in the Stern layer are counterions (e.g. bromide ions in the case of CTAB); the aggregation number \(\mathrm{n}\) describes the number of cations which form each micelle. The total charge on the micelle is determined the aggregation number and a quantity \(\beta\), the latter being the fraction of charge of aggregated ions forming the micelle neutralized by the micelle bound counter ions. The remaining fraction of counter ions exists as ‘free’ ions in aqueous solution. Both \(mathrm{n}\) and \(\beta\) are characteristic of a given surfactant system, and are obtained from analysis of experimental data. The properties of ionic surfactants have been extensively studied. Here we examine four thermodynamic descriptions of these systems.We consider a dilute aqueous solution of an ionic surfactant; e.g. \(\mathrm{AM}^{+} \mathrm{Br}^{-}\). As more surfactant is added a trace amount of micelles appear in the solution when the concentration of surfactant just exceeds the cmc. The trace amount of surfactant is present as micelles constituting a micellar phase. At defined \(\mathrm{T}\) and \(\mathrm{p}\), the following equilibrium is established in the case of the model surfactant \(\mathrm{AM}^{+} \mathrm{Br}^{-}\); \[\mathrm{AM}^{+} \mathrm{Br}^{-}(\mathrm{aq}) \Leftrightarrow \mathrm{AM}^{+} \mathrm{Br}^{-}(\mathrm{mic})\]Then, \[\mu^{\mathrm{cq}}\left[\mathrm{AM}^{+} \mathrm{Br}^{-}(\mathrm{aq})\right]=\mu^{\mathrm{cq}}\left[\mathrm{AM}^{+} \mathrm{Br}^{-}(\mathrm{mic})\right]\]We assume that the micelles carry no charge. The chemical potential of the surfactant in aqueous solution is related to the cmc using the following equation where \(\mathrm{y}_{\pm}\) is the mean ionic activity coefficient. We set \(\mu^{\mathrm{eq}}\left[\mathrm{AM}^{+} \mathrm{Br}^{-} \text {(mic) }\right]\) equal to the chemical potential of the surfactant in the pure micellar state, \(\mu^{*}\left[\mathrm{AM}^{+} \mathrm{Br}^{-} \text {(mic) }\right]\). \[\begin{aligned} \mu^{0}\left(\mathrm{AM}^{+} \mathrm{Br}^{-} ; \mathrm{aq} ; \mathrm{c}-\mathrm{scale}\right)+2 \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{cmc} \, \mathrm{y}_{\pm} \, / \mathrm{c}_{\mathrm{r}}\right) \\ &=\mu^{*}\left(\mathrm{AM}^{+} \mathrm{Br}^{-} ; \text {micellar phase }\right) \end{aligned}\]Here \(\mu^{0}\left(\mathrm{AM}^{+} \mathrm{Br}^{-} ; \mathrm{aq} ; \mathrm{c}-\text { scale }\right)\) is the chemical potential of the salt \(\mathrm{AM}^{+} \mathrm{Br}^{-}\) in aqueous solution at unit concentration where the properties of the salt are ideal. Thus \(\mathrm{y}_{\pm}\) describes the role of ion-ion interactions in the solution having salt concentration cmc. Because the model states that there is only a trace amount of micelles in the system, we do not take account of salt-micelle interactions. Then \[\Delta_{\text {mic }} \mathrm{G}^{0}=\mu^{*}\left(\text { micellar phase; } \mathrm{AM}^{+} \mathrm{Br}^{-}\right)-\mu^{0}\left(\mathrm{AM}^{+} \mathrm{Br}^{-} ; \text {aq; } \mathrm{c}-\text { scale }\right)\]Hence, \[\Delta_{\text {mic }} G^{0}(\mathrm{aq} ; \mathrm{c}-\text { scale })=2 \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{cmc} \, \mathrm{y}_{\pm} / \mathrm{c}_{\mathrm{r}}\right)\]If the salt concentration in the aqueous solution at the cmc is quite low, a useful assumption sets \(\mathrm{y}_{\pm}\) equal to unity. Then, \[\Delta_{\text {mic }} \mathrm{G}^{0}(\mathrm{aq} ; \mathrm{c}-\mathrm{scale})=2 \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{cmc} / \mathrm{c}_{\mathrm{r}}\right)\]The latter equation leads to the calculation of the standard increase in Gibbs energy when one mole of salt \(\mathrm{AM}^{+} \mathrm{Br}^{-}\) passes from the ideal solution, concentration \(1 \mathrm{mol dm}^{-3}\) to the micellar phase.There is a modest problem with the latter equation which can raise conceptual problems. As normally stated the cmc for a given salt is expressed using the unit ‘\(\mathrm{mol dm}^{-3}\)‘ so that \(\mathrm{c}_{\mathrm{r}} = 1 \mathrm{~mol dm}^{-3}\). This means that when \(\mathrm{cmc} > 1 \mathrm{~mol dm}^{-3}\), \(\Delta_{\operatorname{mic}} G^{0}(\mathrm{aq} ; \mathrm{c}-\text { scale })\) is positive. For solutes where \(\mathrm{cmc} < 1 \mathrm{~mol dm}^{-3}\), the derived quantity is negative.Another approach expresses the cmc using the mole fractions, cmx such that equation (c) is written as follows. \[\begin{gathered} \mu^{0}\left(\mathrm{AM}^{+} \mathrm{Br}^{-} ; \text {aq } ; \mathrm{x}-\mathrm{scale}\right)+2 \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{cmx} \, \mathrm{f}_{\pm}^{*}\right) \\ =\mu^{*}\left(\mathrm{AM}^{+} \mathrm{Br}^{-} ; \text {micellar phase }\right) \end{gathered}\]Here \(\mu^{0}\left(\mathrm{AM}^{+} \mathrm{Br}^{-} ; \mathrm{aq} ; \mathrm{x}-\text { scale }\right)\) is the chemical potential of the salt \(\mathrm{AM}^{+} \mathrm{Br}^{-}\) in an ideal solution where the (asymmetric) activity coefficient \(\mathrm{f}_{\pm}^{*}=1.0\) and \(\mathrm{cmx} = 1.0\). .By definition \(\operatorname{limit}\left[x\left(\mathrm{AM}^{+} \mathrm{Br}^{-}\right) \rightarrow 0\right] \mathrm{f}_{\pm}^{*}=1.0 \text { at all } \mathrm{T} \text { and } \mathrm{p}\).} The analogue of equation (f) takes the following form. \[\Delta_{\text {mic }} \mathrm{G}^{0}(\mathrm{aq} ; \mathrm{x}-\text { scale })=2 \, \mathrm{R} \, \mathrm{T} \, \ln (\mathrm{cm} \mathrm{x})\]Because cmx is always less than unity, \(\Delta_{\text {mic }} G^{0}(a q ; x-\text { scale })\) is always negative. It is important in these calculations to note the definitions of reference and standard states for solutes and micelles otherwise false conclusions can be drawn. The analysis proceeds to use the Gibbs-Helmholtz equation. Hence, \[\Delta_{\text {mic }} \mathrm{H}^{0}(\mathrm{aq} ; \mathrm{x}-\text { scale })=-2 \, \mathrm{R} \, \mathrm{T}^{2} \,\{\partial \ln (\mathrm{cmx}) / \partial \mathrm{T}\}_{\mathrm{p}}\]The term \(\left\{\partial \ln (\mathrm{cmx}) / \partial \mathrm{T}_{\mathrm{P}}\right.\) is conveniently obtained by expressing the dependence of cmx on temperature using the following polynomial. \[\ln (c m x)=a_{1}+a_{2} \, T+a_{3} \, T^{2}+\ldots\]Equation (h) is straightforward, the stoichiometric factor ‘2’ emerging from the fact that each mole of salt \(\mathrm{AM}^{+} \mathrm{Br}^{-}\) produces on complete dissociation 2 moles of ions. A key assumption in this analysis is that the micelles carry no electric charge. In other words a micelle is formed by \(\mathrm{n}\) moles of cation \(\mathrm{AM}^{+}\), \(\mathrm{n}\) moles of counter ions \(\mathrm{Br}^{-}\) being bound within the Stern layer such that the charge on each micelle is zero. This model is a little unrealistic.A cationic surfactant \(\mathrm{AM}^{+} \mathrm{Br}^{-}\) in aqueous solution forms micelles when \(\mathrm{n}\) cations come together to form a micellar phase. Bearing in mind that \(\mathrm{n}\) might be greater than 20, the idea that there exists micro-phases of macro-cations in a system with an electric charge at least +20 is not attractive. In practice the charge is partially neutralised by bromide ions in the Stern layer. The quantity \(\beta\) refers to the fraction of counter ions bound to cations. Thus the formal charge number on each micelle is \([n \,(1-\beta)]\). In the model developed here we represent the formation of the micro-phase comprising the micelles as follows where n is the number of cation monomers which cluster, the remaining bromide ions being present in the aqueous solution (phase). \[\begin{aligned} \mathrm{nAM}^{+}(\mathrm{aq}) &+\mathrm{n} \,(1-\beta+\beta) \mathrm{Br}^{-}(\mathrm{aq}) \\ & \Leftrightarrow\left[\mathrm{nAM}^{+} \mathrm{n} \, \beta \mathrm{Br}^{-}\right]^{\mathrm{n} \,(1-\beta)}(\mathrm{mic})+\mathrm{n} \,(1-\beta) \mathrm{Br}^{-}(\mathrm{aq}) \end{aligned}\]We re-express this equilibrium in terms of equilibrium chemical potentials for a system at fixed \(\mathrm{T}\) and \(\mathrm{p}\). \[\begin{aligned} &\mathrm{n} \, \mu^{\mathrm{cq}}\left(\mathrm{AM}^{+} ; \mathrm{aq}\right)+\mathrm{n} \,(1-\beta+\beta) \, \mu^{\mathrm{eq}}\left(\mathrm{Br}^{-} ; \mathrm{aq}\right) \\ &=\mu^{\mathrm{eq}}\left\{\left[\mathrm{nAM}^{+} \mathrm{n} \, \beta \mathrm{Br}^{-}\right]^{\mathrm{n}(1-\beta)} ; \text { micelle }\right\}+\mathrm{n} \,(1-\beta) \mu^{\mathrm{cq}}\left(\mathrm{Br}^{-} ; \mathrm{aq}\right) \end{aligned}\]We define the chemical potential of the micelle microphase which contains 1 mole of \(\mathrm{AM}^{+}\). This is a key extrathermodynamic step. We also describe the micelle as a pure ‘phase’. \[\begin{aligned} \mu^{\mathrm{eq}}\left\{\left[\mathrm{AM}^{+} \beta \mathrm{Br}^{*}\right]^{(1-\beta)} ; \text { micelle }\right\} \\ &=\mu^{\mathrm{cq}}\left\{\left[\mathrm{nAM}^{+} \mathrm{n} \, \beta \mathrm{Br}^{-}\right]^{\mathrm{n}(1-\beta)} ; \text { micelle }\right\} / \mathrm{n} \end{aligned}\]Hence, \[\begin{aligned} &\mu^{\mathrm{eq}}\left(\mathrm{AM}^{+} ; \mathrm{aq}\right)+\mu^{\mathrm{eq}}\left(\mathrm{Br}^{-} ; \mathrm{aq}\right) \\ &=\mu^{*}\left\{\left[\mathrm{AM}^{+} \beta \mathrm{Br}^{*}\right]^{(1-\beta)} ; \text { micelle }\right\}+(1-\beta) \mu^{\mathrm{eq}}\left(\mathrm{Br}^{-} ; \mathrm{aq}\right) \end{aligned}\]Or, \[\begin{aligned} &\mu^{\mathrm{cq}}\left(\mathrm{AM}^{+} \mathrm{Br}^{-1} ; \mathrm{aq}\right)= \\ &\mu^{\mathrm{eq}}\left\{\left[\mathrm{AM}^{+} \beta \mathrm{Br}^{-}\right]^{(1-\beta)} ; \text { micelle\} }+(1-\beta) \mu^{\mathrm{cq}}\left(\mathrm{Br}^{-} ; \mathrm{aq}\right)\right. \end{aligned}\]The term \(\mu^{थ}\left(\mathrm{AM}^{+} \mathrm{Br}^{-1} ; \mathrm{aq}\right)\) is the equilibrium chemical potential of a 1:1 salt in solution at the cmc. The term \(\mu^{\mathrm{eq}}\left(\mathrm{Br}^{-} ; \mathrm{aq}\right)\) is the equilibrium chemical potential of the bromide ion in the solution at the cmc of the surfactant. In any event the system is electrically neutral. \[\begin{aligned} &\mu^{0}\left(\mathrm{AM}^{+} \mathrm{Br}^{-1} ; \mathrm{aq}\right)+2 \, \mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{cmc} \, \mathrm{y}\left(\mathrm{AM}^{+} \mathrm{Br}^{-}\right) / \mathrm{c}_{\mathrm{r}}\right] \\ &=\mu^{*}\left\{\left[\mathrm{AM}^{+} \beta \mathrm{Br}^{*}\right]^{(1-\beta)} ; \text { micelle }\right\} \\ &+(1-\beta) \,\left\{\mu^{0}\left(\mathrm{Br}{ }^{-} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{cmc} \, \mathrm{y}\left(\mathrm{Br}^{-}\right) / \mathrm{c}_{\mathrm{r}}\right]\right\} \end{aligned}\]By definition, \[\begin{aligned} \Delta_{\text {mic }} \mathrm{G}^{0}=\mu^{*} &\left\{\left[\mathrm{AM}^{+} \beta \mathrm{Br}^{*}\right]^{(1-\beta)} ; \text { micelle }\right\} \\ &+(1-\beta) \, \mu^{0}\left(\mathrm{Br}^{-} ; \mathrm{aq}\right)-\mu^{0}\left(\mathrm{AM}^{+} \mathrm{Br}^{-1} ; \mathrm{aq}\right) \end{aligned}\]Assuming both \(\mathrm{y}\left(\mathrm{AM}^{+} \mathrm{Br}^{-}\right)\) and \(\mathrm{y}\left(\mathrm{Br}^{-}\right)\) are unity, \[\Delta_{\text {mic }} \mathrm{G}^{0}=2 \, \mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{cmc} / \mathrm{c}_{\mathrm{r}}\right]-(1-\beta) \, \mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{cmc} / \mathrm{c}_{\mathrm{r}}\right]\]or \[\Delta_{\text {mic }} \mathrm{G}^{0}=(1+\beta) \, \mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{cmc} / \mathrm{c}_{\mathrm{r}}\right]\]The latter equation closely resembles that for non-ionic surfactants for which \(\beta\) is unity. For ionic surfactants it is not justified to assume that \(\beta\) is also unity.As more ionic surfactant is added to a solution having the concentration of surfactant equal to the cmc, so the solution increasingly resembles a mixed salt solution, simple salt, charged micelles and counter ions. Analysis of the properties of such solutions was described by Burchfield and Woolley. We might develop the analysis from equation (k). An advantage of writing the equation in this form stems from the observation that both sides of the equation describe an electrically neutral system. Woolley and co-- workers prefer a form which removes a contribution \(\mathrm{n} \,(1-\beta) \mathrm{Br}^{-}(\mathrm{aq})\) from each side of equation (k). \[\mathrm{nAM}^{+}(\mathrm{aq})+\mathrm{n} \, \beta \mathrm{Br}^{-}(\mathrm{aq}) \Leftrightarrow\left[\mathrm{nAM}^{+} \mathrm{n} \, \beta \mathrm{Br}^{-}\right]^{\mathrm{a} \,(1-\beta)}(\mathrm{aq})\]Nevertheless one might argue that equation (k) does have the merit in comparing two salts whereas equation (t) describes the links between three ions. In terms of equation (k) , there are two salts in solution.\[\begin{aligned} &\mu\left(\mathrm{AM}^{+} \mathrm{Br}^{-}\right)= \\ &\mu^{0}\left(\mathrm{AM}^{+} \mathrm{Br}^{-}\right)+2 \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{c}\left(\mathrm{AM}^{+} \mathrm{Br}^{-}\right) \, \mathrm{y}_{\pm}\left(\mathrm{AM}^{+} \mathrm{Br}^{-}\right) / \mathrm{c}_{\mathrm{r}}\right) \end{aligned}\]\[\begin{aligned} &\mu(\text { mic. salt }) \\ &\left.=\mu^{0} \text { (mic. salt }\right)+[n \,(1-\beta)+1] \, R \, T \, \ln \left[Q \, c(\text { mic.salt }) \, y_{\pm} / c_{r}\right] \end{aligned}\]At equilibrium, \[\mathrm{n} \, \mu^{\mathrm{eq}}\left(\mathrm{AM}^{+} \mathrm{Br}^{-} ; \mathrm{aq}\right)=\mu^{\mathrm{eq}}(\text { mic. salt; aq })\]Hence, \[\Delta_{\text {mics slt }} \mathrm{G}^{0}=-\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{K}_{0}\right)=\mu^{0}(\text { mic.salt })-\mathrm{n} \, \mu^{0}\left(\mathrm{AM}^{+} \mathrm{Br}^{-}\right)\]The total concentration of salt ctot in the system is given by equation (y). \[\operatorname{ctot}\left(\mathrm{AM}^{+} \mathrm{Br}^{-} ; \text {system }\right)=\mathrm{n} \, \mathrm{c}(\text { ch arg ed micelles })+\mathrm{c}\left(\mathrm{AM}^{+} \mathrm{Br}^{-} ; \mathrm{aq}\right)\]The analysis makes no explicit reference to a cmc. Instead the micellar system is described as a mixed salt solution. Application of these equations requires careful computer –based curve fitting for multi-parametric equations. The latter include equations relating mean ionic activity coefficients for salts to the composition of a given solution. A shielding factor \(\delta\) was use by Burchfield and Woolley to reduce the impact of micellar charge of the cationic micelles on calculated ionic strength. Thus the effective charge on the cationic micelles was written as \(n \,(1-\beta) \, \delta\) where \(\delta\) is approx. 0.5.In general terms the equilibrium between surfactant monomers \(\mathrm{Z}^{+}\), counter anions \(\mathrm{X}^{-}\) and micelles \(\mathrm{M}\) can be represented by the following equation. \[\mathrm{n} Z^{+}(\mathrm{aq})+\mathrm{mX} \mathrm{X}^{-}(\mathrm{aq}) \Leftrightarrow \mathrm{M}^{(\mathrm{n}-\mathrm{m})+}(\mathrm{aq})\]Then in terms of the mass action model, the concentration equilibrium constant, \[\mathrm{K}_{\mathrm{c}}^{0}=\left[\mathrm{M}^{(\mathrm{n}-\mathrm{m})+}\right] /\left\{\left[\mathrm{Z}^{+}\right]^{\mathrm{n}} \,\left[\mathrm{X}^{-}\right]^{\mathrm{m}}\right\}\]By definition, \[\Delta_{\text {mic }} \mathrm{G}^{0}=-(\mathrm{n})^{-1} \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{K}_{\mathrm{c}}^{0}\right)\]Then, \[\Delta_{\text {mic }} \mathrm{G}^{0} /(\mathrm{R} \, \mathrm{T})=-(\mathrm{n})^{-1} \, \ln \left[\mathrm{M}^{(\mathrm{n}-\mathrm{m})+}\right]+\ln \left[\mathrm{Z}^{+}\right]+(\mathrm{m} / \mathrm{n}) \, \ln \left[\mathrm{X}^{-}\right]\] N. M. van Os, J. R. Haak and L. A. M. Rupert, Physico – Chemical Properties of Selected Anionic, Cationic and Non-ionic Surfactants, Elsevier, Amsterdam 1993. T. E. Burchfield, and E. M. Woolley, J. Phys. Chem.,1984,88,2149. T. E. Burchfield and E. M. Woolley, in Surfactants in Solution, ed. K. L. Mittal and P. Bothorel, Plenum Press, New Yok, 1987, volume 4, 69. E. M. Woolley and T. E. Burchfield, J. Phys. Chem.,1984,88,2155. T. E. Burchfield and E.M.Wooley, Fluid Phase Equilib., 1985,20,207. D. F.Evans, M. Allen, B.W. Ninham and A. Fouda, J. Solution Chem.,1984,13,87. D. G. Archer, J. Solution Chem.,1986,15,727 M. J. Blandamer, P. M. Cullis, L. G. Soldi and M. C. S. Subha, J. Therm. Anal.,1996,46,1583. R. Zana, Langmuir, 1996,12,1208. M. J. Blandamer, K. Bijma, J. B. F. N. Engberts, P. M. Cullis, P. M. Last, K. D. Irlam and L. G. Soldi, J. Chem.Soc. Faraday Trans.,1997,93,1579; and references therein. M. J. Blandamer, W. Posthumnus, J. B. F. N. Engberts and K. Bijma, J. Mol. Liq., 1997, 73-74,91. R. DeLisi, E. Fiscaro, S. Milioto, E. Pelizetti and P. Savarino, J. Solution Chem.,1990,19, 247. M. J. Blandamer, P. M. Cullis, L. G. Soldi, J. B. F. N. Engberts, A. Kacperska, N. M. van Os and M. C. S. Subha, Adv. Colloid Interface Sci.,1995,58,171. For further references concerning the Stern Layer, see N. J. Buurma, P. Serena, M. J. Blandamer and J. B. F. N. Engberts, J. Org. Chem., 2004, 69, 3899.This page titled 1.20.4: Surfactants and Miceles: Ionics is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.20.5: Surfactants and Micelles: Mixed

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In many industrial and commercial applications, mixed surfactant systems are used. An extensive literature examines the properties of these systems.A given aqueous solution contains two surfactants \(\mathrm{X}\) and \(\mathrm{Y}\) at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\). The critical micellar concentrations are \(\mathrm{cmc}_{\mathrm{X}}^{0}\) and \(\mathrm{cmc}_{\mathrm{Y}}^{0}\). For a solution containing both surfactants \(\mathrm{X}\) and \(\mathrm{Y}\), the critical micellar concentration of the mixed surfactant is cmc(mix). Here we use a pseudo-separate phase model for the micelles. The system under consideration comprises \(\mathrm{n}_{\mathrm{X}}^{0}\) and \(\mathrm{n}_{\mathrm{Y}}^{0}\) moles of the two surfactants. [Here the superscript ‘zero’ refers to the composition of the solution as prepared using the two pure surfactants.] A property \(\mathrm{r}\) is the ratio of the concentration of surfactant \(\mathrm{Y}\) to the total concentration of the two surfactants in the solution. Thus, \[\mathrm{r}=\frac{\mathrm{n}_{\mathrm{Y}}^{0}}{\mathrm{n}_{\mathrm{X}}^{0}+\mathrm{n}_{\mathrm{Y}}^{0}}=\frac{\mathrm{c}_{\mathrm{Y}}^{0}}{\mathrm{c}_{\mathrm{X}}^{0}+\mathrm{c}_{\mathrm{Y}}^{0}}\]Here \(\mathrm{c}_{\mathrm{X}}^{0}\) and \(\mathrm{c}_{\mathrm{Y}}^{0}\) are the concentrations of the two surfactants in solution. We define a model system where cmc(mix) is a linear function of the property \(\mathrm{r}\); equation (b). \[c m c(\operatorname{mix})=\left(\mathrm{cmc}_{\mathrm{Y}}^{0}-\mathrm{cmc}_{\mathrm{X}}^{0}\right) \, \mathrm{r}+\mathrm{cmc}_{\mathrm{X}}^{0}\]Or, \[\mathrm{cmc}(\mathrm{mix})=\mathrm{r} \, \mathrm{cmc}_{\mathrm{Y}}^{0}+\mathrm{cmc}_{\mathrm{X}}^{0} \,(1-\mathrm{r})\]Hence the critical micellar concentration of the two surfactants in a given solution, \(\mathrm{cmc}_{\mathrm{X}}\) and \(\mathrm{cmc}_{\mathrm{Y}}\), depend on parameter \(\mathrm{r}\). \[\mathrm{cmc}_{\mathrm{Y}}=\mathrm{cmc}_{\mathrm{Y}}^{0} \, \mathrm{r}\]\[\mathrm{cmc}_{\mathrm{X}}=\mathrm{cmc}_{\mathrm{X}}^{0} \,(1-\mathrm{r})\]Clearly in the absence of surfactant \(\mathrm{Y}\), micelles are not formed by surfactant \(\mathrm{X}\) until the concentration exceeds \(\mathrm{cmc}_{\mathrm{X}}^{0}\). If surfactant \(\mathrm{Y}\) is added to the solution, the \(\mathrm{cmc}_{\mathrm{X}}\) changes. In other words the properties of surfactants \(\mathrm{X}\) and \(\mathrm{Y}\) in a given solution are linked. We anticipate that for real system cmc(mix) is a function of \(\mathrm{c}_{\mathrm{X}}^{0}\) and \(\mathrm{c}_{\mathrm{Y}}^{0}\) so that cmc(mix) is a function of ratio \(\mathrm{r}\) and a quantity \(\theta\). The latter takes account of surfactant-surfactant interactions in the micellar pseudophase. Then \[\mathrm{cmc}(\operatorname{mix})=\mathrm{r} \,\left(\mathrm{cmc}_{\mathrm{Y}}^{0}-\mathrm{cmc}_{\mathrm{X}}^{0}\right) \, \exp [-\theta \,(1-\mathrm{r})]+\mathrm{cmc}_{\mathrm{X}}^{0}\]For the surfactants \(\mathrm{X}\) and \(\mathrm{Y}\), \[\mathrm{cmc}_{\mathrm{Y}}(\mathrm{mix})=\mathrm{r} \, \mathrm{cmc}_{\mathrm{Y}}^{0} \, \exp [-\theta \,(1-\mathrm{r})]\]and \[\mathrm{cmc}_{\mathrm{x}}(\mathrm{mix})=\mathrm{cmc}_{\mathrm{x}}^{0} \,\{1-\mathrm{r} \, \exp [-\theta \,(1-\mathrm{r})]\}\]Therefore we envisage that the cmc of solutions containing a mixture of surfactants differs from that for model systems.We turn attention to the enthalpies of mixed surfactant solutions. In the case of a mixed aqueous solution containing surfactants \(\mathrm{X}\) and \(\mathrm{Y}\), the partial molar enthalpies of the surfactants are anticipated to depend on their concentrations. We characterize a given system by a single enthalpic interaction parameter, \(\mathrm{h}(\text {int})\). \[\mathrm{H}_{\mathrm{x}}(\mathrm{aq})=\mathrm{H}_{\mathrm{x}}^{\infty}(\mathrm{aq})+\mathrm{h}(\text {int}) \,\left(\mathrm{c}_{\mathrm{x}} / \mathrm{c}_{\mathrm{r}}\right)\]\[\mathrm{H}_{\mathrm{Y}}(\mathrm{aq})=\mathrm{H}_{\mathrm{Y}}^{\infty}(\mathrm{aq})+\mathrm{h}(\text { int }) \,\left(\mathrm{c}_{\mathrm{Y}} / \mathrm{c}_{\mathrm{r}}\right)\]Here \(\mathrm{H}_{\mathrm{x}}^{\infty}(\mathrm{aq})\) and \(\mathrm{H}_{\mathrm{Y}}^{\infty}(\mathrm{aq})\) are the ideal (infinite dilution) partial molar enthalpies of the two monomeric surfactants in aqueous solutions at defined \(\mathrm{T}\) and \(\mathrm{p}\).The micellar pseudo-separate phase comprises two surfactants amounts \(\mathrm{n}_{X}(\text { mic })\) and \(\mathrm{n}_{Y}(\text { mic })\). The mole fractions \(\mathrm{x}_{\mathrm{X}}(\mathrm{mic})\) and \(\mathrm{x}_{\mathrm{X}}(\mathrm{mic}) \left[=\left(1-\mathrm{x}_{\mathrm{X}}(\mathrm{mic})\right]\) are given by equation (k). \[\mathrm{x}_{\mathrm{X}}(\mathrm{mic})=\mathrm{n}_{\mathrm{X}}(\mathrm{mic}) /\left[\mathrm{n}_{\mathrm{x}}(\mathrm{mic})+\mathrm{n}_{\mathrm{Y}}(\mathrm{mic})\right]=1-\mathrm{x}_{\mathrm{Y}}(\mathrm{mic})\]We relate the partial molar enthalpies \(\mathrm{H}_{\mathrm{X}}(\text {mic})\) and \(\mathrm{H}_{\mathrm{Y}}(\text {mic})\) in the mixed pseudo-separate phase to the molar enthalpies of surfactants \(\mathrm{X}\) and \(\mathrm{Y}\) in pure pseudo-separate micellar phases, \(\mathrm{H}_{\mathrm{X}}^{*}(\mathrm{mic})\) and \(\mathrm{H}_{\mathrm{Y}}^{*} \text { (mic) }\) using equations (l) and (m) where \(\mathrm{U}\) is a surfactant-surfactant interaction parameter. \[\mathrm{H}_{\mathrm{X}}(\text { mic })=\mathrm{H}_{\mathrm{X}}^{*}(\mathrm{mic})+\left[1-\mathrm{x}_{\mathrm{X}}(\mathrm{mic})\right]^{2} \, \mathrm{U}\]\[\mathrm{H}_{\mathrm{Y}}(\mathrm{mic})=\mathrm{H}_{\mathrm{Y}}^{*}(\text { mic })+\left[\mathrm{x}_{\mathrm{X}}(\mathrm{mic})\right]^{2} \, \mathrm{U}\] J. H. Clint, J. Chem. Soc. Faraday Trans.,1, 1975,71,1327. P. M .Holland, Adv. Colloid Interface Sci.,1986,26,111; and references therein. A. H. Roux, D. Hetu, G. Perron and J. E. Desnoyers, J. Solution Chem.,1984,13,1. M. J. Hey, J. W. MacTaggart and C. H. Rochester, J. Chem. Soc. Faraday Trans.1, 1985,81,207. J. L. Lopez-Fontan, M. J. Suarez, V. Mosquera and F. Sarmiento, Phys. Chem. Chem. Phys.,1999,1,3583. R. DeLisi, A. Inglese, S. Milioto and A. Pellerito, Langmuir, 1997,13,192. M. J. Blandamer, B. Briggs, P. M. Cullis and J. B. F. N. Engberts, Phys. Chem. Chem. Phys.,2000,2,5146. J. F. Rathman and J. F. Scamehorn, Langmuir, 1988,4,474. A. H. Roux, D. Hetu, G. Perron and J. E. Desnoyers, J. Solution Chem.,1984,13,1.This page titled 1.20.5: Surfactants and Micelles: Mixed is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.21.1: Thermodynamics and Mathematics

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Thermodynamics provides a basis for the mathematical description of important phenomena such as chemical equilibria, solubilities, densities, and heats of reaction. Chemists have confidence in this approach to chemistry. However for many chemists it is somewhat of a shock to discover that at the heart of mathematics there is serious flaw. In 1931 K. Godel showed that there is a fundamental inconsistency in mathematics. In other words mathematics is incomplete. Nevertheless chemists do not ‘throw out the baby with the bathwater’. Atkins notes that it would be foolish to discard mathematics even though there are treacherous regions deep inside its structure. An interesting account is given by D. R. Hofstadter in Godel, Esher and Bach;An Extended Golden Braid, Vintage, New York, 1980. P. W. Atkins, Galileo’s Finger, Oxford, 2003, chapter 10.This page titled 1.21.1: Thermodynamics and Mathematics is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.21.2: Thermodynamic Energy

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The thermodynamic energy \(\mathrm{U}\) of a closed system increases when work \(\mathrm{w}\) is done by the surroundings on the system and heat \(\mathrm{q}\) flows from the surroundings into the system. \[\Delta \mathrm{U}=\mathrm{q}+\mathrm{w}\]Equation (a) uses the acquisitive convention. In effect we record all changes from the point of view of the system.This page titled 1.21.2: Thermodynamic Energy is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.21.3: Thermodynamic Energy: Potential Function

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The Master Equation states that the change in thermodynamic energy of a closed system is given by equation (a). \[\mathrm{dU}=\mathrm{T} \, \mathrm{dS}-\mathrm{p} \, \mathrm{dV}-\mathrm{A} \, \mathrm{d} \xi ; \quad \mathrm{A} \, \mathrm{d} \xi \geq 0\]At constant entropy (i.e. \(\mathrm{dS} = 0\)) and constant volume (i.e. \(\mathrm{dV} = 0\)), equation (a) leads to equation (b). \[\mathrm{dU}=-\mathrm{A} \, \mathrm{d} \xi\]But \[A \, d \xi \geq 0\]Therefore all spontaneous processes at constant \(\mathrm{S}\) and constant \(\mathrm{V}\) take place in a direction for which the thermodynamic energy decreases.The latter statement shows the power of thermodynamics in that it is quite general; we have not stated the nature of the spontaneous process. Of course chemists are interested in those cases where the spontaneous process is chemical reaction. Thus we have a signal of what happens to the energy of the system; the key word here is spontaneous.In the context of most chemists interests, equation (b) is not terribly helpful. Chemists do not normally run their experiments at constant \(\mathrm{S}\) and constant \(\mathrm{V}\). In fact it is not obvious how one might do this. Nevertheless equation (b) is important finding its application when we turn to other thermodynamic variables which can be used as thermodynamic potentials; e.g. Gibbs energy.This page titled 1.21.3: Thermodynamic Energy: Potential Function is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.21.4: Thermodynamic Potentials

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The following important equations describe changes in thermodynamic energy, enthalpy, Helmholtz energy and Gibbs energy of a closed system. \[p\mathrm{dU}=\mathrm{T} \, \mathrm{dS}-\mathrm{p} \, \mathrm{dV}+\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{k}} \mu_{\mathrm{j}} \, \mathrm{dn} \mathrm{j}_{\mathrm{j}}\]\[\mathrm{dH}=\mathrm{T} \, \mathrm{dS}+\mathrm{V} \, \mathrm{dp}+\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{k}} \mu_{\mathrm{j}} \, \mathrm{dn} \mathrm{j}_{\mathrm{j}}\]\[\mathrm{dF}=-\mathrm{S} \, \mathrm{dT}-\mathrm{p} \, \mathrm{dV}+\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{k}} \mu_{\mathrm{j}} \, \mathrm{dn} \mathrm{j}_{\mathrm{j}}\]\[\mathrm{dG}=-\mathrm{S} \, \mathrm{dT}+\mathrm{V} \, \mathrm{dp}+\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{k}} \mu_{\mathrm{j}} \, \mathrm{dn} \mathrm{j}_{\mathrm{j}}\]These four differential equations relate, for example, the change in \(\mathrm{U}, \mathrm{~H}, \mathrm{~F} \text { and } \mathrm{G}\) with the change in amount of each chemical substance, \(\mathrm{dn}_{j}\). These four equations are integrated to yield the following four equations. \[\mathrm{U}=\mathrm{T} \, \mathrm{S}-\mathrm{p} \, \mathrm{V}+\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{k}} \mathrm{n}_{\mathrm{j}} \, \mu_{\mathrm{j}}\]\[\mathrm{H}=\mathrm{T} \, \mathrm{S} +\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{k}} \mathrm{n}_{\mathrm{j}} \, \mu_{\mathrm{j}}\]\[\mathrm{F}=-\mathrm{p} \, \mathrm{V}+\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{k}} \mathrm{n}_{\mathrm{j}} \, \mu_{\mathrm{j}}\]\[G=\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{k}} \mathrm{n}_{\mathrm{j}} \, \mu_{\mathrm{j}}\]The latter equation is particularly useful because it signals that the total Gibbs energy of a system is given by the sum of the products of amounts and chemical potentials of all substances in the system. In the case of an aqueous solution containing \(\mathrm{n}_{1}\) moles of water and \(\mathrm{n}_{j}\) moles of chemical substance \(j\), the Gibbs energy of the solution is given by equation (i). \[\mathrm{G}(\mathrm{aq})=\mathrm{n}_{1} \, \mu_{1}(\mathrm{aq})+\mathrm{n}_{\mathrm{j}} \, \mu_{\mathrm{j}}(\mathrm{aq})\]In conjunction with equation (i) we do not have to attach the phrase ‘at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\)’. Similarly the volume of the solution is given by equation (j) where \(\mathrm{V}_{1}(\mathrm{aq})\) and \(\mathrm{V}_{j}(\mathrm{aq})\) are the partial molar volumes of solvent and solute respectively. \[\mathrm{V}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{V}_{1}(\mathrm{aq})+\mathrm{n}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}(\mathrm{aq})\]The same argument applies in the case of a solution prepared using \(\mathrm{n}_{1}\) moles of solvent water, \(\mathrm{n}_{\mathrm{x}}\) moles of solute \(\mathrm{X}\) and \(\mathrm{n}_{\mathrm{y}}\) moles of solute \(\mathrm{Y}\). Then, for example, \[\mathrm{G}(\mathrm{aq})=\mathrm{n}_{1} \, \mu_{1}(\mathrm{aq})+\mathrm{n}_{\mathrm{x}} \, \mu_{\mathrm{x}}(\mathrm{aq})+\mathrm{n}_{\mathrm{y}} \, \mu_{\mathrm{y}}(\mathrm{aq})\]The analogue of equation (j) also follows but only if \(\mathrm{n}_{\mathrm{x}}\) and \(\mathrm{n}_{\mathrm{y}}\) are independent of pressure. If these two solutes are in chemical equilibrium [eg. \(\mathrm{X}(\mathrm{aq}) \Leftrightarrow \mathrm{Y}(\mathrm{aq})\), amounts \(\mathrm{n}_{\mathrm{x}}^{\mathrm{eq}}\) and \(\mathrm{n}_{\mathrm{y}}^{\mathrm{eq}}\) respectively], then account must be taken of the dependences of \(\mathrm{n}_{\mathrm{x}}^{\mathrm{eq}}\) and \(\mathrm{n}_{\mathrm{y}}^{\mathrm{eq}}\) on pressure at fixed temperature and (with reference to entropies and enthalpies) on temperature at fixed pressure.The simple form of equations (i) and (k) emerge from equation (h) because other than the composition variables, the other differential terms \(\mathrm{dT}\) and \(\mathrm{dp}\) in equation (d) refer to change in intensive variables. For this reason chemists find it advantageous to describe chemical properties in the \(\mathrm{T}-\mathrm{p}\)-composition domain.The relationships between thermodynamic potentials are described as Legendre transforms. The product term \(\mathrm{T} \, \mathrm{S}\) may be called bounded energy. Then the Helmholtz energy (\(\mathrm{F}=\mathrm{U}-\mathrm{T} \, \mathrm{S}\)) is the free internal energy and the Gibbs energy (\(\mathrm{G}=\mathrm{H}-\mathrm{T} \, \mathrm{S}\)) is the free enthalpy. These comments help to understand the old designations of free Helmholtz energy (together with symbol \(\mathrm{F}\)), free Gibbs energy and the still currently used (in the French scientific literature) free enthalpy.Footnotes The term “integrated” in this context deserves comment. Within the set of variables, \(\mathrm{p}-\mathrm{V}-\mathrm{T}-\mathrm{S}\), \(\mathrm{p}\) and \(\mathrm{T}\) are intensive whereas \(\mathrm{V}\) and \(\mathrm{S}\) are extensive variables. Similarly \(\mu_{j}\) is intensive whereas \(\mathrm{n}_{j}\) is extensive. These are the conditions for operating Euler’s integration method. Still the word “integrate” in the present context has been used in subtle arguments when Euler’s theorem is not invoked.E. F. Caldin [Chemical Thermodynamics, Oxford, 1958, p. 166] identifies \(\mathrm{T}, \mathrm{~p} \text { and } \mu_{j}\) as intensive and then integrates by gradual increments of the amount of each chemical substance, keeping the relative amounts constant.K. Denbigh [The Principles of Chemical Equilibrium, Cambridge, 1971, 3rd edn. p. 93] uses a similar argument, but comments that development of the equations (e) to (h) is not mathematical in the sense that the variables are simple. Rather we use our physical knowledge in that intensive variables do not depend on the state of the system.E. A. Guggenheim [Thermodynamics, North-Holland, Amsterdam, 1950, 2nd edn. p. 23] states that the equations (a) to (d) can be integrated by following the artifice when \(\mathrm{dT} = 0, \mathrm{~dp = 0\) and each \(\mathrm{n}_{j}\) is changed by the same proportions as are the extensive variables \(\mathrm{S}\) and \(\mathrm{V}\).{The term artifice is used here to mean a ‘device’, skill rather than “trickery” or “something intended to deceive”; Pocket Oxford Dictionary, Oxford 1942, 4th edn. and Cambridge International Dictionary of English, Cambridge, 1995. \[G=\sum_{j=1}^{j=k} n_{j} \, \mu_{j}\] Then (cf. definition of \(\mathrm{G}\)) \[\mathrm{G}=\mathrm{U}-\mathrm{T} \, \mathrm{S}+\mathrm{p} \, \mathrm{V}\] Then from (a), \[\mathrm{F}=\mathrm{U}-\mathrm{T} \, \mathrm{S}\] Then from (a), \[\mathrm{H}=\mathrm{U}+\mathrm{p} \, \mathrm{V}\]This page titled 1.21.4: Thermodynamic Potentials is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.21.5: Thermodynamic Stability: Chemical Equilibria

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At fixed temperature and pressure, all spontaneous processes lower the Gibbs energy of a closed system. Thermodynamic equilibrium corresponds to the state where \(\mathrm{G}\) is a minimum and the affinity for spontaneous change is zero. The equilibrium is stable. The condition for chemical thermodynamic stability is that \[(\partial \mathrm{A} / \partial \xi)_{\mathrm{T}, \mathrm{p}}<0\]This page titled 1.21.5: Thermodynamic Stability: Chemical Equilibria is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.21.6: Thermodynamic Stability: Thermal, Diffusional and Hydrostatic

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On the bench in front of a chemist is a stoppered flask containing a liquid mixture, ethanol + water, at temperature \(\mathrm{T}\) and ambient pressure \(\mathrm{p}\). The chemist might wonder why the homogeneous liquid does not spontaneously separate into two liquids, say water-rich and alcohol-rich mixtures. The chemist might also wonder why the mixture does not spontaneously produce a system which comprises a warm liquid mixture and a cold liquid mixture. Yet the fact that these changes do not occur spontaneously leads to the conclusion that the conditions in operation which forbid these changes can be traced to the Second Law of Thermodynamics, prompted by the word ‘spontaneously’ used above.Initially a given closed system has thermodynamic energy \(2\mathrm{U}\) and volume \(2\mathrm{V}\). We imagine that the mixture does in fact separate into two liquids, both at equilibrium, having energy \(\mathrm{U} + \delta \mathrm{U}\) with volume \(\mathrm{V}\), and energy \(\mathrm{U} - \delta \mathrm{U}\) also with volume \(\mathrm{V}\). The overall change in entropy at constant overall composition is given by equation (a). \[\delta \mathrm{S}=\mathrm{S}(\mathrm{U}+\delta \mathrm{U}, \mathrm{V})+\mathrm{S}(\mathrm{U}-\delta \mathrm{U}, \mathrm{V})-\mathrm{S}(2 \mathrm{U}, 2 \mathrm{~V})\]The change in entropy can be understood in terms of a Taylor expansion for a change at constant volume \(\mathrm{V}\). Thus \[\delta S=\left(\frac{\partial^{2} S}{\partial U^{2}}\right)_{V, \xi} \,(\delta U)^{2}\]However at constant \(\mathrm{V}\) and composition \(\xi\), the Second Law of Thermodynamics requires that \(\delta \mathrm{S}\) is positive for all spontaneous processes. The fact that such a change is not observed requires that \(\left(\frac{\partial^{2} \mathrm{~S}}{\partial \mathrm{U}^{2}}\right)\) is negative. But \[\left(\frac{\partial \mathrm{S}}{\partial \mathrm{U}}\right)_{\mathrm{V}, \xi}=\mathrm{T}^{-1}\]Then \[\left(\frac{\partial^{2} \mathrm{~S}}{\partial \mathrm{U}^{2}}\right)_{\mathrm{V}, \xi}=\frac{\partial}{\partial \mathrm{U}}\left(\mathrm{T}^{-1}\right)=-\frac{1}{\mathrm{~T}^{2}} \,\left(\frac{\partial \mathrm{T}}{\partial \mathrm{U}}\right)_{\mathrm{V}, \xi}=-\frac{1}{\mathrm{~T}^{2} \, \mathrm{C}_{\mathrm{V} \xi}}\]In order for the latter condition to hold, \(\mathrm{C}_{\mathrm{V}\xi}\) must be positive. This is therefore the condition for thermal stability. In other words we will not observe spontaneous separation into hot and cold domains in that heat capacities are positive variables.A given system at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\) contains \(2 \, \mathrm{n}_{\mathrm{i}}\) moles of each i-chemical substance, for \(\mathrm{i} = 1, 2, 3 \ldots\). The system is divided into two parts such that each part at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\) contains \(\mathrm{n}_{\mathrm{i}}\) moles of each chemical substance for \(\mathrm{i} = 2, 3, 4, \ldots\). However one part contains \(\mathrm{n}_{1} + \Delta \mathrm{n}_{1}\) moles and the other part contains \(\mathrm{n}_{1} - \Delta \mathrm{n}_{1}\) moles of chemical substance 1. Then the change in Gibbs energy \(\delta \mathrm{G}\) is given by equation (e). \[\begin{gathered} \delta \mathrm{G}=\mathrm{G}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}+\Delta \mathrm{n}_{1}, \mathrm{n}_{2}, \mathrm{n}_{3} . .\right]+\mathrm{G}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}-\Delta \mathrm{n}_{1}, \mathrm{n}_{2}, \mathrm{n}_{3} . .\right] \\ -\mathrm{G}\left[\mathrm{T}, \mathrm{p}, 2 \, \mathrm{n}_{1}, 2 \, \mathrm{n}_{2}, 2 \, \mathrm{n}_{3} . .\right] \end{gathered}\]In terms of a Taylor expansions, \[\delta G=\left(\frac{\partial^{2} G}{\partial n_{1}^{2}}\right)_{T, p, n, n . .} \,\left(\delta n_{1}\right)^{2}\]But chemical potential, \[\mu_{1}=\left(\frac{\partial \mathrm{G}}{\partial \mathrm{n}_{1}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}, \mathrm{n} \ldots}\]Then \[\left(\frac{\partial^{2} \mathrm{G}}{\partial \mathrm{n}_{1}^{2}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}, \mathrm{n} \ldots}=\left(\frac{\partial \mu_{1}}{\partial \mathrm{n}_{1}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}, \mathrm{n} \ldots .}\]We conclude that spontaneous separation of the system into parts rich and depleted in chemical substance-1 would occur if \(\left(\frac{\partial \mu_{1}}{\partial n_{1}}\right)_{T, p, n, n}\) is negative. But this process is never observed. Hence the condition for diffusional (or material) stability is that \(\left(\frac{\partial \mu_{1}}{\partial n_{1}}\right)_{T, p, n, n \ldots .}>0\). Further if we add \(\delta \mathrm{n}_{1}\) moles of chemical substance to a closed system at fixed \(\mathrm{T}, \mathrm{p}, \mathrm{n}_{2}, \mathrm{n}_{3} \ldots\), chemical potential µ1 must increase.A given system at temperature \(\mathrm{T}\) and chemical composition \(\xi\) has volume \(2\mathrm{V}\). We imagine that a infinitely thin partition exists separating the system into two parts having equal volumes \(\mathrm{V}\). The Helmholtz energy of the system volume \(2\mathrm{V}\) is given by equation (i). \[\mathrm{F}=\mathrm{F}[\mathrm{T}, 2 \mathrm{~V}, \xi]\]The Helmholtz energy of the two parts, volume \(\mathrm{V}\) is given by equation (j). \[\mathrm{F}=\mathrm{F}[\mathrm{T}, \mathrm{V}, \xi]\]The partition is envisaged as moving to produce two parts having volumes \((\mathrm{V}+\delta \mathrm{V})\) and \((\mathrm{V}-\delta \mathrm{V})\). Then at constant composition the change in Helmholtz energy is given by equation (k). \[\delta \mathrm{F}=\mathrm{F}[\mathrm{T}, \mathrm{V}+\delta \mathrm{V}]+\mathrm{F}[\mathrm{T}, \mathrm{V}-\delta \mathrm{V}]-\mathrm{F}[\mathrm{T}, 2 \mathrm{~V}]\]Then using Taylor’s theorem, \[\delta \mathrm{F}=\left(\partial^{2} \mathrm{~F} / \partial \mathrm{V}^{2}\right)_{\mathrm{T}} \,(\partial \mathrm{V})^{2}\]Hence \[\left(\partial^{2} \mathrm{~F} / \partial \mathrm{V}^{2}\right)_{\mathrm{T}}=-(\partial \mathrm{p} / \partial \mathrm{V})_{\mathrm{T}, \xi}\]Irrespective of the sign of (\(\delta \mathrm{V}\)), we conclude that \(\delta \mathrm{F}\) would be negative in the event that \((\partial \mathrm{p} / \partial \mathrm{V})_{\mathrm{T}, \xi}\) is positive. But we never witness such a spontaneous separation of a system into two parts. In other words \((\partial \mathrm{p} / \partial \mathrm{V})_{\mathrm{T}, \xi}<0\). Hence, \[\kappa_{\mathrm{T}}=-\left(\frac{1}{\mathrm{~V}}\right) \,\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \xi}>0\]Therefore for a system at fixed composition and temperature, if we as observers of this system increase the pressure \(\mathrm{p}\), the volume of the system decreases. This is the condition of hydrostatic (or, mechanical) stability.Taken together, conditions for thermal stability \(\left(\mathrm{C}_{\mathrm{V} \xi}>0\right)\) and for mechanical stability \(\left(\kappa_{\mathrm{T}}>0\right)\) have further consequences. Since, \[\sigma=\mathrm{C}_{\mathrm{V} \xi} / \mathrm{V}+\frac{\mathrm{T} \,\left(\alpha_{\mathrm{p}}\right)^{2}}{\kappa_{\mathrm{T}}}\]Then isobaric heat capacities and heat capacitances , \(\sigma\left(=\mathrm{C}_{\mathrm{pg}} / \mathrm{V}\right)\) are positive for stable phases.Furthermore, heat capacities and compressibilities are related by equation (p). \[\frac{\mathrm{K}_{\mathrm{S}}}{\mathrm{K}_{\mathrm{T}}}=\frac{\mathrm{C}_{\mathrm{V}}}{\mathrm{C}_{\mathrm{p}}}\]Hence the isentropic compressibility \(\kappa_{\mathrm{S}}\) of a stable phase must also be positive. In summary, \[\mathrm{C}_{\mathrm{pm}} \geq \mathrm{C}_{\mathrm{Vm}_{\mathrm{m}}}>0\]\[\kappa_{\mathrm{T}} \geq \kappa_{\mathrm{S}}>0\] M. L. McGlashan, Chemical Thermodynamics, Academic Press, London, 1979, chapter 7. By definition, \(\mathrm{F}=\mathrm{F}[\mathrm{T}, \mathrm{V}, \xi]\). The total differential of the latter equation is as follows. \[\mathrm{dF}=\left(\frac{\partial \mathrm{F}}{\partial \mathrm{T}}\right)_{\mathrm{V}, \xi} \, \mathrm{dT}+\left(\frac{\partial \mathrm{F}}{\partial \mathrm{V}}\right)_{\mathrm{T}, \xi} \, \mathrm{dV}+\left(\frac{\partial \mathrm{F}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{V}} \, \mathrm{d} \xi\]But (the ‘all-minus’ equation) \[\mathrm{dF}=-\mathrm{S} \, \mathrm{dT}-\mathrm{p} \, \mathrm{dV}-\mathrm{A} \, \mathrm{d} \xi\]Then, \[\left(\frac{\partial \mathrm{F}}{\partial \mathrm{V}}\right)_{\mathrm{T}, \xi}=-\mathrm{p}\]Or, \[\left(\frac{\partial^{2} \mathrm{~F}}{\partial \mathrm{V}^{2}}\right)_{\mathrm{T}, \xi}=-\left(\frac{\partial \mathrm{p}}{\partial \mathrm{V}}\right)_{\mathrm{T}, \xi}\] H. B. Callen, Thermodynamics and an Introduction to Thermostatistics, Wiley, New York, 2nd edn., 1985, pp.209-210.This page titled 1.21.6: Thermodynamic Stability: Thermal, Diffusional and Hydrostatic is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.21.7: Thermodynamics and Kinetics

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A given system at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\) is prepared using \(\mathrm{n}_{1}\) moles of water(\(\ell\)), the solvent, together with \(\mathrm{n}_{\mathrm{X}}^{0}\) and \(\mathrm{n}_{\mathrm{Y}}^{0}\) moles of chemical substances \(\mathrm{X}\) and \(\mathrm{Y}\) respectively at time ‘\(\mathrm{t} = 0\)’. The molalities of these solutes are \(\mathrm{m}_{\mathrm{X}}^{0}\left(=\mathrm{n}_{\mathrm{X}}^{0} / \mathrm{n}_{1} \, \mathrm{M}_{1}=\mathrm{n}_{\mathrm{X}}^{0} / \mathrm{w}_{1}\right)\) and \(\mathrm{m}_{\mathrm{Y}}^{0}\left(=\mathrm{n}_{\mathrm{Y}}^{0} / \mathrm{n}_{1} \, \mathrm{M}_{1}=\mathrm{n}_{\mathrm{Y}}^{0} / \mathrm{w}_{1}\right)\) respectively at time ‘\(\mathrm{t} = 0\)’; the concentrations are \(\mathrm{c}_{\mathrm{X}}^{0}\left(=\mathrm{n}_{\mathrm{XA}}^{0} / \mathrm{V}\right)\) and \(\mathrm{c}_{\mathrm{Y}}^{0}\left(=\mathrm{n}_{\mathrm{Y}}^{0} / \mathrm{V}\right)\) respectively.Spontaneous chemical reaction leads to the formation of product \(\mathrm{Z}\). Here we consider this spontaneous change from the standpoints of chemical thermodynamics and chemical kinetics.The spontaneous chemical reaction is driven by the affinity for chemical reaction, \(\mathrm{A}\). At each stage of the reaction the composition is described by the extent of reaction \(\xi\). The affinity \(\mathrm{A}\) is defined by the thermodynamic independent variables, \(\mathrm{T}, \mathrm{~p} \text { and } \xi\). Thus \[\mathrm{A}=\mathrm{A}[\mathrm{T}, \mathrm{p}, \xi]\]Therefore \[\mathrm{dA}=\left(\frac{\partial \mathrm{A}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \xi} \, \mathrm{dT}+\left(\frac{\partial \mathrm{A}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \xi} \, \mathrm{dp}+\left(\frac{\partial \mathrm{A}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}} \, \mathrm{d} \xi\]At constant \(\mathrm{T}\) and \(\mathrm{p}\), \[\mathrm{dA}=\left(\frac{\partial \mathrm{A}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}} \, \mathrm{d} \xi\]In terms of thermodynamics, the reference point is thermodynamic equilibrium where the affinity for spontaneous change is zero and the composition is \(\xi^{\mathrm{eq}}\).In the context of chemical reaction in solution, the system under study is, conventionally, a very dilute solution so that from a macroscopic standpoint the system at ‘\(\mathrm{t} = 0\)’ is slightly displaced from equilibrium where \(\mathrm{A}\) is zero. Thus chemists exploit their skill in monitoring for a solution the change with time of the absorbance at fixed wavelength, electrical conductivity, pH…. In a key assumption, the rate of change of composition \(\mathrm{d}\xi / \mathrm{~dt}\) is proportional to the affinity \(\mathrm{A}\) for spontaneous change; \[\mathrm{d} \xi / \mathrm{dt}=\mathrm{L} \, \mathrm{A}\]Here \(\mathrm{L}\) is a phenomenological constant describing, in the present context, the phenomenon of spontaneous chemical reaction.In general terms for processes at fixed temperature and pressure, the phenomenological property \(\mathrm{L}\) is related the isobaric – isothermal dependence of affinity \(\mathrm{A}\) on extent of chemical reaction by a relaxation time \(\tau_{\mathrm{T}, \mathrm{p}}\). Thus \[\mathrm{L}^{-1}=-\left(\frac{\partial \mathrm{A}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}} \, \tau_{\mathrm{T}, \mathrm{p}}\]Relaxation time \(\tau_{\mathrm{T},\mathrm{p}\) is a macroscopic property of a given system (at defined \(\mathrm{T}\) and \(\mathrm{p}\)) which chemists understand in terms of spontaneous chemical reaction (in a closed system). The task for chemists is to identify the actual chemical reaction in a given closed system.In most treatments of chemical reactions the reference state is chemical equilibrium where away from equilibrium the property \(\mathrm{dA}\) equals the affinity \(\mathrm{A}\) on the grounds that at equilibrium, \(\mathrm{A}\) is zero; \(\mathrm{A}=\mathrm{A}-\mathrm{A}^{\mathrm{eq}}=\mathrm{A}-0\). Hence combination of equations (c), (d) and (e) yields the key kinetic-thermodynamic equation. \[\frac{\mathrm{d} \xi}{\mathrm{dt}}=-\mathrm{A} \,\left(\tau_{\mathrm{T}, \mathrm{p}}\right)^{-1} \,\left(\frac{\partial \mathrm{A}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}^{-1}\]Equation relates the rate of change of composition to the affinity for chemical reaction and relaxation time \(\tau_{\mathrm{T}, \mathrm{p}}\). Equation (f) is therefore the key equation describing spontaneous chemical reaction in a closed system. In this context we stress the importance of equation (f).Equation (f) is an interesting and important description of the kinetics of chemical reaction. In fact the link between the rate of chemical reaction (\(\mathrm{d} \xi / \mathrm{dt}\)) and the affinity for spontaneous change \(\mathrm{A}\) is intuitively attractive. However while one may monitor the dependence of composition on time, \(\mathrm{d} \xi / \mathrm{dt}\), it is not immediately obvious how one might estimate the affinity \(\mathrm{A}\) and the property \(\left(\frac{\partial \mathrm{A}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}\) at time \(\mathrm{t}\). The Law of Mass Action offers a way forward although this law does not emerge from either the First or Second Laws of Thermodynamics. As Hammett notes the Law of Mass Action was ‘first derived from limited observations’ and became ‘established through accumulation of observations with the principle and in the absence of contradictory evidence’. After the ‘Dark Ages’, came the renaissance and ‘Bartlett and Ingold and Peterson… accepting without question or comment the validity of the law of mass action’.The link back to thermodynamics was constructed using Transition State Theory developed by Eyring and described by Glasstone, Laidler and Eyring. Therefore the phenomenological Law of Mass Action was brought into the fold of thermodynamics by offering a language which allowed activation parameters to be understood in terms of, for example, standard enthalpies and standard isobaric heat capacities of activation. I. Prigogine and R. Defay, Chemical Thermodynamics, (transl. D. H. Everett) Longmans Green, London, 1954. See for example, M. J. Blandamer, Introduction to Chemical Ultrasonics, Academic Press, London, 1973. E. F. Caldin, Fast Reactions in Solution, Blackwell, Oxford, 1964. L. P. Hammett, Physical Organic Chemistry, McGraw-Hill, New York, McGraw-Hill, New York, 2nd. edition, 1970, p.94. S. Glasstone, K. J. Laidler and H. Eyring, The Theory of Rate Processes, McGraw-Hill, New York,1941.This page titled 1.21.7: Thermodynamics and Kinetics is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.21.8: Third Law of Thermodynamics

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The Third Law of Thermodynamics states that the entropy and the heat capacity of a perfect crystal vanish in the limit of zero kelvin.When a system is heated, energy is stored in the form of molecular vibrations leading to an increase in entropy. The isochoric heat capacity is related to the differential dependence of energy on temperature at equilibrium.A more detailed analysis is required if phase changes and inter-component mixing is involved. Here there are additional contributions to the change in entropy accompanying irreversible processes which Gurney describes as the cratic part of the entropy. In a certain sense an isentropic process is ‘a draw between Maxwell’s demon and a natural process’.Footnotes. K. S. Pitzer, Thermodynamics, McGraw-Hill, New York, 3rd. edn.,1965. R. W. Gurney, Ionic Processes in Solution, McGraw-Hill, New York, 1953. A. B. Pippard, The Elements of Classical Thermodynamics, University Press, Cambridge ,1957, p.99.This page titled 1.21.8: Third Law of Thermodynamics is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.22.1: Volume: Partial Molar: General Analysis

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At temperature \(\mathrm{T}\), pressure \(\mathrm{p}\) and equilibrium, the volume of a closed system containing i-chemical substances where the amounts can be independently varied, is defined by the following equation. \[\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{2}, \ldots \ldots \mathrm{n}_{\mathrm{i}}\right]\]Or, in general terms according to Euler’s theorem, \[\mathrm{V}=\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{n}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}\]where \[\mathrm{V}_{\mathrm{j}}=\left(\frac{\partial \mathrm{V}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}_{\mathrm{i} \neq \mathrm{j}}}\]The general differential of equation (b) has the following form. \[\mathrm{dV}=\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{n}_{\mathrm{j}} \, \mathrm{dV} \mathrm{V}_{\mathrm{j}}+\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{V}_{\mathrm{j}} \, \mathrm{dn} \mathrm{n}_{\mathrm{j}}\]The general differential of equation (a) has the following form \[\mathrm{dV}=\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \mathrm{n}_{\mathrm{i}}} \, \mathrm{dT}+\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{n}_{\mathrm{i}}} \, \mathrm{dp}+\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}}\left(\frac{\partial \mathrm{V}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}_{\mathrm{i} \neq \mathrm{j}}} \, \mathrm{dn}_{\mathrm{j}}\]Comparison of equations (d) and (e) shows that \[0=-\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \mathrm{n}_{\mathrm{i}}} \, \mathrm{dT}-\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{n}_{\mathrm{i}}} \, \mathrm{dp}+\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{n}_{\mathrm{j}} \, \mathrm{dV} \mathrm{j}_{\mathrm{j}}\]Equation (f) is the Gibbs-Duhem Equation with respect to the volumetric properties of a closed system at equilibrium.A given closed system contains \(\mathrm{n}_{1}\) moles of solvent (water) and \(\mathrm{n}_{j}\) moles of solute \(j\) at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\). The system is at equilibrium where \(\mathrm{G}\) is a minimum, the affinity for spontaneous change \(\mathrm{A}\) is zero and the composition-organisation \(\xi^{\mathrm{eq}}\). The dependent variable volume \(\mathrm{V}\) is defined using a set of independent variables; equation (g).\[\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}\right]\]Equation (k) has an interesting property. If we multiply the extensive variables \(\mathrm{n}_{1}\) and \(\mathrm{n}_{j}\) by a factor \(\mathrm{k}\), the volume of the system equals (\(\mathrm{V}. \mathrm{~k}\)). In terms of Euler’s Theorem, the variable \(\mathrm{V}\) linked to the variables \(\mathrm{n}_{1}\) and \(\mathrm{n}_{j}\) is a homogeneous function of the first degree. The important consequence is the following key relation.\[\mathrm{V}=\mathrm{n}_{1} \, \mathrm{V}_{1}+\mathrm{n}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}\]where\[\mathrm{V}_{1}=\left(\frac{\partial \mathrm{V}}{\partial \mathrm{n}_{1}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{j})}\]and\[\mathrm{V}_{\mathrm{j}}=\left(\frac{\partial \mathrm{V}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}}\]We do not have to specify the conditions ‘at constant \(\mathrm{T}\) and \(\mathrm{p}\)’ in conjunction with equation (h) which is a mathematical identity.Footnote Degree of HomogeneityAt temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\), the volume of a closed system containing \(\mathrm{n}_{j}\) moles of each chemical substance \(j\) is given by\[\mathrm{V}=\mathrm{V}\left[\mathrm{n}_{1}, \mathrm{n}_{2} \ldots \ldots \ldots \ldots . \ldots \mathrm{n}_{\mathrm{k}}\right]\]The property volume has unit degree of homogeneity. That is to say – if the amount of each substance is increased by a factor \(\lambda\) then the volume increases by the same factor. Thus\[\mathrm{V}\left[\lambda \mathrm{n}_{1}, \lambda \mathrm{n}_{2} \ldots \ldots \ldots \ldots . . \lambda \mathrm{n}_{\mathrm{k}}\right]=\lambda \, \mathrm{V}\left[\mathrm{n}_{1}, \mathrm{n}_{2} \ldots \ldots \ldots \ldots . \ldots \mathrm{n}_{\mathrm{k}}\right]\]This page titled 1.22.1: Volume: Partial Molar: General Analysis is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.22.10: Volume: Aqueous Binary Liquid Mixtures

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For binary aqueous mixtures (at ambient pressure and fixed temperature) there are two interesting reference points.In the latter case we imagine that each molecule of liquid 2 is surrounded by an infinite expanse of water. With gradual increase in \(\mathrm{x}_{2}\), so (on average) the molecules of liquid 2 move closer together.For these systems \(\left[\mathrm{V}_{2}^{\infty}(\operatorname{mix})-\mathrm{V}_{2}^{*}(\lambda)\right]\) is negative. But this pattern is not unique to aqueous systems. The unique feature is the decrease in \(\left[\mathrm{V}_{2}(\operatorname{mix})-\mathrm{V}_{2}^{*}(\lambda)\right]\) with increase in \(\mathrm{x}_{2}\) at low \(\mathrm{x}_{2}\). In fact with increase in hydrophobicity of chemical substance 2, the decrease is more striking and the minimum in \(\left[\mathrm{V}_{2}(\operatorname{mix})-\mathrm{V}_{2}^{*}(\lambda)\right]\) occurs at lower \(\mathrm{x}_{2}\). At mole fractions beyond \(\mathrm{x}_{2}\left[\mathrm{~V}_{2}(\mathrm{mix})-\mathrm{V}_{2}^{*}(\lambda)\right]\) increases with increase in \(\mathrm{x}_{2}\). Many explanations have been offered for this complicated pattern. The following is one explanation.The negative \(\left[\mathrm{V}_{2}^{\infty}(\operatorname{mix})-\mathrm{V}_{2}^{*}(\lambda)\right]\) is accounted for in terms of a liquid clathrate in which part of the hydrophobic group ‘occupies’ a guest site in the liquid water ‘lattice’. The decrease in \(\left[\mathrm{V}_{2}(\operatorname{mix})-\mathrm{V}_{2}^{*}(\lambda)\right]\) is accounted for in terms of an increasing tendency towards a liquid clathrate hydrate structure. With increase in \(\mathrm{x}_{2}\) there comes a point where there is insufficient water to construct the liquid clathrate host. Hence \(\left[\mathrm{V}_{2}(\operatorname{mix})-\mathrm{V}_{2}^{*}(\lambda)\right]\) increases.Although \(\left[\mathrm{V}_{2}^{\infty}(\operatorname{mix})-\mathrm{V}_{2}^{*}(\lambda)\right]\) is negative no minimum is observed in \(\left[\mathrm{V}_{2}(\operatorname{mix})-\mathrm{V}_{2}^{*}(\lambda)\right]\). See for example; fluoroalcohols(aq); C. H. Rochester and J. R. Symonds, J. Fluorine Chem.,1974,4,141. F. Franks, Ann. N. Y. Acad. Sci.,1955,125,277. For many binary aqueous mixtures the patterns in volume related properties often identify transition points at ‘structurally interesting compositions’; G. Roux, D. Roberts, G. Perron and J. E. Desnoyers, J. Solution Chem.,1980,9,629.This page titled 1.22.10: Volume: Aqueous Binary Liquid Mixtures is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.22.11: Volumes: Liquid Mixtures: Binary: Method of Tangents

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The ‘Method of Tangents’ is an important technique which is readily illustrated using the volumetric properties of binary liquid mixtures. The starting point is (as always?) the Gibbs - Duhem Equation which leads to equation (a) for systems at fixed temperature and pressure. \[\mathrm{n}_{1} \, \mathrm{dV}_{1}(\operatorname{mix})+\mathrm{n}_{2} \, d \mathrm{dV}_{2}(\operatorname{mix})=0\]Dividing by \(\left(\mathrm{n}_{1} + \mathrm{~n}_{2}\right)\), \[\mathrm{x}_{1} \, d \mathrm{~V}_{1}(\operatorname{mix})+\mathrm{x}_{2} \, \mathrm{dV}_{2}(\operatorname{mix})=0\]The molar volume is given by equation (c). \[\mathrm{V}_{\mathrm{m}}(\mathrm{mix})=\mathrm{x}_{1} \, \mathrm{V}_{1}(\mathrm{mix})+\mathrm{x}_{2} \, \mathrm{V}_{2}(\mathrm{mix})\]Hence (at equilibrium, fixed temperature and pressure) the differential dependence of \(\mathrm{V}_{\mathrm{m}}(\operatorname{mix})\) on mole fraction \(\mathrm{x}_{1}\) is given by equation (d). \[\begin{aligned} \frac{\mathrm{dV}_{\mathrm{m}}(\mathrm{mix})}{\mathrm{dx}_{1}}=\mathrm{V}_{1}(\mathrm{mix}) &+\mathrm{x}_{1} \,\left[\frac{\mathrm{dV}_{1}(\mathrm{mix})}{\mathrm{dx}_{1}}\right] \\ &+\mathrm{V}_{2}(\mathrm{mix}) \,\left[\frac{\mathrm{dx}_{2}}{\mathrm{dx}}\right]+\mathrm{x}_{2} \,\left[\frac{\mathrm{dV}_{2}(\mathrm{mix})}{\mathrm{dx}_{1}}\right] \end{aligned}\]From the Gibbs-Duhem equation, \[\mathrm{x}_{1} \, \frac{\mathrm{dV}_{1}(\mathrm{mix})}{\mathrm{dx}_{1}}+\mathrm{x}_{2} \, \frac{\mathrm{dV}_{2}(\mathrm{mix})}{\mathrm{dx}_{1}}=0\][Note the common denominator.] Also \(\mathrm{x}_{1} + \mathrm{~x}_{2} = 1\). And so, \[\mathrm{dx}_{1}=-\mathrm{dx}_{2}\]Therefore, \[\frac{\mathrm{dV} \mathrm{m}_{\mathrm{m}}(\mathrm{mix})}{\mathrm{dx}_{1}}=\mathrm{V}_{1}(\operatorname{mix})-\mathrm{V}_{2}(\operatorname{mix})\]Combination of equations (c) and (g) yields the following equation. \[\mathrm{V}_{\mathrm{m}}(\mathrm{mix})=\mathrm{x}_{1} \, \mathrm{V}_{1}(\mathrm{mix})+\mathrm{x}_{2} \,\left[\mathrm{V}_{1}(\mathrm{mix})-\frac{\mathrm{dV}_{\mathrm{m}}(\mathrm{mix})}{\mathrm{dx}_{1}}\right]\]Further, \[\mathrm{x}_{1} \, \mathrm{V}_{1}(\operatorname{mix})+\mathrm{x}_{2} \, \mathrm{V}_{1}(\mathrm{mix})=\mathrm{V}_{1}(\mathrm{mix})\]Hence, \[\mathrm{V}_{1}(\mathrm{mix})=\mathrm{V}_{\mathrm{m}}(\mathrm{mix})+\left(1-\mathrm{x}_{1}\right) \, \frac{\mathrm{dV}_{\mathrm{m}}(\mathrm{mix})}{\mathrm{dx}_{1}}\]At a given mole fraction, we determine the molar volume of the mixture \(\mathrm{V}_{\mathrm{m}}(\mathrm{mix})\) and its dependence on mole fraction. \(\left[\mathrm{dV} \mathrm{m}_{\mathrm{m}}(\operatorname{mix}) / \mathrm{dx}_{1}\right]\) is the gradient of the tangent at mole fraction \(\mathrm{x}_{1}\) to the curve recording the dependence of \(\mathrm{V}_{\mathrm{m}}(\operatorname{mix})\) on \(\mathrm{x}_{1}\); hence the name of this method of data analysis. This analysis is relevant because, as commented above, we can determine the variables \(\mathrm{V}_{1}^{*}(\ell), \mathrm{V}_{2}^{*}(\ell) \text { and } \mathrm{V}_{\mathrm{m}}(\mathrm{mix})\).Another approach is based on excess molar volumes \(\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}\) and their dependence on mole fraction at fixed temperature and pressure. Since \[\mathrm{V}_{\mathrm{m}}(\mathrm{id})=\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}(\ell)\]And \[\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}=\mathrm{V}_{\mathrm{m}}(\mathrm{mix})-\mathrm{V}_{\mathrm{m}}(\mathrm{id})\]From equations (c), and (k), \[\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}=\mathrm{x}_{1} \,\left[\mathrm{V}_{1}(\mathrm{mix})-\mathrm{V}_{1}^{*}(\ell)\right]-\mathrm{x}_{2} \,\left[\mathrm{V}_{2}(\mathrm{mix})-\mathrm{V}_{2}^{*}(\ell)\right]\]We define excess partial molar volumes; \[\mathrm{V}_{1}^{\mathrm{E}}(\operatorname{mix})=\mathrm{V}_{1}(\operatorname{mix})-\mathrm{V}_{1}^{*}(\ell)\]and \[\mathrm{V}_{2}^{\mathrm{E}}(\operatorname{mix})=\mathrm{V}_{2}(\operatorname{mix})-\mathrm{V}_{2}^{*}(\ell)\]Hence the excess molar volume of the mixture is related to two excess partial molar volumes. \[\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}=\mathrm{x}_{1} \, \mathrm{V}_{1}^{\mathrm{E}}(\operatorname{mix})+\mathrm{x}_{2} \, \mathrm{V}_{2}^{\mathrm{E}}(\operatorname{mix})\]We use equation (m) to obtain the differential dependence of \(\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}\) on mole fraction \(\mathrm{x}_{1}\). \[\begin{aligned} \mathrm{dV}_{\mathrm{m}}^{\mathrm{E}}=& {\left[\mathrm{V}_{1}(\mathrm{mix})-\mathrm{V}_{1}^{*}(\ell)\right]+\mathrm{x}_{1} \,\left[\frac{\mathrm{dV}_{1}(\mathrm{mix})}{\mathrm{dx} \mathrm{x}_{1}}\right] } \\ &+\left[\frac{\mathrm{dx}_{2}}{\mathrm{dx} \mathrm{x}_{1}}\right] \,\left[\mathrm{V}_{2}(\mathrm{mix})-\mathrm{V}_{2}^{*}(\ell)\right]+\mathrm{x}_{2} \,\left[\frac{\mathrm{dV}_{2}(\mathrm{mix})}{\mathrm{dx}_{1}}\right] \end{aligned}\]We write the Gibbs - Duhem equation in the form shown in equation (e) together with equation (p). Hence, \[\mathrm{dV}_{\mathrm{m}}^{\mathrm{E}} / \mathrm{dx}_{1}=\mathrm{V}_{1}^{\mathrm{E}}-\mathrm{V}_{2}^{\mathrm{E}}\]or, \[\mathrm{V}_{2}^{\mathrm{E}}=\mathrm{V}_{1}^{\mathrm{E}}-\mathrm{dV}_{\mathrm{m}}^{\mathrm{E}} / \mathrm{dx}_{1}\]Hence using equation (o), \[\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}=\mathrm{x}_{1} \, \mathrm{V}_{1}^{\mathrm{E}}+\mathrm{x}_{2} \, \mathrm{V}_{1}^{\mathrm{E}}-\mathrm{x}_{2} \, \mathrm{dV} \mathrm{V}_{\mathrm{m}}^{\mathrm{E}} / \mathrm{dx}_{1}\]Thus, \[\mathrm{V}_{1}^{\mathrm{E}}=\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}+\left(1-\mathrm{x}_{1}\right) \, d \mathrm{~V}_{\mathrm{m}}^{\mathrm{E}} / \mathrm{dx}_{1}\]Equation (t) is the excess form of equation (j). A plot of \(\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}\) against \(\mathrm{x}_{1}\) shows a curve passing through '\(\mathrm{V}_{\mathrm{m}}^{\mathrm{E}} = 0\)' at \(\mathrm{x}_{1} = 0\) and \(\mathrm{x}_{1} = 1\). Other than these two reference points, thermodynamics does not define the shape of the plot of \(\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}\) against \(\mathrm{x}_{1}\). Thermodynamics does not define the shape of the plot of \(\mathrm{V}_{1}^{\mathrm{E}}\) against \(\mathrm{x}_{1}\) other than to require that at \(\mathrm{x}_{1} = 1\), \(\mathrm{V}_{1}^{\mathrm{E}}\) is zero. An interesting feature is the sign and magnitude of \(\mathrm{V}_{1}^{\mathrm{E}}\) in the limit that \(\mathrm{x}_{1} = 0\); i.e. at \(\mathrm{x}_{2} = 1\).The volumetric properties of a binary liquid (homogeneous) mixture is summarized by a plot of excess molar volume \(\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}\) against, for example, mole fraction \(\mathrm{x}_{1}\). In fact this type of plot is used for many excess molar properties including \(\mathrm{G}_{\mathrm{m}}^{\mathrm{E}}\) and \(\mathrm{H}_{\mathrm{m}}^{\mathrm{E}}\). Here we consider a general excess molar property \(\mathrm{X}_{\mathrm{m}}^{\mathrm{E}}\). The corresponding excess partial molar property of chemical substance 1 is \(\mathrm{X}_{\mathrm{1}}^{\mathrm{E}}\) which is related to \(\mathrm{X}_{\mathrm{m}}^{\mathrm{E}}\) and the dependence of \(\mathrm{X}_{\mathrm{m}}^{\mathrm{E}}\) on \(\mathrm{x}_{1}\) at mole fraction \(\mathrm{x}_{1}\), \[\mathrm{X}_{1}^{\mathrm{E}}=\mathrm{X}_{\mathrm{m}}^{\mathrm{E}}+\left(1-\mathrm{x}_{1}\right) \, \mathrm{dX} \mathrm{X}_{\mathrm{m}}^{\mathrm{E}} / \mathrm{dx}_{1}\]Calculation of \(\mathrm{X}_{\mathrm{1}}^{\mathrm{E}}\) requires the gradient \(\mathrm{dX} \mathrm{m} / \mathrm{dx}_{1}\) as a function of mole fraction composition. The way forward involves fitting the dependence of \(\mathrm{X}_{\mathrm{m}}^{\mathrm{E}}\) on \(\mathrm{x}_{1}\) to a general equation and then calculating \(\mathrm{dX} \mathrm{m} / \mathrm{dx}_{1}\) using the derived parameters. C. W. Bale and A. D. Pelton, Metallurg. Trans.,1974,5,2323. C. Jambon and R. Philippe, J.Chem.Thermodyn.,1975,7,479. M. J. Blandamer, N. J. Blundell, J. Burgess, H. J. Cowles and I. M. Horn, J. Chem. Soc. Faraday Trans.,1990,86,277. A description of a useful procedure for non-linear least squares analysis is given by W. E. Wentworth, J.Chem.Educ.,1965,42,96.This page titled 1.22.11: Volumes: Liquid Mixtures: Binary: Method of Tangents is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.22.12: Volume of Reaction: Dependence on Pressure

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Consider a chemical equilibrium between two solute \(\mathrm{X}(\mathrm{aq})\) and \(\mathrm{Y}(\mathrm{aq})\) in aqueous solution at fixed \(\mathrm{T}\) and \(\mathrm{p}\). We assume that the thermodynamic properties of the two solutes are ideal. The chemical equilibrium is be expressed as follows. \[\mathrm{X}(\mathrm{aq}) \Leftrightarrow \mathrm{Y}(\mathrm{aq})\]The (dimensionless intensive) degree of reaction \(\alpha\) is related to the equilibrium constant \(\mathrm{K}^{0}\) using equation (b). \[\alpha=\mathrm{K}^{0} /\left(1+\mathrm{K}^{0}\right)\]At fixed temperature, \[\frac{\mathrm{d} \alpha}{\mathrm{dp}}=\frac{\mathrm{K}^{0}}{\left(1+\mathrm{K}^{0}\right)^{2}} \, \frac{\mathrm{d} \ln \left(\mathrm{K}^{0}\right)}{\mathrm{dp}}\]Or, \[\frac{\mathrm{d} \alpha}{\mathrm{dp}}=-\frac{\mathrm{K}^{0}}{\left(1+\mathrm{K}^{0}\right)^{2}} \, \frac{\Delta_{\mathrm{r}} \mathrm{V}^{0}(\mathrm{aq})}{\mathrm{R} \, \mathrm{T}}\]\(\Delta_{\mathrm{r}} \mathrm{V}^{0}(\mathrm{aq})\) is the limiting volume of reaction. The (equilibrium) volume of the system at a defined \(\mathrm{T}\) and \(\mathrm{p}\) is given by equation (e). \[\mathrm{V}(\mathrm{aq})=\mathrm{n}_{\mathrm{x}} \, \mathrm{V}_{\mathrm{x}}^{\infty}(\mathrm{aq})+\mathrm{n}_{\mathrm{Y}} \, \mathrm{V}_{\mathrm{Y}}^{\infty}(\mathrm{aq})+\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)\]\(\mathrm{V}_{1}^{*}(\ell)\) is the molar volume of solvent, water. If \(\mathrm{n}_{\mathrm{x}}^{0}\) is total amount of solute, (i.e. \(\mathrm{X}\) and \(\mathrm{Y}\)) in the system, \[\mathrm{V}(\mathrm{aq})=(1-\alpha) \, \mathrm{n}_{\mathrm{X}}^{0} \, \mathrm{V}_{\mathrm{x}}^{\infty}(\mathrm{aq})+\alpha \, \mathrm{n}_{\mathrm{X}}^{0} \, \mathrm{V}_{\mathrm{Y}}^{\infty}(\mathrm{aq})+\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)\]Or, \[\mathrm{V}(\mathrm{aq})=\mathrm{n}_{\mathrm{X}}^{0} \, \mathrm{V}_{\mathrm{x}}^{\infty}(\mathrm{aq})+\alpha \, \mathrm{n}_{\mathrm{X}}^{0} \, \Delta_{\mathrm{r}} \mathrm{V}^{\infty}(\mathrm{aq})+\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)\]\(\Delta_{\mathrm{r}} \mathrm{V}^{\infty}(\mathrm{aq})\) is the limiting volume of reaction. We assume that at temperature \(\mathrm{T}\), the properties \(\mathrm{V}_{\mathrm{x}}^{\infty}(\mathrm{aq}), \Delta_{\mathrm{r}} \mathrm{V}^{\infty}(\mathrm{aq}) \text { and } \mathrm{V}_{1}^{*}(\ell)\) are independent of pressure. Hence using equations (d) and (g), \[\left(\frac{\partial \mathrm{V}(\mathrm{aq})}{\partial \mathrm{p}}\right)_{\mathrm{T}}=-\mathrm{n}_{\mathrm{X}}^{0} \, \frac{\left[\Delta_{\mathrm{r}} \mathrm{V}^{\infty}(\mathrm{aq})\right]^{2}}{\mathrm{R} \, \mathrm{T}} \, \frac{\mathrm{K}^{0}}{\left[1+\mathrm{K}^{0}\right]^{2}}\]We have taken account of the fact that, \[\frac{\mathrm{d} \ln \left(\mathrm{K}^{0}\right)}{\mathrm{dp}}=-\frac{\Delta_{\mathrm{r}} \mathrm{V}^{\infty}(\mathrm{aq})}{\mathrm{R} \, \mathrm{T}}\]Similarly \[\left(\frac{\partial \mathrm{V}(\mathrm{aq})}{\partial \mathrm{T}}\right)_{\mathrm{p}}=\mathrm{n}_{\mathrm{X}}^{0} \, \frac{\Delta_{\mathrm{r}} \mathrm{V}^{\infty}(\mathrm{aq}) \, \Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\mathrm{aq})}{\mathrm{R} \, \mathrm{T}^{2}} \, \frac{\mathrm{K}^{0}}{\left[1+\mathrm{K}^{0}\right]^{2}}\]Equation (h) shows that irrespective of the sign of \(\Delta_{\mathrm{r}} \mathrm{V}^{\infty}(\mathrm{aq})\), the contribution to \(\left(\frac{\partial V(a q)}{\partial p}\right)_{\mathrm{T}}\) is always negative. No such generalisation emerges with respect to equation (j). A closely related subject concerns the dependence of rate constants on pressure leading to volumes of activation. From equation (a) \[\begin{aligned} &\mathrm{dV} / \mathrm{dp}=\left[\mathrm{m}^{3}\right] /\left[\mathrm{N} \mathrm{m}^{-2}\right]=\left[\mathrm{m}^{5} \mathrm{~N}^{-1}\right]\\ &=\frac{\left[\mathrm{m}^{6}\right]}{[\mathrm{N} \mathrm{m}]}=\left[\mathrm{m}^{5} \mathrm{~N}^{-1}\right] \end{aligned}\] \[\begin{aligned} &\frac{\mathrm{dV}}{\mathrm{dT}}=\frac{\left[\mathrm{m}^{3}\right]}{[\mathrm{K}]}\\ &\mathrm{n}_{\mathrm{X}}^{0} \, \frac{\Delta_{\mathrm{r}} \mathrm{V}^{\infty}(\mathrm{aq}) \, \Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\mathrm{aq})}{\mathrm{R} \, \mathrm{T}^{2}} \, \frac{\mathrm{K}^{0}}{\left[1+\mathrm{K}^{0}\right]^{2}}\\ &=[\mathrm{mol}] \, \frac{\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right] \,\left[\mathrm{J} \mathrm{mol}^{-1}\right]}{\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}]^{2}}=\left[\mathrm{m}^{3} \mathrm{~K}^{-1}\right] \end{aligned}\] See for example, J. E. Desnoyers, Pure Appl.Chem.,1982, 54,1469. (i) W. J. leNoble, J. Chem. Educ.,1967,44,729. (ii) B. S. El’yanov and S. D. Hamann, Aust. J Chem.,1975,28,945. (iii) B.S. El’yanov and M. G. Gonikberg, Russian J. Phys. Chem., 1972, 46, 856.This page titled 1.22.12: Volume of Reaction: Dependence on Pressure is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.22.2: Volume: Components

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.22%3A_Volume/1.22.2%3A_Volume%3A_Components

For a system containing one chemical substance we define the volume as follows, \[\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}\right]\]The variables in the square brackets are called the INDEPENDENT VARIABLES. The term independent means that within limits, we can change \(\mathrm{T}\) independently of the pressure and \(\mathrm{n}_{1}\); change \(\mathrm{p}\) independently of \(\mathrm{T}\) and \(\mathrm{n}_{1}\); change \(\mathrm{n}_{1}\) independently of \(\mathrm{T}\) and \(\mathrm{p}\). There are some restrictions in our choice of independent variables. At least one of the variables must define the amount of all chemical substances in the system and one variable must define the degree of ‘hotness’ of the system.If the composition of a given closed system is specified in terms of the amounts of two chemical substances, 1 and 2, four independent variables \(\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{2}\right]\) define the independent variable \(\mathrm{V}\). \[\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{2}\right]\]For a system containing i - chemical substances where the amounts can be independently varied, the dependent variable \(\mathrm{V}\) is defined by the following equation. \[\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{2}, \ldots . \mathrm{n}_{\mathrm{i}}\right]\]This page titled 1.22.2: Volume: Components is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.22.3: Volume: Partial and Apparent Molar

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.22%3A_Volume/1.22.3%3A_Volume%3A_Partial_and_Apparent_Molar

In descriptions of the volumetric properties of solutions, two terms are extensively used. We refer to the partial molar volume of solute \(j\) in, for example, an aqueous solution \(\mathrm{V}_{j}(\mathrm{aq})\) and the corresponding apparent molar volume \(\phi\left(\mathrm{V}_{j}\right)\). Here we explore how these terms are related. We consider an aqueous solution prepared using water, \(1 \mathrm{~kg}\), and \(\mathrm{m}_{j}\) moles of solute \(j\). The volume of this solution at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\) is given by equation (a). \[\mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{V}_{1}(\mathrm{aq})+\mathrm{m}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}(\mathrm{aq})\]The chemical potential of solvent, water, in the aqueous solution \(\mu_{1}(\mathrm{aq})\) is related to the molality \(\mathrm{m}_{j}\) using equation (b) where \(\mu_{1}^{*}(\ell)\) is the chemical potential of pure water(\(\ell\)), molar mass \(\mathrm{M}_{1}\), at the same \(\mathrm{T}\) and \(\mathrm{p}\). \[\mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\ell)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\]Here practical osmotic coefficient \(\phi\) is defined by equation (c). \[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi=1 \quad \text { at all T and } \mathrm{p}\]But \[\mathrm{V}_{1}(\mathrm{aq})=\left[\partial \mu_{1}(\mathrm{aq}) / \partial \mathrm{p}\right]_{\mathrm{T}}\]Then \[\mathrm{V}_{1}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\mathrm{V}_{1}^{*}(\ell)-\mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,(\partial \phi / \partial \mathrm{p})_{\mathrm{T}}\]For the solute, the chemical potential \(\mu_{j}(\mathrm{aq})\) is related to the molality of solute \(\mathrm{m}_{j}\) using equation (f) where pressure \(\mathrm{p}\) is close to the standard pressure. \[\mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)\]where \[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{j}=1 \quad \text { at all } \mathrm{T} \text { and } \mathrm{p}\]Then \[\mathrm{V}_{\mathrm{j}}(\mathrm{aq})=\mathrm{V}_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \,\left[\partial \ln \left(\gamma_{\mathrm{j}}\right) / \partial \mathrm{p}\right]_{\mathrm{T}}\]\[\operatorname{Limit}\left(m_{j} \rightarrow 0\right) V_{j}(a q)=V_{j}^{0}(a q)=V_{j}^{\infty}(a q)\]Combination of equations (a), (e) and (h) yields equation (j). \[\begin{gathered} \mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \,\left[\mathrm{V}_{1}^{*}(\ell)-\mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,(\partial \phi / \partial \mathrm{p})_{\mathrm{T}}\right] \\ \quad+\mathrm{m}_{\mathrm{j}} \,\left\{\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \,\left[\partial \ln \left(\gamma_{\mathrm{j}}\right) / \partial \mathrm{p}\right]_{\mathrm{T}}\right\} \end{gathered}\]An important point emerges if we re-arrange equation (j). \[\begin{aligned} &\mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{V}_{1}^{*}(\ell) \\ &\quad+\mathrm{m}_{\mathrm{j}} \,\left\{\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \,\left[\partial \ln \left(\gamma_{\mathrm{j}}\right) / \partial \mathrm{p}\right]_{\mathrm{T}}-\mathrm{R} \, \mathrm{T} \,(\partial \phi / \partial \mathrm{p})_{\mathrm{T}}\right\} \end{aligned}\]Equation (k) has an interesting form in that the brackets {….} contain \(\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq})\) and terms describing the extent to which the volumetric properties of the solution are not ideal in a thermodynamic sense. It is therefore convenient to define an apparent molar volume of solute \(j\), \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)\) using equation (l). \[\phi\left(\mathrm{V}_{\mathrm{j}}\right)=\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \,\left[\partial \ln \left(\gamma_{\mathrm{j}}\right) / \partial \mathrm{p}\right]_{\mathrm{T}}-\mathrm{R} \, \mathrm{T} \,(\partial \phi / \partial \mathrm{p})_{\mathrm{T}}\]Then \[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi\left(\mathrm{V}_{\mathrm{j}}\right)=\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq})\]Therefore we obtain equation (n). \[\mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\]Interest in equation (n) arises from the fact that the for a given solution \(\mathrm{V}(\mathrm{aq})\) can be measured {using the density \(\rho(\mathrm{aq})\)} and hence knowing \(\mathrm{V}_{1}^{*}(\ell)\), \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)\) is obtained. If we measure \(\phi\left(\mathrm{V}_{j}\right)\) as a function of \(\mathrm{m}_{j}\), equation (m) indicates how one obtains \(\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq})\). Moreover the difference \(\left[\phi\left(\mathrm{V}_{\mathrm{j}}\right)-\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq})\right]\) signals the role of solute - solute interactions. \[\begin{array}{r} \mathrm{R} \, \mathrm{T} \,\left[\frac{\partial \phi}{\partial \mathrm{p}}\right]=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}] \,\left[\frac{1}{\left[\mathrm{~N} \mathrm{~m}^{-2}\right]}\right] \\ =\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right] \end{array}\] \[\begin{aligned} \mathrm{R} \, \mathrm{T} \,\left[\frac{\partial \ln \left(\gamma_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right]=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}] \,\left[\frac{1}{\left[\mathrm{~N} \mathrm{~m}^{-2}\right]}\right] \\ =& {[\mathrm{N} \mathrm{m} \mathrm{mol}] \,\left[\frac{1}{\left[\mathrm{~N} \mathrm{~m}^{-2}\right.}\right]=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right] } \end{aligned}\]This page titled 1.22.3: Volume: Partial and Apparent Molar is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.22.4: Volume: Partial Molar: Frozen and Equilibrium

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.22%3A_Volume/1.22.4%3A_Volume%3A_Partial_Molar%3A_Frozen_and_Equilibrium

Consider the volume of a closed system defined by equation (a). \[\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}\right]\]This system is displaced to a neighboring state by addition of a small amount of substance \(j\), \(\delta\mathrm{n}_{j}\). The change in volume at fixed affinity \(\mathrm{A}\) is related to the change in volume at fixed composition or organization. At fixed temperature, fixed pressure, and fixed \(\mathrm{n}_{1}\), \[\left(\frac{\partial \mathrm{V}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{A}}=\left(\frac{\partial \mathrm{V}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\xi}-\left(\frac{\partial \mathrm{A}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\xi} \,\left(\frac{\partial \xi}{\partial \mathrm{A}}\right)_{\mathrm{n}_{\mathrm{j}}} \,\left(\frac{\partial \mathrm{V}}{\partial \xi}\right)_{\mathrm{n}_{\mathrm{j}}}\]For a system at equilibrium (\(\mathrm{A} = 0 \text { and } \xi = \xi^{\mathrm{eq}}\)), the triple product term on the R.H.S. of equation (b) is not zero. Hence we distinguish between two properties; \(\left(\frac{\partial \mathrm{V}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{~A}=0} \text { and } \left(\frac{\partial \mathrm{V}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{eq}^{\mathrm{eq}}}\). By convention the first of these two terms is called the partial molar volume of substance \(j\) in the system.A given aqueous solution is prepared by dissolving \(\mathrm{n}_{j}\) moles of urea in \(\mathrm{n}_{1}\) moles of water at \(298.2 \mathrm{~K}\) and ambient pressure. This system has volume \(\mathrm{V}(\mathrm{aq})\) which is determined in part by water-water, water-urea and urea-urea interactions. We add \(\delta \mathrm{n}_{j}\) moles of urea to this system but stipulate that the water-water, water-urea and urea-urea interactions remain unchanged; i.e. frozen. The property \(\left(\partial \mathrm{V} / \partial \mathrm{n}_{\mathrm{j}}\right)_{\mathrm{T} ; \mathrm{p} ; \mathrm{n}, \xi}\) is a frozen partial molar volume of urea in the aqueous solution. On the other hand, if we stipulate that the water-water, water-urea and urea-urea interactions re-adjust in order that the system is at a minimum in Gibbs energy, the property \(\left(\partial \mathrm{V} / \partial \mathrm{n}_{\mathrm{j}}\right)_{\mathrm{T}, \mathrm{p} ; \mathrm{n}, \mathrm{A}=0}\) is the equilibrium partial molar volume for urea in this aqueous solution.We consider an aqueous solution containing \(\mathrm{n}_{j}\) moles of ethanoic acid in \(\mathrm{n}_{1}\) moles of water at defined temperature and defined pressure. Conventionally, the chemical equilibrium operating in the system is expressed in the following form. \[\mathrm{HA}(\mathrm{aq}) \leftrightarrow \mathrm{H}^{+}(\mathrm{aq})+\mathrm{A}^{-}(\mathrm{aq})\]The volume of this system \(\mathrm{V}(\mathrm{HA} ; \mathrm{aq} ; \mathrm{T} ; \mathrm{p})\) is a state variable. We add \(\delta \mathrm{δ}(\mathrm{HA})\) moles of substance \(\mathrm{HA}\) to the system. In the frozen limit, the amounts of \(\mathrm{H}^{+}(\mathrm{aq})\) and \(\mathrm{A}^{-}(\mathrm{aq})\) in the solution do not change. In terms of composition all that happens is the amount of \(\mathrm{HA}(\mathrm{aq})\) increases. Hence, \(\left(\frac{\delta \mathrm{V}}{\delta \mathrm{n}(\mathrm{HA})}\right)\) is a measure of the ‘frozen partial molar volume’ of \(\mathrm{HA}\) in the system. If we remove the frozen restriction and allow chemical equilibrium to be re-established, the derived quantity is the equilibrium partial molar volume for \(\mathrm{HA}\) in this aqueous solution, part of added \(\delta\mathrm{P}mathrm{n}(\mathrm{HA})\) having dissociated in order that the resulting solution has zero affinity for spontaneous change. We use quotation marks ‘….’ Around the phrase ‘frozen partial molar volume’ to make the point that this property is not a proper equilibrium thermodynamic property.This page titled 1.22.4: Volume: Partial Molar: Frozen and Equilibrium is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.22.5: Volumes: Solutions: Apparent and Partial Molar Volumes: Determination

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.22%3A_Volume/1.22.5%3A_Volumes%3A_Solutions%3A_Apparent_and_Partial_Molar_Volumes%3A_Determination

An aqueous solution is prepared using \(\mathrm{n}_{1}\) moles of water and \(\mathrm{n}_{j}\) moles of solute. Thus, \[\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}\right]\]The density of this solution \(\rho(\mathrm{aq})\) can be accurately measured at the specified temperature and pressure together with the density of the pure solvent, \(\rho_{1}^{*}(\ell)\). The molar mass of the solute is \(\mathrm{M}_{j} \mathrm{~kg mol}^{-1}\). Two equations are encountered in the literature depending on the method used to describe the composition of the solution. Molality Scale \[\phi\left(V_{j}\right)=\left[m_{j} \, \rho(\mathrm{aq}) \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\rho_{1}^{*}(\ell)-\rho(\mathrm{aq})\right]+\mathrm{M}_{\mathrm{j}} / \rho(\mathrm{aq})\]Concentration Scale \[\phi\left(\mathrm{V}_{\mathrm{j}}\right)=\left[\mathrm{c}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\rho_{1}^{*}(\ell)-\rho(\mathrm{aq})\right]+\mathrm{M}_{\mathrm{j}} / \rho_{1}^{*}(\ell)\]Equation (b) using molalities and (c) using concentrations yield the same property of the solute, namely the apparent molar volume of solute \(j\), \(\phi\left(\mathrm{V}_{j}\right)\). Equations (b) and (c) are exact. The equations are readily distinguished by the difference in the denominators of the last terms. In any event the trick in deriving these equations is to seek an equation having the form, {[Property of solvent] minus [Property of solute]}.The subject is slightly complicated because the concentration of solute j can be expressed using either the unit ‘\(\mathrm{mol m}^{-3}\)’ or the unit ‘\(\mathrm{mol dm}^{-3}\)’, the latter being the most common. There is also a problem over the unit used for densities. Some authors use the unit ‘\(\mathrm{kg m}^{-3}\)‘ whereas other authors use the unit ‘\(\mathrm{g cm}^{-3}\)‘. The latter practice accounts for the numerical factor \(10^{3}\) which often appears in many published equations of the form shown in equations (b) and (c).Partial molar and partial molal properties are often identified. The two terms are synonymous in the case of partial molar volumes and partial molal volumes of solutes in aqueous solutions. IUPAC recommends the use of the term ‘partial molar volume’.Significantly we can never know the absolute value of the chemical potential of a solute in a given solution but we can determine the partial molar volume, the differential dependence of chemical potential on pressure. Indeed the challenge of understanding patterns in partial molar volumes seems less awesome than the task of understanding other thermodynamic properties of solutes.Equations (b) and (c) do not describe how \(\phi\left(\mathrm{V}_{j}\right)\) for a given solute depends on either \(\mathrm{m}_{j}\) or \(\mathrm{c}_{j}\). This dependence is characteristic of a solute (at fixed \(\mathrm{T}\) and \(\mathrm{p}\)) and reflects the role of solute - solute interactions. In many cases where solute \(j\) is a simple neutral solute, \(\phi\left(\mathrm{V}_{j}\right)\) for dilute solutions is often satisfactorily accounted for by an equation in which \(\phi\left(\mathrm{V}_{j}\right)\) is a linear function of \(\mathrm{m}_{j}\). The slope \(\mathrm{S}\) is characteristic of the solute (at fixed \(\mathrm{T}\) and \(\mathrm{p}\)) [4d,6]. \[\phi\left(V_{\mathrm{j}}\right)=\phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}+\mathrm{S} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)\]\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi\left(\mathrm{V}_{\mathrm{j}}\right)=\phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}=\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq})\]In the case of urea(aq) at \(298.2 \mathrm{~K}\) and ambient pressure the dependence of \(\phi\left(\mathrm{V}_{j}\right)\) on \(\mathrm{m}_{j}\) is described by the following quadratic equation. \[\phi\left(\mathrm{V}_{\mathrm{j}}\right) / \mathrm{cm}^{3} \mathrm{~mol}^{-1}=44.20+0.126 \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)-0.004 \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2}\]In general terms therefore \(\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq})\) {and \(\mathrm{V}_{\mathrm{j}}^{\infty}\) for solute \(j\) in other solvents} characterizes solute - solvent interactions and the dependences of \(\mathrm{V}_{\mathrm{j}}(\mathrm{aq})\) on \(\mathrm{m}_{j}\) characterizes solute - solute interactions. Of course the partial molar volume of solute-\(j\) in solution is not the actual volume of solute-\(j\). Instead \(\mathrm{V}_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\) measures the differential change in the volume of an aqueous solution when \(\delta \mathrm{n}_{j}\) moles of substance-\(j\) are added. We emphasize the importance of an approach using the molalities of solutes. The reasons are straightforward. If we compare \(\phi\left(\mathrm{V}_{j}\right)\) for a solute in solutions containing \(0.1\) and \(0.01 \mathrm{~mol kg}^{-1}\), in this comparison, the mass of solvent remains the same. If on the other hand we compare \(\phi\left(\mathrm{V}_{j}\right)\) for solute in solutions where \(\mathrm{c}_{j} / \mathrm{~mol dm}^{-3} = 0.1\) and \(0.01\), the amounts of solvent are not defined. Nevertheless many treatments of the properties of solutions examine \(\phi\left(\mathrm{V}_{j}\right)\) as a function of concentration. In fact chemists tend to think in terms of concentrations and hence in terms of distances between solute molecules. So in these terms concentration might be thought of as the 'natural' scale. Just as in life, one is more interested in the distance between two people rather than their mass. No rule forbids one to fit the dependences of \(\phi\left(\mathrm{V}_{j}\right)\) on \(\mathrm{c}_{j}\) using an equation of the following form. \[\phi\left(V_{j}\right)=a_{1}+a_{2} \, c_{j}+a_{3} \, c_{j}^{2}+\ldots\]But if \(\phi\left(\mathrm{V}_{j}\right)\) is a linear function of \(\mathrm{m}_{j}\), \(\phi\left(\mathrm{V}_{j}\right)\) is not a linear function of \(\mathrm{c}_{j}\). Of course \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)=\phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}\) in both \(\operatorname{limit}\left(\mathrm{m}_{j} \rightarrow 0 \right)\) and \(\operatorname{limit} \left(\mathrm{c}_{j} \rightarrow 0 \right)\).Granted the outcome of an experiment is the dependence of \(\phi\left(\mathrm{V}_{j}\right)\) on \(\mathrm{m}_{j}\), the partial molar volume \(\mathrm{V}_{\mathrm{j}}(\mathrm{aq})\) of solute \(j\) is readily calculated; equation (h). \[\mathrm{V}_{\mathrm{j}}(\mathrm{aq})=\phi\left(\mathrm{V}_{\mathrm{j}}\right)+\mathrm{m}_{\mathrm{j}} \,\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{m}_{\mathrm{j}}}\right)\]We note important features in the context of two plots;Then \(\mathrm{V}_{j}(\mathrm{aq})\) is the gradient of the tangent to the curve in plot (i) at the specified molality; \(\phi\left(\mathrm{V}_{j}\right)\) is the gradient of the line in plot(ii) joining the origin and \(\left[\mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)-\left(1 / \mathrm{M}_{1}\right) \, \mathrm{V}_{1}^{*}(\ell)\right]\) at the specified molality. For the solution volume \(\mathrm{V}(\mathrm{aq})\), \[\mathrm{V}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\]If the molar mass of the solvent is \(\mathrm{M}_{1} \mathrm{~kg mol}^{-1}\), \(\mathrm{V}(\mathrm{aq})=\left[\mathrm{n}_{1} \, \mathrm{M}_{1}+\mathrm{n}_{\mathrm{j}} \, \mathrm{M}_{\mathrm{j}}\right] / \rho(\mathrm{aq})\) and \(\mathrm{V}_{1}^{*}(\ell)=\mathrm{M}_{1} / \rho_{1}^{*}(\ell)\). From equation (a), \[\frac{\mathrm{n}_{1} \, \mathrm{M}_{1}}{\rho(\mathrm{aq})}+\frac{\mathrm{n}_{\mathrm{j}} \, \mathrm{M}_{\mathrm{j}}}{\rho(\mathrm{aq})}=\mathrm{n}_{1} \, \frac{\mathrm{M}_{1}}{\rho_{1}^{*}(\ell)}+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\]Divide by \(\mathrm{n}_{1} \, \mathrm{~M}_{1}\) and rearrange; \[\frac{n_{j}}{n_{1} \, M_{1}} \, \phi\left(V_{j}\right)=\frac{1}{\rho(a q)}-\frac{1}{\rho_{1}^{*}(\ell)}+\frac{n_{j} \, M_{j}}{n_{1} \, M_{1} \, \rho(a q)}\]But molality \(\mathrm{m}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} / \mathrm{n}_{1} \, \mathrm{M}_{1}\). Then, \[\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)=\left[\frac{1}{\rho(\mathrm{aq})}-\frac{1}{\rho_{1}^{*}(\ell)}\right]+\frac{\mathrm{m}_{\mathrm{j}} \, \mathrm{M}_{\mathrm{j}}}{\rho(\mathrm{aq})}\]or, \[\phi\left(V_{j}\right)=\left[m_{j} \, \rho(a q) \, \rho_{1}^{*}\right]^{-1} \,\left[\rho_{1}^{*}(\ell)-\rho(a q)\right]+\frac{M_{j}}{\rho(a q)}\]Thus, \[\phi\left(\mathrm{V}_{\mathrm{j}}\right)=\left[\frac{1}{\left[\mathrm{~mol} \mathrm{~kg}^{-1}\right] \,\left[\mathrm{kg} \mathrm{m}^{-3}\right]^{2}}\right] \,\left[\mathrm{kg} \mathrm{m}^{-3}\right]+\frac{\left[\mathrm{kg} \mathrm{mol}^{-1}\right]}{\left[\mathrm{kg} \mathrm{m}^{-3}\right.}=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]\] At fixed \(\mathrm{T}\) and \(\mathrm{p}\), \[\mathrm{c}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} / \mathrm{V}(\mathrm{aq})=\mathrm{n}_{\mathrm{j}} /\left[\mathrm{n}_{\mathrm{1}} \, \mathrm{V}_{1}^{*}(\mathrm{l})+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\right]\]But, \(\mathrm{m}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} / \mathrm{n}_{\mathrm{l}} \, \mathrm{M}_{1}\). Then, \[\mathrm{c}_{\mathrm{j}}=\frac{\mathrm{m}_{\mathrm{j}} \, \mathrm{n}_{1} \, \mathrm{M}_{1}}{\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)}\]Invert. \(\frac{1}{\mathrm{c}_{\mathrm{j}}}=\frac{\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)}{\mathrm{m}_{\mathrm{j}} \, \mathrm{n}_{1} \, \mathrm{M}_{1}}+\frac{\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\mathrm{m}_{\mathrm{j}} \, \mathrm{n}_{1} \, \mathrm{M}_{1}}\) But \(\mathrm{n}_{\mathrm{j}} / \mathrm{m}_{\mathrm{j}} \, \mathrm{n}_{1} \, \mathrm{M}_{1}=1\) Then, \[\frac{1}{\mathrm{c}_{\mathrm{j}}}=\frac{1}{\mathrm{~m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)}+\phi\left(\mathrm{V}_{\mathrm{j}}\right)\]or, \[\frac{1}{\mathrm{~m}_{\mathrm{j}}}=\frac{\rho_{1}^{*}(\ell)}{\mathrm{c}_{\mathrm{j}}}-\rho_{1}^{*}(\ell) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\]The latter equation links \(\mathrm{m}_{j}\) and \(\mathrm{c}_{j}\). From, \[\phi\left(\mathrm{V}_{\mathrm{j}}\right)=\frac{1}{\mathrm{~m}_{\mathrm{j}}} \,\left[\frac{1}{\rho(\mathrm{aq})}-\frac{1}{\rho_{1}^{*}(\ell)}\right]+\frac{\mathrm{M}_{\mathrm{j}}}{\rho(\mathrm{aq})}\]Then, from, \[\phi\left(V_{j}\right)=\left[\frac{\rho_{1}^{*}(\ell)}{c_{j}}-\rho_{1}^{*}(\ell) \, \phi\left(V_{j}\right)\right] \,\left[\frac{1}{\rho(\mathrm{aq})}-\frac{1}{\rho_{1}^{*}(\ell)}\right]+\frac{M_{j}}{\rho(\mathrm{aq})}\]Hence, \[\phi\left(\mathrm{V}_{\mathrm{j}}\right)=\frac{\rho_{1}^{*}(\ell)}{\mathrm{c}_{\mathrm{j}} \, \rho(\mathrm{aq})}-\frac{1}{\mathrm{c}_{\mathrm{j}}}-\frac{\rho_{1}^{*}(\ell) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\rho(\mathrm{aq})}+\phi\left(\mathrm{V}_{\mathrm{j}}\right)+\mathrm{M}_{\mathrm{j}} / \rho(\mathrm{aq})\]Then, \(\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \frac{\rho_{1}^{*}(\ell)}{\rho(\mathrm{aq})}=\frac{\rho_{1}^{*}(\ell)}{\mathrm{c}_{\mathrm{j}} \, \rho(\mathrm{aq})}-\frac{1}{\mathrm{c}_{\mathrm{j}}}+\frac{\mathrm{M}_{\mathrm{j}}}{\rho(\mathrm{aq})}\) \[\phi\left(\mathrm{V}_{\mathrm{j}}\right)=\frac{1}{\mathrm{c}_{\mathrm{j}}}-\frac{1}{\mathrm{c}_{\mathrm{j}}} \, \frac{\rho(\mathrm{aq})}{\rho_{1}^{*}(\ell)}+\frac{\mathrm{M}_{\mathrm{j}}}{\rho_{1}^{*}(\ell)}\]or, \[\phi\left(\mathrm{V}_{\mathrm{j}}\right)=\left[\mathrm{c}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\rho_{1}^{*}(\ell)-\rho(\mathrm{aq})\right]+\mathrm{M}_{\mathrm{j}} / \rho_{1}^{*}(\ell)\]Thus, \[\phi\left(\mathrm{V}_{\mathrm{j}}\right)=\frac{1}{\left[\mathrm{~mol} \mathrm{~m}^{-3}\right]} \, \frac{1}{\left[\mathrm{~kg} \mathrm{~m}^{-3}\right]^{-1}} \,\left[\mathrm{kg} \mathrm{m}^{-3}\right]+\frac{\left[\mathrm{kg} \mathrm{mol}^{-1}\right]}{\left[\mathrm{kg} \mathrm{m}^{-3}\right]}=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]\] The following publications use equation (b) based on the molality composition scale. Manual of Symbols and Terminology for Physicochemical Quantities and Units, IUPAC, Pergamon, Oxford, 1979. F. Franks and H.T. Smith, Trans. Faraday Soc., 1968, 64, 2962. D. Hamilton and R. H. Stokes, J. Solution Chem., 1972,1, 213. R. H. Stokes, Aust . J. Chem., 1967,20, 2087. From, \(\mathrm{m}_{\mathrm{j}}=\mathrm{c}_{\mathrm{j}} /\left[\rho_{1}^{*}(\ell)-\rho_{1}^{*}(\ell) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \mathrm{c}_{\mathrm{j}}\right]\). If \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)=\mathrm{a}_{1}+\mathrm{a}_{2} \, \mathrm{m}_{\mathrm{j}}\) Then, \[\phi\left(\mathrm{V}_{\mathrm{j}}\right)=\mathrm{a}_{1}+\left\{\mathrm{a}_{2} / \rho_{1}^{*}(\ell) \,\left[1-\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \mathrm{c}_{\mathrm{j}}\right]\right\} \, \mathrm{c}_{\mathrm{j}}\]i.e. the slope depends on the product \(\phi\left(\mathrm{V}_{j}\right) \, \mathrm{c}_{j}\). From \[\mathrm{V}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\]At constant \(\mathrm{n}_{1}\), \[\left(\frac{\partial \mathrm{V}}{\partial \mathrm{n}_{\mathrm{j}}}\right)=\phi\left(\mathrm{V}_{\mathrm{j}}\right)+\mathrm{n}_{\mathrm{j}} \,\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{n}_{\mathrm{j}}}\right)\]Or, \[\mathrm{V}_{\mathrm{j}}=\phi\left(\mathrm{V}_{\mathrm{j}}\right)+\mathrm{m}_{\mathrm{j}} \,\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{m}_{\mathrm{j}}}\right)\]This page titled 1.22.5: Volumes: Solutions: Apparent and Partial Molar Volumes: Determination is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.22.6: Volumes: Apparent Molar and Excess Volumes

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For a solution prepared using \(1 \mathrm{~kg}\) of water, the volume is related to the apparent molar volume of the solute \(\phi \left(\mathrm{V}_{j}\right)\) using equation (a). \[\mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\]If the thermodynamic properties of this solution are ideal, \[\mathrm{V}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}\]Here \(V_{\mathrm{j}}^{\infty}(\mathrm{aq}) \equiv \phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}\). The difference between \(\mathrm{V}(\mathrm{aq})\) and \(\mathrm{V}(\mathrm{aq}: \mathrm{id})\) defines an excess volume \(\mathrm{V}^{\mathrm{E}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)\). Thus, \[\mathrm{V}^{\mathrm{E}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)-\mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}: \mathrm{id}\right)\]Hence, \[\mathrm{V}^{\mathrm{E}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\mathrm{m}_{\mathrm{j}} \,\left[\phi\left(\mathrm{V}_{\mathrm{j}}\right)-\phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}\right]\]This page titled 1.22.6: Volumes: Apparent Molar and Excess Volumes is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.22.7: Volumes: Neutral Solutes: Limiting Partial Molar Volumes

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A given aqueous solution at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\) contains solute \(j\), having molality \(\mathrm{m}_{j}\). The chemical potential of solute \(j\), \(\mu_{j}(\mathrm{aq})\) is related to \(\mathrm{m}_{j}\) using equation (a). \[\begin{aligned} &\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p}) \\ &=\mu_{\mathrm{j}}^{0}\left(\mathrm{aq} ; \mathrm{T} ; \mathrm{p}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)+\int_{\mathrm{p}^{0}}^{\mathrm{p}} \mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq} ; \mathrm{T}) \, \mathrm{dp} \end{aligned}\]But \[\mathrm{V}_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\left(\frac{\partial \mu_{\mathrm{j}}}{\partial \mathrm{p}}\right)_{\mathrm{T}}\]Also \(\mu_{\mathrm{j}}^{0}\left(\mathrm{aq} ; \mathrm{T} ; \mathrm{p}^{0}\right)\) is, by definition, independent of pressure. From equation (a), \[\mathrm{V}_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})+\mathrm{R} \, \mathrm{T} \,\left(\frac{\partial \ln \left(\gamma_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right)_{\mathrm{T}}\]In equation (c), there is no term explicitly in terms of molaity \(\mathrm{m}_{j}\). From the definition of \(\gamma_{j}\), \[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{V}_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\]\(\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\) is the limiting partial molar volume of solute \(j\) in aqueous solution at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\). In other words \(\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\) is the partial molar volume of solute \(j\) in the (ideal) solution where there are no solute-solute interactions and characterizes solute-water interactions. Because \(\gamma_{j}\) tends to unity as \(\mathrm{m}_{j}\) tends to zero, \(\gamma_{j}\) is sometimes called an asymmetric activity coefficient. [Contrast rational activity coefficients where \(\mathrm{f}_{1} \rightarrow 1 \text { as } \mathrm{x}_{1} \rightarrow 1\).]At the risk of being repetitive we distinguish between the two possible reference states for substance \(j\) such as urea. One reference state is the pure solid chemical substance \(j\) at ambient pressure and \(298.2 \mathrm{~K}\). Another reference state is the ideal solution where \(\mathrm{m}_{\mathrm{j}}=1 \mathrm{~mol} \mathrm{~kg}^{-1}\) at ambient pressure and \(298.2 \mathrm{~K}\). The properties of urea in the two states, pure solid and solution standard state are clearly quite different. Indeed, we can compare \(\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq} ; 298.2 \mathrm{~K} ; \text { ambient p) }\) and \(\mathrm{V}_{\mathrm{j}}^{*}(\mathrm{~s} ; 298.2 \mathrm{~K} ; \text { ambient } \mathrm{p})\). We can also compare, for example, \(\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{j}=\text { urea; sln} 298.2 \mathrm{~K} ; \text { ambient p) }\) in a range of solvents. These points are also nicely illustrated by the volumetric properties of water [3,]. At \(298.2 \mathrm{~K}\) and ambient pressure \(\mathrm{V}_{1}^{*}\left(\ell ; \mathrm{H}_{2} \mathrm{O}\right)\) is \(18.07 \mathrm{~cm}^{3} \mathrm{~mol}^{-1}\) but for water as a solute in three solvents, \(\mathrm{V}^{\infty}\left(\mathrm{H}_{2} \mathrm{O} ; \operatorname{sln}\right)=18.47(\mathrm{MeOH}), 14.42(\mathrm{EtOH}) \text { and } 17.00(\mathrm{THF}) \mathrm{cm}^{3} \mathrm{~mol}^{-1}\). There is, of course, no reason why we should expect anything different. A water molecule in liquid water is surrounded by many millions of other water molecules. But a water molecule at infinite dilution in solvent ethanol is surrounded by many millions of ethanol molecules.In the analysis of experimental results , we may express the composition of the solution in terms of mole fraction of solute \(\mathrm{x}_{j}\). Then \[\begin{aligned} &\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p}) \\ &=\mu_{\mathrm{j}}^{0}\left(\mathrm{aq} ; \mathrm{T} ; \mathrm{p}^{0} ; \mathrm{x}-\mathrm{scale}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{\mathrm{j}} \, \mathrm{f}_{\mathrm{j}}^{*}\right)+\int_{\mathrm{p}^{0}}^{\mathrm{p}} \mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq} ; \mathrm{T}) \, \mathrm{dp} \end{aligned}\]But mole fraction \(\mathrm{x}_{j}\) is independent of pressure. \[\mathrm{V}_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})+\mathrm{R} \, \mathrm{T} \,\left(\frac{\partial \ln \left(\mathrm{f}_{\mathrm{j}}^{*}\right)}{\partial \mathrm{p}}\right)_{\mathrm{T}}\]From the definition of \(\mathrm{f}_{\mathrm{j}}^{*}\), \[\operatorname{limit}\left(\mathrm{x}_{\mathrm{j}} \rightarrow 0\right) \mathrm{V}_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\]The limiting value of \(\mathrm{V}_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\) is identical on the molality and mole fraction scales. If we use the concentration scale a problem arises in that the concentration of solute \(j\), \(\mathrm{c}_{j}\) is dependent on pressure because the volume of the solution is pressure dependent.Footnote W. L. Masterton and H. K. Seiler, J. Phys. Chem., 1968,72, 4257. For \(j =\) urea at \(298.2 \mathrm{~K}\) and ambient pressure, \(\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{s} \ln ) / \mathrm{cm}^{3} \mathrm{~mol}^{-1}=\) 44.24 (water), 36.97 (methanol), 40.75 (ethanol) and 41.86 (DMSO). \(\mathrm{V}_{\mathrm{j}}^{\infty}(298.15 \mathrm{~K} ; \mathrm{j}=\text { water })=18.57 \mathrm{~cm}^{3} \mathrm{~mol}^{-1}(\text { solvent }=\text { octan-1-ol })\) and \(31.3 \mathrm{cm}^{3} \mathrm{~mol}^{-1} \text { (solvent }= \mathrm{CCl}_{4})\); P. Berti, S. Cabani and V. Mollica, Fluid PhaseEquilib., 1987,32 , 1. M. Sakurai and T. Nakagawa, Bull. Chem. Soc. Jpn., 1984, 55, 195; J. Chem. Thermodyn., 1982, 14, 269; 1984, 16 , 171. A similar contrast exists (H. Itsuki, S. Terasawa, K. Shinohara and H. Ikezwa, J. Chem. Thermodyn., 1987,19, 555) between the molar volume of a hydrocarbon and its limiting partial molar volume in another hydrocarbon; \(\mathrm{V}^{*}\left(\ell ; \mathrm{C}_{6} \mathrm{H}_{14}\right)=131.61 \mathrm{~cm}^{3} \mathrm{~mol}^{-1}\), but \(\mathrm{V}^{\infty}\left(\mathrm{C}_{6} \mathrm{H}_{14} ; \text { sln; solvent }=\mathrm{C}_{16} \mathrm{H}_{34}\right)=130.2 \mathrm{~cm}^{3} \mathrm{~mol}^{-1}\) at \(298 \mathrm{~K}\) and ambient pressure.In this context the limiting enthalpies of solution water in monohydric alcohols depend on the alcohol at \(298.2 \mathrm{~K}\); (S.-O. Nilsson, J. Chem. Thermodyn., 1986, 18, 1115). The partial molar volumes of fullerene in solution is \(401 \mathrm{~cm}^{3} \mathrm{~mol}^{-1}\) in cis-decalin and \(389 \mathrm{~cm}^{3} \mathrm}{mol}^{-1}\) in 1,2-dichlorobenzene both values being significantly less than the predicted volume of the pure liquid \(\mathrm{C}_{60}\); (P. Ruelle, A. Farina-Cuendet and U. W. Kesselring, J. Chem. Soc. Chem. Commun., 1995, 1161).This page titled 1.22.7: Volumes: Neutral Solutes: Limiting Partial Molar Volumes is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.22.8: Volume: Salt Solutions: Born-Drude-Nernst Equation

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Differentiation of the Born Equation with respect to pressure (at fixed temperature) yields the Born-Drude-Nernst Equation which describes the difference in partial molar volumes of ion \(j\) in the gas phase and in solution. The simplest model assumes that the radius \(\mathrm{r}_{j}\) is independent of pressure. \[\begin{aligned} \Delta(\mathrm{pfg}&\rightarrow \mathrm{s} \ln ) \mathrm{V}_{\mathrm{j}}\left(\mathrm{c}_{\mathrm{j}}=1 \mathrm{~mol} \mathrm{dm} \mathrm{m}^{-3} ; \mathrm{id} ; \mathrm{p}, \mathrm{T}\right)=\\ &-\mathrm{N}_{\mathrm{A}} \,\left(\mathrm{z}_{\mathrm{j}} \, \mathrm{e}\right)^{2} \,\left[\frac{1}{\varepsilon_{\mathrm{r}}} \,\left(\frac{\partial \varepsilon_{\mathrm{r}}}{\partial \mathrm{p}}\right)_{\mathrm{T}}\right] \, \frac{1}{8 \, \pi \, \mathrm{r}_{\mathrm{j}} \, \varepsilon_{0}} \end{aligned}\]A more complicated equation emerges if radius \(\mathrm{r}_{j}\) is assumed to depend on pressure, but there seems little merit in taking account of such a dependence.Footnote \[\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]=\left[\mathrm{mol}^{-1}\right] \,\left[\mathrm{A}^{2} \mathrm{~s}^{2}\right] \,\left[\mathrm{N} \mathrm{m}^{-2}\right]^{-1} \,^{-1} \,^{-1} \,\left[\mathrm{m}^{-1}\right] \,\left[\mathrm{F} \mathrm{m}^{-1}\right]^{-}\]where, \(\left[\mathrm{F} \mathrm{m}^{-1}\right]=\left[\mathrm{A}^{2} \mathrm{~s}^{4} \mathrm{~kg}^{-1} \mathrm{~m}^{-3}\right]\)This page titled 1.22.8: Volume: Salt Solutions: Born-Drude-Nernst Equation is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.22.9: Volume: Liquid Mixtures

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A given binary liquid mixture is prepared using \(\mathrm{n}_{1}\) moles of liquid 1 and \(\mathrm{n}_{2}\) moles of liquid 2 at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\). The term ‘mixture’ usually means that a homogeneous single liquid phase is spontaneously formed on mixing characterized by a minimum in Gibbs energy \(\mathrm{G}\) where the molecular organization is characterized by \(\xi^{\mathrm{eq}}\). \(\mathrm{G}_{\mathrm{m}}^{\mathrm{E}}\) measures the extent to which \(\mathrm{G} /\left(\mathrm{n}_{1}+\mathrm{n}_{2}\right)\) differs from this ratio in the event that the mixing is, in a thermodynamic sense, ideal.A given binary liquid mixture is displaced to a neighboring state by a change in pressure at constant temperature. The overall composition remains at \(\left(\mathrm{n}_{1} + \mathrm{n}_{2}\right)\) but the organization changes to a new value for \(\xi^{\mathrm{eq}}\) where ‘\(\mathrm{A} = 0\)’. The differential dependence of \(\mathrm{G}_{\mathrm{m}}^{\mathrm{E}}\) on pressure at constant temperature \(\mathrm{T}\) is the excess molar volume \(\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}\). \[\mathrm{V}(\operatorname{mix})=\left(\frac{\partial \mathrm{G}(\operatorname{mix})}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{A}=0}\]\[\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}=\left(\frac{\partial \mathrm{G}_{\mathrm{m}}^{\mathrm{E}}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{A}=0}\]For the molar volume, \[\mathrm{V}_{\mathrm{m}}=\left(\frac{\partial \mathrm{G}_{\mathrm{m}}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{A}=0}\]The quantities, \(\mathrm{V}(\mathrm{mix})\), \(\mathrm{V}_{\mathrm{m}}\) and \(\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}\) are interesting because they can be determined whereas the same cannot be said for \(\mathrm{G}(\mathrm{mix})\) and \(\mathrm{G}_{\mathrm{m}}\) although \(\mathrm{G}_{\mathrm{m}}^{\mathrm{E}}\) can be obtained m from vapor pressures of mixtures and pure components. \[\mathrm{V}_{\mathrm{m}}=\mathrm{V}(\mathrm{mix}) /\left(\mathrm{n}_{1}+\mathrm{n}_{2}\right)\]Density, \[\rho(\operatorname{mix})=\left(\mathrm{n}_{1} \, \mathrm{M}_{1}+\mathrm{n}_{2} \, \mathrm{M}_{2}\right) / \mathrm{V}(\operatorname{mix})\]\(\mathrm{M}_{1}\) and \(\mathrm{M}_{2} are the molar masses of the two liquid components. By measuring \(\rho(\mathrm{mix})\) as a function of mixture composition, we form a plot of molar volume Vm as a function of mole fraction composition. The plot has two limits; \[\operatorname{limit}\left(\mathrm{x}_{1} \rightarrow 1\right) \mathrm{V}_{\mathrm{m}}=\mathrm{V}_{1}^{*}(\ell)\]\[\operatorname{limit}\left(\mathrm{x}_{2} \rightarrow 1\right) \mathrm{V}_{\mathrm{m}}=\mathrm{V}_{2}^{*}(\ell)\]If the thermodynamic properties of the binary liquid mixture are ideal (i.e. \(\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}=0\)), \[\mathrm{V}_{\mathrm{m}}(\mathrm{id})=\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}(\ell)\]Or, \[\mathrm{V}_{\mathrm{m}}(\mathrm{id})=\left(1-\mathrm{x}_{2}\right) \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}(\ell)\]Hence, \[\mathrm{V}_{\mathrm{m}}(\mathrm{id})=\mathrm{V}_{1}^{*}(\ell)+\mathrm{x}_{2} \,\left[\mathrm{V}_{2}^{*}(\ell)-\mathrm{V}_{1}^{*}(\ell)\right]\]The latter is an equation for a straight line. The molar volume of a real binary liquid mixture is usually less than \(\mathrm{V}_{m}(\mathrm{id})\). For a real binary liquid mixture, \[\mathrm{V}_{\mathrm{m}}(\operatorname{mix})=\mathrm{x}_{1} \, \mathrm{V}_{1}(\operatorname{mix})+\mathrm{x}_{2} \, \mathrm{V}_{2}(\mathrm{mix})\]The difference between the molar volume of real and ideal binary liquid mixture is the excess molar volume \(\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}\). \[\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}=\mathrm{x}_{1} \,\left[\mathrm{V}_{1}(\operatorname{mix})-\mathrm{V}_{1}^{*}(\ell)\right]+\mathrm{x}_{2} \,\left[\mathrm{V}_{2}(\operatorname{mix})-\mathrm{V}_{2}^{*}(\ell)\right]\]Or, \[V_{m}^{E}=x_{1} \, V_{1}^{E}(\operatorname{mix})+x_{2} \, V_{2}^{E}(\operatorname{mix}\]A given mixture, mole fraction \(\mathrm{x}_{2}\) at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\), is perturbed by addition of \(\delta \mathrm{n}_{2}\) moles of chemical substance 2. The system can be perturbed either at constant organization \(\xi\) or constant affinity \(\mathrm{A}\). Here we are concerned with \(\left(\frac{\partial \mathrm{V}}{\partial \mathrm{n}_{2}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}, \mathrm{A}=0}\), the (equilibrium) partial molar volume of substance 2 in the mixture, \(\mathrm{V}_{2}(\mathrm{mix})\).The condition '\(\mathrm{A} = 0\)' implies that there is a change in organization \(\xi\) in order to hold the system in the equilibrium state. A similar argument is formulated for the (equilibrium) partial molar volume \(\mathrm{V}_{1}(\mathrm{mix})\). Moreover according to the Gibbs-Duhem equation (at constant temperature and pressure), \[\mathrm{n}_{1} \, d \mathrm{~V}_{1}(\operatorname{mix})+\mathrm{n}_{2} \, d \mathrm{~V}_{2}(\operatorname{mix})=0\]Further, \[\mathrm{V}(\operatorname{mix})=\mathrm{n}_{1} \, \mathrm{V}_{1}(\operatorname{mix})+\mathrm{n}_{2} \, \mathrm{V}_{2}(\operatorname{mix})\]The property \(\mathrm{V}(\mathrm{mix})\) is directly determined from the density \(\rho(\mathrm{mix})\). \[\mathrm{V}(\operatorname{mix})=\left(\mathrm{n}_{1} \, \mathrm{M}_{1}+\mathrm{n}_{2} \, \mathrm{M}_{2}\right) / \rho(\operatorname{mix})\]The important point is that the thermodynamic extensive property \(\mathrm{V}(\mathrm{mix})\) is directly determined by experiment whereas we cannot for example measure the enthalpy \(\mathrm{H}(\mathrm{mix})\). The excess molar volume is given by equation (l). \[\begin{aligned} \frac{\mathrm{dV}_{\mathrm{m}}^{\mathrm{E}}}{d \mathrm{x}_{1}}=& {\left[\mathrm{V}_{1}(\mathrm{mix})-\mathrm{V}_{1}^{*}(\ell)\right]+\mathrm{x}_{1} \, \frac{\mathrm{dV} \mathrm{V}_{1}(\mathrm{mix})}{\mathrm{dx}_{1}} } \\ &-\left[\mathrm{V}_{2}(\mathrm{mix})-\mathrm{V}_{2}^{*}(\ell)\right]+\mathrm{x}_{2} \, \frac{\mathrm{dV}_{2}(\mathrm{mix})}{\mathrm{dx}_{1}} \end{aligned}\]Using the Gibbs -Duhem equation, \[\frac{\mathrm{dV}_{\mathrm{m}}^{\mathrm{E}}}{\mathrm{dx}_{1}}=\left[\mathrm{V}_{1}(\mathrm{mix})-\mathrm{V}_{1}^{*}(\ell)\right]-\left[\mathrm{V}_{2}(\mathrm{mix})-\mathrm{V}_{2}^{*}(\ell)\right]\]Or, \[\left[\mathrm{V}_{1}(\mathrm{mix})-\mathrm{V}_{1}^{*}(\ell)\right]=\frac{\mathrm{dV}_{\mathrm{m}}^{\mathrm{E}}}{\mathrm{dx}_{1}}+\left[\mathrm{V}_{2}(\mathrm{mix})-\mathrm{V}_{2}^{*}(\ell)\right]\]From equation (l), \[\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}=\mathrm{x}_{1} \, \frac{\mathrm{dV}_{\mathrm{m}}^{\mathrm{E}}}{\mathrm{dx}_{1}}+\mathrm{x}_{1} \,\left[\mathrm{V}_{2}(\operatorname{mix})-\mathrm{V}_{2}^{*}(\ell)\right]+\mathrm{x}_{2} \,\left[\mathrm{V}_{2}(\operatorname{mix})-\mathrm{V}_{2}^{*}(\ell)\right]\]Hence, \[\left[\mathrm{V}_{2}(\mathrm{mix})-\mathrm{V}_{2}^{*}(\ell)\right]=\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}-\mathrm{x}_{1} \, \frac{\mathrm{dV}_{\mathrm{m}}^{\mathrm{E}}}{\mathrm{dx}_{1}}\]The derivation leading up to equation (s) is the 'Method of Tangents'. Moreover at the mole fraction composition where \(\frac{\mathrm{dV}_{\mathrm{m}}^{\mathrm{E}}}{\mathrm{dx}_{1}}\) is zero, \(\left[\mathrm{V}_{2}(\operatorname{mix})-\mathrm{V}_{2}^{*}(\ell)\right]\) equals \(\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}\).For liquid component 1 the chemical potential in the liquid mixture is related to the mole fraction composition (at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\)). \[\mu_{1}(\operatorname{mix}, \mathrm{T}, \mathrm{p})=\mu_{1}^{*}(\ell, \mathrm{T})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right)+\int_{\mathrm{p}^{0}}^{\mathrm{p}} \mathrm{V}_{1}^{*}(\ell ; \mathrm{T}) \, \mathrm{dp}\]At temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\), \[\mathrm{V}_{1}(\operatorname{mix})=\mathrm{V}_{1}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \,\left[\partial \ln \left(\mathrm{f}_{1}\right) / \partial \mathrm{p}\right]_{\mathrm{T}}\]But \[\mathrm{V}_{1}^{\mathrm{E}}(\operatorname{mix})=\mathrm{V}_{1}(\operatorname{mix})-\mathrm{V}_{1}^{*}(\ell)\]Hence, \[\mathrm{V}_{1}^{\mathrm{E}}(\operatorname{mix})=\mathrm{R} \, \mathrm{T} \,\left[\partial \ln \left(\mathrm{f}_{1}\right) / \partial \mathrm{p}\right]_{\mathrm{T}}\]Then, \[\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}(\mathrm{mix})=\mathrm{R} \, \mathrm{T} \,\left\{\mathrm{x}_{1} \,\left[\partial \ln \left(\mathrm{f}_{1}\right) / \partial \mathrm{p}\right]_{\mathrm{T}}+\mathrm{x}_{2} \,\left[\partial \ln \left(\mathrm{f}_{2}\right) / \partial \mathrm{p}\right]_{\mathrm{T}}\right\}\]Footnote R. Battino, Chem.Rev.,1971,71,5.This page titled 1.22.9: Volume: Liquid Mixtures is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.23.1: Water

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There can be little doubt that water(\(\ell\)) has been all-pervasive in the development of thermodynamics and, of course, solution chemistry. Chemical laboratories throughout the whole world have this liquid ‘on tap’. Speculation about intelligent life forms (cf. are we alone?) in the rest of the universe often start with comments concerning the presence of water(\(\ell\)) on distant planets.Footnotes Bill Bryson, A Short History of Nearly Everything, Doubleday, New York, 2003, chapter 16.This page titled 1.23.1: Water is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.23.2: Water: Molar Volume

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The molar volume \(\mathrm{V}_{1}^{*}(\ell)\) and density \(\rho_{1}^{*}(\ell)\) of water(\(\ell\)) are intensive properties. The \(\mathrm{p}-\mathrm{V}-\mathrm{T}\) properties of water are perhaps the most extensively studied. Two properties are almost universally known;At ambient pressure the temperature of maximum density for water, \(\mathrm{TMD} = 3.98 \text { Celsius}\); for \(\mathrm{D}_{2}\mathrm{O}\), \(\mathrm{TMD} = 11.44 \text { Celsius}\). The \(\mathrm{TMD}\) for \(\mathrm{SiO}_{2}\) is around 15270 Celsius, the dependence of density on temperature being less marked about the \(\mathrm{TMD}\) than that for water.The dependence of \(\mathrm{V}^{*}\left(\mathrm{H}_{2} \mathrm{O} ; \ell\right)\) on temperature and pressure is reported by many laboratories. Most accounts cite the study reported by Kell and Whalley in 1965, later extended in 1978. Kell has examined the results in detail. The \(\mathrm{TMD}\) has, of course, attracted considerable attention. Nevertheless the \(\mathrm{TMD}\) has no deep significance in the context of understanding the properties of water(\(\ell\)). Other properties of water(\(\ell\)) show extrema at other temperatures; e.g. isothermal compressibility near \(300 \mathrm{~K}\). C. A. Angell and H. Kanno, Science, 1976,193,1121. G. S. Kell and E. Whalley, Philos. Trans. R. Soc. London, 1965,258,565. G. S. Kell, G. M. McLaurin and E.Whalley, Proc. R. Soc. London, Ser. A,1978,360,389. G. S. Kell, J. Chem. Eng. Data, 1967,12,66; 1975,20,97; 1970,15,119. R.A. Fine and F. J. Millero, J.Chem.Phys.,1975,63,89; 1973,59,5529. D.-P. Wang and F. J. Millero, J.Geophys. Res., 1973,78,7122. F. J. Millero, R. W. Curry and W. Drost-Hansen, J. Chem. Eng. Data, 1969, 14,422.This page titled 1.23.2: Water: Molar Volume is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.23.3: Water: Hydrogen Ions

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Chemists are often faced with the situation where on adding salt \(\mathrm{MX}\) to water(\(\ell\)) experimental evidence shows that the cation exists as a hydrate \(\mathrm{M}\left(\mathrm{H}_{2} \mathrm{O}\right)_{\mathrm{n}}\). For example, adding \(\mathrm{CuSO}_{4}(\mathrm{s})\), a white powder, to water produces a blue solution containing \(\left[\mathrm{Cu}\left(\mathrm{H}_{2}\mathrm{O}\right)_{4} \right]^{2+}\). If solute molecules bind solvent molecules to produce new solute molecules, one can imagine a limiting situation where, as the depletion of solvent continues, there is little ‘solvent’ as such left in the system.An important example of the problems linked to description concerns hydrogen ions in aqueous solution. Two common descriptions areAs a starting point, we assume that the system comprises \(\mathrm{n}_{j}\) moles of solute \(\mathrm{HX}\) and \(\mathrm{n}_{1}\) moles of waterA simple description of hydrogen ions is in terms of \(\mathrm{H}^{+} (\mathrm{aq})\) although intuitively the idea of protons as ions in aqueous solutions is not attractive. Arguably a more satisfactory description of hydrogen ions in solution is in terms of \(\mathrm{H}_{3}\mathrm{O}^{+}\) ions. Description of hydrogen ions in aqueous solution as \(\mathrm{H}_{3}\mathrm{O}^{+}\) finds general support. Adam recalls the experiment conducted by Bagster and Cooling. The latter authors observed that the electrical conductivity of a solution of \(\mathrm{HCl}\) in \(\mathrm{SO}_{2} (\ell)\) is low but increases dramatically when 1 mole of water(\(\ell\)) is added for each mole of \(\mathrm{HCl}\). Moreover on electrolysis, hydrogen ions and water are liberated at the cathode; water(\(\ell\)) drips from this electrode.In the chemistry of aqueous solutions, two ions \(\mathrm{H}^{+}\) and \(\mathrm{OH}^{-}\) command interest. Hydrogen ions are also called hydronium ions when written as \(\mathrm{H}_{3}\mathrm{O}^{+}\). The latter ion is a flat pyramid with \(\mathrm{d}(\mathrm{O}-\mathrm{H}) = 96.3 \mathrm{~pm}\); the \(\mathrm{HOH}\) angle = 110-1120. A case for writing the formula \(\mathrm{H}_{3}\mathrm{O}^{+}\) is based on the existence of isomorphous solids, + − \(\mathrm{NH}_{4}^{+}\mathrm{ClO}_{4}^{-} \text { and } \mathrm{H}_{3}\mathrm{O}^{+}\mathrm{ClO}_{4}^{-}\). The mass spectra of \(\mathrm{H}^{+}\left(\mathrm{H}_{2} \mathrm{O}\right)_{\mathrm{n}}\) have been observed for \(1 \leg \mathrm{n} \leq 8\). Neutron-scattering and X-ray scattering data show that for \(\mathrm{D}_{3}\mathrm{O}^{+}\) ions in solution , \(\mathrm{d}(\mathrm{O}-\mathrm{D}) = 101.7 \mathrm{~pm}\). In an ‘isolated’ \(\mathrm{H}_{3}\mathrm{O}^{+}\) ion the \(\mathrm{O}-\mathrm{H}\) bond length is \(97 \mathrm{~pm}\) and the \(\mathrm{HOH}\) angle is 110 - 112.In aqueous solution, \(\mathrm{H}_{3}\mathrm{O}^{+}\) ions do not exist as solutes comparable to \(\mathrm{Na}^{+}\) ions in \(\mathrm{NaCl}(\mathrm{aq})\). Instead a given \(\mathrm{H}_{3}\mathrm{O}^{+}\) ion transfers a proton to a neighboring water molecule. The time taken for the transfer is very short, approx. 10-13 seconds granted that the receiving water molecule has the correct orientation. But a given \(\mathrm{H}_{3}\mathrm{O}^{+}\) ion has a finite lifetime, sufficient to be characterized by thermodynamic and spectroscopic properties The rate determining step in proton migration involves reorientation of neighboring water molecules, thereby accounting for the increase in molar conductance \(\lambda^{\infty}\left(\mathrm{H}^{+};\mathrm{aq})\) with increase in \(\mathrm{T}\) and \(\mathrm{p}\) [7 - 10]. [In ice, proton transfer is rate determining and so the mobilities of \(\mathrm{H}^{+}\) and \(\mathrm{OH}^{-}\) ions are higher in ice than in water(\(\ell\)).} The high mobility of protons in aqueous solution involves a series of isomerizations between \(\mathrm{H}_{9}\mathrm{O}_{4}^{+}\) and \(\mathrm{H}_{5}\mahtrm{O}_{2}^{+}\), the first triggered by hydrogen bond cleavage of a second shell water molecule and the second by the reverse, hydrogen bond formation. An iconoclastic approach, expressed by Hertz and co-workers, argues against the existence of \(\mathrm{H}^{+}\) ions as such except in so far as this symbol describes as dynamical property of a solution. In their view \(\mathrm{H}_{3}\mathrm{O}^{+}\) is ephemeral. Among other interesting ions discussed in this context are \(\mathrm{H}_{5}\mathrm{O}_{2}^{+}\).Comparisons are often drawn between \(\mathrm{NH}_{4}^{+}(\mathrm{aq})\) and \(\mathrm{H}_{3}\mathrm{O}^{+} (\mathrm{aq})\) ions but they are not really related. For example the electrical mobility of \(\mathrm{NH}_{4}^{+}(\mathrm{aq})\) is not exceptional.The chemical potential of \(\mathrm{H}^{+} (\mathrm{aq})\) describes the change in Gibbs energy when \(\delta \mathrm{n}\left(\mathrm{H}^{+}\right)\) moles are added at constant \(\mathrm{n}\left(\mathrm{H}_{2}\mathrm{O}\right), \mathrm{~n}\left(\mathrm{X}^{-}\right)\), \(\mathrm{T}\) and \(\mathrm{p}\). The chemical potential of \(\mathrm{H}^{+} (\mathrm{aq})\) is related to the molality \(\mathrm{m}\left(\mathrm{H}^{+}\right) \left[= \mathrm{n}\left(\mathrm{H}^{+}\right) / \mathrm{n}_{1} \, \mathrm{M}_{1}\right]\) and the single activity coefficient \(\gamma\left(\mathrm{H}^{+}\right)\).The Gibbs energy of an aqueous solution \(\mathrm{G}(\mathrm{aq})\) prepared using \(\mathrm{n}_{1}\) moles of water and \(\mathrm{n}_{j}\) moles of acid \(\mathrm{H}^{+} \mathrm{~X}^{-}\) is given by the equation (a) (for the solution at defined \(\mathrm{T}\) and \(\mathrm{p}\), which we assume is close to the standard pressure \(\mathrm{p}^{0}\)). \[\mathrm{G}(\mathrm{aq})=\mathrm{n}_{1} \, \mu_{1}\left(\mathrm{H}_{2} \mathrm{O} ; \mathrm{aq}\right)+\mathrm{n}\left(\mathrm{H}^{+}\right) \, \mu\left(\mathrm{H}^{+} ; \mathrm{aq}\right)+\mathrm{n}\left(\mathrm{X}^{-}\right) \, \mu\left(\mathrm{X}^{-} ; \mathrm{aq}\right)\]where \[\mu\left(\mathrm{H}^{+}\right)=\left[\partial \mathrm{G} / \partial \mathrm{n}\left(\mathrm{H}^{+}\right)\right]_{\mathrm{n}\left(\mathrm{H}_{2} \mathrm{O}\right), \mathrm{n}\left(\mathrm{X}^{-}\right)}\]and \[\mu\left(\mathrm{H}^{+} ; \mathrm{aq}\right)=\mu^{0}\left(\mathrm{H}^{+} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{m}\left(\mathrm{H}^{+}\right) \, \gamma\left(\mathrm{H}^{+}\right) / \mathrm{m}^{0}\right]\]For the electrolyte \(\mathrm{H}^{+} \mathrm{~X}^{-}\) with \(ν = 2\), \[\mu\left(\mathrm{H}^{+} \mathrm{X}^{-} ; \mathrm{aq}\right)=\mu^{0}\left(\mathrm{H}^{+} \mathrm{X}^{-} ; \mathrm{aq}\right)+2 \, \mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{m}\left(\mathrm{H}^{+} \mathrm{X}^{-}\right) \, \gamma_{\pm}\left(\mathrm{H}^{+} \mathrm{X}^{-}\right) / \mathrm{m}^{0}\right]\]and, all \(\mathrm{T}\) and \(\mathrm{p}\), \[\operatorname{limit}\left[\mathrm{m}\left(\mathrm{H}^{+} \mathrm{X}^{-}\right) \rightarrow 0\right] \gamma_{\pm}=1.0\]In a similar fashion, we define the partial molar volume, enthalpy and isobaric heat capacity for \(\mathrm{H}^{+}\) in aqueous solution. \[\mathrm{V}\left(\mathrm{H}^{+} ; \mathrm{aq} ; \mathrm{T} ; \mathrm{p}\right)=\left[\partial \mathrm{V} / \partial \mathrm{n}\left(\mathrm{H}^{+}\right)\right]_{\mathrm{T}, \mathrm{p}, \mathrm{n}\left(\mathrm{H}_{2} \mathrm{O}\right), \mathrm{n}\left(\mathrm{X}^{-}\right)}\]\[\mathrm{H}\left(\mathrm{H}^{+} ; \mathrm{aq} ; \mathrm{T} ; \mathrm{p}\right)=\left[\partial \mathrm{H} / \partial \mathrm{n}\left(\mathrm{H}^{+}\right)\right]_{\mathrm{T}, \mathrm{p}, \mathrm{n}\left(\mathrm{H}_{2} \mathrm{O}\right), \mathrm{n}\left(\mathrm{X}^{-}\right)}\]For electrolyte \(\mathrm{H}^{+} \mathrm{~X}^{-}\), \[\mathrm{V}^{\infty}\left(\mathrm{H}^{+} \mathrm{X}^{-} ; \mathrm{aq} ; \mathrm{T} ; \mathrm{p}\right)=\mathrm{V}^{\infty}\left(\mathrm{H}^{+} ; \mathrm{aq} ; \mathrm{T} ; \mathrm{p}\right)+\mathrm{V}^{\infty}\left(\mathrm{X}^{-} ; \mathrm{aq} ; \mathrm{T} ; \mathrm{p}\right)\]\[\mathrm{H}^{\infty}\left(\mathrm{H}^{+} \mathrm{X} ; \mathrm{aq} ; \mathrm{T} ; \mathrm{p}\right)=\mathrm{H}^{\infty}\left(\mathrm{H}^{+} ; \mathrm{aq} ; \mathrm{T} ; \mathrm{p}\right)+\mathrm{H}^{\infty}\left(\mathrm{X}^{-} ; \mathrm{aq} ; \mathrm{T} ; \mathrm{p}\right)\]and \[\mathrm{C}_{\mathrm{p}}^{\infty}\left(\mathrm{H}^{+} \mathrm{X}^{-} ; \mathrm{aq} ; \mathrm{T} ; \mathrm{p}\right)=\mathrm{C}_{\mathrm{p}}^{\infty}\left(\mathrm{H}^{+} ; \mathrm{aq} ; \mathrm{T} ; \mathrm{p}\right)+\mathrm{C}_{\mathrm{p}}^{\infty}\left(\mathrm{X}^{-} ; \mathrm{aq} ; \mathrm{T} ; \mathrm{p}\right)\]The chemical potential of \(\mathrm{H}_{3}\mathrm{O}^{+}\) ions in aqueous solution describes the change in Gibbs energy when \(\delta \mathrm{n}\left(\mathrm{H}_{3}\mathrm{O}^{+}\right)\) ions are added at constant \(\mathrm{n}\left(\mathrm{H}_{2}\mathrm{O}\right), \mathrm{~n}\left(\mathrm{X}^{-}\right)\), \(\mathrm{T}\) and \(\mathrm{p}\). At defined \(\mathrm{T}\) and \(\mathrm{p}\), \[\mu\left(\mathrm{H}_{3} \mathrm{O}^{+} ; \mathrm{aq}\right)=\left[\partial \mathrm{G} / \partial \mathrm{n}\left(\mathrm{H}_{3} \mathrm{O}^{+}\right)\right]_{\mathrm{n}\left(\mathrm{H}_{2} \mathrm{O}\right) ; \mathrm{n}\left(\mathrm{X}^{-}\right)}\]The chemical potential of the electrolyte \(\mathrm{H}_{3}\mathrm{O}^{+}\mathrm{X}^{-}\) in aqueous solution is described by the following equation. \[\begin{aligned} &\mu\left(\mathrm{H}_{3} \mathrm{O}^{+} \mathrm{X}^{-} ; \mathrm{aq}\right) \\ &=\mu^{0}\left(\mathrm{H}_{3} \mathrm{O}^{+} \mathrm{X}^{-} ; \mathrm{aq}\right)+2 \, \mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{m}\left(\mathrm{H}_{3} \mathrm{O}^{+} \mathrm{X}^{-}\right) \, \gamma_{\pm}\left(\mathrm{H}_{3} \mathrm{O}^{+} \mathrm{X}^{-}\right) / \mathrm{m}^{0}\right] \end{aligned}\]where \[\mathrm{m}\left(\mathrm{H}_{3} \mathrm{O}^{+} \mathrm{X}^{-}\right)=\mathrm{n}\left(\mathrm{H}_{3} \mathrm{O}^{+} \mathrm{X}^{-}\right) / \mathrm{n}\left(\mathrm{H}_{2} 0\right) \, \mathrm{M}_{1}\]\[\mathrm{n}\left(\mathrm{H}_{2} \mathrm{O}\right)=\mathrm{n}_{1}-\mathrm{n}\left(\mathrm{H}_{3} 0^{+}\right)\]and where \[\operatorname{limit}\left[\mathrm{m}\left(\mathrm{H}_{3} \mathrm{O}^{+} \mathrm{X}^{-}\right) \rightarrow 0\right) \gamma_{\pm}\left(\mathrm{H}_{3} \mathrm{O}^{+} \mathrm{X}^{-}\right)=1.0 \text { at all } \mathrm{T} \text { and } \mathrm{p}\]Thus the chemical potential \(\mu\left(\mathrm{H}_{3} \mathrm{O}^{+} \mathrm{~X}^{-} ; \mathrm{aq} ; \mathrm{T} ; \mathrm{p}\right)\) of the electrolyte \(\mathrm{H}_{3} \mathrm{O}^{+} \mathrm{~X}^{-}\) is related to the molality \(\mathrm{m}\left(\mathrm{H}_{3}\mathrm{O}^{+} \mathrm{~X}^{-}\right)\) and the mean ionic activity coefficient \(\gamma_{\pm} \left(\mathrm{H}_{3}\mathrm{O}^{+} \mathrm{~X}^{-}\right)\). We compare two descriptions of the same system. In description I, the system is an aqueous solution containing \(\mathrm{H}^{+}\) and \(\mathrm{X}^{-}\) ions whereas in description II the system is an aqueous solution containing \(\mathrm{H}_{3}\mathrm{O}^{+}\) and \(\mathrm{X}^{-}\) ions.Description I; \(\mathrm{n}_{j}\) moles of \(\mathrm{H}^{+} \mathrm{~X}^{-}\); \[\mathrm{G}(\mathrm{aq} ; \mathrm{I})=\mathrm{n}_{1} \, \mu_{1}(\mathrm{aq} ; \mathrm{I})+\mathrm{n}_{\mathrm{j}} \, \mu\left(\mathrm{H}^{+} ; \mathrm{aq} ; \mathrm{I}\right)+\mathrm{n}_{\mathrm{j}} \, \mu\left(\mathrm{X}^{-} ; \mathrm{aq} ; \mathrm{I}\right)\]Description II \[\mathrm{G}(\mathrm{aq} ; \mathrm{II})=\left(\mathrm{n}_{1}-\mathrm{n}_{\mathrm{j}}\right) \, \mu_{1}(\mathrm{aq} ; \mathrm{II})+\mathrm{n}_{\mathrm{j}} \, \mu\left(\mathrm{H}_{3} \mathrm{O}^{+} ; \mathrm{aq}\right)+\mathrm{n}_{\mathrm{j}} \, \mu\left(\mathrm{X}^{-} ; \mathrm{aq} ; \mathrm{II}\right)\]At equilibrium,At equilibrium (at defined \(\mathrm{T}\) and \(\mathrm{p}\)) \[\mu\left(\mathrm{H}^{+} ; \mathrm{aq}\right)+\mu_{1}(\mathrm{aq})=\mu\left(\mathrm{H}_{3} \mathrm{O}^{+} ; \mathrm{aq}\right)\]In effect we shifted a mole of water for each mole of \(\mathrm{H}^{+}\) ions from consideration as part of the solvent in description I to part of the solute in description II forming \(\mathrm{H}_{3}\mathrm{O}^{+}\) ions. The link between these descriptions is achieved through two formulations of \(\mathrm{G}^{\mathrm{eq}}(\mathrm{aq})\) which is identical for both systems (as are \(\mathrm{V}^{\mathrm{eq}}, \mathrm{~S}^{\mathrm{eq}} \text { and } \mathrm{H}^{\mathrm{eq}}\)). The equality of the total Gibbs function and equilibrium chemical potentials of substances common to both descriptions leads to equation (u) relating \(\mu\left(\mathrm{H}^{+} ; \mathrm{aq}\right)\) and \(\mu\left(\mathrm{H}_{3} \mathrm{O}^{+} ; \mathrm{aq}\right)\). We take the analysis a stage further and use the equations relating chemical potential and composition for the two ions \(\mathrm{H}^{+}\) and \(\mathrm{H}_{3}\mathrm{O}^{+}\) in aqueous solution at defined \(\mathrm{T}\) and \(\mathrm{p}\).Description I \[\mathrm{m}\left(\mathrm{H}^{+}\right)=\mathrm{n}\left(\mathrm{H}^{+}\right) / \mathrm{n}_{1} \, \mathrm{M}_{1} \quad \mathrm{~m}\left(\mathrm{X}^{-} ; \mathrm{I}\right)=\mathrm{n}\left(\mathrm{X}^{-}\right) / \mathrm{n}_{1} \, \mathrm{M}_{1}\]Description II \[\mathrm{m}\left(\mathrm{H}_{3} \mathrm{O}^{+}\right)=\mathrm{n}\left(\mathrm{H}_{3} 0^{+}\right) /\left(\mathrm{n}_{1}-\mathrm{n}_{\mathrm{j}}\right) \, \mathrm{M}_{1}\]and \[\mathrm{m}\left(\mathrm{X}^{-} ; \mathrm{II}\right)=\mathrm{n}\left(\mathrm{X}^{-}\right) /\left(\mathrm{n}_{1}-\mathrm{n}_{\mathrm{j}}\right) \, \mathrm{M}_{1}\]Also, \[\mu^{e q}\left(\mathrm{H}_{3} \mathrm{O}^{+} ; \mathrm{aq}\right)=\mu^{e q}\left(\mathrm{H}^{+} ; \mathrm{aq}\right)+\mu_{1}^{\mathrm{eq}}(\mathrm{aq})\]Hence, \[\begin{array}{r} \mu^{0}\left(\mathrm{H}_{3} \mathrm{O}^{+} \mathrm{X} ; \mathrm{aq}\right)+2 \, \mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{m}\left(\mathrm{H}_{3} \mathrm{O}^{+} \mathrm{X}\right) \, \gamma_{\pm}\left(\mathrm{H}_{3} \mathrm{O}^{+} \mathrm{X}^{-}\right) / \mathrm{m}^{0}\right] \\ =\mu^{0}\left(\mathrm{H}^{+} \mathrm{X} ; \mathrm{aq}\right)+2 \, \mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{m}\left(\mathrm{H}^{+} \mathrm{X}^{-}\right) \, \gamma_{\pm}\left(\mathrm{H}^{+} \mathrm{X}\right) / \mathrm{m}^{0}\right] \\ \left.\quad+\mu_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})-2 \, \phi(\mathrm{I}) \, \mathrm{R} \, \mathrm{T} \, \mathrm{m}^{*} \mathrm{H}^{+} \mathrm{X}^{\circ}\right) \, \mathrm{M}_{1} \end{array}\]But \[\begin{aligned} &\operatorname{limit}\left(\mathrm{n}_{\mathrm{j}} \rightarrow 0\right) \mathrm{m}\left(\mathrm{H}^{+} \mathrm{X}^{-}\right) / \mathrm{m}\left(\mathrm{H}_{3} \mathrm{O}^{+}\right)=1.0 \\ &\gamma_{\pm}\left(\mathrm{H}^{+} \mathrm{X}^{-}\right) / \gamma_{\pm}\left(\mathrm{H}_{3} \mathrm{O}^{+} \mathrm{X}^{-}\right)=1 \quad \phi(\mathrm{I})=1.0 \end{aligned}\]Hence, \[\mu^{\prime \prime}\left(\mathrm{H}_{3} \mathrm{O}^{+} ; \mathrm{aq}\right)=\mu^{\prime \prime}\left(\mathrm{H}^{+} ; \mathrm{aq}\right)+\mu_{1}^{*}\left(\mathrm{H}_{2} \mathrm{O} ; \ell\right)\]Also, \[\mathrm{V}^{\infty}\left(\mathrm{H}_{3} \mathrm{O}^{+} ; \mathrm{aq}\right)=\mathrm{V}^{\infty}\left(\mathrm{H}^{+} ; \mathrm{aq}\right)+\mathrm{V}_{1}^{*}\left(\mathrm{H}_{2} \mathrm{O} ; \ell\right)\]And \[\mathrm{C}_{\mathrm{p}}^{\infty}\left(\mathrm{H}_{3} \mathrm{O}^{+} ; \mathrm{aq}\right)=\mathrm{C}_{\mathrm{p}}^{\infty}\left(\mathrm{H}^{+} ; \mathrm{aq}\right)+\mathrm{C}_{\mathrm{pl}}^{*}\left(\mathrm{H}_{2} \mathrm{O} ; \ell\right)\]Interestingly the difference in reference chemical potentials of \(\mathrm{H}^{+}(\mathrm{aq})\) and \(\mathrm{H}_{3}\mathrm{O}^{+}(\mathrm{aq})\) equals the chemical potential of water(\(\ell\)) at the same \(\mathrm{T}\) and \(\mathrm{p}\). We combine equations (z) and (za) to obtain an equation relating the mean ionic activity coefficients \(\gamma_{\pm}\left(\mathrm{H}^{+} \mathrm{X}^{-}\right)\) and \(\gamma_{\pm}\left(\mathrm{H}_{3} \mathrm{O}^{+} \mathrm{X}^{-}\right)\). Thus (at defined \(\mathrm{T}\) and \(\mathrm{p}\)) \[\begin{aligned} \ln \left[\gamma_{\pm}\left(\mathrm{H}_{3} \mathrm{O}^{+} \mathrm{X}^{-}\right)\right] &=\ln \left[\gamma_{\pm}\left(\mathrm{H}^{+} \mathrm{X}^{-}\right)\right]+\ln \left[\mathrm{m}\left(\mathrm{H}^{+} \mathrm{X}^{-}\right) / \mathrm{m}\left(\mathrm{H}_{3} \mathrm{O}^{+} \mathrm{X}^{-}\right)\right] \\ &-\phi(\mathrm{I}) \, \mathrm{m}\left(\mathrm{H}^{+} \mathrm{X}^{-}\right) \, \mathrm{M}_{1} \\ +& {[1 / 2 \, \mathrm{R} \, \mathrm{T}] \,\left[\mu^{0}\left(\mathrm{H}^{+} \mathrm{X}^{-} ; \mathrm{aq}\right)+\mu_{1}^{*}(\ell)-\mu^{0}\left(\mathrm{H}_{3} \mathrm{O}^{+} \mathrm{X}^{-} ; \mathrm{aq}\right)\right] } \end{aligned}\]Then \[\begin{aligned} \ln \left[\gamma_{\pm}\left(\mathrm{H}_{3} \mathrm{O}^{+} \mathrm{X}^{-}\right)\right] &=\ln \left[\gamma_{\pm}\left(\mathrm{H}^{+} \mathrm{X}^{-}\right)\right]+\ln \left[\mathrm{m}\left(\mathrm{H}^{+} \mathrm{X}^{-}\right) / \mathrm{m}\left(\mathrm{H}_{3} \mathrm{O}^{+} \mathrm{X}^{-}\right)\right] \\ &-\phi(\mathrm{I}) \, \mathrm{m}\left(\mathrm{H}^{+} \mathrm{X}^{-}\right) \, \mathrm{M}_{1} \end{aligned}\]Also from equations (v) and (w), \[\ln \left[\mathrm{m}\left(\mathrm{H}^{+} \mathrm{X}^{-}\right) / \mathrm{m}\left(\mathrm{H}_{3} \mathrm{O}^{+} \mathrm{X}^{-}\right)=\ln \left[1-\mathrm{M}_{1} \, \mathrm{m}\left(\mathrm{H}^{+} \mathrm{X}^{-}\right)\right]\right.\]Clearly in dilute aqueous solutions where \(\left\{\mathrm{m}\left(\mathrm{H}^{+} \mathrm{X}^{-}\right) / \mathrm{m}\left(\mathrm{H}_{3} \mathrm{O}^{+} \mathrm{X}-\right)\right\}\) is approximately unity and \(\phi(\mathrm{I}) \, \mathrm{m}\left(\mathrm{H}^{+} \mathrm{X}^{-}\right) \, \mathrm{M}_{1}\) is negligibly small, the two mean ionic activity coefficients are equal but this approximation becomes less acceptable with increase in the ratio \(\mathrm{n}\left(\mathrm{H}^{+} \mathrm{X}^{-}\right) / \mathrm{n}\left(\mathrm{H}_{2} \mathrm{O}\right)\). H. L. Clever, J. Chem.Educ.,1963,40,637. Neil Kensington Adam, Physical Chemistry, Oxford, 1956, pp. 376. L.S.Bagster and G.Cooling, J. Chem. Soc., 1920,117,693. R. Triolo and A. H. Narten, J. Chem.Phys.,1975, 63, 3264. A. J. C. Cunningham, J. D. Payzant and P. Kebarle, J. Am. Chem. Soc.,1972,94,7627. P. A. Kollman and C. F. Bender, Chem. Phys. Lett., 1973,21,271. M. Eigen and L. De Maeyer, Proc. R. Soc. London,Ser. A, 1958,247,505. B. E. Conway, J. O’M. Bockris and H. Linton, J. Chem.Phys.,1956, 24 ,834. G. J. Hills, P. J. Ovenden and D. R. Whitehouse. Discuss. Faraday Soc.. 1965,39, 207. E.U. Franck, D. Hartmann and F. Hensel, Discuss. Faraday Soc., 1965,39, 200. N. Agmon, Chem. Phys. Lett., 1995, 244,456. H. G. Hertz, Chemica Scripta,1987,27,479. H. G. Hertz, B. M. Braun, K. J. Muller and R. Maurer, J. Chem.Educ.,1987,64,777. J.-O. Lundgren and I. Olovsson, Acta Cryst., 1967,23,966,971.This page titled 1.23.3: Water: Hydrogen Ions is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.23.4: Water: Relative Permittivity

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Parallel with interest in the \(\mathrm{p}-\mathrm{V}-\mathrm{T}\) properties of water(\(\ell\)), enormous interest has been shown in the relative permittivity of water(\(\ell\)) as a function of \(\mathrm{T}\) and \(\mathrm{p}\). Owen and Brinkley showed that a type of Tait equation could be used to express the \(\mathrm{T}-\mathrm{p}\) dependence of the relative permittivity of many liquids, including water(l). In 1980, Uematsu and Franck reviewed published information concerning the permittivity of water(\(\ell\)) and steam published over a period of 100 years.A careful determination was reported by Deul in 1984. At \(298.15 \mathrm{~K}\) and ambient pressure, the relative permittivities of water(\(\ell\)) and \(\mathrm{D}_{2}\mathrm{O}(\ell\)) are 78.39 and 78.06 respectively. A report by Owen and co-workers in 1961 stated a value of 78.358 for water(\(\ell\)) at \(298.15 \mathrm{~K}\). Bradley and Pitzer survey published relative permittivities of water(\(\ell\)) over an extensive temperature range. B. B. Owen and S. R. Brinkley, Phys. Rev.,1943,64,32. M. Uematsu and E. U. Franck, J. Phys. Chem. Ref. Data, 1980, 9,1291. K. Heger, M. Uematsu and E. U. Franck, Ber. Bunsenges, Phys.Chem.,1980,84,758. R. Deul, PhD Thesis, Faculty of Chemistry, University of Karlsruhe, 1984. G. A. Vidulich, D. F. Evans and R. L. Kay, J.Phys.Chem.,1967,71,656. B. B. Owen, R. C. Miller, C. E. Milner and H. L. Cogan, J. Phys. Chem., 1961, 65, 2065. D. J. Bradley and K. S. Pitzer, J Phys.Chem.,1979,83,1599.This page titled 1.23.4: Water: Relative Permittivity is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.23.5: Water: Self-Dissociation

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A significant contribution to the chemistry of aqueous solutions stems from the self dissociation of water(\(\ell\))(see also \(\mathrm{D}_{2}\mathrm{O}(\ell\))). At \(298.15 \mathrm{~K}\) and ambient pressure, \(\mathrm{pK}_{\mathrm{a}} equals 14.004. Olofsson and Hepler recommended a ‘best value’ for the standard enthalpy of self dissociation at ambient pressure and \(298.15 \mathrm{~K}\) equal to \(55.81 \mathrm{~kJ mol}^{-1}\). Hepler and colleagues recommend a best value for\(\Delta_{\mathrm{d}} C_{\mathrm{p}}^{0}\) equal to \(- 215 \mathrm{~J K}^{-1} \mathrm{~mol}^{-1}\); see also together with for details characterizing \(\mathrm{D}_{2}\mathrm{O}\).The standard volume of self-dissociation for water(\(\ell\)) at \(298 \mathrm{~K}\) is negative, approx. \(- 20 \mathrm{cm}^{3} \mathrm{~mol}^{-1}\), decreasing with increase in temperature.An extensive literature describes the thermodynamics of self-dissociation of water in binary liquid mixtures; see also for \(\mathrm{D}_{2}\mathrm{O}\). A. K. Covington, R. A. Robinson and R. G. Bates, J. Phys. Chem., 1966, 70, 3820. A. K. Covington, M. I. A. Ferra and R. A. Robinson, J. Chem. Soc. Faraday Trans.,1,1977,73,1721. G. Olofsson and L. G. Hepler, J. Solution Chem.,1975,4,127. G. Olofsson and I. Olofsson, J. Chem. Thermodyn.,1973, 5, 533; 1977, 9, 65. O. Enea, P. P. Singh, E. M. Woolley, K. G. McCurdy and L. G. Hepler, J. Chem. Thermodyn., 1977, 9,731. G. C. Allred and E. M. Woolley, J. Chem. Thermodyn..,1981, 13, 147. P. P. Singh, K. G. McCurdy, E. M. Woolley and L. G. Hepler, J. Solution Chem., 1977,6,327. J. J. Christensen, G. L .Kimball, H. D. Johnston and R. M. Izatt, Thermochim. Acta,1972,4,141. J. P. Hershey, R. Damasceno and F. J. Millero, J. Solution Chem.,1984,13,825. R. E.George and E. M. Woolley, J. Solution Chem.,1972,1,279.This page titled 1.23.5: Water: Self-Dissociation is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.23.6: Water: (Shear) Viscosity

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When the properties of water(\(\ell\)) are reviewed, general practice is to identify the importance of intermolecular hydrogen bonding as a molecular cohesive force. In these terms it is perhaps a surprise to discover that water(\(\ell\)) pours quite smoothly and freely, certainly more freely than, say, glycerol(\(\ell\)). Indeed the viscosity of water(\(\ell\)) is quite modest; \(0.8903 \mathrm{~cP}\) at \(298.15 \mathrm{~K}\).Nevertheless there are indications of complexity because below \(230 \mathrm{~K}\) the viscosity of water(\(\ell\)) decreases with increase in pressure before increasing. Good agreement exists between the results reported by many laboratories for viscosities of water(\(\ell\)) at low pressures but disagreement at high pressure. \[\mathrm{P}=10^{-1} \mathrm{~Pa} \mathrm{~s}=10^{-1} \mathrm{~J} \mathrm{~m}^{-3} \mathrm{~s}\] G. S. Kell, in Water; A Comprehensive Treatise, ed. F. Franks, McGraw-Hill, New York, 1973, volume 1, chapter 10. S. D. Hamann, Physics and Chemistry of the Earth, 1981,13,89.This page titled 1.23.6: Water: (Shear) Viscosity is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.24: Misc

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1.14.1: Adiabatic1.14.2: Adsorption- Langmuir Adsorption Isotherm- One Adsorbate1.14.3: Absorption Isotherms - Two Absorbates1.14.4: Apparent Molar Properties- Solutions- Background1.14.5: Apparent Molar Properties- Solutions- General1.14.6: Axioms1.14.7: Boundary1.14.8: Calculus1.14.9: Clausius - Clapeyron Equation1.14.10: Gibbs - Helmholtz Equation1.14.11: Guggenheim-Scatchard Equation / Redlich-Kister Equation1.14.12: Legendre Transformations1.14.13: Closed System1.14.14: Cohesive Energy Density1.14.15: Degree of Dissociation1.14.16: Energy and First Law of Thermodynamics1.14.17: Energy and Entropy1.14.18: Electrochemical Units1.14.19: Electrical Units1.14.20: Electric Conductivities of Salt Solutions- Dependence on Composition1.14.21: Donnan Membrane Equilibria1.14.22: Descriptions of Systems1.14.23: Enzyme-Substrate Interaction1.14.24: Equation of State- General Thermodynamics1.14.25: Equation of State- Perfect Gas1.14.26: Equation of State - Real Gases, van der Waals, and Other Equations1.14.27: Euler's Theorem1.14.28: First Law of Thermodynamics1.14.29: Functions of State1.14.30: Heat, Work, and Energy1.14.31: Helmholtz Energy1.14.32: Hildebrand Solubility Parameter1.14.33: Infinite Dilution1.14.34: Internal Pressure1.14.35: Internal Pressure: Liquid Mixtures: Excess Property1.14.36: Irreversible Thermodynamics1.14.37: Irreversible Thermodynamics: Onsager Phenomenological Equations1.14.38: Joule-Thomson Coefficient1.14.39: Kinetic Salt Effects1.14.40: Laws of Thermodynamics1.14.41: Lewisian Variables1.14.42: L'Hospital's Rule1.14.43: Master Equation1.14.44: Maxwell Equations1.14.45: Moderation1.14.46: Molality and Mole Fraction1.14.47: Newton-Laplace Equation1.14.48: Open System1.14.49: Osmotic Coefficient1.14.50: Osmotic Pressure1.14.51: Partial Molar Properties: General1.14.52: Partial Molar Properties: Definitions1.14.53: Phase Rule1.14.54: Poynting Relation1.14.55: Process1.14.56: Properties: Equilibrium and Frozen1.14.57: Reversible Change1.14.58: Reversible Chemical Reactions1.14.59: Salting-In and Salting-Out1.14.60: Second Law of Thermodynamics1.14.61: Solubility Products1.14.62: Solubilities of Gases in Liquids1.14.63: Solubilities of Solids in Liquids1.14.64: Solutions: Solute and Solvent1.14.65: Spontaneous Change: Isothermal and Isobaric1.14.66: Spontaneous Chemical Reaction1.14.67: Standard States: Reference States: Processes1.14.68: Surroundings and System1.14.69: Temperature of Maximum Density: Aqueous Solutions1.14.70: Solutions: Neutral Solutes: Inter-Solute Distances1.14.71: Time and Thermodynamics (Timenote)1.14.72: Variables: Independent and Dependent1.14.73: Variables: Gibbsian and Non-Gibbsian1.14.74: Vaporization1.14.75: Viscosities: Salt Solutions1.14.76: Work This page titled 1.24: Misc is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.3.1: Calorimeter- Isobaric

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An isobaric calorimeter is designed to measure the heat accompanying the progress of a closed system from state (I) to state (II) at constant pressure. It follows from the first law that if only ‘\(p-\mathrm{V}\)’ work is involved, \[\Delta \mathrm{U}=\mathrm{q}-\mathrm{p} \, \Delta \mathrm{V}\]By definition the enthalpy \(\mathrm{H}\) of a closed system is given by equation (b); \[\mathrm{H}=\mathrm{U}+\mathrm{p} \, \mathrm{V}\]\[\text { Then, } \Delta \mathrm{H}=\Delta \mathrm{U}+\mathrm{p} \, \Delta \mathrm{V}+\Delta \mathrm{p} \, \mathrm{V}\]Hence from equations (a) and (c), at constant pressure, \[\Delta \mathrm{H}=\mathrm{q}\]\[\text { Thus at constant pressure, } \Delta \mathrm{H}=\mathrm{H}(\mathrm{II})-\mathrm{H}(\mathrm{I})=\mathrm{q}\]Hence if we record the heat (exothermic or endothermic) at constant pressure we have the change in enthalpy, \(\Delta \mathrm{H}\). Equation (e) highlights the optimum thermodynamic equation. On one side of the equation is a measured property/change and on the other side of the equation is a change in a property of the system which we judge to be informative about the chemical properties of a system; e.g. \(\Delta \mathrm{H}\). The problem is that the derived property is not the actual change in energy, \(\Delta \mathrm{U}\).Footnotes W. Zielenkiewicz, J.Therm. Anal.,1988, 33, 7. Hess’ Law. This law is a consequence of the observation that the enthalpy of a closed system is a state variable. \(\Delta \mathrm{H}\) accompanying the change from state I to state II is independent of the number of intermediary states and of the general path between the two states and the rate of change. Isothermal calorimetryThis page titled 1.3.1: Calorimeter- Isobaric is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.3.2: Calorimetry- Isobaric- General Operation

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Calorimetry, particularly isobaric calorimetry, is a key technique in chemical thermodynamics, for studying the properties of liquid mixtures and solutions. Numerous designs for calorimeters have been published. The operation of a classic calorimeter involves two key steps.Step 1. Known amounts of two liquids (e.g. solvent and solution) are mixed in a thermally insulated reaction vessel at constant pressure. The rise in temperature is recorded.Step 2. A known electric current is passed for a recorded length of time through an electric resistance in the reaction vessel to produce a comparable rise in temperature.By proportion the required amount of energy to produce the measured rise in temperature in step 1 is obtained. [Complications emerge by the need to take account of spontaneous cooling in both steps when the temperature of the calorimeter exceeds ambient temperature; cf. Newton’s Law of Cooling.]Another type of isobaric calorimeter involves injecting aliquots of one liquid (solution or solvent) into sample cell containing another liquid, recording the rise in temperature accompanying injection of each aliquot. The calorimeter is again calibrated electrically.Footnotes M. L. McGlashan, Chemical Thermodynamics, Academic Press, London, 1979, chapter 4.This page titled 1.3.2: Calorimetry- Isobaric- General Operation is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.3.3: Calorimetry- Solutions- Isobaric

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Classic (isobaric) calorimetric experiments often centre on the determination of the change in enthalpy ∆H for a given well-defined process. For example, the heat accompanying the mixing of known amounts of two liquids [e.g. water(\(\lambda\)) and ethanol(\(\lambda\))] to form a binary liquid mixture yields the enthalpy of mixing, \(\Delta_{\mathrm{mix}}\mathrm{H}\). Similarly enthalpies of solution are obtained by recording the heat accompanying the solution of a known amount of solute (e.g. urea) in a known amount of solvent; e.g. water(\(\lambda\)). Key equations emerge from the following analysis.The enthalpy \(\mathrm{H}\) of a closed system is an extensive function of state which for a closed system is defined by the set of independent variables, \(\mathrm{T}\), \(p\) and \(\xi\) where \(\xi\) represents the chemical composition. \[\mathrm{H}=\mathrm{H}[\mathrm{T}, \mathrm{p}, \xi] \label{a}\]Equation \ref{b} is the complete differential of Equation \ref{a}. \[\mathrm{dH}=\left(\frac{\partial \mathrm{H}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \xi} \, \mathrm{dT}+\left(\frac{\partial \mathrm{H}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \xi} \, \mathrm{dp}+\left(\frac{\partial \mathrm{H}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}} \, \mathrm{d} \xi \label{b}\]If the closed system is held at constant pressure (e.g. ambient) the differential enthalpy \(\mathrm{dH}\) equals the heat \(\mathrm{dq}\). \[T\mathrm{dq}=\left(\frac{\partial \mathrm{H}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \xi} \, \mathrm{dT}+\left(\frac{\partial \mathrm{H}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}} \, \mathrm{d} \xi\]Here \(\left(\frac{\partial \mathrm{H}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \xi}\), is the differential dependence of enthalpy \(\mathrm{H}\) on temperature at constant pressure and composition whereas \(\left(\frac{\partial H}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}\), is the differential dependence of enthalpy \(\mathrm{H}\) on composition at fixed temperature and pressure.This page titled 1.3.3: Calorimetry- Solutions- Isobaric is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.3.4: Calorimetry- Solutions- Adiabatic

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A general equation describes heat \(q\) in terms of changes in temperature and composition at constant pressure; \(\mathrm{dH} = q\).\[\mathrm{dq}=\left(\frac{\partial \mathrm{H}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \xi} \, \mathrm{dT}+\left(\frac{\partial \mathrm{H}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}} \, \mathrm{d} \xi \label{a}\]In this application of Equation \ref{a}, the system is thermally insulated; i.e. \(q\) is zero. An aliquot of solution containing a small amount of chemical substance \(j\) is added to a solution held in a thermally insulated container. A rearranged Equation \ref{a} takes the following form.\[\mathrm{dT}=-\frac{(\partial \mathrm{H} / \partial \xi)_{\mathrm{T}, \mathrm{p}}}{(\partial \mathrm{H} / \partial \mathrm{T})_{\mathrm{p}, \xi}} \, \mathrm{d} \xi\label{b}\]Chemical reaction occurs in the sample cell, the rate of chemical reaction being governed by the composition of the solution and appropriate rate constants. The differential isobaric dependence of temperature on time, \(\mathrm{dT} / \mathrm{dt}\) is given by Equation \ref{c}.\[\frac{\mathrm{dT}}{\mathrm{dt}}=-\frac{(\partial \mathrm{H} / \partial \xi)_{\mathrm{T}, \mathrm{p}}}{(\partial \mathrm{H} / \partial \mathrm{T})_{\mathrm{p}, \xi}} \, \frac{\mathrm{d} \xi}{\mathrm{dt}}\label{c}\]The calorimeter records the dependence of temperature on time. An equation based on the Law of Mass Action yields the rate of change of composition \(\mathrm{d}\xi / \mathrm{dt}\). The integrated form of Equation \ref{c} yields a calculated dependence of \(\mathrm{T}\) on time which can be compared with the recorded dependence. This subject is important in the context of thermal imaging calorimetry.Footnotes M. J. Blandamer, P. M. Cullis, and P. T. Gleeson, Phys. Chem. Chem. Phys.,2002,4,765. B. Jandeleit, D. J. Schaefer, T. S. Powers, H. W. Turner and W. H. Weinbereg, Angew. Chem. Int. Ed. Engl.,1999,38,2495. M. T. Reetz, M. H. Becker, K. M. Kuling and A. Holzwarth, Angew. Chem. Int. Ed. Engl.,1998, 37,2647. G. C. Davies, R. S. Hutton, N. Millot, S. J. F. Macdonald, M. S. Hansom and I. B. Campbell, Phys. Chem. Chem.Phys.,2002,4,1791.This page titled 1.3.4: Calorimetry- Solutions- Adiabatic is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.3.5: Calorimetry- Solutions - Heat Flow

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At temperature \(\mathrm{T}\) and pressure \(p\), the enthalpy of a closed system having composition \(\xi\) can be defined by Equation \ref{a}. \[\mathrm{H}=\mathrm{H}[\mathrm{T}, \mathrm{p}, \xi] \label{a}\]The general differential of Equation \ref{a} takes the following form. \[\mathrm{dH}=\left(\frac{\partial \mathrm{H}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \xi} \, \mathrm{dT}+\left(\frac{\partial \mathrm{H}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \xi} \, \mathrm{dp}+\left(\frac{\partial \mathrm{H}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}} \, \mathrm{d} \xi\]If the pressure is constant at, for example, ambient pressure, \(\mathrm{dH}\) equals the differential heat \(\mathrm{dq}\) passing between system and surroundings. In the application considered here, the temperature is held constant.The following equation describes heat \(\mathrm{dq}\) in terms of changes in composition at constant pressure and constant temperature.Thus\[\mathrm{dq}=\left(\frac{\partial \mathrm{H}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}} \, \mathrm{d} \xi\]Moreover \(\mathrm{d}\xi\) is the extent of chemical reaction in the time period \(\mathrm{dt}\). \[\text {Then } \left(\frac{\mathrm{dq}}{\mathrm{dt}}\right)=\left(\frac{\partial \mathrm{H}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}} \,\left(\frac{\mathrm{d} \xi}{\mathrm{dt}}\right)\]If the chemical reaction in the sample cell involves a single chemical reaction, \((\partial \mathrm{H} / \partial \xi)_{\mathrm{T}, \mathrm{p}}\) is the enthalpy of reaction \(\Delta_{\mathrm{r}}\mathrm{H}\). \[\text { Therefore, }\left(\frac{\mathrm{dq}}{\mathrm{dt}}\right)=\Delta_{\mathrm{r}} \mathrm{H} \,\left(\frac{\mathrm{d} \xi}{\mathrm{dt}}\right)\]In Heat Flow Calorimetry, a small closed reaction vessel is in contact with a heat sink so that the reaction vessel is held at constant temperature. The flow of heat between sample cell and heat sink is monitored such that the recorded quantity is the thermal power, the rate of heat production (\(\mathrm{dq} / \mathrm{dt}\)) as a result of chemical reaction. The property (\(\mathrm{dq} / \mathrm{dt}\)) is recorded as a function of time; also as the amount of reactants decreases, \(\operatorname{limit}(\mathrm{t} \rightarrow \infty)(\mathrm{dq} / \mathrm{dt})\) is zero. Nevertheless because \((\mathrm{d} \xi / \mathrm{dt})\) is a function of time, \((\mathrm{dq} / \mathrm{dt})\) effectively monitors the progress of chemical reaction. Intuitively it is apparent that for an exothermic reaction \((\mathrm{dq} / \mathrm{dt})\) at time zero is also zero, rises rapidly and then decreases to zero as all reactants are consumed.The Law of Mass Action relates \((\mathrm{d} \xi / \mathrm{dt})\) to the composition of the system at time \(\mathrm{t}\). Because \((\mathrm{d} \xi / \mathrm{dt})\) depends on time, \((\mathrm{dq}/\mathrm{dt})\) also depends on time, approaching zero as reactants are consumed.For example, in the case of a simple chemical reaction of the form \(\mathrm{X} \rightarrow \mathrm{Y}\) where at \(\mathrm{t} = 0\) the amount of chemical substance \(\mathrm{X}\) is \(n_{\mathrm{X}}^{0}\), the amounts of \(\mathrm{X}\) and \(\mathrm{Y}\) at time \(t\) are \(\left(\mathrm{n}_{\mathrm{X}}^{0}-\xi\right)\) and \(\xi\) moles respectively. If the volume of the sample cell is \(\mathrm{V}\),\[(1 / \mathrm{V}) \, \mathrm{d} \xi / \mathrm{dt}=(1 / \mathrm{V}) \, \mathrm{k} \,\left(\mathrm{n}_{\mathrm{X}}^{0}-\xi\right)\]\[\text { Or, } \quad \mathrm{d} \xi / \mathrm{dt}=\mathrm{k} \, \mathrm{V} \,\left[\mathrm{c}_{\mathrm{x}}^{0}-(\xi / \mathrm{V})\right]\]\[\text { Or, } \quad \mathrm{d} \xi / \mathrm{dt}=\mathrm{k} \, \mathrm{V} \, \mathrm{c}_{\mathrm{X}}^{0} \, \exp (-\mathrm{k} \, \mathrm{t})\]Hence using equation (e) and for a dilute solution, \[\mathrm{dq} / \mathrm{dt}=\Delta_{\mathrm{r}} \mathrm{H}^{0} \, \mathrm{k} \, \mathrm{V} \, \mathrm{c}_{\mathrm{X}}^{0} \, \exp (-\mathrm{k} \, \mathrm{t})\]The integral of equation (i) between \(\mathrm{t} = 0\) and time \(\mathrm{t}\) yields the amount of heat passing between system and heat sink. \[\text { Thus, } \quad \int_{0}^{\mathrm{t}} \mathrm{dq}=\Delta_{\mathrm{r}} \mathrm{H}^{\circ} \, \mathrm{k} \, \mathrm{V} \, \mathrm{c}_{\mathrm{X}}^{0} \,[1-\exp (-\mathrm{k} \, \mathrm{t})]\]Hence the measured dependence of (\(\mathrm{dq} / \mathrm{dt}\)) is compared with that calculated using equations (i) and (j). The analysis is readily extended to second order reactions.The technique of heat flow calorimetry has been applied across a wide range of subjects (e.g. screening of catalystsand characterising complex reactions) and subject to different analytical approaches.Footnotes I. Prigogine and R. Defay, Chemical Thermodynamics, trans. D. H. Everett, Longmans Green, London, 1954. I. Wadso, Chem. Soc. Rev.,1997,26,79. P. Backman, M. Bastos, D. Hallen, P. Lombro and I. Wadso, J. Biochem. Biophysic. Methods, 1994,28,85. M. J. Blandamer, P. M. Cullis and P. T. Gleeson, J. Phys. Org. Chem.,2002,15,343. A. Thiblin, J. Phys. Org. Chem.,2002,15,233. D. G. Blackmond, T. Rosner and A. Pfaltz, Organic Process Research and Development,1999,3,275. C. Le Blond, J . Wang, R. D. Larsen, C. J. Orella, A. L. Forman, R. N. Landau, J. Lequidara, J. R. Sowa Jr., D. G. Blackmond and Y.-K. Sun.,Thermochim Acta,1996,289.189. A. E. Beezer, A. C. Morris, M. A.A. O’Neil, R. J. Willson, A. K. Hills, J. C. Mitchell and J. A. O’Connor, J.Phys.Chem.B.,2001,105,1212. R. J. Willson, A. E. Beezer, J. C. Mitchell and W. Loh, J. Phys. Chem., 1995, 99.7108.This page titled 1.3.5: Calorimetry- Solutions - Heat Flow is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.3.6: Calorimetry - Titration Microcalorimetry

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In a common type of calorimeter, aliquots of one liquid (solution or solvent) are injected into a sample cell containing another liquid. The rise in temperature accompanying injection of each aliquot is recorded. The calorimeter is calibrated electrically.With advances in microelectronics and calorimeter design the volume of liquids required in titration calorimetry has dropped so that only micro-litres of aliquots are injected into a sample cell having a volume of the order \(1 \mathrm{~cm}^{3}\). Operation of the calorimeter is under the control of a mini-computer. The sensitivity of these calorimeters is such that recorded heats are of the order of \(10^{–6} \mathrm{~J}\). In a typical experiment sample and reference cells, held in an evacuated enclosure, are heated such that the temperatures of both cells increase at the rate of a few micro-kelvin per second. The electronic heaters and thermistors are coupled so that these temperatures (plus that of the adiabatic shield) stay in step. Under computer control, aliquots of a given solution from a micro-syringe are injected into the sample cell at predetermined intervals. The operation of the calorimeter is readily understood where the chemical processes in the sample cell following injection of an aliquot are exothermic. In this case the temperature of the solution in the sample cell increases so heating of this cell is stopped. The reference cell continues to be heated until at some stage the temperatures of both sample and reference cells are again equal, when again both cells are heated in preparation for the next injection of an aliquot. The computer records how much heat was produced by the electric heaters in the reference cell to recover the situation of equal temperatures. This amount of heat must have been produced effectively by chemical processes in the sample cell.Titration microcalorimetry has had a major impact in biochemistry with respect to the study of enzyme - substrate binding.The starting point of the thermodynamic analysis is the definition of the extensive variable enthalpy \(\mathrm{H}\) of a closed system in terms of temperature, pressure and composition; equation (a). \[\mathrm{H}=\mathrm{H}[\mathrm{T}, \mathrm{p}, \xi]\]The complete differential of equation (a) takes the following form. \[\mathrm{dH}=\left(\frac{\partial \mathrm{H}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \xi} \, \mathrm{dT}+\left(\frac{\partial \mathrm{H}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \xi} \, \mathrm{dp}+\left(\frac{\partial \mathrm{H}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}} \, \mathrm{d} \xi\]The key term in the present context is the last term in equation (b) which describes a change in enthalpy at constant \(\mathrm{T}\) and \(p\). \[\mathrm{dH}=\left(\frac{\partial \mathrm{H}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}} \, \mathrm{d} \xi\]In the present context the change in composition/organisation \(\mathrm{d}\xi\) refers to the contents of the sample cell accompanying injection of an aliquot from the syringe. Heat \(q\) is recorded following injection of \(\mathrm{d}{n_{j}}^{0}\) moles of chemical substance \(j\) into the sample cell on going from injection number \(\mathrm{I}\) to injection number \(\mathrm{I}+1\). \[\left[\frac{\mathrm{q}}{\mathrm{dn}_{\mathrm{j}}^{0}}\right]_{1}^{1+1}=\left[\left(\frac{\partial \mathrm{H}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}} \, \frac{\mathrm{d} \xi}{\mathrm{dn}_{\mathrm{j}}^{0}}\right]_{1}^{1+1}\]Equation (d) is the key to titration microcalorimetry. The recorded quantity \(q\) on the left-hand side of equation (d) is the recorded heat at injection number \(\mathrm{I}+1\) when further \(\mathrm{d}{n_{j}}^{0}\) moles of chemical substance \(j\) are injected into the sample cell. The right-hand-side shows that the recorded ratio \[\left[\frac{\mathrm{q}}{\mathrm{dn}_{\mathrm{j}}^{0}}\right]_{\mathrm{l}}^{\mathrm{l}+1}\] is related to the dependence of enthalpy \(\mathrm{H}\) on composition, \(\left(\frac{\partial \mathrm{H}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}\), and the dependence of composition/organisation on the amount of substance \(j\) injected. Plots of \(\left[\frac{\mathrm{q}}{\mathrm{dn}_{\mathrm{j}}^{0}}\right]_{\mathrm{l}}^{\mathrm{l}+1}\) as a function of injection number are called enthalpograms.Equation (d) highlights an underlying problem in the analysis of experimental results. The recorded quantity is heat \(q\) and no information immediately emerges concerning the chemical processes responsible although we note that the sign of heat \(q\) is not predetermined; i.e. processes can be exo- or endo- thermic. The r.h.s. of equation (d) involves the product of two quantities, \(\left(\frac{\partial H}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}\) and \(\frac{\mathrm{d} \xi}{\mathrm{dn}_{\mathrm{j}}^{0}}\). We have no ‘a priori’ indication concerning how to pull these terms apart. In other words we require a model for the chemical processes in the sample cell.Footnotes T. S. Wiseman, S. Williston, J.F.Brandts and Z.-N. Lim, Anal.Biochem., 1979, 179,131. Biocalorimetry, ed. J. E. Ladbury and B. Z. Chowdhry, Wiley Chichester, 1998. M. J. Blandamer, P. M. Cullis and J. B. F. N. Engberts, J. Chem. Soc. Faraday Trans., 1998, 94, 2261. J. Ladbury and B. Z. Chowdhry, Chemistry and Biology, 1996,3,79. M. J. Blandamer, P. M. Cullis and J. B. F. N. Engberts, Pure Appl. Chem.,1996,68,1577.This page titled 1.3.6: Calorimetry - Titration Microcalorimetry is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.3.7: Calorimeter- Titration Microcalorimetry- Enzyme-Substrate Interaction

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.03%3A_Calorimeter/1.3.07%3A_Calorimeter-_Titration_Microcalorimetry-_Enzyme-Substrate_Interaction

The technique of titration microcalorimetry was developed with the aim of probing enzyme-substrate interactions. At the start of the experiment, the sample cell contains an aqueous solution containing a known amount of a macromolecular enzyme \(\mathrm{M}(\mathrm{aq})\). The injected aliquots contain a known amount of substrate \(\mathrm{X}\) such that during the experiment the composition of the sample cell is described in terms of the following chemical equilibrium.\[\mathrm{M}(\mathrm{aq})+\mathrm{X}(\mathrm{aq}) \Leftrightarrow \mathrm{MX}(\mathrm{aq})\][It is important to note that the term substrate refers to the chemical substance which is bound (adsorbed?) by the macromolecular enzyme. In treatments of adsorption, the macromolecular substrate adsorbs small molecules.] In the limit of strong binding (see below), most of the injected substrate at the start of the experiment is bound by the enzyme. But gradually as more substrate is added, the number of free binding sites decreases until eventually all sites are occupied and hence no heat \(q\) is recorded. The plot of \(\left[\mathrm{q} / \mathrm{dn}_{\mathrm{X}}^{0}\right]\) against injection number in the textbook case is sigmoidal.The equilibrium established in the sample cell is described as follows.\[\begin{array}{llccc} & \mathrm{M}(\mathrm{aq}) & \mathrm{X}(\mathrm{aq}) \quad<==> & \mathrm{MX}(\mathrm{aq}) & \\ \text { At } \mathrm{t}=0 & \mathrm{n}^{0}(\mathrm{M}) & \mathrm{n}^{0}(\mathrm{X}) & 0 & \mathrm{~mol} \\ \text { At } \mathrm{t}=\infty & \mathrm{n}^{0}(\mathrm{M})-\xi & \mathrm{n}^{0}(\mathrm{X})-\xi & \xi & \mathrm{mol} \\ & {\left[\mathrm{n}^{0}(\mathrm{M})-\xi\right] / \mathrm{V}} & {\left[\mathrm{n}^{0}(\mathrm{X})-\xi\right] / \mathrm{V}} & \xi / \mathrm{V} & \mathrm{mol} \mathrm{dm} \end{array}\]\(\mathrm{V}\) is the volume of the sample cell. The analysis uses equilibrium constants defined in terms of the concentrations of chemical substances in the system. There are advantages in using equilibrium constants having the following form where \(\mathrm{c}_{\mathrm{r} = 1 \mathrm{~mol dm}^{-3}\).\[\mathrm{K}=(\xi / \mathrm{V}) \, \mathrm{c}_{\mathrm{r}} /\left\{\left[\mathrm{n}^{0}(\mathrm{M})-\xi\right] / \mathrm{V}\right\} \,\left\{\left[\mathrm{n}^{0}(\mathrm{X})-\xi\right] / \mathrm{V}\right\}\]The latter is a quadratic in the extent of reaction, \(\xi\).\[\xi^{2}+b \, \xi+c=0\]where\[ b=-n^{0}(M)-n^{0}(X)-V \, c_{r} \, K^{-1}\]and\[\mathrm{c}=\mathrm{n}^{0}(\mathrm{M}) \, \mathrm{n}^{0}(\mathrm{X})\]Therefore\[\xi=-(b / 2) \pm(1 / 2) \,\left(b^{2}-4 \, c\right)^{1 / 2}\]The negative root of the quadratic yields the required solution on the grounds that, with increase in \(n^{0}(\mathrm{X}\)\) in the sample cell, more substrate is bound by the enzyme. The required quantity is \(\left(\mathrm{d} \xi / \operatorname{dn}_{\mathrm{X}}^{0}\right)\). We note that from equations (g) and (h), \[\mathrm{db} / \mathrm{dn}_{\mathrm{X}}^{0}=-1 ; \quad \mathrm{dc} / \mathrm{dn}_{\mathrm{X}}^{0}=\mathrm{n}_{\mathrm{M}}^{0}\]In the experiment we control the ratio of total amounts of substrate to enzyme, \(\mathrm{n}^{0}(\mathrm{X}) / \mathrm{n}^{0}(\mathrm{M})\), which increases as more substrate is added to the sample cell. A measure of the ‘tightness of binding’ is the fraction of substrate bound when this ratio is unity. \[\text { By definition, } \quad X_{r}=n^{0}(X) / n^{0}(M)=\left[X_{\text {total }}\right] /\left[M_{\text {total }}\right]\]We define two variables \(\mathrm{r}\) and \(\mathrm{C}\); (note uppercase). \[\mathrm{V} \, \mathrm{c}_{\mathrm{r}} / \mathrm{K} \, \mathrm{n}^{0}(\mathrm{M})=\mathrm{c}_{\mathrm{r}} / \mathrm{K} \,\left[\mathrm{M}_{\text {total }}\right]=\mathrm{r}=1 / \mathrm{C}\]\[\text { From equation (e), }[\mathrm{X}]^{\mathrm{eq}} /[\mathrm{MX}]^{\mathrm{eq}}=\mathrm{c}_{\mathrm{r}} / \mathrm{K} \,[\mathrm{M}]^{\mathrm{eq}}\]If \(\mathrm{K}\) is small, then \(\mathrm{r}\) is large and only a small amount of the injected substrate is bound to the enzyme. \([\mathrm{X}]^{\mathrm{eq}}\) is, in relative terms, large and \([\mathrm{MX}]^{\mathrm{eq}\) is small. If \(\mathrm{r}\) is large, \(\mathrm{C}\) is small.If on the other hand \(\mathrm{K}\) is large, \(\mathrm{r}\) is small, \(\mathrm{C}\) is large and in the limit all substance \(\mathrm{X}\) is bound to the enzyme \(\mathrm{M}\).We return to equation (i) because in order to calculate \(\xi\) we require \(b\). \[\mathrm{b}^{2}-4 \, \mathrm{c}=\left[\mathrm{n}^{0}(\mathrm{M})\right]^{2} \,\left[\mathrm{X}_{\mathrm{r}}^{2}-2 \, \mathrm{X}_{\mathrm{r}} \,(1-\mathrm{r})+(1+\mathrm{r})^{2}\right]\]We also require \(\left[\mathrm{d} \xi / \operatorname{dn}^{0}(\mathrm{X})\right]\) which describes the dependence of extent of substrate binding on total amount of \(\mathrm{X}\) in the sample cell, noting that we can control the latter through the concentration of the injected aliquots. We return to equation (i) making use of equation (n). \[\frac{\mathrm{d} \xi}{\mathrm{dn}_{\mathrm{X}}^{0}}=\frac{1}{2}+\frac{\left[1-(1 / 2) \,(1+\mathrm{r})-\mathrm{X}_{\mathrm{r}} / 2\right]}{\left[\mathrm{X}_{\mathrm{r}}^{2}-2 \, \mathrm{X}_{\mathrm{r}} \,(1-\mathrm{r})+(1+\mathrm{r})^{2}\right]^{1 / 2}}\]The enthalpy of the solution in the sample cell (assuming the thermodynamic properties of the solution are ideal) is given by equation (p). \[\begin{aligned} \mathrm{H}(\mathrm{aq} ; \mathrm{id})=\mathrm{n}_{1}(\lambda) \, \mathrm{H}_{1}^{*}(\lambda)+\left[\mathrm{n}^{0}(\mathrm{M})-\xi\right] \, \mathrm{H}^{\infty}(\mathrm{M} ; \mathrm{aq}) \\ &+\left[\mathrm{n}_{\mathrm{X}}^{0}-\xi\right] \, \mathrm{H}^{\infty}(\mathrm{X} ; \mathrm{aq})+\xi \, \mathrm{H}^{\infty}(\mathrm{MX} ; \mathrm{aq}) \end{aligned}\]\[[\partial \mathrm{H}(\mathrm{aq} ; \mathrm{id}) / \partial \xi \xi]=-\mathrm{H}^{\infty}(\mathrm{M} ; \mathrm{aq})-\mathrm{H}^{\infty}(\mathrm{X} ; \mathrm{aq})+\mathrm{H}^{\infty}(\mathrm{MX} ; \mathrm{aq})=\Delta_{\mathrm{B}} \mathrm{H}^{\infty}\]Hence the dependence of the ratio \(\left[\mathrm{q} / \mathrm{dn}_{\mathrm{X}}^{0}\right]\) on composition of the sample cell is given by equation (r). \[\frac{\mathrm{q}}{\mathrm{dn}_{\mathrm{X}}^{0}}=\Delta_{\mathrm{B}} \mathrm{H}^{\infty} \,\left[\frac{1}{2}+\frac{\left[1-(1 / 2) \,(1+\mathrm{r})-\mathrm{X}_{\mathrm{r}} / 2\right]}{\left[\mathrm{X}_{\mathrm{r}}^{2}-2 \, \mathrm{X}_{\mathrm{r}} \,(1-\mathrm{r})+(1+\mathrm{r})^{2}\right]^{1 / 2}}\right]\]The latter key equation describes the recorded enthalpogram. The latter falls into one of three general classes determined by the quantity \(\mathrm{C}\) in equation (\(\lambda\)).Footnotes T. S. Wiseman, S. Williston, J.F.Brandts and Z.-N. Lim, Anal. Biochem., 1979,179,131. M. J. Blandamer, in Biocalorimetry, ed. J. E. Ladbury and B. Z. Chowdhry, Wiley Chichester, 1998, p5. \(\mathrm{n}^{0}(\mathrm{M}) \, \mathrm{n}^{0}(\mathrm{X})-\xi \,\left[\mathrm{n}^{0}(\mathrm{M})+\mathrm{n}^{0}(\mathrm{X})\right]+\xi^{2}=\mathrm{V} \, \mathrm{c}_{\mathrm{r}} \, \mathrm{K}^{-1} \, \xi\) \[b^{2}=\left[-n^{0}(M)-n^{0}(X)-V \, c_{r} \, K^{-1}\right]^{2}\]\[\text { or, } b^{2}=\left[n^{0}(M)\right]^{2} \,\left[1+n^{0}(X) / n^{0}(M)+V \, c_{r} / K \, n^{0}(M)\right]^{2}\]\[\begin{aligned} \mathrm{b}^{2}-4 \, \mathrm{c}=\left[\mathrm{n}^{0}(\mathrm{M})\right]^{2} \, & {\left[\mathrm{n}^{0}(\mathrm{X}) / \mathrm{n}^{0}(\mathrm{M})+1\right.} \\ &\left.+\mathrm{V} \, \mathrm{c}_{\mathrm{r}} / \mathrm{K} \, \mathrm{n}^{0}(\mathrm{M})\right]^{2}-4 \, \mathrm{n}^{0}(\mathrm{M}) \, \mathrm{n}^{0}(\mathrm{X}) \end{aligned}\]\[\text { Or, } b^{2}-4 \, c=\left[n^{0}(M)\right]^{2} \,\left[X_{r}+1+r\right]^{2}-4 \, n^{0}(M) \, n^{0}(X)\]\[\mathrm{b}^{2}-4 \, \mathrm{c}=\left[\mathrm{n}^{0}(\mathrm{M})\right]^{2} \,\left[\left(\mathrm{X}_{\mathrm{r}}+1\right)^{2}+\mathrm{r}^{2}+2 \, \mathrm{r} \,\left(\mathrm{X}_{\mathrm{r}}+1\right)\right]-4 \, \mathrm{n}^{0}(\mathrm{M}) \, \mathrm{n}^{0}(\mathrm{X})\]\[\mathrm{b}^{2}-4 \, \mathrm{c}=\left[\mathrm{n}^{0}(\mathrm{M})\right]^{2} \,\left[\mathrm{X}_{\mathrm{r}}^{2}+2 \, \mathrm{X}_{\mathrm{r}}+1+\mathrm{r}^{2}+2 \, \mathrm{r} \, \mathrm{X}_{\mathrm{r}}+2 \, \mathrm{r}-4 \, \mathrm{X}_{\mathrm{r}}\right]\] From equation (i) \[\frac{\mathrm{d} \xi}{\mathrm{dn}_{\mathrm{x}}^{0}}=-\frac{1}{2} \frac{\mathrm{db}}{\mathrm{dn}_{\mathrm{x}}^{0}}-\frac{1}{2} \, \frac{1}{2} \, \frac{1}{\left(\mathrm{~b}^{2}-4 \, \mathrm{c}\right)^{1 / 2}} \,\left[2 \, \mathrm{b} \, \frac{\mathrm{db}}{\mathrm{dn}_{\mathrm{x}}^{0}}-4 \, \frac{\mathrm{dc}}{\mathrm{dn}_{\mathrm{x}}^{0}}\right]\]\[\text { Or, } \quad \frac{\mathrm{d} \xi}{\mathrm{dn}_{\mathrm{X}}^{0}}=-\frac{1}{2} \frac{\mathrm{db}}{\mathrm{dn}_{\mathrm{X}}^{0}}-\frac{1}{2} \, \frac{1}{\left(\mathrm{~b}^{2}-4 \, \mathrm{c}\right)^{1 / 2}} \,\left[\mathrm{b} \, \frac{\mathrm{db}}{\mathrm{dn}_{\mathrm{X}}^{0}}-2 \, \frac{\mathrm{dc}}{\mathrm{dn}_{\mathrm{X}}^{0}}\right]\]We ignore for the moment the term \(\left(b^{2}-4 \, c\right)^{1 / 2}\) and concentrate attention on the two derivatives \(\frac{\mathrm{db}}{\mathrm{dn}_{\mathrm{X}}^{0}}\) and \(\frac{\mathrm{dc}}{\mathrm{dn}_{\mathrm{X}}^{0}}\); equation (j). \[\frac{\mathrm{d} \xi}{\operatorname{dn}_{\mathrm{X}}^{0}}=\frac{1}{2}-\frac{1}{2} \, \frac{1}{\left(\mathrm{~b}^{2}-4 \, \mathrm{c}\right)^{1 / 2}} \,\left[-1 \,\left\{-\mathrm{n}^{0}(\mathrm{M})-\mathrm{n}^{0}(\mathrm{X})-\mathrm{V} \, \mathrm{c}_{\mathrm{r}} \, \mathrm{K}^{-1}\right\}-2 \, \mathrm{n}^{0}(\mathrm{M})\right]\]\[\text { Or, } \frac{\mathrm{d} \xi}{\mathrm{dn}_{\mathrm{X}}^{0}}=\frac{1}{2}+\frac{1}{2} \, \frac{1}{\left(\mathrm{~b}^{2}-4 \, \mathrm{c}\right)^{1 / 2}} \,\left[\mathrm{n}^{0}(\mathrm{M})-\mathrm{n}^{0}(\mathrm{X})-\mathrm{V} \, \mathrm{c}_{\mathrm{r}} \, \mathrm{K}^{-1}\right]\]\[\begin{aligned} &\text { Or, } \\ &\frac{\mathrm{d} \xi}{\operatorname{dn}_{\mathrm{X}}^{0}}=\frac{1}{2}+\frac{1}{2} \, \frac{1}{\left(\mathrm{~b}^{2}-4 \, \mathrm{c}\right)^{1 / 2}} \, \mathrm{n}^{0}(\mathrm{M}) \,\left[1-\mathrm{n}^{0}(\mathrm{X}) / \mathrm{n}^{0}(\mathrm{M})-\mathrm{V} \, \mathrm{c}_{\mathrm{r}} / \mathrm{K} \, \mathrm{n}^{0}(\mathrm{M})\right] \end{aligned}\]\[\begin{aligned} &\text { Or, } \\ &\frac{\mathrm{d} \xi}{\mathrm{dn}_{\mathrm{x}}^{0}}=\frac{1}{2}+\frac{1}{2} \, \frac{1}{\left(\mathrm{~b}^{2}-4 \, \mathrm{c}\right)^{1 / 2}} \, \mathrm{n}^{0}(\mathrm{M}) \,\left[2-\left\{1+\mathrm{V} \, \mathrm{c}_{\mathrm{r}} / \mathrm{K} \, \mathrm{n}^{0}(\mathrm{M})\right\}-\mathrm{n}^{0}(\mathrm{X}) / \mathrm{n}^{0}(\mathrm{M})\right] \end{aligned}\]\[\begin{aligned} &\text { Or } \\ &\frac{d \xi}{\operatorname{dn}_{\mathrm{X}}^{0}}=\frac{1}{2} \\ &+\frac{1}{\left(\mathrm{~b}^{2}-4 \, \mathrm{c}\right)^{1 / 2}} \, \mathrm{n}^{0}(\mathrm{M}) \,\left[1-(1 / 2) \,\left\{1+\mathrm{V} \, \mathrm{c}_{\mathrm{r}} / \mathrm{K} \, \mathrm{n}^{0}(\mathrm{M})\right\}-\left\{\mathrm{n}^{0}(\mathrm{X}) / 2 \, \mathrm{n}^{0}(\mathrm{M})\right\}\right] \end{aligned}\]Now consider the term \(\left(b^{2}-4 \, c\right)^{1 / 2}\). From equation (n) \[b^{2}-4 \, c=\left[n^{0}(M)\right]^{2} \,\left[X_{r}^{2}-2 \, X_{r} \,(1-r)+(1+r)^{2}\right]\]\[\text { But } \frac{\mathrm{d} \xi}{\mathrm{dn}_{\mathrm{x}}^{0}}=\frac{1}{2}+\frac{1}{\left(\mathrm{~b}^{2}-4 \, \mathrm{c}\right)^{1 / 2}} \, \mathrm{n}^{0}(\mathrm{M}) \,\left[1-(1 / 2) \,(1+\mathrm{r})-\mathrm{X}_{\mathrm{r}} / 2\right]\] J. E. Ladbury and B. Z.Chowdhry, Chemistry and Biology, 1996, 3, 791.This page titled 1.3.7: Calorimeter- Titration Microcalorimetry- Enzyme-Substrate Interaction is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.3.8: Calorimetry- Titration Microcalorimetry- Micelle Deaggregation

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.03%3A_Calorimeter/1.3.08%3A_Calorimetry-_Titration_Microcalorimetry-_Micelle_Deaggregation

Aliquots of a concentrated surfactant solution are injected into the sample cell of a titration microcalorimeter. The sample cell initially contains water(\(\lambda\)). As more surfactant solution is injected into the sample cell a stage is reached where the concentration of surfactant in the sample cells exceeds the critical micellar concentration, \(\mathrm{cmc}\). The magnitude of the recorded heat changes dramatically, leading to estimates of both the cmc and the enthalpy of micelle formation.This calorimetric techniques has proved important in studies of ionic surfactants; e.g. hexadecyltrimethylammonium bromide (CTAB). For these surfactants the microcalorimeter signals a marked difference in recorded heats as the concentration of the surfactant changes from below to above the cmc. Titration microcalorimetric results for non-ionic surfactants are unfortunately not so readily interpreted. In addition to micelle formation, the monomers cluster in small aggregates below the cmc and the micelles cluster above the cmc.The volume of injected aliquot \(\operatorname{vinj}\) is significantly less than the volume of the sample cell. The amount of surfactant in each aliquot is \(\operatorname{ninj}\), the concentration of surfactant being \(\operatorname{cinj}[=\operatorname{ninj} / \text { vinj }]\). If \(\operatorname{cinj}\) is significantly above the \(\mathrm{cmc}\), the contribution of the surfactant to the enthalpy of the injected aliquot is \(\operatorname{ninj} \, \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{mic})\) where \(\mathrm{H}_{\mathrm{j}}^{0}(\mathrm{mic})\) is the contribution of one mole of monomer to the molar enthalpy of a micelle. If the concentration of solution in the sample cell is below the cmc, the contribution of each monomer to the enthalpy of the solution equals \(\mathrm{H}_{\mathrm{j}}^{0}(\text { mon })\). We concentrate attention on the contribution of the surfactant to the enthalpies of injected solution and the solution in the sample cell.Enthalpy of the injected aliquot, \[\mathrm{H}(\mathrm{inj})=\operatorname{ninj} \, \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{mic})\]The contribution of the surfactant to the enthalpy of the solution in the sample cell at injection number I is given by equation (b).Enthalpy of solution in the sample cell at injection number I, \[\mathrm{H}(\mathrm{I})=\mathrm{I} \, \operatorname{ninj} \, \mathrm{H}_{\mathrm{j}}^{0} \text { (mon) }\]Enthalpy of solution in the sample cell at injection number (I+1), \[\mathrm{H}(\mathrm{I}+1)=(\mathrm{I}+1) \, \operatorname{ninj} \, \mathrm{H}_{\mathrm{j}}^{0} \text { (mon) }\]Recorded heat \[\mathrm{q}=\mathrm{H}(!+1)-\mathrm{H}(\mathrm{I})-\mathrm{H}(\text { inj })\]\[\text { or, }[\mathrm{q} / \text { ninj }]_{\text {lowl }}=\mathrm{H}_{\mathrm{j}}^{0}(\mathrm{mon})-\mathrm{H}_{\mathrm{j}}^{0}(\mathrm{mic})=-\Delta_{\text {mic }} \mathrm{H}^{0}\]At high injection numbers the enthalpies of solution in the sample cell are \(\mathrm{I} \, \operatorname{ninj} \, \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{mic})\) and \((\mathrm{I}+1) \, \operatorname{ninj} \, \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{mic})\). \[[\mathrm{q} / \text { ninj }]_{\text {highl }}=(\mathrm{I}+1) \, \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{mic})-\mathrm{I} \, \mathrm{H}_{\mathrm{j}}^{0}(\text { mic })-\mathrm{H}_{\mathrm{j}}^{0}(\text { mic })=0\]At low injection numbers the recorded \((q/\operatorname{ninj}\)) is effectively the enthalpy of micelle formation. The recorded ratio \([q/\operatorname{ninj}]\) is effectively zero at high injection numbers, the switch in pattern of \((q/\operatorname{ninj}\)) from \(\Delta_{\operatorname{mic}} \mathrm{H}^{0}\) to zero marking the cmc of the surfactant.This page titled 1.3.8: Calorimetry- Titration Microcalorimetry- Micelle Deaggregation is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.3.9: Calorimetry- Scanning

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.03%3A_Calorimeter/1.3.09%3A_Calorimetry-_Scanning

A given closed system is prepared using \(n_{1}\) moles of water (\(\lambda\)) and \(n_{\mathrm{X}}^0\) moles of solute \(\mathrm{X}\) at pressure \(p\) (\(\cong \mathrm{p}^{0}\), the standard pressure) and temperature \(\mathrm{T}\). The thermodynamic properties of the solution are ideal such that, at some low temperature, the enthalpy of the solution \(\mathrm{H}(\mathrm{aq} ; \text { low } \mathrm{T})\) is given by equation (a). \[\mathrm{H}(\mathrm{aq} ; \text { low } \mathrm{T})=\mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\lambda ; \text { low } \mathrm{T})+\mathrm{n}_{\mathrm{X}}^{0} \, \mathrm{H}_{\mathrm{X}}^{\infty}(\text { aq } ; \text { low } \mathrm{T})\]The temperature of the solution is raised to high temperature such that the solution contains only solute \(\mathrm{Y}\), all solute \(\mathrm{X}\) having been converted to \(\mathrm{Y}\). \[\mathrm{H}(\mathrm{aq} ; \text { high } \mathrm{T})=\mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\lambda ; \text { high } \mathrm{T})+\mathrm{n}_{\mathrm{X}}^{0} \, \mathrm{H}_{\mathrm{Y}}^{\infty}(\text { aq; high } \mathrm{T})\]At intermediate temperatures, a chemical equilibrium exists between solutes \(\mathrm{X}\) and \(\mathrm{Y}\). At temperature \(\mathrm{T}\), the chemical composition of the solution is characterised by extent of reaction \(\xi(\mathrm{T})\). \[\begin{array}{llcc} & \mathrm{X}(\mathrm{aq})<\overline{=} & \mathrm{Y}(\mathrm{aq}) & \\ \text { At low } \mathrm{T} & \mathrm{n}_{\mathrm{X}}^{0} & 0 & \mathrm{~mol} \\ \text { At high } \mathrm{T} & 0 & \mathrm{n}_{\mathrm{x}}^{0} & \mathrm{~mol} \\ \text { At intermediate } \mathrm{T} & \mathrm{n}_{\mathrm{X}}^{0}-\xi^{\mathrm{eq}}(\mathrm{T}) & \xi^{\mathrm{eq}}(\mathrm{T}) & \mathrm{mol} \end{array}\]For a solution where the thermodynamic properties are ideal, we define an equilibrium constant \(\mathrm{K}(\mathrm{T})\), at temperature \(\mathrm{T}\). \[\mathrm{K}(\mathrm{T})=\xi^{\mathrm{eq}}(\mathrm{T}) /\left[\mathrm{n}_{\mathrm{X}}^{0}-\xi^{\mathrm{eq}}(\mathrm{T})\right]\]By definition, the degree of reaction, \(\alpha(T)=\xi^{e q}(T) / n_{x}^{0}\). \[\mathrm{K}(\mathrm{T})=\alpha(\mathrm{T}) \, \mathrm{n}_{\mathrm{X}}^{0} /\left[\mathrm{n}_{\mathrm{X}}^{0}-\mathrm{n}_{\mathrm{X}}^{0} \, \alpha(\mathrm{T})\right]\]\[\text { Therefore, } \quad \alpha(\mathrm{T})=\mathrm{K}(\mathrm{T}) /[1+\mathrm{K}(\mathrm{T})]\]At temperature \(\mathrm{T}\), the enthalpy of the aqueous solution is given by equation (f) where \(\mathrm{H}_{1}^{*}(\lambda ; \mathrm{T})\) is the molar enthalpy of water(\(\lambda\)) in the aqueous solution again assuming that the thermodynamic properties of the solution are ideal. \[\begin{array}{r} \mathrm{H}(\mathrm{aq} ; \mathrm{T})=\mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\lambda ; \mathrm{T})+\mathrm{n}_{\mathrm{X}}^{0} \,[1-\alpha(\mathrm{T})] \, \mathrm{H}_{\mathrm{X}}^{\infty}(\mathrm{aq} ; \mathrm{T}) \\ +\mathrm{n}_{\mathrm{X}}^{0} \, \alpha(\mathrm{T}) \, \mathrm{H}_{\mathrm{Y}}^{0}(\mathrm{aq} ; \mathrm{T}) \end{array}\]The limiting enthalpy of reaction, \(\Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\mathrm{aq} ; \mathrm{T})\) is given by equation (g). \[\Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\mathrm{aq} ; \mathrm{T})=\mathrm{H}_{\mathrm{Y}}^{\infty}(\mathrm{aq} ; \mathrm{T})-\mathrm{H}_{\mathrm{X}}^{\infty}(\mathrm{aq} ; \mathrm{T})\]From equation (f), \[\begin{aligned} \mathrm{H}(\mathrm{aq} ; \mathrm{T})=\mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\lambda ; \mathrm{T})+\mathrm{n}_{\mathrm{X}}^{0} \, & \mathrm{H}_{\mathrm{X}}^{\infty}(\mathrm{aq} ; \mathrm{T}) \\ &+\mathrm{n}_{\mathrm{x}}^{0} \, \alpha(\mathrm{T}) \, \Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\mathrm{aq} ; \mathrm{T}) \end{aligned}\]We assume that \(\Delta_{r} H^{\infty}(\mathrm{aq})\) is independent of temperature. The differential of equation (h) with respect to temperature yields the isobaric heat capacity of the solution.\[\mathrm{C}_{\mathrm{p}}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{C}_{\mathrm{p} 1}^{*}(\lambda)+\mathrm{n}_{\mathrm{X}}^{0} \, \mathrm{C}_{\mathrm{pX}}^{\infty}(\mathrm{aq}) +\mathrm{n}_{\mathrm{X}}^{0} \, \Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\mathrm{aq}) \, \mathrm{d} \alpha / \mathrm{dT}\]The term \((\mathrm{d} \alpha / \mathrm{dT})\) signals the contribution of the change of composition with temperature to the isobaric heat capacity of the system, the ‘relaxational’ isobaric heat capacity \(\mathrm{C}_{\mathrm{p}}(\text {relax})\). Thus\[\mathrm{C}_{\mathrm{p}}(\text { relax })=\mathrm{n}_{\mathrm{X}}^{0} \, \Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\mathrm{aq}) \, \mathrm{d} \alpha / \mathrm{dT}\]\[\text { From equation (e) } \frac{\mathrm{d} \alpha(\mathrm{T})}{\mathrm{dT}}=\frac{\mathrm{K}(\mathrm{T})}{[1+\mathrm{K}(\mathrm{T})]^{2}} \, \frac{\mathrm{d} \ln [\mathrm{K}(\mathrm{T})]}{\mathrm{dT}}\]But according to the van’t Hoff Equation, \[\frac{\mathrm{d} \ln \mathrm{K}(\mathrm{T})}{\mathrm{dT}}=\frac{\Delta_{\mathrm{r}} \mathrm{H}^{\infty}}{\mathrm{R} \, \mathrm{T}^{2}}\]\[\text { Hence, } \mathrm{C}_{\mathrm{p}}(\text { relax })=\mathrm{n}_{\mathrm{x}}^{0} \, \frac{\left[\Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\mathrm{aq})\right]^{2}}{\mathrm{R} \, \mathrm{T}^{2}} \, \frac{\mathrm{K}(\mathrm{T})}{[1+\mathrm{K}(\mathrm{T})]^{2}}\]The contribution to the molar isobaric heat capacity of the system from the change in composition of the solution is given by equation (n). \[C_{p m}(\text { relax })=\frac{\left[\Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\mathrm{aq})\right]^{2}}{\mathrm{R} \, \mathrm{T}^{2}} \, \frac{\mathrm{K}(\mathrm{T})}{[1+\mathrm{K}(\mathrm{T})]^{2}}\]The dependence of \(C_{p m}(\text {relax})\) on temperature has the following characteristics.The tertiary structures of enzymes in aqueous solution are very sensitive to temperature. In the general case, an enzyme changes from, say, active to inactive form as the temperature is raised; i.e. the enzyme denatures. The change from active to inactive form is characterised by a ‘melting temperature’. The explanation is centred on the role of hydrophobic interactions in stabilising the structure of the active form. However the strength of hydrophobic bonding is very sensitive to temperature. Hence equation (n) forms the basis of an important application of modern differential scanning calorimeters into structural reorganisation in biopolymers on changing the temperature. The scans may also identify domains within a given biopolymer which undergo structural transitions at different temperatures.Indeed there is evidence that a given enzyme is characterised by a temperature range within which the active form is stable. Outside this range, both at low and high temperatures the active form is not stable. In other words the structure of an enzyme may change to an inactive form on lowering the temperature. The pattern can be understood in terms of the dependence of \(\left[\mu_{\mathrm{j}}^{0} / \mathrm{T}\right]\) on temperature where \(\left[\mu_{\mathrm{j}}^{0}\) is the reference chemical potentials for solute \(j\). In this case we consider the case where in turn solute j represents the active and inactive forms of the enzyme. There is a strong possibility that the plots of two dependences intersect at two temperatures. The active form is stable in the window between the two temperatures.The analysis leading to equation (n) is readily extended to systems involving coupled equilibria. The impact of changes in composition is also an important consideration in analysing the dependence on temperature of the properties of weak acids in solution.Footnotes \(\begin{aligned} \frac{\mathrm{d} \alpha}{\mathrm{dT}}=\left[\frac{1}{1+\mathrm{K}}\right.&\left.-\frac{1}{(1+\mathrm{K})^{2}}\right] \, \frac{\mathrm{dK}}{\mathrm{dT}}=\left[\frac{1+\mathrm{K}-\mathrm{K}}{(1+\mathrm{K})^{2}}\right] \, \frac{\mathrm{dK}}{\mathrm{dT}}=\left[\frac{1}{(1+\mathrm{K})^{2}}\right] \, \frac{\mathrm{dK}}{\mathrm{dT}} \\ &=\left[\frac{\mathrm{K}}{(1+\mathrm{K})^{2}}\right] \, \frac{\mathrm{d} \ln \mathrm{K}}{\mathrm{dT}} \end{aligned}\) \(\mathrm{C}_{\mathrm{p}}(\text { relax })=[\mathrm{mol}] \, \frac{\left[\mathrm{J} \mathrm{mol}^{-1}\right]^{2}}{\left[\mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}]^{2}} \, \frac{}{}=\left[\mathrm{J} \mathrm{K}^{-1}\right]\) V. V. Plotnikov, J. M. Brandts, L-V. Lin and J. F. Brandts, Anal. Biochem.,1997,250,237. C. O.Pabo, R. T. Sauer, J. M. Sturtevant and M. Ptashne, Proc. Natl. Acad.Sci. USA,1979, 76,1608. S. Mabrey and J. M. Sturtevant, Proc. Natl. Acad.Sci. USA,1976,73,3802. T. Ackerman, Angew. Chem. Int.Ed. Engl.,1989,28,981. J. M. Sturtevant, Annu. Rev.Phys.Chem.,1987,38,463. M. J. Blandamer, B. Briggs, P. M. Cullis, A. P Jackson, A. Maxwell and R. J. Reece, Biochemistry, 1994,33,7510. F. Franks, R. M. Hately and H. L. Friedman, Biophys.Chem.,1988,31,307. F. Franks and T. Wakabashi, Z. Phys. Chem., 1987,155,171. M. J. Blandamer, B. Briggs, J. Burgess and P. M. Cullis, J. Chem. Soc. Faraday Trans.,1990,86,1437. G. J. Mains, J. W. Larson and L. G. Hepler, J. Phys Chem.,1984,88,1257. J. K. Hovey and L.G. Hepler, J. Chem. Soc. Faraday Trans.,1990,86,2831. E. M. Woolley and L. G. Hepler, Can. J.Chem.,1977,55,158. J. K. Hovey and L.G. Hepler, J. Phys. Chem.,1990,94,7821.This page titled 1.3.9: Calorimetry- Scanning is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.3.10: Calorimetry- Solutions- Flow Microcalorimetry

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.03%3A_Calorimeter/1.3.10%3A_Calorimetry-_Solutions-_Flow_Microcalorimetry

An important event in experimental calorimetry was the development of the Picker flow microcalorimeter. In this calorimeter, two liquids [e.g. water(\(\lambda\)) and an aqueous solution] at the same temperature flow through two cells. The liquids are heated, the calorimeter recording the difference in power required to keep both liquids at the same temperature. The recorded difference is a function of the difference in isobaric heat capacities per unit volume. The isobaric heat capacity of the solvent [e.g. water(\(\lambda\))] per unit volume (or, heat capacitance) , \(\sigma^{*}(\lambda)\) is the reference. The technique has been extended to measure enthalpies and rates of reaction.Footnotes P. Picker, P-A. Leduc, P. R.Philip and J. E. Desnoyers, J. Chem. Thermodyn., 1971, 3, 631. J.-L. Fortier, P.-A. Leduc, P. Picker and J. E. Desnoyers, J. Solution Chem., 1973, 2, 467. J. E. Desnoyers, C. de Visser, G. Peron and P. Picker, J.Solution Chem., 1976, 5, 605. A. Roux, G. Peron, P. Picker and J. E. Desnoyers, J. Solution Chem.,1980,9,59.This page titled 1.3.10: Calorimetry- Solutions- Flow Microcalorimetry is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.4.1: Chemical Equilibria- Solutions

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.04%3A_Chemical_Equilibria/1.4.01%3A_Chemical_Equilibria-_Solutions

The conditions for chemical equilibrium in a closed system at fixed temperature and pressure are as follows.\[\sum_{j=1}^{j=i} v_{j} \, \mu_{j}^{e q}=0\]If all \(i\)-chemical substances are solutes in aqueous solution at temperature \(\mathrm{T}\) and pressure \(p\), the latter being close to the standard pressure \(p^{0}\), the equilibrium chemical potentials are related to the composition of the system. Hence, \[A^{\mathrm{cq}}=-\left(\frac{\partial \mathrm{G}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}^{\mathrm{eq}}=0=-\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{v}_{\mathrm{j}} \,\left[\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)\right]^{\mathrm{eq}}\]\text { Hence } \quad \sum_{j=1}^{j=i} v_{j} \, \mu_{j}^{0}(a q)=-\sum_{j=1}^{j=i} v_{j} \, R \, T \, \ln \left(m_{j} \, \gamma_{j} / m^{0}\right)^{e q}\]The left-hand-side of equation (c) defines the standard Gibbs energy of reaction, \(\Delta_{\mathrm{r}} \mathrm{G}^{0}\) which in turn leads to the definition of an equilibrium constant \(\mathrm{K}^{0}\). \[\Delta_{\mathrm{r}} \mathrm{G}^{0}=-\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{K}^{0}\right)=\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{v}_{\mathrm{j}} \, \mu_{\mathrm{j}}^{0}(\mathrm{aq})\]Combination of equation (c) and equation (d) yields an equation for \(\mathrm{K}^{0}\) in terms of the equilibrium composition of the system. \[\mathrm{K}^{0}=\left[\Pi_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}}\left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)^{\mathrm{v}(\mathrm{j})}\right]^{\mathrm{eq}}\]Equation (e) is remarkable. The right hand side describes the stoichiometry of the chemical equilibrium and the composition of the closed system at defined temperature and pressure. The left-hand-side in the form of \(\mathrm{K}^{0}\) defined using equation (d) is related to the ideal thermodynamic process in terms of reference chemical potentials of reactants and products. If the solutes are non-ionic and the solution is dilute then a reasonable assumption sets \(" \gamma_{j}^{e q}=1^{\prime \prime}\) for all \(i\)-solutes.Footnotes From a thermodynamic standpoint, an equilibrium constant emerges from the idea of zero affinity for chemical reaction at a minimum in Gibbs energy. Accounts which treat equilibrium constants as the ratio of rate constants are unsatisfactory. The equations set out here describe the general case where substance \(j\) is one of \(i\)-simple solutes in solution. In some cases one or more of the solutes are ionic and the solvent (e.g. water) is directly involved in the chemical reaction. In each case we assume that the systems have been assayed such that the composition of the system at equilibrium is known together with the stoichiometries. In general terms for a systems at pressure \(p\), \[\mathrm{A}^{\mathrm{eq}}=-\left(\frac{\partial G}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}^{\mathrm{eq}}=0\] Hence for a chemical equilibrium involving \(i\)-solutes in aqueous solution the following condition holds. \[0=\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{v}_{\mathrm{j}} \,\left[\mu_{\mathrm{j}}^{0}\left(\mathrm{aq} ; \mathrm{T} ; \mathrm{p}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)+\int_{\mathrm{p}^{0}}^{p} \mathrm{~V}_{\mathrm{j}}^{\infty}(\mathrm{aq}) \, \mathrm{dp}\right]^{\mathrm{eq}}\]\[\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} v_{\mathrm{j}} \, \mu_{\mathrm{j}}^{0}\left(\mathrm{aq} ; \mathrm{T} ; \mathrm{p}^{0}\right)=-\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} v_{\mathrm{j}} \,\left[\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)+\int_{\mathrm{p}^{0}}^{\mathrm{p}} \mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq}) \, \mathrm{dp}\right]^{\mathrm{eq}}\]\[\text { where } \Delta_{\mathrm{r}} \mathrm{G}^{0}\left(\mathrm{~T} ; \mathrm{p}^{0}\right)=-\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{K}^{0}\right)=\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} v_{\mathrm{j}} \, \mu_{\mathrm{j}}^{0}\left(\mathrm{aq} ; \mathrm{T} ; \mathrm{p}^{0}\right)\]This page titled 1.4.1: Chemical Equilibria- Solutions is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.4.2: Chemical Equilibria- Solutions- Derived Thermodynamic Parameters

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.04%3A_Chemical_Equilibria/1.4.02%3A_Chemical_Equilibria-_Solutions-_Derived_Thermodynamic_Parameters

A given closed system at fixed temperature and fixed pressure contains a number of chemical substances in chemical equilibrium. The composition of the system depends on temperature and pressure. Key equations describe the dependences of the equilibrium Gibbs energy on temperature and pressure. \(\mathrm{H}^{\mathrm{eq}}=-\mathrm{T}^{2} \,\left(\frac{\partial(\mathrm{G} / \mathrm{T})}{\partial \mathrm{T}}\right)_{\mathrm{p}}^{\mathrm{eq}} ; \quad \mathrm{V}^{\mathrm{eq}}=\left(\frac{\partial \mathrm{G}}{\partial \mathrm{p}}\right)_{\mathrm{T}}^{\mathrm{eq}}\) The situation is complicated by the fact that both \(\mathrm{H}^{\mathrm{eq}}\) and \(\mathrm{V}^{\mathrm{eq}}\) depend on the equilibrium composition of the system, \(\xi^{\mathrm{eq}}\); \(\left(\frac{\partial \mathrm{H}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}^{\mathrm{eq}}\) and \(\left(\frac{\partial \mathrm{V}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}^{\mathrm{eq}}\). We consider the case where the chemical substances involved in the chemical equilibrium are solutes. Both partial derivatives are re-expressed in terms of the partial molar properties of each solute in the system. \[\left(\frac{\partial H}{\partial \xi}\right)_{T, p}^{e q}=\sum_{j=1}^{j=i}\left(\frac{\partial H}{\partial n_{j}}\right)_{T, p, n(i \neq j)}^{e q} \,\left(\frac{\partial n_{j}}{\partial \xi}\right)^{e q}]\[\text { But partial molar enthalpy, } \quad \mathrm{H}_{\mathrm{j}}^{\mathrm{eq}}=\left(\frac{\partial \mathrm{H}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{i} \neq \mathrm{j})}^{\mathrm{eq}}\]\[\text { Further, } \quad\left(\partial \mathrm{n}_{\mathrm{j}} / \partial \xi\right)^{\mathrm{eq}}=\mathrm{v}_{\mathrm{j}}\]Here \(ν_{j}\) is the stoichiometry associated with chemical substance \(j\), being positive for products and negative for reactants. \[\text { Therefore, } \quad\left(\frac{\partial H}{\partial \xi}\right)_{T, p}^{e q}=\sum_{j=1}^{j=i} v_{j} \, H_{j}^{e q}\]\[\text { Therefore, } \quad \left(\frac{\partial \mathrm{V}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}^{\mathrm{eq}}=\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{V}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}^{\mathrm{eq}}\]\({\mathrm{V}_{j}}^{\mathrm{eq}}\) is the partial molar volume of substance \(j\) in the solution at equilibrium. The partial molar enthalpy of solute \(j\) can be expressed in terms of a limiting molar enthalpy \({\mathrm{H}_{j}}^{\infty}\) and the dependence of activity coefficient \(\gamma_{j}\) on temperature. \[\text { Therefore, } \quad\left(\frac{\partial H}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}^{\mathrm{eq}}=\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{v}_{\mathrm{j}} \,\left[\mathrm{H}_{\mathrm{j}}^{\infty}-\mathrm{R} \, \mathrm{T}^{2} \,\left(\frac{\partial \ln \left(\gamma_{\mathrm{j}}\right)}{\partial \mathrm{T}}\right)_{\mathrm{p}}^{\mathrm{eq}}\right]\]In other words the dependence of enthalpy of the system on composition at equilibrium is a function of the limiting molar enthalpies of all chemical substances involved in the equilibrium and the dependences on temperature of their activity coefficients.By definition, the limiting molar enthalpy of reaction, \[\Delta_{\mathrm{r}} \mathrm{H}^{\infty}=\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{v}_{\mathrm{j}} \, \mathrm{H}_{\mathrm{j}}^{\infty}\]\[\text { Then } \quad \left(\frac{\partial H}{\partial \xi}\right)_{T, p}^{e q}=\Delta_{r} H^{\infty}-\sum_{j=1}^{j=i} v_{j} \, R \, T^{2} \,\left(\frac{\partial \ln \gamma_{j}}{\partial T}\right)_{p}^{e q}\]In some applications, the solutions are quite dilute and the assumption is made that at all temperatures and pressures \(\gamma_{j}\) for chemical substance \(j\) is unity. \[\text { Hence, }\left(\frac{\partial H}{\partial \xi}\right)_{T, p}^{\text {eq }}=\Delta_{\mathrm{r}} H^{\infty}\]A similar analysis is possible in terms of partial molar volumes. From equation (e), we obtain the following equation for the volume of reaction. \[\left(\frac{\partial V}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}^{e q}=\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{V}_{\mathrm{j}} \,\left[\mathrm{V}_{\mathrm{j}}^{\infty}+\mathrm{R} \, \mathrm{T} \,\left(\frac{\partial \ln \left(\gamma_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right)_{\mathrm{T}}^{\mathrm{eq}}\right]\]\[\text { The limiting volume of reaction, } \Delta_{\mathrm{r}} \mathrm{V}^{\infty}=\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{v}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}^{\infty}\]\[\text { Thus, } \quad \left(\frac{\partial V}{\partial \xi}\right)_{T, p}^{e q}=\Delta_{r} V^{\infty}+\sum_{j=1}^{j=i} v_{j} \, R \, T \,\left(\frac{\partial \ln \left(\gamma_{j}\right)}{\partial p}\right)_{T}^{e q}\]If the solution is dilute, it can often be assumed that the activity coefficient of each chemical substance is independent of pressure. Then, \[\left(\frac{\partial \mathrm{V}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}^{\mathrm{eq}}=\Delta_{\mathrm{r}} \mathrm{V}^{\infty}\]A slight complication to these general equations arises if one of the substances involved in the chemical equilibrium is the solvent. As an example we consider the following equilibrium. \[\mathrm{X}(\mathrm{aq})+\mathrm{H}_{2} \mathrm{O}(\mathrm{aq}) \rightleftharpoons \mathrm{Y}(\mathrm{aq})\]\[\begin{aligned} &\text { Then, } \quad(\partial \mathrm{H} / \partial \xi)_{\mathrm{T}, \mathrm{p}}^{\mathrm{eq}}=\left[\mathrm{H}_{\mathrm{Y}}^{\infty}(\mathrm{aq})-\mathrm{R} \, \mathrm{T}^{2} \,\left(\partial \ln \gamma_{\mathrm{Y}} / \partial \mathrm{T}\right)_{\mathrm{p}}^{\mathrm{eq}}\right] \\ &-\left[\mathrm{H}_{\mathrm{X}}^{\infty}(\mathrm{aq})-\mathrm{R} \, \mathrm{T}^{2} \,\left(\partial \ln \gamma_{\mathrm{X}} / \partial \mathrm{T}\right)_{\mathrm{p}}^{\mathrm{eq}}\right] \\ &-\left[\mathrm{H}_{1}^{*}(\lambda)+\mathrm{R} \, \mathrm{T}^{2} \, \mathrm{M}_{1}\left(\mathrm{~m}_{\mathrm{X}}+\mathrm{m}_{\mathrm{Y}}\right)_{\mathrm{p}}^{\mathrm{eq}} \,(\partial \phi / \partial \mathrm{T})_{\mathrm{p}}^{\mathrm{eq}}\right] \end{aligned}\]If the properties of the solution are ideal (e.g. very dilute), equation (o) is written in the following form. \[(\partial \mathrm{H} / \partial \xi)_{\mathrm{T}, \mathrm{p}}^{\mathrm{eq}}=\mathrm{H}_{\mathrm{Y}}^{\infty}(\mathrm{aq})-\mathrm{H}_{\mathrm{X}}^{\infty}(\mathrm{aq})-\mathrm{H}_{1}^{*}(\lambda)\]With reference to the limiting volume of reaction, the analogue of equation (p) is as follows. \[(\partial \mathrm{V} / \partial \xi)_{\mathrm{T}, \mathrm{p}}^{\mathrm{eq}}=\mathrm{V}_{\mathrm{Y}}^{\infty}(\mathrm{aq})-\mathrm{V}_{\mathrm{X}}^{\infty}(\mathrm{aq})-\mathrm{V}_{1}^{*}(\lambda)\]Footnotes \(\left(\frac{\partial \mathrm{H}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}=\left[\mathrm{J} \mathrm{mol}^{-1}\right]+\left[ \,\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}]^{2} \,\left(\frac{}{[\mathrm{K}]}\right)\right]=\left[\mathrm{J} \mathrm{mol}^{-1}\right]\) \(\left(\frac{\partial \mathrm{V}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]+\left[ \,\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}] \,\left[\frac{}{\left[\mathrm{N} \mathrm{m}^{-2}\right]}\right]\right]=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]\)This page titled 1.4.2: Chemical Equilibria- Solutions- Derived Thermodynamic Parameters is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.4.3: Chemical Equilibria- Solutions- Simple Solutes

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.04%3A_Chemical_Equilibria/1.4.03%3A_Chemical_Equilibria-_Solutions-_Simple_Solutes

A given chemical equilibrium involves association of two solutes \(\mathrm{X}(\mathrm{aq})\) and \(\mathrm{Y}(\mathrm{aq})\) to form solute \(\mathrm{Z}(\mathrm{aq})\). \[2 X(a q)+Y(a q) \rightleftharpoons 4 Z(a q)\]Phase Rule. The aqueous solution is prepared using two chemical substances: substance \(\mathrm{Z}\) and solvent water. Hence \(\mathrm{C} = 2\). There are 2 phases: vapour and solution so \(\mathrm{P} = 2\). Then \(\mathrm{F} = 2\). Hence at fixed temperature and in a system prepared using mole fraction \(x_{\mathrm{Z}}\) of substance \(\mathrm{Z}\) (an intensive composition variable), the equilibrium vapour pressure and the equilibrium amounts of \(\mathrm{X}(\mathrm{aq})\), \(\mathrm{Y}(\mathrm{aq})\) and \(\mathrm{Z}(\mathrm{aq})\) are unique. \[\text { At equilibrium, } 2 \, \mu_{\mathrm{X}}^{\mathrm{eq}}(\mathrm{aq})+\mu_{\mathrm{Y}}^{\mathrm{eq}}(\mathrm{aq})=4 \, \mu_{\mathrm{Z}}^{\mathrm{eq}}(\mathrm{aq})\]At fixed \(\mathrm{T}\) and \(p\), assuming ambient pressure is close to the standard pressure \(p^{0}\), \[\begin{aligned} &2 \,\left[\mu_{\mathrm{X}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{X}} \, \gamma_{\mathrm{X}} / \mathrm{m}^{0}\right)^{\mathrm{eq}}\right] \\ &\quad+\left[\mu_{\mathrm{Y}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{Y}} \, \gamma_{\mathrm{Y}} / \mathrm{m}^{0}\right)^{\mathrm{eq}}\right] \\ &\quad=4 \,\left[\mu_{\mathrm{Z}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{Z}} \, \gamma_{\mathrm{Z}} / \mathrm{m}^{0}\right)^{\mathrm{eq}}\right] \end{aligned}\]\[\text { where } \Delta_{\mathrm{r}} \mathrm{G}^{0}=-\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{K}^{0}\right)=4 \, \mu_{\mathrm{Z}}^{0}(\mathrm{aq})-2 \, \mu_{\mathrm{X}}^{0}(\mathrm{aq})-\mu_{\mathrm{Y}}^{0}(\mathrm{aq})\]For this equilibrium at temperature \(\mathrm{T}\) and pressure \(p\), \[\mathrm{K}^{0}=\frac{\left(\mathrm{m}_{\mathrm{Z}}^{\mathrm{eq}} \, \gamma_{\mathrm{Z}}^{\mathrm{eq}} / \mathrm{m}^{0}\right)^{4}}{\left(\mathrm{~m}_{\mathrm{X}}^{\mathrm{eq}} \, \gamma_{\mathrm{X}}^{\mathrm{eq}} / \mathrm{m}^{0}\right)^{2} \,\left(\mathrm{m}_{\mathrm{Y}}^{\mathrm{eq}} \, \gamma_{\mathrm{Y}}^{\mathrm{eq}} / \mathrm{m}^{0}\right)}\]If the solution is quite dilute, \(\gamma_{\mathrm{X}}^{\mathrm{eq}}\), \(\gamma_{\mathrm{Y}}^{\mathrm{eq}}\) and \gamma_{\mathrm{Z}}^{\mathrm{eq}} are effectively unity in the real solution at equilibrium. Then \[\mathrm{K}^{0}=\left(\mathrm{m}_{\mathrm{eq}}^{\mathrm{eq}} / \mathrm{m}^{0}\right)^{4} /\left(\mathrm{m}_{\mathrm{X}}^{\mathrm{eq}} / \mathrm{m}^{0}\right)^{2} \,\left(\mathrm{m}_{\mathrm{Y}}^{\mathrm{eq}} / \mathrm{m}^{0}\right)\]\(\mathrm{K}^{0}\) is dimensionless. But the latter statement signals a common problem in this subject because chemists find it more convenient and informative to define a quantity \({\mathrm{K}_{\mathrm{m}}}^{0}\) in which the m0 terms in equation (f) [or its equivalent] have been removed. \[\text { Thus, } \mathrm{K}_{\mathrm{m}}^{0}=\left(\mathrm{m}_{\mathrm{Z}}^{\mathrm{eq}}\right)^{4} /\left(\mathrm{m}_{\mathrm{X}}^{\mathrm{eq}}\right)^{2} \,\left(\mathrm{m}_{\mathrm{Y}}^{\mathrm{eq}}\right)\]Hence the units for \({\mathrm{K}_{\mathrm{m}}}^{0}\) signal the stoichiometry of the equilibrium whereas the dimensionless \(\mathrm{K}^{0}\) does not.Footnotes As a consequence of the removal of the \(\mathrm{m}^{0}\) terms, \(\mathrm{K}^{0}\) quantities have units unless the equation for the chemical equilibrium is stoichiometrically balanced: e.g. n-moles of reactants form n-moles of products.But from equation (g), \(\mathrm{K}_{\mathrm{m}}^{0}=\left[\mathrm{mol} \mathrm{kg}^{-1}\right]^{4} \,\left[\mathrm{mol} \mathrm{kg}{ }^{-1}\right]^{-2} \,\left[\mathrm{mol} \mathrm{kg}^{-1}\right]^{-1}\) or \(\mathrm{K}_{\mathrm{m}}^{0}=\left[\mathrm{mol} \mathrm{} \mathrm{kg}^{-1}\right]\)If we write, \(\Delta_{\mathrm{r}} \mathrm{G}^{0}=-\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{K}_{\mathrm{m}}^{0}\right)\)Then \(\Delta_{\mathrm{r}} \mathrm{G}^{0}=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}] \, \ln \left[\mathrm{mol} \mathrm{kg}{ }^{-1}\right]\)ln There is clearly a slight problem in handling a logarithm of a composition unit. There are two approaches to this problem. The first approach ignores the problem, which is unsatisfactory practice. The second approach is to ask - what happened to the composition unit and trace the problem back through the equations. The impact of composition units in quantitative analysis of data was addressed by E. A. Guggenheim, Trans. Faraday Soc., 1937,33,607.This page titled 1.4.3: Chemical Equilibria- Solutions- Simple Solutes is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.4.4: Chemical Equilibria- Solutions- Ion Association

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.04%3A_Chemical_Equilibria/1.4.04%3A_Chemical_Equilibria-_Solutions-_Ion_Association

A given equilibrium in aqueous solution involves association of two ions to form a neutral solute. \[\text { Thus, } \quad \mathrm{M}^{+}(\mathrm{aq})+\mathrm{X}^{-}(\mathrm{aq}) \rightleftharpoons \mathrm{Z}(\mathrm{aq})\]The chemical equilibrium is described in terms of chemical potentials using the following equation in which we recognise that the reactant is a 1:1 salt. \[\mu^{0}\left(\mathrm{M}^{+} \mathrm{X}^{-} ; \mathrm{aq}\right)+2 \, \mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{m}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right) \, \gamma_{\pm}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right) / \mathrm{m}^{0}\right]^{\mathrm{eq}} = \mu^{0}(Z ; \mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{m}(\mathrm{Z}) \, \gamma(\mathrm{Z}) / \mathrm{m}^{0}\right]^{\mathrm{cq}}\]In the latter equation, \(\gamma_{\pm}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right)\) is the mean ionic activity for salt \(\mathrm{M}^{+} \mathrm{X}^{-}\) in the aqueous solution.By definition, at fixed temperature \(\mathrm{T}\) and pressure \(p\) where this pressure is ambient and hence close to the standard pressure \(p^{0}\), \[\Delta_{\mathrm{r}} \mathrm{G}^{0}=\mu^{0}(\mathrm{Z} ; \mathrm{aq})-\mu^{0}\left(\mathrm{M}^{+} \mathrm{X}^{-} ; \mathrm{aq}\right)=-\mathrm{R} \, \mathrm{T} \, \ln \mathrm{K}^{0}\]where, \[\mathrm{K}^{0}=\left[\frac{\mathrm{m}(\mathrm{Z}) \, \gamma(\mathrm{Z}) / \mathrm{m}^{0}}{\mathrm{~m}^{2}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right) \, \gamma_{\pm}^{2}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right) /\left(\mathrm{m}^{0}\right)^{2}}\right]^{e q}\]In many cases, particularly for dilute solutions \(\gamma(\mathrm{Z})\) is approximately unity but rarely can one ignore the term \(\gamma_{\pm}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right)\). \[\text { By definition, } \mathrm{K}^{0}(\mathrm{app})=\left\{\mathrm{m}(\mathrm{Z}) \, \mathrm{m}^{0} /\left[\mathrm{m}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right)\right]^{2}\right\}^{\mathrm{eq}}\]\[\text { Then, } \ln \mathrm{K}^{0}(\text { app })=\ln \mathrm{K}^{0}+2 \ln \left[\gamma_{\pm}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right)\right]\]The solution may be so dilute that the mean ionic activity coefficient can be calculated using the Debye-Huckel Limiting Law (DHLL). \[\text { Hence } \quad \ln \mathrm{K}^{0}(\mathrm{app})=\ln \mathrm{K}^{0}-2 \, \mathrm{S}_{\gamma} \,\left[\mathrm{m}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right) / \mathrm{m}^{0}\right] 11 / 2\]In this equation the negative sign signals that in real solutions the extent of ion association to form \(\mathrm{Z}(\mathrm{aq})\) is less than in the corresponding ideal solution because charge - charge interactions in real solutions stabilise the ions.This page titled 1.4.4: Chemical Equilibria- Solutions- Ion Association is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.4.5: Chemical Equilibria- Solutions- Sparingly Soluble Salt

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.04%3A_Chemical_Equilibria/1.4.05%3A_Chemical_Equilibria-_Solutions-_Sparingly_Soluble_Salt

A given aqueous solution contains a sparingly soluble 1:1 salt [e.g. \(\operatorname{AgCl}(\mathrm{s})\)] at fixed temperature and pressure. The following phase equilibrium is established.\[\mathrm{M}^{+} \mathrm{X}^{-}(\mathrm{s}) \rightleftharpoons \mathrm{M}^{+}(\mathrm{aq})+\mathrm{X}^{-}(\mathrm{aq}) \label{a}\]In terms of the Phase Rule, the system contains two components, water and sparingly soluble substance \(\mathrm{MX}\); \(\mathrm{C} = 2\). There are three phases: solution, vapour and solid. Then \(\mathrm{F} = 1\). Hence if we define the temperature, the vapour pressure and the equilibrium composition of the liquid phase are defined.A thermodynamic description of equilibrium (Equation \ref{a}) is based on equality of chemical potentials of reactants and products. The key point is that the solid, \(\mathrm{M}^{+} \mathrm{X}^{-}(\mathrm{s})\) is a reference state. \[\text { Hence, } \mu^{0}\left(\mathrm{M}^{+} \mathrm{X}^{-} ; \mathrm{s}\right)=\mu^{\mathrm{eq}}\left(\mathrm{M}^{+} \mathrm{X}^{-} ; \mathrm{aq}\right)\]Noting that \(\mathrm{M}^{+} \mathrm{X}^{-}\) is a 1:1 salt, \[\begin{aligned} &\mu^{0}(\mathrm{MX} ; \mathrm{s})= \\ &\mu^{0}\left(\mathrm{M}^{+} \mathrm{X}^{-} ; \mathrm{aq}\right)+2 \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right) \, \gamma_{\pm}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right) / \mathrm{m}^{0}\right)^{\mathrm{eq}} \end{aligned}\]The solubility product for salt \(\mathrm{M}^{+} \mathrm{X}^{-}\), \(\mathrm{K}_{\mathrm{s}}^{0}\) is defined as follows. [\(\mathrm{K}_{\mathrm{s}}^{0}\) is dimensionless.] \[\Delta_{\mathrm{s}} \mathrm{G}^{0}=\mu^{0}\left(\mathrm{M}^{+} \mathrm{X}^{-} ; \mathrm{aq}\right)-\mu^{0}\left(\mathrm{M}^{+} \mathrm{X}^{-} ; \mathrm{s}\right)=-\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{K}_{\mathrm{s}}^{0}\right)\]\[\text { Hence }, \mathrm{K}_{\mathrm{s}}^{0}=\left[\operatorname{Sol}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right)^{\mathrm{eq}} \, \gamma_{\pm}^{\mathrm{eq}}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right) / \mathrm{m}^{0}\right]^{2}\]\(\operatorname{Sol}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right)^{\mathrm{eq}}\) is the (equilibrium) solubility, a quantity obtained experimentally. \[\ln \left[\operatorname{Sol}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right)^{\mathrm{eq}} / \mathrm{m}^{0}\right]=-\ln \left(\gamma_{\pm}^{\mathrm{eq}}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right)\right)+(1 / 2) \, \ln \left(\mathrm{K}_{\mathrm{s}}^{0}\right)\]In many cases salt \(\mathrm{M}^{+} \mathrm{X}^{-}(\mathrm{s}))\) is so sparingly soluble that \(\ln \left(\gamma_{\pm}^{\mathrm{eq}}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right)\right)\) can be calculated using the Debye-Huckel Limiting Law (DHLL). The DHLL relates \(\ln \left(\gamma_{\pm}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right)\right)^{e q}\) to the ionic strength of the solution, I. The ionic strength is controlled by adding a second soluble salt \(\mathrm{N}^{+} \mathrm{Y}^{-}\). \[\ln \left[\operatorname{Sol}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right)^{\mathrm{eq}} / \mathrm{m}^{0}\right]=\mathrm{S}_{\gamma} \,\left(\mathrm{I} / \mathrm{m}^{0}\right)^{1 / 2}+(1 / 2) \, \ln \left(\mathrm{K}_{\mathrm{s}}^{0}\right)\]This is a classic equation because in many cases \(\ln \left[\mathrm{Sol}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right)^{\mathrm{eq}} / \mathrm{m}^{0}\right]\) is a linear function of \(\left(\mathrm{I} / \mathrm{m}^{0}\right)^{1 / 2}\) so that \(\ln \left(\mathrm{K}_{\mathrm{S}}^{0}\right)\) is obtained from the corresponding intercept. We understand the form of equation (g) in terms of increasing stabilisation of the ions \(\mathrm{M}^{+}(\mathrm{aq})\) and \(\mathrm{X}^{-}(\mathrm{aq})\) in solution by the ion-ion interactions in the real solution which are enhanced when the ionic strength is increased by adding soluble salt \(\mathrm{N}^{+} \mathrm{Y}^{-}\).Footnotes \(\mathrm{K}_{\mathrm{s}}^{0}\) is dimensionless. However in many reports a quantity \(\mathrm{K}_{\mathrm{m}}^{0}\) is defined as follows. \[\mathrm{K}_{\mathrm{m}}^{0}=\left[\operatorname{Sol}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right)^{e q} \, \gamma_{\pm}^{\mathrm{eq}}\right]^{2}\]For a 1:1 salt, \(\mathrm{K}_{\mathrm{m}}^{0}\) has units, \(\left(\mathrm{mol kg}^{-1}\right)^{2}\). Or \(\mathrm{K}_{\mathrm{s}}^{0}=\mathrm{K}_{\mathrm{m}}^{0} /\left(\mathrm{m}^{0}\right)^{2}\) All three terms in equation (g) are dimensionless.This page titled 1.4.5: Chemical Equilibria- Solutions- Sparingly Soluble Salt is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.4.6: Chemical Equilibria- Cratic and Unitary Quantities

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.04%3A_Chemical_Equilibria/1.4.06%3A_Chemical_Equilibria-_Cratic_and_Unitary_Quantities

We comment on the term ‘equilibrium constant‘ where the composition of the solution under examination is expressed in terms of solute molalities, solute concentrations and solute mole fractions. We consider a closed system in which the solvent is water(\(\lambda\)) at defined temperature and pressure where the pressure is ambient and close to the standard pressure \(p^{0}\). The solution contains \(n_{1}\) moles of water and \({n_{j}}^{\mathrm{eq}}\) moles of each chemical substance \(j\), solutes, where \(j\) the composition is described in terms of a chemical equilibrium. The latter is described in the following general terms where \(ν_{j}\) is the stoichiometry for chemical substance \(j\), being positive for products and negative for reactants.\[\sum_{j=1}^{j=i} v_{j} \, \mu_{j}^{e q}(a q)=0\]The sum is taken over all \(i\)-solutes in solution with respect to the equilibrium chemical potentials of each chemical substance \(j\).The equilibrium molality of solute \(j\) is given by the ratio \(\left(\mathrm{n}_{\mathrm{j}} / \mathrm{n}_{1} \, \mathrm{M}_{1}\right)\) where \(\mathrm{M}_{1}\) is the molar mass of water. From equation (a), \[\sum_{j=1}^{j=i} v_{j} \,\left[\mu_{j}^{0}(a q ; T)+R \, T \, \ln \left(m_{j}^{\mathrm{eq}} \, \gamma_{j}^{e q} / m^{0}\right)\right]=0\]By definition for each solute \(j\), \[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{\mathrm{j}}=1.0 \text { at all } \mathrm{T} \text { and } \mathrm{p}\]A standard equilibrium constant \({\mathrm{K}_{\mathrm{m}}}^{0}\) is defined using equation (d) where the \(\mathrm{m}\) subscript ‘m’ is a reminder that we are using molalities to express the composition of the solution under examination. \[\Delta_{\mathrm{r}} \mathrm{G}^{0}(\mathrm{~T} ; \mathrm{m}-\text { scale })=-\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{K}_{\mathrm{m}}^{0}(\mathrm{~T})\right]=\sum_{\mathrm{j}=1}^{\mathrm{ji}} \mathrm{v}_{\mathrm{j}} \, \mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T})\]At temperature \(\mathrm{T}\), \(\mathrm{K}_{\mathrm{m}}^{0}(\mathrm{~T})\) is related to the equilibrium composition of the solution using equation (e). \[\mathrm{K}_{\mathrm{m}}^{0}(\mathrm{~T})=\prod_{\mathrm{j}=1}^{\mathrm{ji}}\left(\mathrm{m}_{\mathrm{j}}^{\mathrm{eq}} \, \gamma_{\mathrm{j}}^{\mathrm{eq}} / \mathrm{m}^{0}\right)^{v(\mathrm{j})}\]\[\text { Also by definition, } \mathrm{pK}_{\mathrm{m}}^{0}(\mathrm{~T})=-\lg \left[\mathrm{K}_{\mathrm{m}}^{0}(\mathrm{~T})\right]\]From the Gibbs –Helmholtz Equation , \[\Delta_{\mathrm{r}} \mathrm{H}_{\mathrm{m}}^{0}=-\mathrm{T}^{2} \,\left[\frac{\partial}{\partial \mathrm{T}}\left(\frac{\Delta_{\mathrm{r}} \mathrm{G}_{\mathrm{m}}^{0}}{\mathrm{~T}}\right)\right]_{\mathrm{p}}=\mathrm{R} \, \mathrm{T}^{2} \,\left[\frac{\partial \ln \left(\mathrm{K}_{\mathrm{m}}^{0}\right)}{\partial \mathrm{T}}\right]_{\mathrm{p}}=-\mathrm{R} \,\left[\frac{\partial \ln \left(\mathrm{K}_{\mathrm{m}}^{0}\right)}{\partial \mathrm{T}^{-1}}\right]_{\mathrm{p}}\]\[\Delta_{\mathrm{r}} \mathrm{C}_{\mathrm{pm}}^{0}(\mathrm{~T})=\left[\partial \Delta_{\mathrm{r}} \mathrm{H}_{\mathrm{m}}^{0} / \partial \mathrm{T}\right]_{\mathrm{p}}\]\[\Delta_{\mathrm{r}} \mathrm{S}_{\mathrm{m}}^{0}=\mathrm{T}^{-1} \,\left[\Delta_{\mathrm{r}} \mathrm{H}_{\mathrm{m}}^{0}-\Delta_{\mathrm{r}} \mathrm{G}_{\mathrm{m}}^{0}\right]\]The algebra is a little tortuous but the points are clearly made if we confine attention to chemical equilibria in solutions having thermodynamic properties which are ideal. For a chemical equilibrium involving \(j\) chemical substances in solution where the solvent is chemical substance 1, at fixed temperature and pressure, \[\mathrm{K}^{0}=\prod_{\mathrm{j}=2}^{\mathrm{j}=\mathrm{i}}\left(\mathrm{m}_{\mathrm{j}}^{\mathrm{eq}} / \mathrm{m}^{0}\right)^{\mathrm{v}_{\mathrm{i}}}\]\[\text {Also } \quad \mathrm{K}_{\mathrm{m}}^{0}=\prod_{\mathrm{j}=2}^{\mathrm{j}=\mathrm{i}}\left(\mathrm{m}_{\mathrm{j}}^{\mathrm{eq}}\right)^{v_{j}}\]\[\Delta_{\mathrm{r}} \mathrm{G}^{0}=-\mathrm{R} \, \mathrm{T} \, \ln \mathrm{K}^{0}\]\[\Delta_{\mathrm{r}} \mathrm{G}^{0}=-\mathrm{R} \, \mathrm{T} \, \ln \left[\prod_{\mathrm{j}=2}^{\mathrm{j}=\mathrm{i}}\left(\frac{\mathrm{m}_{\mathrm{j}}^{\mathrm{eq}}}{\mathrm{m}^{0}}\right)^{\mathrm{v}_{\mathrm{j}}}\right]\]\[\Delta_{\mathrm{r}} \mathrm{G}^{0}=-\mathrm{R} \, \mathrm{T} \, \ln \left[\prod_{\mathrm{j}=2}^{\mathrm{j}=\mathrm{i}}\left(\mathrm{m}_{\mathrm{j}}^{\mathrm{eq}}\right)^{\mathrm{v}_{\mathrm{j}}}\right]+\mathrm{v} \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}^{0}\right)\]\[\Delta_{\mathrm{r}} \mathrm{G}^{0}=-\mathrm{R} \, \mathrm{T} \, \ln \mathrm{K}_{\mathrm{m}}^{0}+\mathrm{v} \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}^{0}\right)\]\[\Delta_{\mathrm{r}} \mathrm{G}^{0}=\Delta_{\mathrm{r}} \mathrm{G}_{\mathrm{m}}^{0}+\mathrm{V} \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}^{0}\right)\]\[\Delta_{\mathrm{r}} \mathrm{G}_{\mathrm{m}}^{0}=-\mathrm{R} \, \mathrm{T} \, \ln \mathrm{K}_{\mathrm{m}}^{0}\]This rather dull analysis has merit in showing that the lost units in the equation \(\Delta_{\mathrm{r}} \mathrm{G}^{0}=-\mathrm{R} \, \mathrm{T} \, \ln \mathrm{K}_{\mathrm{m}}^{0}\) are found in the term \(\left[V \, R \, T \, \ln \left(m^{0}\right)\right]\) where \(V=\sum_{j=2}^{j=i} v_{j}\). This concern arises because for correct dimensions the logarithm operation should operate on a pure number. No such problems emerge if \(ν\) is zero as is the case for a stoichiometrically balanced equilibrium. Moreover if we probe the dependence of \(\mathrm{K}^{0}\) or \({\mathrm{K}_{\mathrm{m}}}^{0}\) on temperature we have that, \[\frac{\mathrm{d}}{\mathrm{dT}} \,\left[\frac{\mathrm{v} \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}^{0}\right)}{\mathrm{T}}\right]=0\]If we are interested in the dependence of either \(\mathrm{K}^{0}\) or \({\mathrm{K}_{\mathrm{m}}}^{0}\) on pressure, we have that, \[\frac{\mathrm{d}}{\mathrm{dp}} \,\left[\mathrm{v} \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}^{0}\right)\right]=0\]The total amount of all chemical substances in the closed system, an aqueous solution, at chemical equilibrium is given by equation (t). \[n_{\mathrm{T}}^{\mathrm{eq}}=\mathrm{n}_{1}+\sum \mathrm{n}_{\mathrm{j}}^{\mathrm{eq}}\]For a given chemical substance, solute \(k\) \[\mathrm{x}_{\mathrm{k}}^{\mathrm{eq}}=\mathrm{n}_{\mathrm{k}}^{\mathrm{eq}} /\left[\mathrm{n}_{1}+\sum \mathrm{n}_{\mathrm{j}}^{e q}\right]\]In terms of mole fractions, the equilibrium chemical potential for solute \(j\), \(\mu_{\mathrm{j}}^{\mathrm{eq}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\) is related to the equilibrium mole fraction ) \(\mathrm{x}_{\mathrm{j}}^{e q}(\mathrm{aq})\) using equation (v). \[\mu_{\mathrm{j}}^{\mathrm{eq}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{\mathrm{j}}^{0}(\mathrm{x}-\mathrm{scale} ; \mathrm{aq} ; \mathrm{T})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{\mathrm{j}}^{\mathrm{eq}} \, \mathrm{f}_{\mathrm{j}}^{*_{\mathrm{eq}}}\right)\]\[\text { By definition, at all } \mathrm{T} \text { and } \mathrm{p} \text {, } \operatorname{limit}\left(\mathrm{x}_{\mathrm{j}} \rightarrow 0\right) \mathrm{f}_{\mathrm{j}}^{*}=1.0 \text {. }\]Further, \[\mu_{j}^{0}(\mathrm{x}-\text { scale; aq; } \mathrm{T})\] is the chemical potential of substance \(j\) in aqueous solution at temperature \(\mathrm{T}\) in a solution where the mole fraction of solute \(j\) is unity. Here therefore \(\mu_{\mathrm{j}}^{0}(\mathrm{x}-\text { scale; aq; } \mathrm{T})\) is the reference chemical potential. From equation (a), \[\sum_{j=1}^{\mathrm{j}=\mathrm{i}} \mathrm{v}_{\mathrm{j}} \,\left[\mu_{\mathrm{j}}^{0}(\mathrm{x}-\text { scale; aq; } \mathrm{T})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{\mathrm{j}}^{\mathrm{eq}} \, \mathrm{f}_{\mathrm{j}}^{* \mathrm{eq}}\right)\right]=0\]\[\Delta_{\mathrm{r}} \mathrm{G}_{\mathrm{x}}^{0}(\mathrm{~T})=-\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{K}_{\mathrm{x}}^{0}(\mathrm{~T})\right]=\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} v_{\mathrm{j}} \, \mu_{\mathrm{j}}^{0}(\mathrm{x}-\text { scale; aq; } \mathrm{T})\](T) R T ln[K (T)] (x scale;aq;T) At temperature \(\mathrm{T}\), \(\mathrm{K}_{\mathrm{x}}^{0}(\mathrm{~T})\) is related to the equilibrium mole fractions of the solutes. \[\mathrm{K}_{\mathrm{x}}^{0}(\mathrm{~T})=\prod_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}}\left(\mathrm{x}_{\mathrm{j}}^{\mathrm{eq}} \, \mathrm{f}_{\mathrm{j}}^{* \mathrm{eq}}\right)^{v(\mathrm{j})}\]From the Gibbs –Helmholtz Equation, \[\Delta_{\mathrm{r}} \mathrm{H}_{\mathrm{x}}^{0}=-\mathrm{T}^{2} \,\left[\frac{\partial}{\partial \mathrm{T}}\left(\frac{\Delta_{\mathrm{r}} \mathrm{G}_{\mathrm{x}}^{0}}{\mathrm{~T}}\right)\right]_{\mathrm{p}}=\mathrm{R} \, \mathrm{T}^{2} \,\left[\frac{\partial \ln \left(\mathrm{K}_{\mathrm{x}}^{0}\right)}{\partial \mathrm{T}}\right]_{\mathrm{p}}=-\mathrm{R} \,\left[\frac{\partial \ln \left(\mathrm{K}_{\mathrm{x}}^{0}\right)}{\partial \mathrm{T}^{-1}}\right]_{\mathrm{p}}\]\[\Delta_{\mathrm{r}} \mathrm{C}_{\mathrm{px}}^{0}(\mathrm{~T})=\left[\partial \Delta_{\mathrm{r}} \mathrm{H}_{\mathrm{x}}^{0} / \partial \mathrm{T}\right]_{\mathrm{p}}\]\[\Delta_{\mathrm{r}} \mathrm{S}_{\mathrm{x}}^{0}=\mathrm{T}^{-1} \,\left[\Delta_{\mathrm{r}} \mathrm{H}_{\mathrm{x}}^{0}-\Delta_{\mathrm{r}} \mathrm{G}_{\mathrm{x}}^{0}\right]\]We note a complication. We suppose that the composition of a given closed system, an aqueous solution, is described in terms of the formation of a dimer by a solute \(\mathrm{Z}\) in aqueous solution at defined \(\mathrm{T}\) and \(p\) \[2 Z(a q) \Longleftrightarrow Z_{2}(a q)\]At equilibrium, the solution contains n1 moles of water, \(\mathrm{n}^{\mathrm{eq}}(\mathrm{Z})\) moles of monomer and \(\mathrm{n}^{\mathrm{eq}}\left(Z_{2}\right)\) moles of dimer. \[x(Z)^{e q}=n(Z)^{e q} /\left[n_{1}+n(Z)^{e q}+n\left(Z_{2}\right)^{e q}\right]\]\[x\left(Z_{2}\right)^{\text {eq }}=n\left(Z_{2}\right)^{\text {eq }} /\left[n_{1}+n(Z)^{e q}+n\left(Z_{2}\right)^{e q}\right]\]\[\text { Further, } x_{1}^{e q}=n_{1} /\left[n_{1}+n(Z)^{e q}+n\left(Z_{2}\right)^{e q}\right]\]As a result of a change in temperature \(\mathrm{n}^{\mathrm{eq}}(\mathrm{Z})\) and \(\mathrm{n}^{\mathrm{eq}}\left(\mathrm{Z}_{2}\right)\) change; \(n_{1}\) does not. For example \(x(Z)^{e q}\) changes as a result of changes in both numerator and denominator in equation (ze). This unwelcome complication is not encountered if we use the molality scale. The way forward is to confine attention to dilute solutions such that at all temperatures, \(\sum_{j=2}^{j=i} n_{j}^{e q}<Footnotes L. Hepler, Thermochim Acta,1981,50,69. E. A. Guggenheim, Trans. Faraday Soc.,1937,33,607. E. Euranto, J.J.Kankare and N.J.Cleve, J. Chem. Eng. Data,1969, 14, 455. R. W. Gurney, Ionic Processes in Solution , McGraw-Hill, New York, 1952. As Guggenheim remarked the units of \(\ln (\mathrm{V})\) are \(\ln \left(\mathrm{m}^{3}\right)\). If \(\mathrm{V}=100 \mathrm{~m}^{3}\), then \(\log (\mathrm{V})=\log+\log \left(\mathrm{m}^{3}\right)\), \(\log (\mathrm{V})-\log \left(\mathrm{m}^{3}\right)=2\); \(\log \left(\mathrm{V} / \mathrm{m}^{3}\right)=2\). M. H. Abraham and A. Nasehzadeh, J. Chem. Soc. Chem. Commun., 1981, 905.This page titled 1.4.6: Chemical Equilibria- Cratic and Unitary Quantities is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.4.7: Chemical Equilibria- Composition- Temperature and Pressure Dependence

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The variable \(\xi\) describes in quite general terms molecular composition – molecular organisation. For a given closed system at fixed \(\mathrm{T}\) and \(p\) there exists a composition-organisation \(\xi_{\mathrm{eq}\) corresponding to a minimum in Gibbs energy where the affinity for spontaneous change is zero. In general terms there exists an extent of reaction \(\xi\) corresponding to a given affinity \(\mathrm{A}\) at defined \(\mathrm{T}\) and \(p\). In fact we can express \(\xi\) as a dependent variable defined by the independent variables, \(\mathrm{T}\), \(p\) and \(\mathrm{A}\). \[\xi=\xi[T, p, A]\]The general differential of equation (a) takes the following form. \[\mathrm{d} \xi=\left(\frac{\partial \xi}{\partial T}\right)_{\mathrm{p}, \mathrm{A}} \, \mathrm{dT}+\left(\frac{\partial \xi}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{A}} \, \mathrm{dp}+\left(\frac{\partial \xi}{\partial \mathrm{A}}\right)_{\mathrm{T}, \mathrm{p}} \, \mathrm{dA}\]Equation (b) describes the dependence of extent of reaction on changes in \(\mathrm{T}\), \(p\) and affinity \(\mathrm{A}\). \[\text { Moreover, }\left(\frac{\partial \xi}{\partial T}\right)_{\mathrm{p}, \mathrm{A}}=-\left(\frac{\partial \xi}{\partial \mathrm{A}}\right)_{\mathrm{T}, \mathrm{p}} \,\left[\frac{1}{\mathrm{~T}} \,\left(\frac{\partial \mathrm{H}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}+\frac{\mathrm{A}}{\mathrm{T}}\right]\]At equilibrium where ‘\(\mathrm{A} = 0\)’ and \((\partial \mathrm{A} / \partial \xi)_{\mathrm{T}, \mathrm{p}}<0\), then \[\left(\frac{\partial \xi}{\partial T}\right)_{p, A=0} \text { takes the sign of }\left[\frac{1}{T} \,\left(\frac{\partial H}{\partial \xi}\right)_{T, p}^{e q}\right]=\left[\frac{1}{T} \, \sum_{j=1}^{j=i} v_{j} \, H_{j}^{e q}\right]\]\[\text { Similarly, }\left(\frac{\partial \xi}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{A}=0} \text { takes the sign of }\left(\frac{\partial \mathrm{V}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}} \,\left(\frac{\partial \xi}{\partial \mathrm{A}}\right)_{\mathrm{T}, \mathrm{p}}\]Again at equilibrium where ‘\(\mathrm{A} = 0\)’ and \((\partial \mathrm{A} / \partial \xi)_{\mathrm{T}, \mathrm{p}}<0\), then \[\left(\frac{\partial \xi}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{A}=0} \text { takes the sign of }\left(\frac{\partial \mathrm{V}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}^{\mathrm{eq}}=-\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{v}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}^{\mathrm{eq}}\]Equations (d) and (f) are important being universally valid and forming the basis of important generalisations, the Laws of Moderation.Equation (d) shows that the differential dependence of composition on temperature is related to the enthalpy of reaction. If the chemical reaction is exothermic {i.e. \(\left(\frac{\partial H}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}^{\mathrm{eq}}\) is negative}, the chemical equilibrium shifts to favour an increase in the amount of reactants. Whereas if the reaction is endothermic , the composition swings in a direction to favour the products.In another experiment, the equilibrium system is perturbed by an increase in pressure. Equation (f) shows that the equilibrium composition swings to favour the reactants if the volume of reaction is positive. Alternatively if the volume of reaction is negative, the composition of the system changes to favour products.Footnotes The conclusions reached here are called ‘Theorems of Moderation’. MJB was taught that the outcome is ‘Nature’s Laws of Cussedness’ [ = obstinacy]. An exothermic reaction generates heat to raise the temperature of the system, so the system responds, when the temperature is raised, by shifting the equilibrium in the direction for which the process is endothermic. The line of argument is not good thermodynamics but it makes the point.This page titled 1.4.7: Chemical Equilibria- Composition- Temperature and Pressure Dependence is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.4.8: Chemical Equilibrium Constants- Dependence on Temperature at Fixed Pressure

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A given set of data reports the dependence on temperature (at fixed pressure \(p\), which is close to the standard pressure \(p^{0}\)) of \(\mathrm{K}^{0}\) for a given chemical equilibrium.[1 - 3] \[\Delta_{\mathrm{r}} \mathrm{G}^{0}=-\mathrm{R} \mathrm{T} \ln \mathrm{K}^{0}=\Delta \mathrm{H}^{0}-\mathrm{T} \Delta_{\mathrm{r}} \mathrm{S}^{0}\]If we confine our attention to systems where the chemical equilibria involve solutes in dilute solution in a given solvent, we can replace \(\Delta_{\mathrm{r}} \mathrm{H}^{0}\) in this equation with the limiting enthalpy of reaction, \(\Delta_{\mathrm{r}} \mathrm{H}^{\infty}\). According to the Gibbs - Helmholtz Equation, at fixed pressure,\[ \dfrac{ \mathrm{d}\left[\Delta_{\mathrm{r}} \mathrm{G}^{0} / \mathrm{T}\right] }{ \mathrm{dT}} =- \dfrac{ \Delta_{\mathrm{r}} \mathrm{H}^{\infty} }{ \mathrm{T}^{2}} \]Hence\[\dfrac{ \mathrm{d} \ln \left(\mathrm{K}^{0}\right) }{\mathrm{dT}} = \dfrac{ \Delta_{\mathrm{r}} \mathrm{H}^{\infty} }{ \mathrm{R} \mathrm{T}^{2}} \] or,\[ \dfrac{ \mathrm{d} \ln \mathrm{K}^{0} }{\mathrm{dT}^{-1}} =- \dfrac{\Delta_{\mathrm{r}} \mathrm{H}^{\infty} }{ \mathrm{R} }\]The latter two equations are equivalent forms of the van ’t Hoff Equation expressing the dependence of \(\mathrm{K}^{0}\) on temperature. This equation does not predict how equilibrium constants depend on temperature. For example the van’t Hoff equation does not require that \(\ln \left(\mathrm{K}^{0}\right)\) is a linear function of \(\mathrm{T}-1\). In fact for simple carboxylic acids, the plots of \(\ln (\text {acid dissociation constant})\) against temperature show maxima. For example, \(\ln \left(\mathrm{K}^{0}\right)\) for the acid dissociation constant of ethanoic acid in aqueous solution at ambient pressure increases with increase in temperature, passes through a maximum near \(295 \mathrm{~K}\) and then decreases. At the temperature where \(\mathrm{K}^{0}\) is a maximum, the limiting enthalpy of dissociation is zero. This pattern is possibly surprising at first sight but can be understood in terms of a balance between the standard enthalpy of heterolytic fission of the \(\mathrm{O}-\mathrm{H}\) group in the carboxylic acid group and the standard enthalpies of hydration of the resulting hydrogen and carboxylate ions.Thus the dependence of \(\mathrm{K}^{0}\) on temperature can be obtained experimentally, the dependence being unique for each system. Nevertheless these equations signal how the dependence forms the basis for determining limiting enthalpies of reaction. The analysis also recognises that \(\Delta_{\mathrm{r}} \mathrm{H}^{0}\) is likely to depend on temperature. There is merit in expressing the dependence of \(\mathrm{K}^{0}\) on temperature about a reference temperature \(\theta\), chosen near the middle of the experimental temperature range. Over the experimental temperature range straddling \(\theta\), we express the dependence of \(\mathrm{K}^{0}\) on temperature using the integrated form of equation (c). \[\ln \left[\mathrm{K}^{0}(\mathrm{~T})\right]=\ln \left[\mathrm{K}^{0}(\theta)\right]+\int_{\theta}^{\mathrm{T}}\left[\frac{\Delta_{\mathrm{r}} \mathrm{H}^{\omega}}{\mathrm{RT}^{2}}\right] \mathrm{dT}\]By definition, the limiting isobaric heat capacity of reaction \(\Delta_{\mathrm{r}} C_{\mathrm{p}}^{\infty}\) is given by equation (f).\[\Delta_{\mathrm{r}} \mathrm{C}_{\mathrm{p}}^{\infty}=\left( \dfrac{ \mathrm{d} \Delta_{\mathrm{r}} \mathrm{H}^{\infty} }{ \mathrm{dT}}\right)_{\mathrm{p}}\]The analysis becomes complicated because we recognise that \(\Delta_{\mathrm{r}} C_{\mathrm{p}}^{\infty}\) depends on temperature. In fact only in rare instances are experimental results sufficiently precise to warrant taking such a dependence into account. A reasonable assumption is that \(\Delta_{r} C_{p}^{\infty}\) is independent of temperature such that \(\Delta_{\mathrm{r}} \mathrm{H}^{\infty}\) is a linear function of temperature over the experimental temperature range.\[\Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\mathrm{T})=\Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\theta)+\Delta_{\mathrm{r}} \mathrm{C}_{\mathrm{p}}^{\infty}(\mathrm{T}-\theta)\]Hence,\[\begin{aligned} &\ln \left[\mathrm{K}^{0}(\mathrm{~T})\right]= \\ &\ln \left[\mathrm{K}^{0}(\theta)\right]+\frac{1}{\mathrm{R}} \int_{\theta}^{\mathrm{T}}\left[\frac{\Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\theta)}{\mathrm{T}^{2}}+\Delta_{\mathrm{r}} \mathrm{C}_{\mathrm{p}}^{\infty} \left(\frac{1}{\mathrm{~T}}-\frac{\theta}{\mathrm{T}^{2}}\right)\right] \mathrm{dT} \end{aligned}\]Hence,\[\begin{aligned} &\ln \left[\mathrm{K}^{0}(\mathrm{~T})\right]= \\ &\ln \left[\mathrm{K}^{0}(\theta)\right]+\frac{\Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\theta)}{\mathrm{R}} \left[\frac{1}{\theta}-\frac{1}{\mathrm{~T}}\right]+\frac{\Delta_{\mathrm{r}} \mathrm{C}_{\mathrm{p}}^{\infty}}{\mathrm{R}} \left[\frac{\theta}{\mathrm{T}}-1+\ln \left(\frac{\mathrm{T}}{\theta}\right)\right] \end{aligned}\]Numerical analysis uses linear least squares procedures with reference to the dependence of \(\ln K^{0}(T)\) on temperature about the reference temperature \(\theta\) in order to obtain estimates of \(\Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\theta)\) and \(\Delta_{r} C_{p}^{\infty}\). The coupling of estimates of derived parameters is minimal if θ is chosen near the centre of the measured temperature range. Granted that the analysis yields \(\Delta_{\mathrm{r}} \mathrm{H}^{\infty}\) at a given temperature and pressure, combination with the corresponding \(\Delta_{\mathrm{r}} \mathrm{G}^{0}\) yields the entropy term, \(\Delta_{\mathrm{r}} \mathrm{S}^{0}\).Other methods of data analysis in this context use (a) orthogonal polynomials, and (b) sigma plots.An extensive literature describes the thermodynamics of acid dissociation in alcohol + water mixtures. In these solvent systems the standard enthalpies and other thermodynamic parameters pass through extrema as the mole fraction composition of the solvent is changed.Perlmutter-Hayman examines the related problem of the dependence on temperature of activation energies.Enthalpies of dissociation for weak acids in aqueous solution can be obtained calorimetrically.Footnotes R. W. Ramette, J. Chem. Educ.,1977,54,280 M. J. Blandamer, J. Burgess, R. E. Robertson and J. M. W. Scott, Chem. Rev., 1982, 82,259. M. J. Blandamer, Chemical Equilibria in Solution, Ellis Horwood PTR Prentice Hall, New York,1992. \(\mathrm{d} \ln \mathrm{K}^{0} / \mathrm{dT}^{-1}=\left[\mathrm{J} \mathrm{mol}^{-1}\right] /\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right]=[\mathrm{K}]\) H. S. Harned and N. D. Embree, J. Am. Chem. Soc.,1934,56,1050. H. S. Harned and R. W. Ehlers, J.Am.Chem.Soc.,1932,54,1350. See also ethanoic acid in D2O; M. Paabo, R. G. Bates and R. A. Robinson, J. Phys. Chem., 1966,70,2073; and references therein. Substituted benzoic acids(aq); L. E. Strong, C. L. Brummel and P. Lindower, J. Solution Chem., 1987, 16, 105; and references therein. E. C. W. Clarke and D. N. Glew, Trans. Faraday Soc.,1966,62,539. H. F. Halliwell and L. E. Strong, J. Phys. Chem.,1985,89,4137. \(\Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\theta)=\left[\mathrm{J} \mathrm{mol}^{-1}\right]+\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] [\mathrm{K}]\) D. J. G. Ives and P. D. Marsden, J. Chem. Soc.,1965,649 and 2798. D. J. G. Ives, P. G. N. Moseley, J. Chem. Soc. Faraday Trans.1, 1976,72,1132. Anilinium ions in EtOH+water mixtures;W. van der Poel, Bull. Soc. Chim. Belges., 1971,80,401; and references therein. Enthalpies of transfer for carboxylic acids in water+ 2-methyl propan-2-ol mixtures; L. Avedikian, J. Juillard and J.-P. Morel, Thermochim. Acta, 1973,6,283. Benzoic acid in DMSO + water mixtures; F. Rodante, F. Rallo and P. Fiordiponti, Thermochim. Acta, 1974, 9,269. Tris in water + methanol mixtures; C. A. Vega, R. A. Butler, B. Perez and C. Torres, J. Chem. Eng. Data, 1985,30,376. F. J. Millero, C-h. Wu and L. G. Hepler, J. Phys. Chem., 1969, 73,2453. B. Perlmutter-Hayman, Prog. Inorg. Chem.,1976,20,229. F. Rodante, G. Ceccaroni and F. Fantauzzi, Thermochim. Acta,1983,70,91.This page titled 1.4.8: Chemical Equilibrium Constants- Dependence on Temperature at Fixed Pressure is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.4.9: Chemical Equilibria- Dependence on Pressure at Fixed Temperature

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A key quantity in the description of a chemical equilibrium is the equilibrium constant. In the majority of cases the symbol used is \(\mathrm{K}^{0}\) indicating with the superscript ‘0’ a standard property. This symbol is used because, again in the majority of cases an equilibrium constant refers to a system at ambient pressure which is close to the standard pressure; i.e. \(10^{5} \mathrm{~Pa}\). In reporting \(\mathrm{K}^{0}\) therefore the temperature is stated but by definition \(\mathrm{K}^{0}\) is not dependent on pressure. However the equilibrium composition of a closed system generally depends on pressure at fixed temperature \(\mathrm{T}\). This problem over symbols and nomenclature is resolved as follows.An aqueous solution contains \(i\)-chemical substances, solutes, in chemical equilibrium. For a given solute–\(j\) the dependence of chemical potential \(\mu_{j}(a q ; T ; p)\) on molality \(\mathrm{m}_{j}\) is given by equation (a). \[\begin{aligned} &\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})= \\ &\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)+\int_{\mathrm{p}^{\circ}}^{\mathrm{p}} \mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq}) \, \mathrm{dp} \end{aligned}\]We define a reference chemical potential for solute-\(j\) \(\mu_{j}^{*}\) at temperature \(\mathrm{T}\) and pressure \(p\) using equation (b). \[\mu_{\mathrm{j}}^{\#}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T})+\int_{\mathrm{p}^{0}}^{\mathrm{p}} \mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq}) \, \mathrm{dp}\]Combination of equations (a) and (b) yields equation (c). \[\mu_{j}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{\mathrm{j}}^{\#}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)\]XKHere \(\mu_{j}^{H}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\) is the chemical potential of solute-j in an ideal solution (i.e. \(\gamma_{j}=1\)) having unit molality (i.e. \(\mathrm{m}_{\mathrm{j}}=1 \mathrm{~mol} \mathrm{~kg}{ }^{-1}\)) at specified \(\mathrm{T}\) and \(p\). At equilibrium at pressure \(p\) and temperature \(\mathrm{T}\), \[\sum_{j=1}^{j=i} v_{j} \, \mu_{j}^{\mathrm{eq}}(\mathrm{aq} ; T ; p)=0\]By definition, \[\Delta_{\mathrm{r}} \mathrm{G}^{*}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\sum \mathrm{v}_{\mathrm{j}} \, \mu_{\mathrm{j}}^{*}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=-\mathrm{R} \, \mathrm{T} \, \ln \mathrm{K}^{*}(\mathrm{~T}, \mathrm{p})\]\[\text { and } \mathrm{K}^{\#}(\mathrm{~T}, \mathrm{p})=\prod_{\mathrm{j}=1}^{\mathrm{j}=1}\left[\left(\mathrm{~m}_{\mathrm{j}}^{\mathrm{eq}} / \mathrm{m}^{0}\right) \, \gamma_{j}^{e q}\right]^{v(j)}\]The differential dependence on pressure of \(\mathrm{K}^{\#}(\mathrm{~T}, \mathrm{p})\) yields the limiting volume of reaction, \(\Delta_{\mathrm{r}} \mathrm{V}^{\infty}\). \[\Delta_{r} V^{\infty}=\sum_{j=1}^{j=i} V_{j} \, V_{j}^{\infty}(a q ; T ; p)=0\]\[\text { and [c.f. V } \left.=[\partial \mathrm{G} / \partial \mathrm{p}]_{\mathrm{T}}\right] \mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\left[\partial \mu_{\mathrm{j}}^{*}(\mathrm{aq}) / \partial \mathrm{p}\right]_{\mathrm{T}}\]\[\text { Hence at pressure } p,\left(\frac{\partial \Delta_{\mathrm{r}} \mathrm{G}^{*}(\mathrm{~T})}{\partial \mathrm{p}}\right)_{\mathrm{T}}=\Delta_{\mathrm{r}} \mathrm{V}^{\infty}(\mathrm{T}, \mathrm{p})\]\[\text { or, }\left(\frac{\partial \ln \mathrm{K}^{\prime \prime}(\mathrm{T})}{\partial \mathrm{p}}\right)_{\mathrm{T}}=-\frac{\Delta_{\mathrm{r}} \mathrm{V}^{\infty}(\mathrm{T}, \mathrm{p})}{\mathrm{R} \, \mathrm{T}}\]The negative sign in equation (j) means that if \(\ln \mathrm{K}^{\#}\) for a given chemical equilibrium increases with increases in pressure then \(\Delta_{\mathrm{r}} \mathrm{V}^{\infty}\) is negative. But thermodynamics does not define how a given equilibrium constant depends on pressure. This dependence must be measured. Moreover we cannot assume that the limiting volume of reaction \(\Delta_{\mathrm{r}} \mathrm{V}^{\infty}\) is independent of pressure. This dependence is described by the limiting isothermal compressions of reaction, \(\Delta_{\mathrm{r}} \mathrm{K}_{\mathrm{T}}^{\infty}\). \[\Delta_{\mathrm{r}} \mathrm{K}_{\mathrm{T}}^{\infty}=-\left[\frac{\mathrm{d} \Delta_{\mathrm{r}} \mathrm{V}^{\infty}}{\mathrm{dp}}\right]_{\mathrm{T}}\]Indeed we cannot assume that \(\mathrm{K}_{\mathrm{T}}^{\infty}\) is independent of pressure but in most cases the precision of the data is insufficient to obtain a meaningful estimate of this dependence. Hence we are often justified in assuming that \(\Delta_{\mathrm{r}} \mathrm{V}^{\infty}\) is a linear function of pressure about a reference pressure \(\pi\), the latter usually chosen as ambient pressure. \[\Delta_{\mathrm{r}} \mathrm{V}^{\infty}(\mathrm{p})=\Delta_{\mathrm{r}} \mathrm{V}^{\infty}(\pi)-\Delta_{\mathrm{r}} \mathrm{K}_{\mathrm{T}}^{\infty}(\mathrm{p}-\pi)\]Hence, \[\ln \left(\mathrm{K}^{\#}(\mathrm{p})\right)=\ln \left(\mathrm{K}^{\#}(\pi)\right)-(\mathrm{R} \, \mathrm{T})^{-1} \, \Delta_{\mathrm{r}} \mathrm{V}^{\infty}(\pi) \,(\mathrm{p}-\pi) -(2 \, R \, T)^{-1} \, \Delta_{r} K_{T}^{\infty} \,\left((p-\pi)^{2}\right)\]Thus, \(\ln \mathrm{K}^{\#}(\mathrm{p})\) is a quadratic in \((\mathrm{p}-\pi)\).Alternatively we may express the dependence of \(\ln \left[\mathrm{K}^{\#}(\mathrm{p}) / \mathrm{K}^{\#}(\pi)\right]\) on pressure using the following equation. \[\begin{aligned} &\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{K}^{\#}(\mathrm{p}) / \mathrm{K}^{\#}(\pi)\right]= \\ &\quad-\Delta_{\mathrm{r}} \mathrm{V}^{\infty}(\pi) \,(\mathrm{p}-\pi)-0.5 \, \Delta_{\mathrm{r}} \mathrm{K}_{\mathrm{T}}^{\infty} \,(\mathrm{p}-\pi)^{2} \end{aligned}\]This equation shows how \(\left[\mathrm{K}^{\#}(\mathrm{p}) / \mathrm{K}^{\#}(\pi)\right]\) may be calculated from estimates of \(\Delta_{\mathrm{r}} \mathrm{V}^{\infty}(\pi)\) obtained from independently obtained estimates of partial molar volumes and partial molar isothermal compressions of the chemical substances involved in the chemical equilibrium; e.g. acid dissociation of boric acid.Another approach expresses the ratio \(\left[\mathrm{K}^{\#}(\mathrm{p}) / \mathrm{K}^{\#}(\pi)\right]\) as a function of solvent density at pressure \(p\), \(\rho(p)\) together with density \(\rho(\pi)\) at pressure \(\pi\) and a parameter \(\beta\) using equation (o). \[\ln \left[K^{\#}(\mathrm{p}) / \mathrm{K}^{\#}(\pi)\right]=(\beta-1) \, \ln [\rho(\pi) / \rho(\mathrm{p})]\]This approach is closely linked to numerical analysis based on equation (p). \[\ln \left[K^{\#}(p) / K^{\#}(\pi)\right]=-\left[\Delta_{r} V^{\infty}(\pi) / R \, T\right] \,[p /(1+b \, p)]\]A rather different approach for chemical equilibria between solutes in aqueous solutions refers to equation (q). \(\mathrm{A}\) and \(\mathrm{B}\) are constants independent of pressure but dependent on temperature; these constants describe the dependence of the molar volume of water on pressure at fixed temperature; Tait’s isotherm. \[\mathrm{V}_{1}^{*}(\mathrm{p})=\mathrm{V}_{1}^{*}(\pi) \,\left[1-\mathrm{A} \, \ln \left(\frac{\mathrm{B}+\mathrm{p}}{\mathrm{B}+\pi}\right)\right]\]There are a few case where the experimental data warrant consideration of the dependence on pressure of \(\Delta_{r} K^{\infty}\). Under these circumstances the Owen-Brinkley equation has the following form.\[\begin{aligned} \mathrm{R} \, \mathrm{T} \, \ln \left[\frac{\mathrm{K}^{\#}(\mathrm{p})}{\mathrm{K}^{\#}(\pi)}\right]=-\Delta_{\mathrm{r}} \mathrm{V}^{\infty}(\pi) \,(\mathrm{p}-\pi) \\ &+\Delta_{\mathrm{r}} \mathrm{K}_{\mathrm{T}}^{\infty}(\pi) \,\left[(\mathrm{B}+\pi) \,(\mathrm{p}-\pi)-(\mathrm{B}+\pi)^{2} \, \ln \left(\frac{\mathrm{B}+\mathrm{p}}{\mathrm{B}+\pi}\right)\right] \end{aligned}\]Footnotes M. J. Blandamer, Chemical Equilibria in Solution, Ellis Horwood PTR, Prentice Hall, New York,1992. B. B. Owen and S. R. Brinkley, Chem. Rev.,1941,29,401. S. W. Benson and J. A. Person, J. Amer. Chem. Soc.,1962,84,152. S. D. Hamann, J. Solution Chem.,1982,11,63; and references therein. N. A. North, J.Phys.Chem.,1973,77,931. B. S. El’yanov and E. M. Vasylvitskaya, Rev. Phys. Chem. Jpn, 1980, 50, 169; and references therein. There is a strong link between this subject and analysis of the dependence of rate constants for chemical reactions on pressure at fixed temperature; By definition the standard equilibrium constant \({\mathrm{K}}_{\mathrm{m}}}^{0}\) describes the case where at temperature \(\mathrm{T}\), the pressure is the standard pressure. \(\frac{\mathrm{d} \ln \mathrm{K}^{*}}{\mathrm{dp}}=\frac{\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]}{\left[\mathrm{J} \mathrm{mol}^{-1} \mathrm{~K}^{-1}\right] \,[\mathrm{K}]}=\left[\mathrm{N} \mathrm{m}^{-2}\right]^{-1}\) \(\Delta_{\mathrm{r}} \mathrm{V}^{\infty}(\mathrm{p})=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]+\left(\frac{\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]}{\left[\mathrm{N} \mathrm{m}^{-2}\right]}\right) \,\left[\mathrm{N} \mathrm{m}^{-2}\right]\) \(\ln \left(\mathrm{K}^{\#}(\mathrm{p})\right)=+\frac{1}{\left[\mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\right]} \,\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right] \,\left[\mathrm{N} \mathrm{m}^{-2}\right] +\frac{1}{ \,\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}]} \,\left[\mathrm{m}^{3} \mathrm{~mol}^{-1} \mathrm{~N}^{-1} \mathrm{~m}^{2}\right] \,\left[\mathrm{N} \mathrm{m}^{-2}\right]^{2}\) D. A. Lown, H. R. Thirsk and Lord Wynne-Jones, Trans. Faraday Soc., 1968, 64, 2073. A. J. Read {J. Solution Chem.,1982, 11, 649;1988, 17, 213} uses a simpler form of the equation which has the general form, \(y=m . x+c\). Thus, \(\left[\frac{\mathrm{R} \, \mathrm{T}}{\mathrm{p}-\pi}\right] \, \ln \left[\frac{\mathrm{K}^{\#}(\mathrm{p})}{\mathrm{K}^{\#}(\pi)}\right]=-\Delta_{\mathrm{r}} \mathrm{V}^{\infty}(\pi)-0.5 \,(\mathrm{p}-\pi) \, \Delta_{\mathrm{r}} \mathrm{K}_{\mathrm{T}}^{\infty}\) G. K. Ward and F. J. Millero, J. Solution Chem.,1974,3,417. See also pH and pOH; Y. Kitamura and T. Itoh, J. Solution Chem., 1987, 16, 715. W. L. Marshall and R. E. Mesmer, J. Solution Chem., 1984, 13, 383; and references therein. B. S. El’yanov and S. D. Hamann, Aust. J. Chem.,1975,28,945. R. E. Gibson, J.Am.Chem.Soc.,1934,56,4. S. D. Hamann and F. E. Smith, Aust. J. Chem.,1971,24,2431. G. A. Neece and D. R. Squire, J. Phys. Chem.,1968,72,128. K. E. Weale, Chemical Reactions at High Pressure, Spon, London,1967; and references therein.This page titled 1.4.9: Chemical Equilibria- Dependence on Pressure at Fixed Temperature is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.5.1: Chemical Potentials, Composition and the Gas Constant

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.05%3A_Chemical_Potentials/1.5.01%3A_Chemical_Potentials_Composition_and_the_Gas_Constant

In many Topics describing the thermodynamic properties of liquid mixtures and solutions, key equations relate the chemical potentials of components to the composition of a given system. For example in the case of a binary aqueous mixture the chemical potential of water \(\mu_{1}(\mathrm{~T}, \mathrm{p}, \mathrm{mix})\) is related to the mole fraction of water \(x_{1}\) at temperature \(\mathrm{T}\) and pressure \(p\) using equation (a). \[\mu_{1}(\mathrm{~T}, \mathrm{p}, \operatorname{mix})=\mu_{1}^{*}(\mathrm{~T}, \mathrm{p}, \ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right)\]\[\text { By definition, limit }\left(\mathrm{x}_{1} \rightarrow 1\right) \mathrm{f}_{1}=1.0\]Here \(\mu_{1}^{*}(\mathrm{~T}, \mathrm{p}, \ell)\) is the chemical potential of water(\(\ell\)) at the same \(\mathrm{T}\) and \(p\); \(\mathrm{f}_{1}\) is the rational activity coefficient of water in the mixture.Similarly for solute \(j\) in an aqueous solution at temperature \(\mathrm{T}\) and pressure \(p\), the chemical potential of solute \(j\), \(\mu_{j}(T, p, a q)\) is related to the molality mj using equation (c) where \(\mathrm{m}^{0}=1 \mathrm{~mol} \mathrm{~kg}^{-1}\). \[\mu_{\mathrm{j}}(\mathrm{aq}, \mathrm{T}, \mathrm{p})=\mu_{\mathrm{j}}^{0}(\mathrm{aq}, \mathrm{T}, \mathrm{p})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)\]\[\text { By definition, at all T and } p \text { limit }\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{\mathrm{j}}=1.0\]Here \(\mu_{\mathrm{j}}^{0}(\mathrm{aq}, \mathrm{T}, \mathrm{p})\) is the chemical potentials of solute \(j\) in an aqueous solution at the same \(\mathrm{T}\) and \(p\) where \(\mathrm{m}_{\mathrm{j}}=1.0 \mathrm{~mol} \mathrm{~kg}\) and \(\gamma_{\mathrm{j}}=1.0\).In equations (a) and (c) the parameter \(\mathrm{R}\) is the Gas Constant, \(8.314 \mathrm{~J mol}^{-1} \mathrm{~K}^{-1}\). The word ‘Gas’ in the latter sentence is interesting bearing in mind that equations (a) and (c) describe the properties of liquids, mixtures and solutions. Here we examine how this parameter emerges in these equations.The starting point is a description of a closed system containing \(i\)–chemical substances, the amount of chemical substance \(j\) being \(n_{j}\). \[\text { Then, } \mathrm{G}=\mathrm{G}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{\mathrm{i}}\right]\]The chemical potential \(\mu_{j}(T, p)\) of chemical substance \(j\) is given by equation (f). \[\mu_{\mathrm{j}}(\mathrm{T}, \mathrm{p})=\left(\frac{\partial \mathrm{G}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{i} \neq \mathrm{j})}\]Moreover the partial molar volume \(\mathrm{V}_{j}\) of chemical substance \(j\) is given by equation (g). \[V_{j}=\left(\frac{\partial \mu_{j}}{\partial p}\right)_{T}\]We simplify the argument by considering a system comprising pure chemical substance 1. \[\text { Then } \quad \mathrm{V}_{1}^{*}=\left(\frac{\partial \mu_{1}^{*}}{\partial \mathrm{p}}\right)_{\mathrm{T}}\]Thus \(\mathrm{V}_{1}^{*}(\mathrm{~T}, \mathrm{p})\) is the molar volume of pure substance 1 at temperature \(\mathrm{T}\) and pressure \(p\). In the event that chemical substance 1 is a perfect (ideal) gas, the following equation describes the \(p-\mathrm{V}-\mathrm{T}\) properties. \[p_{1}^{*} \, V_{1}^{*}(g)=R \, T\]We write equation (h) in the following form describing an ideal gas at constant temperature \(\mathrm{T}\). \[d \mu_{1}^{*}(g)=V_{1}^{*}(g) \, d p\]Equations (i) and (j) yield equation (k). \[\mathrm{d} \mu_{1}^{*}(\mathrm{~g})=\mathrm{R} \, \mathrm{T} \, \mathrm{d} \ln \mathrm{p}_{1}^{*}\]We integrate equation (k) between limits \(p_{1}^{*}\) and \(p^{0}\) where \(p^{0}\) is the standard pressure, \(101325 \mathrm{~N m}^{-2}\). \[\text { Hence, at temperature } \mathrm{T}, \mu_{1}^{*}\left(\mathrm{~g} ; \mathrm{T} ; \mathrm{p}_{1}^{*}\right)=\mu_{1}^{*}\left(\mathrm{~g} ; \mathrm{T} ; \mathrm{p}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{p}_{1}^{*} / \mathrm{p}^{0}\right)\]In a more complicated system, the gas phase is a gaseous mixture, comprising two components, component 1 and component 2 with partial pressures \(\mathrm{p}_{1}\) and \(\mathrm{p}_{2}\). We assume the thermodynamic properties of the gas phase in equilibrium with a liquid phase are ideal. Hence equation (l) takes the following form where \(\mu_{1}\left(g ; \text { mix; } p_{1}\right)\) is the chemical potential of gas-1 at partial pressure \(\mathrm{p}_{1}\). \[\mu_{1}^{\mathrm{eq}}\left(\mathrm{g} ; \mathrm{mix} ; \mathrm{id} ; \mathrm{T} ; \mathrm{p}_{1}\right)=\mu_{1}^{*}\left(\mathrm{~g} ; \mathrm{T} ; \mathrm{p}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{p}_{1}^{\mathrm{eq}} / \mathrm{p}^{0}\right)\]A given closed system contains chemical substances 1 and 2, present in two phases, gas and a liquid mixture at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\). Thus \(\mathrm{p}_{1}^{\text {eq }}\) is the equilibrium partial pressure of chemical substance 1 in the gas phase. At equilibrium the chemical potentials of chemical substance 1 in the vapour and liquid mixture phases are equal. \[\mu_{1}^{\mathrm{eq}}(\ell ; \operatorname{mix} ; \mathrm{id} ; \mathrm{p} ; \mathrm{T})=\mu_{1}^{\mathrm{eq}}\left(\mathrm{g} ; \mathrm{mix} ; \mathrm{T} ; \mathrm{p}_{1}^{\mathrm{eq}}\right)\]Thus \(\mathrm{p}_{1}^{\text {eq }}\) is the partial pressure of chemical substance 1 in the gas phase, the superscript ‘eq’ indicating an equilibrium with the liquid phase at pressure \(\mathrm{p}\); the complete system is at temperature \(\mathrm{T}\).Hence using equations (m) and (n) we obtain an equation for the equilibrium chemical potential of chemical substance 1 in an ideal liquid mixture at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\) \[\mu_{1}^{\text {eq }}(\ell ; \operatorname{mix} ; \mathrm{id} ; \mathrm{p} ; \mathrm{T})=\mu_{1}^{*}\left(\mathrm{~g} ; \mathrm{p}^{0} ; \mathrm{T}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{p}_{1}^{\mathrm{eq}} / \mathrm{p}^{0}\right)\]The thermodynamic analysis calls on the results of experiments in which the partial pressure \(\mathrm{p}_{i}\) of chemical substance-\(i\) in a liquid mixture at temperature \(\mathrm{T}\) is measured as a function of mole fraction \(\mathrm{x}_{i}\). It turns out that for nearly all liquid mixtures at fixed temperature, \(\mathrm{p}_{i}\) is approximately a linear function of the mole fraction \(\mathrm{x}_{1}\) at low \(\mathrm{x}_{1}\). We therefore define an ideal liquid mixture. By definition the (equilibrium) vapour pressure of chemical substance \(i\), one component of a liquid mixture, is related to the mole fraction composition at temperature \(\mathrm{T}\) using equation (p). \[\text { Thus } \mathrm{p}_{\mathrm{i}}^{\mathrm{eq}}(\mathrm{T} ; \text { mix } ; \mathrm{id})=\mathrm{x}_{\mathrm{i}} \, \mathrm{p}_{\mathrm{i}}^{*}(\ell ; \mathrm{T})\]Here \(\mathrm{x}_{i}\) is the mole fraction of component-\(i\) in the liquid mixture; \(\mathrm{p}_{\mathrm{i}}^{*}(\ell ; \mathrm{T})\) is the vapour pressure of pure liquid substance 1 at temperature \(\mathrm{T}\).For example if \(\mathrm{x}_{i}\) is \(0.5\), the contribution to the vapour pressure of the (ideal) mixture is one-half of the vapour pressure of the pure liquid-\(i\) at the same temperature. Equation (p) is Raoult’s law, describing the properties of an ideal liquid mixture having ideal thermodynamic properties. We note that the Gas Constant emerges in equation (o) because the r.h.s. of equation (o) describes the properties of chemical substance 1 in the vapour phase.Combination of equations (o) and (p) yields equation (q). \[\mu_{\mathrm{i}}^{\mathrm{eq}}(\ell ; \mathrm{mix} ; \mathrm{id} ; \mathrm{T} ; \mathrm{p})=\mu_{\mathrm{i}}^{*}\left(\mathrm{~g} ; \mathrm{T} ; \mathrm{p}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{x}_{\mathrm{i}} \, \mathrm{p}_{\mathrm{i}}^{*}(\mathrm{~T}) / \mathrm{p}^{0}\right]\]\[\text { Or, } \mu_{\mathrm{i}}^{\mathrm{eq}}(\ell ; \mathrm{mix} ; \mathrm{id} ; \mathrm{p} ; \mathrm{T})=\mu_{\mathrm{i}}^{*}\left(\mathrm{~g} ; \mathrm{p}^{0} ; \mathrm{T}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{p}_{\mathrm{i}}^{*}(\mathrm{~T}) / \mathrm{p}^{0}\right]+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{\mathrm{i}}\right)\]For the pure liquid-\(i\) at pressure \(\mathrm{p}\), \[\mu_{\mathrm{i}}^{*}(\ell ; \mathrm{p} ; \mathrm{T})=\mu_{\mathrm{i}}^{*}\left(\mathrm{~g} ; \mathrm{p}^{0} ; \mathrm{T}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{p}(\mathrm{T}) / \mathrm{p}^{0}\right]\]\[\text { Hence, } \mu_{\mathrm{i}}^{\mathrm{eq}}(\ell ; \text { mix } ; \mathrm{id} ; \mathrm{p} ; \mathrm{T})=\mu_{\mathrm{i}}^{*}(\ell ; \mathrm{p} ; \mathrm{T})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{\mathrm{i}}\right)\]We notice that the Gas Constant in equation (t) emerged from equation (i) describing the properties of an ideal gas.A similar argument is used when we turn our attention to the thermodynamic properties of a solute, chemical substance \(j\). In this case we use Henry’s Law as the link between theory and the properties of solutions. This law relates the equilibrium partial pressure \(\mathrm{p}_{j}\) of solute \(j\) to the molality of solute \(j\), \(\mathrm{m}_{j}|) for a solution at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\). Experiment shows that certainly for dilute solutions, the partial pressure \(\mathrm{p}_{j}\) is close to a linear function of molality \(\mathrm{m}_{j}\). Taking this experimental result as a lead we state that, by definition, in the event that the thermodynamic properties of the solution are ideal, equation (u) relates the partial pressure \(\mathrm{p}_{j}\) to the solute molality \(\mathrm{m}_{j}\); \(\mathrm{m}^{0}=1 \mathrm{~mol} \mathrm{} \mathrm{kg}^{-1}\). \[\text { Thus, } \mathrm{p}_{\mathrm{j}}\left(\mathrm{s} \ln ; \mathrm{T} ; \mathrm{m}_{\mathrm{j}} ; \mathrm{id}\right)=\mathrm{H}_{\mathrm{j}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)\]Here \(\mathrm{H}_{j}\) is Henry’s Law constant characteristic of solute, solvent, \(\mathrm{T}\) and \(\mathrm{p}\). \(\mathrm{H}_{j}\) is a pressure being the partial pressure of solute \(j\) where \(\mathrm{m}_{\mathrm{j}}=1 \mathrm{~mol} \mathrm{~kg}\). In other words equation (u) is not thermodynamic in the sense of being derived from the Laws of Thermodynamics. Rather the basis is experiment. We return to equation (n) but written for the equilibrium for solute in solution and in the vapour phase, a mixture of solute \(j\) and solvent. \[\mu_{j}^{\mathrm{eq}}\left(\mathrm{s} \ln ; \mathrm{m}_{\mathrm{j}} ; \mathrm{T} ; \mathrm{p}\right)=\mu_{\mathrm{j}}^{\mathrm{eq}}\left(\mathrm{g} ; \operatorname{mix} ; \mathrm{T} ; \mathrm{p}_{\mathrm{j}}^{\mathrm{eq}}\right)\]For the vapour phase, \(\mu_{j}^{c q}\left(g ; \operatorname{mix} ; T ; \mathrm{p}_{\mathrm{j}}^{\mathrm{cq}}\right)\) is related to the partial pressure \(\mathrm{p}_{\mathrm{j}}^{\mathrm{cq}}\) using equation (w). \[\mu_{\mathrm{j}}^{\mathrm{eq}}\left(\mathrm{g} ; \mathrm{mix} ; \mathrm{T} ; \mathrm{p}_{\mathrm{j}}^{\mathrm{eq}}\right)=\mu_{\mathrm{j}}^{0}\left(\mathrm{~g} ; \mathrm{T} ; \mathrm{p}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{p}_{\mathrm{j}}^{\mathrm{eq}} / \mathrm{p}^{0}\right)\]Hence using equations (u)-(w), \[\mu_{\mathrm{j}}^{\mathrm{cq}}\left(\mathrm{s} \ln ; \mathrm{m}_{\mathrm{j}} ; \mathrm{T} ; \mathrm{p}\right)=\mu_{\mathrm{j}}^{0}\left(\mathrm{~g} ; \mathrm{T} ; \mathrm{p}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left[\frac{\mathrm{H}_{\mathrm{j}}}{\mathrm{p}^{0}} \, \frac{\mathrm{m}_{\mathrm{j}}}{\mathrm{m}^{0}}\right]\]\[\text { Or, } \mu_{\mathrm{j}}^{\mathrm{eq}}\left(\mathrm{s} \ln ; \mathrm{m}_{\mathrm{j}} ; \mathrm{T} ; \mathrm{p}\right)=\left\{\mu_{\mathrm{j}}^{0}\left(\mathrm{~g} ; \mathrm{T} ; \mathrm{p}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left[\frac{\mathrm{H}_{\mathrm{j}}}{\mathrm{p}^{0}}\right]\right\}+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)\]The term \(\left\{\mu_{j}^{0}\left(g ; T ; p^{0}\right)+R \, T \, \ln \left[\frac{H_{j}}{p^{0}}\right]\right\}\) characterises solute \(j\) in a solution at the same \(\mathrm{T}\) and \(\mathrm{p}\) when \(\mathrm{m}_{\mathrm{j}}=1 \mathrm{~mol} \mathrm{} \mathrm{kg}^{-1}\). Thus we define a reference chemical potential for the solute-\(j\), \[\mu_{\mathrm{j}}^{0}(\mathrm{~s} \ln ; \mathrm{T} ; \mathrm{p}) \text { as given by }\left\{\mu_{\mathrm{j}}^{0}(\mathrm{~g} ; \mathrm{T} ; \mathrm{p})+\mathrm{R} \, \mathrm{T} \, \ln \left[\frac{\mathrm{H}_{\mathrm{j}}}{\mathrm{m}_{\mathrm{j}}^{0}}\right]\right\}\]\[\text { Therefore, } \mu_{\mathrm{j}}^{\mathrm{eq}}\left(\mathrm{s} \ln ; \mathrm{m}_{\mathrm{j}} ; \mathrm{T} ; \mathrm{p}\right)=\mu_{\mathrm{j}}^{0}(\mathrm{~s} \ln ; \mathrm{T} ; \mathrm{p})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)\]Again we can trace the gas constant \(\mathrm{R}\) in equation (za) to a description of the vapour state although the term \(\mu_{j}^{\mathrm{cq}}\left(\mathrm{s} \ln ; \mathrm{m}_{\mathrm{j}} ; \mathrm{T} ; \mathrm{p}\right)\) describes the chemical potential of chemical substance \(j\), the solute, in solution.Finally we should note that for real as opposed to ideal liquid mixtures and ideal solutions, activity coefficients express the extent to which the properties of these systems differ from those defined as ideal.This page titled 1.5.1: Chemical Potentials, Composition and the Gas Constant is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.5.2: Chemical Potentials- Gases

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A given closed system contains gas \(j\) at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\). The chemical potential \(\mu_{\mathrm{j}}(\mathrm{g} ; \mathrm{T} ; \mathrm{p})\) is given by Equation \ref{a} where \(\mathrm{p}^{0}\) is the standard pressure and \(\mathrm{V}_{\mathrm{j}}^{*}(\mathrm{~T}, \mathrm{p})\) is the molar volume of the gas \(j\).\[\mu_{j}(g ; T ; p)= \mu_{j}^{0}(p f g ; T)+ R T \ln \left(\dfrac{p}{p^{0}}\right)+\int_{0}^{p}\left[V_{j}^{*}(T ; p)-\left(\dfrac{R T}{p}\right)\right] d p \label{a}\]\(\mathrm{V}_{\mathrm{j}}^{*}(\mathrm{~T} ; \mathrm{p})\) is the molar volume at pressure \(\mathrm{p}\) and temperature \(\mathrm{T}\). In the event that gas \(j\) has the properties of a perfect gas, the chemical potential is given by Equation \ref{b}.\[\mu_{j}(p f g ; T ; p)=\mu_{j}^{0}(p f g ; T)+R T \ln \left( \dfrac{p}{p^{0}} \right) \label{b}\]If gas \(j\) exists at mole fraction \(\mathrm{x}_{j}\) as one component of a mixture of \(\mathrm{k}\) gases the chemical potential of gas \(j\) is given by Equation \ref{c} where \(\mathrm{x}_{\mathrm{k}}\) is the set of mole fractions defining the composition of the mixture.\[ \mu_{\mathrm{j}}\left(\mathrm{g} ; \mathrm{T} ; \mathrm{p} ; \mathrm{x}_{\mathrm{k}}\right)= \mu_{\mathrm{j}}^{0}(\mathrm{~g} ; \mathrm{T})+\mathrm{R} \mathrm{T} \ln \left(\mathrm{x}_{\mathrm{j}} \mathrm{p} / \mathrm{p}^{0}\right) +\int_{\mathrm{o}}^{\mathrm{p}}\left[\mathrm{V}_{\mathrm{j}}\left(\mathrm{g} ; \mathrm{T} ; \mathrm{p} ; \mathrm{x}_{\mathrm{c}}\right)-\left(\dfrac{RT}{p}\right)\right] \mathrm{dp} \label{c}\]Here \(\mathrm{V}_{\mathrm{j}}\left(\mathrm{g} ; \mathrm{T} ; \mathrm{p} ; \mathrm{x}_{\mathrm{c}}\right)\) is the molar volume of gas \(j\) in the gaseous mixture. M. L. McGlashan, Chemical Thermodynamics, Academic Press, London, 1979, page 184.This page titled 1.5.2: Chemical Potentials- Gases is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.5.3: Chemical Potentials- Solutions- General Properties

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.05%3A_Chemical_Potentials/1.5.03%3A_Chemical_Potentials-_Solutions-_General_Properties

A key quantity in chemical thermodynamics is the chemical potential of chemical substance \(j\), \(\mu_{j}\). The latter is the differential dependence of Gibbs energy on amount of substance \(j\) at fixed \(\mathrm{T}\), \(\mathrm{p}\) and amounts of all other substances in the system.\[\mu_{\mathrm{j}}=\left(\frac{\partial \mathrm{G}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{i} \neq \mathrm{j})}\]An important point to note is that the conditions ‘fixed \(T\) and fixed \(p\)’ on the partial differential refer to intensive variables. These conditions are called Gibbsian in recognition of the development by Gibbs of the concept of thermodynamic potential for changes in the properties of a closed system at fixed \(\mathrm{T}\) and fixed \(\mathrm{p}\).In general terms, the chemical potential of substance \(j\) is defined using analogous partial derivatives of the thermodynamic internal energy \(\mathrm{U}\), enthalpy \(\mathrm{H}\) and Helmholtz energy \(\mathrm{F}\). \[\mu_{\mathrm{j}}=\left(\frac{\partial \mathrm{G}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{i} \neq \mathrm{j})}=\left(\frac{\partial \mathrm{U}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{s}, \mathrm{V}, \mathrm{n}(\mathrm{i} \neq \mathrm{j})}=\left(\frac{\partial \mathrm{H}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{s}, \mathrm{p}, \mathrm{n}(\mathrm{i} \neq \mathrm{j})}=\left(\frac{\partial \mathrm{F}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{V}, \mathrm{n}(\mathrm{i} \neq \mathrm{j})}\]With reference to a given closed system, thermodynamics defines macroscopic properties including volume \(\mathrm{V}\), Gibbs energy \(\mathrm{G}\), enthalpy \(\mathrm{H}\) and entropy \(\mathrm{S}\). Nevertheless we need to “tell” these thermodynamic variables that a given system probably comprises different chemical substances. The analysis is reasonably straightforward if we define the system under consideration by the ‘Gibbsian’ set of independent variables; i.e. \(\mathrm{T}\), \(\mathrm{p}\) and amounts of each chemical substance. The analysis leads to the definition of a chemical potential for each substance \(j\), \(\mu_{j}\), in a closed system. It might be argued that we have switched our attention from closed to open systems because we are considering a change in Gibbs energy when we add \(\partial n_{j}\) moles of substance to the system. This comment is true in part. But what we actually envisage is something a little different. We take a closed system containing \(n_{1}\) and \(n_{j}\) moles of substances \(1\) and \(j\) respectively. We open the system, rapidly pop in \(\delta \mathrm{n}_{\mathrm{j}}\) moles of substance \(j\) and put the lid back on the system to return it to the closed state. Then the closed system contains \(\left(\mathrm{n}_{\mathrm{j}}+\delta \mathrm{n}_{\mathrm{j}}\right)\) moles of substance j so changes in chemical composition and molecular organisation follow producing a change in Gibbs energy at, say, fixed \(\mathrm{T}\) and fixed \(\mathrm{p}\). G. N. Lewis, (with possibly one of the key papers in chemistry)[Comment: Paper (a) is the German translation of paper (b).] The analysis presented here (a) is confined to bulk systems in the absence of magnetic and electric fields and (b) ignores surface effects. To quote E. Grunwald [J. Am. Chem. Soc., 1984, 106, 5414] “any first derivative with respect to any variable of state at equilibrium is isodelphic”; see also E. Grunwald, Thermodynamics of Molecular Species, Wiley, New York, 1997.This page titled 1.5.3: Chemical Potentials- Solutions- General Properties is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.5.4: Chemical Potentials- Solutions- Composition

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A given aqueous solution at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\) (both near ambient) was prepared using \(\mathrm{n}_{1}\) moles of water and \(\mathrm{n}_{j}\) moles of urea (i.e. chemical substance \(j\)). The Gibbs energy \(\mathrm{G}(\mathrm{aq})\), an extensive property (variable), is given by the sum of products of amounts of each chemical substance and chemical potentials.\[\mathrm{G}(\mathrm{aq})=\mathrm{n}_{1} \, \mu_{1}(\mathrm{aq})+\mathrm{n}_{\mathrm{j}} \, \mu_{\mathrm{j}}(\mathrm{aq}) \label{a}\]Equation \ref{a} is key although we cannot put number values to \(\mathrm{G}(\mathrm{aq})\), \(\mu_{1}(\mathrm{aq})\) and \(\mu_{j}(\mathrm{aq}\). The latter two quantities are, respectively, the chemical potentials of the solvent, water and solute \(j\) in the aqueous solution at the same temperature and pressure. Equation \ref{a} seems a strange starting point granted it contains three quantities which we can never know. Matters can only improve.There is merit in turning attention to an intensive property describing the Gibbs energy of a solution prepared using \(1 \mathrm{~kg}\) of solvent, \(\mathrm{G}\left(\mathrm{aq} ; \mathrm{w}_{1}=1.0 \mathrm{~kg}\right)\). Therefore, we do not have to worry about the size of the flask containing the solution. The same descriptor applies to \(0.1 \mathrm{~cm}^{3}\) or \(10 \mathrm{m}^{3}\) of a given solution. \[\text { By definition } \quad \mathrm{G}\left(\mathrm{aq} ; \mathrm{w}_{1}=1.0 \mathrm{~kg}\right)=\mathrm{G}(\mathrm{aq}) / \mathrm{w}_{1}\]\[\mathrm{G}\left(\mathrm{aq} ; \mathrm{w}_{1}=1.0 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mu_{1}(\mathrm{aq})+\mathrm{m}_{\mathrm{j}} \, \mu_{\mathrm{j}}(\mathrm{aq})\]\(\mathrm{M}_{1}\) is the molar mass of solvent, water, and mj is the molality of solute \(j\). Again we cannot put number values to \(\mathrm{G}\left(\mathrm{aq} ; \mathrm{w}_{1}=1.0 \mathrm{~kg}\right)\), \(\mu_{1}(\mathrm{aq})\) and \(\mu_{\mathrm{j}}(\mathrm{aq})\). Faced with this situation, the well-established approach involves an examination of differences. With respect to \(\mu_{1}(a q)\), the properties of water in an aqueous solution are compared with the properties of water at the same temperature and pressure. In these terms, we compare \(\mu_{1}(\mathrm{aq}, \mathrm{T}, \mathrm{p})\) with \(\mu_{1}^{*}(\ell, T, p)\). The superscript * in the latter term indicates that the chemical substance is pure and the symbol '\(\ell\)' indicates that this substance is a liquid. Hence, comparison is drawn with the chemical potential of pure liquid water at the same \(\mathrm{T}\) and \(\mathrm{p}\). In one sense we regard the solute as a controlled impurity perturbing the properties of the solvent. [We use the subscript '1' to indicate chemical substance 1 which in the convention used here refers to the solvent; water in the case of aqueous solutions.]In considering the properties of, for example, urea in this aqueous solution molality \(\mathrm{m}_{j}\), we need a reference state against which to compare the properties of urea in the real solution prepared by dissolving \(\mathrm{n}_{j}\) moles of urea in \(\mathrm{n}_{1}\) moles of water. There is little point in comparing the properties of solute, urea with those of solid urea, a hard crystalline solid. Instead, we identify a reference solution state.In general terms chemists explore how the chemical potentials of solvent and solute in an aqueous solution are related to the composition of the solution. Equations which offer such relationships should satisfy two criteria: in the limit of infinite dilution (i) the partial molar volumes \(\mathrm{V}_{\mathrm{j}}(\mathrm{aq})\) and \(\mathrm{V}_{1}(\mathrm{aq})\) are meaningful and (ii) the partial molar enthalpies \(\mathrm{H}_{\mathrm{j}}(\mathrm{aq}\) and \(\mathrm{H}_{1}(\mathrm{aq})\) are meaningful. In other words, these properties do not approach an asymptotic limit of either \(+ \infty\) or \(– \infty\) with increasing dilution. For this reason physical chemists usually favour expressing the composition of solutions in molalities.In summary analysis of the properties of solutions and liquid mixtures is built around the somewhat abstract concept of the chemical potential introduced by J. Willard Gibbs and by Pierre Duhem. The task of showing chemists the significance and application of this concept was left to Lewis and Randall in their classic monograph published in 1923. Chemical potentials are one example of a class of properties called partial molar which provide the key link between macroscopic thermodynamic descriptions of systems and molecular properties. \(\mathrm{G}(\mathrm{aq})=[\mathrm{mol}] \,\left[\mathrm{J} \mathrm{mol}{ }^{-1}\right]+[\mathrm{mol}] \,\left[\mathrm{J} \mathrm{mol}^{-1}\right]=[\mathrm{J}]\) \(\begin{aligned} &\mathrm{G}\left(\mathrm{aq} ; \mathrm{w}_{1}=1.0 \mathrm{~kg}\right)= \\ &\quad\left[1 / \mathrm{kg} \mathrm{mol}^{-1}\right] \,\left[\mathrm{J} \mathrm{mol}^{-1}\right]+\left[\mathrm{mol} \mathrm{kg}^{-1}\right] \,\left[\mathrm{J} \mathrm{mol}^{-1}\right]=\left[\mathrm{J} \mathrm{kg}^{-1}\right] \end{aligned}\) With reference to equations (a) and (c), we must avoid the temptation to write “at constant temperature and pressure”. This condition is implicit in the description of the system using the independent variables \(\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}\right]\) for the aqueous solution containing the solute with the added condition that \(\mathrm{T}\) and \(\mathrm{p}\) are intensive variables; i.e. the set of independent variables is Gibbsian. Nonetheless there is often merit in using a complete set of descriptions of a system even if we over-define the variable under discussion. In describing the Gibbs energy defined by equation (a), we might write \(\mathrm{G}(\mathrm{T} ; \mathrm{p} ; \mathrm{aq})\). Similarly for the system described by equation (b) it is often helpful to write \(\mathrm{G}\left(\mathrm{T} ; \mathrm{p} ; \mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg} ; \mathrm{m}_{\mathrm{j}}\right)\). In reviewing the properties of solutions our interest, unless otherwise stated, centres on solutions at equilibrium where the affinity \(\mathrm{A}\) is zero and the organisation characteristic of the equilibrium system, \(\xi^{e q}\). We may find it helpful to write \(\mathrm{G}\left(\mathrm{T} ; \mathrm{p} ; \mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg} ; \mathrm{m}_{\mathrm{j}} ; \mathrm{A}=0 ; \xi^{\mathrm{eq}}\right)\), replacing \(\mathrm{G}\) by \(\mathrm{H}\), \(\mathrm{S}\) and \(\mathrm{V}\) for the corresponding enthalpy, entropy and volume of this solution. Clearly this over-definition is somewhat silly. Nevertheless it is often preferable to over-define a system rather than under-define when mistakes can arise. J. E. Garrod and T. M. Herrington, J. Chem. Educ., 1969, 46, 165. G. N. Lewis and M. L. Randall, Thermodynamics and the Free Energy of Chemical Substances, McGraw-Hill, New York, 1923. See also, G. N. Lewis, Proc. Am. Acad. Arts Sci.,1907,43,259. L. Hepler, Thermochim. Acta, 1986, 100, 171.This page titled 1.5.4: Chemical Potentials- Solutions- Composition is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.5.5: Chemical Potentials- Solutions- Partial Molar Properties

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.05%3A_Chemical_Potentials/1.5.05%3A_Chemical_Potentials-_Solutions-_Partial_Molar_Properties

A given solution comprises \(\mathrm{n}_{1}\) moles of solvent, liquid chemical substance 1, and \(\mathrm{n}_{\mathrm{j}}\) moles of solute, chemical substance \(\mathrm{j}\). We ask ---- What contributions are made by the solvent and by the solute to the volume of the solution at defined \(\mathrm{T}\) and \(\mathrm{p}\)? In fact we can only guess at these contributions. This is disappointing. The best that we can do is to probe the sensitivity of the volume of a given solution to the addition of small amounts of solute and of solvent. This approach leads to a set of properties called partial molar. The starting point is the Gibbs energy of a solution. We develop an argument which places the Gibbs energy at the centre from which all other thermodynamic variables develop.A given closed system comprises \(\mathrm{n}_{1}\) moles of solvent (e.g. water) and \(\mathrm{n}_{\mathrm{j}}\) moles of a simple solute j (e.g. urea) at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\). The Gibbs energy of the solution is defined by equation (a).\[\mathrm{G}=\mathrm{G}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{\mathrm{l}}, \mathrm{n}_{\mathrm{j}}\right] \label{a}\]We introduce a partial derivative having the following form: \(\left(\frac{\partial G}{\partial n_{j}}\right)_{T, p, n_{1}}\). The latter partial differential describes the differential dependence of Gibbs energy \(\mathrm{G}\) on the amount of chemical substance \(\mathrm{j}\). By definition, the chemical potential of chemical substance \(\mathrm{j}\),\[\mu_{\mathrm{j}}=\left(\frac{\partial \mathrm{G}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}_{\mathrm{l}}}\]We also envisage that displacement of the system by adding \(\delta n_{j}\) moles of chemical substance \(\mathrm{j}\) from the original state to a neighbouring state produces a change in Gibbs energy at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\). In one class of displacements the system moves along a path of constant affinity for spontaneous reaction \(\mathrm{A}\). In another displacement the system moves along a path at constant organisation/composition, \(\xi\); i.e. frozen. These two pathways are related by the following equation. For the system at fixed \(\mathrm{T}\), \(\mathrm{p}\) and \(\mathrm{n}_{1}\)\[\left[\frac{\partial \mathrm{G}}{\partial \mathrm{n}_{\mathrm{j}}}\right]_{\mathrm{A}}=\left[\frac{\partial \mathrm{G}}{\partial \mathrm{n}_{\mathrm{j}}}\right]_{\xi}-\left[\frac{\partial \mathrm{A}}{\partial \mathrm{n}_{\mathrm{j}}}\right]_{\xi} \,\left[\frac{\partial \xi}{\partial \mathrm{A}}\right]_{\mathrm{n}(\mathrm{j})} \,\left[\frac{\partial \mathrm{G}}{\partial \xi}\right]_{\mathrm{n}(\mathrm{j})}\]The conditions, constant \(\mathrm{T}\) and \(\mathrm{p}\), refer to intensive variables. We direct attention to a closed system at equilibrium where ‘\(\mathrm{A} = 0\)’ and the composition \(\xi=\xi^{\mathrm{eq}}\). Moreover at equilibrium, \((\partial \mathrm{G} / \partial \xi)_{\mathrm{T}, \mathrm{p}}\) is zero. Therefore the chemical potential of chemical substance \(\mathrm{j}\) in a system at equilibrium is defined by the following equation. Hence from equation (c),\[\mu_{\mathrm{j}}=\left(\frac{\partial \mathrm{G}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{~A}=0}=\left(\frac{\partial \mathrm{G}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}_{1},, \mathrm{G}^{c_{q}}}\]A similar argument in the context of chemical substance 1 shows that,\[\mu_{1}=\left(\frac{\partial \mathrm{G}}{\partial \mathrm{n}_{1}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}_{\mathrm{j}}, \mathrm{A}=0}=\left(\frac{\partial \mathrm{G}}{\partial \mathrm{n}_{1}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}_{\mathrm{j}}, \zeta^{\mathrm{eq}}}\]Equations (d) and (e) are key results. Similarly for a closed system at equilibrium at fixed \(\mathrm{T}\) and fixed \(\mathrm{p}\) (at a minimum in \(\mathrm{G}\), \(\mathrm{A} = 0\), \(\xi=\xi^{\mathrm{eq}}\) ), for all \(i\)-substances,\[V_{j}(A=0)=V_{j}\left(\xi^{e q}\right)\]\[\mathrm{S}_{\mathrm{j}}(\mathrm{A}=0)=\mathrm{S}_{\mathrm{j}}\left(\xi^{\mathrm{eq}}\right)\]\[\mathrm{H}_{\mathrm{j}}(\mathrm{A}=0)=\mathrm{H}_{\mathrm{j}}\left(\xi^{e q}\right)\]\[\mu_{j}(A=0)=\mu_{j}\left(\xi^{e q}\right)\]But in the case of, for example, isobaric expansions and isobaric heat capacities, \(\mathrm{E}_{\mathrm{pj}}(\mathrm{A}=0) \neq \mathrm{E}_{\mathrm{pj}}\left(\xi^{\mathrm{eq}}\right)\) and \(\mathrm{C}_{\mathrm{pj}}(\mathrm{A}=0) \neq \mathrm{C}_{\mathrm{pj}}\left(\xi^{e q}\right)\). The identifications, (f) to (i), arise because these variables are first derivatives of the Gibbs energy of a closed system at equilibrium where \((\partial \mathrm{G} / \partial \xi)\) at fixed \(\mathrm{T}\) and \(\mathrm{p}\) is zero.This page titled 1.5.5: Chemical Potentials- Solutions- Partial Molar Properties is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.5.6: Chemical Potentials- Liquid Mixtures- Raoult's Law

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.05%3A_Chemical_Potentials/1.5.06%3A_Chemical_Potentials-_Liquid_Mixtures-_Raoult's_Law

A given closed system contains two volatile miscible liquids. The closed system is connected to a pressure-measuring device which records that at temperature T the pressure is ptot. The composition of the liquid mixture is known; i.e. mole fractions \(\mathrm{x}_{1}\) and \(\mathrm{x}_{2}\) (where \(\mathrm{x}_{2} = 1 - \mathrm{x}_{1}\)). The system contains two components so that in terms of the Phase Rule, \(\mathrm{C} = 2\) There are two phases, vapour and liquid so that \(\mathrm{P} = 2\). From the rule, \(\mathrm{P} + \mathrm{F} = \mathrm{C} + 2\), we have fixed the composition and temperature using up the two degrees of freedom. Hence the pressure ptot is fixed.We imagine that the mixture under examination is a binary aqueous mixture; water is chemical substance 1. If we measure the partial pressure of, say, liquid 1, \(\mathrm{p}_{1}\) we find that \(\mathrm{p}_{1}\) is close to a linear function of mole fraction \(\mathrm{x}_{1}\). \[\text { At equilibrium and temperature } \mathrm{T}, \quad \mathrm{p}_{1}^{\mathrm{eq}} \cong \mathrm{p}_{1}^{*}(\ell) \, \mathrm{x}_{1}\]As the mole fraction \(\mathrm{x}_{1}\) approaches unity (i.e. the composition of the mixture approaches pure water) the equilibrium vapour pressure of water \(\mathrm{p}_{1}^{\mathrm{eq}}\) approaches that of pure liquid water at the same temperature, \(\mathrm{p}_{1}^{*}(\ell)\) We have linked the equilibrium vapour pressure of water to the composition of the liquid mixture.In fact it turns out that as the composition of the mixture approaches pure liquid 1, the latter relationship becomes an equation. We assert that if the thermodynamic properties of the mixture were ideal then \(\mathrm{p}_{1}\) would be related to mole fraction \(\mathrm{x}_{1}\) using the following equation. \[\mathrm{p}_{1}(\operatorname{mix} ; \mathrm{id})=\mathrm{x}_{1} \, \mathrm{p}_{1}^{*}(\ell)\]Returning to experiment, we invariably find that as a real solution becomes more dilute (i.e. as \(\mathrm{x}_{1}\) approaches unity) \(\mathrm{p}_{1}^{\mathrm{eq}}\) for real solutions approaches \(p_{1}^{e q}(a q ; i d)\). Therefore we rewrite equation (b) as an equation for a real solution by introducing a new property called the (rational) activity coefficient \(\mathrm{f}_{1}\). \[\mathrm{p}_{1}(\operatorname{mix})=\mathrm{x}_{1} \, \mathrm{f}_{1} \, \mathrm{p}_{1}^{*}(\ell)\]Here \(\mathrm{f}_{1}\) is the (rational) activity coefficient for liquid component 1 defined as follows. \[\operatorname{limit}\left(x_{1} \rightarrow 1\right) f_{1}=1\]\[\text { Similarly for volatile liquid } 2 ; p_{2}=x_{2} \, f_{2} \, p_{2}^{*}\]\[\operatorname{limit}\left(x_{2} \rightarrow 1\right) f_{2}=1\]Although equations (d) and (f) have simple forms, rational activity coefficients carry a heavy load in terms of information. For a given aqueous system, \(\mathrm{f}_{1}\) describes the extent to which interactions involving water molecules in a real system differ from those in the corresponding ideal system. The challenge of expressing this information in molecular terms is formidable.We carry over these ideas to the task of formulating an equation for the chemical potential of water in the liquid mixture at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\). We make the link between partial pressure and the tendency for liquid 1 to escape to the vapour phase, down a gradient of chemical potentialBy definition (at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\)), \[\mu_{1}(\operatorname{mix})=\mu_{1}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right)\]\[\text { where, at all } \mathrm{T} \text { and } \mathrm{p}, \quad \operatorname{limit}\left(\mathrm{x}_{1} \rightarrow 1.0\right) \mathrm{f}_{1}=1.0\]\(\mu_{1}^{*}(\ell)\) l is the chemical potential of pure liquid water (at the same \(\mathrm{T}\) and \(\mathrm{p}\)). In other words, pure liquid water is the reference state against which we compare the properties of water in an aqueous mixture. For the pure liquid at temperature \(\mathrm{T}\), \(V_{1}^{*}(\ell)=\mathrm{d} \mu_{1}^{*}(\ell) / \mathrm{dp}\). If \(\mathrm{p}^{0}\) is the standard pressure \(\left(10^{5} \mathrm{~N} \mathrm{~m}^{-2}\right)\), \[\int_{\mathrm{p}^{0}}^{\mathrm{p}} \mathrm{d} \mu_{1}^{*}(\ell)=\int_{\mathrm{p}^{0}}^{\mathrm{p}} \mathrm{V}_{1}^{*}(\ell) \, \mathrm{dp}\]\[\text { Then }, \quad \mu_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})-\mu_{1}^{0}(\ell ; \mathrm{T})=\int_{\mathrm{p}^{0}}^{\mathrm{p}} \mathrm{V}_{1}^{*}(\ell) \, \mathrm{dp}\]\(\mu_{1}^{0}(\ell ; \mathrm{T})\) is the standard chemical potential of water(\(\ell\)) at temperature \(\mathrm{T}\). \[\text { Therefore, } \quad \mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{1}^{0}(\ell ; \mathrm{T})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right)+\int_{\mathrm{p}^{0}}^{\mathrm{p}} \mathrm{V}_{1}^{*}(\ell) \, \mathrm{dp}\]This is an important equation although, at this stage, we can go no further. Without information concerning the dependence on pressure of \(\mathrm{V}_{1}^{*}(\ell)\) {or density} we cannot evaluate the integral in equation (k). However, we can comment on possible patterns in these chemical potentials. If the thermodynamic properties of the liquid mixture are ideal then \(\mathrm{f}_{1}\) equals \(1.0\). Hence equation (k) takes the following simple form (at fixed \(\mathrm{T}\) and \(\mathrm{p}\)). \[\mu_{1}(\operatorname{mix} ; \mathrm{id})=\mu_{1}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right)\]In a solution the mole fraction x1 is less than unity and so \(\ln \left(x_{1}\right)<0\). Hence \(\mu_{1}(\operatorname{mix} ; \mathrm{id})<\mu_{1}^{*}(\ell)\) at the same \(\mathrm{T}\) and \(\mathrm{p}\) J.D. Cox, Pure Appl. Chem., 1982,54, 1239; R.D. Freeman, Bull. Chem. Thermodyn., 1982,25, 523. \(\mathrm{V}_{1}^{*}(\ell) \, \mathrm{dp}=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right] \,\left[\mathrm{N} \mathrm{m}^{-2}\right]=\left[\mathrm{N} \mathrm{m} \mathrm{mol}^{-1}\right]=\left[\mathrm{J} \mathrm{mol}^{-1}\right]\) With increase in the amount of component 2 so \(\mathrm{x}_{1}\) tends to zero. In this limit \(\mu_{1}(\mathrm{aq})\) is minus infinity. Note that \(\mathrm{p}_{1}^{\mathrm{eq}}(\operatorname{mix} ; \mathrm{id})-\mathrm{p}_{1}^{*}<0\). Adding a solute lowers the vapour pressure of the water. However the total vapour pressure of a binary liquid mixture can be either increased or decreased by adding a small amount of the second component; G. Bertrand and C. Treiner, J. Solution Chem.,1984,13,43.This page titled 1.5.6: Chemical Potentials- Liquid Mixtures- Raoult's Law is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.5.7: Chemical Potentials- Solutions- Raoult's Law

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.05%3A_Chemical_Potentials/1.5.07%3A_Chemical_Potentials-_Solutions-_Raoult's_Law

A given closed system contains an aqueous solution; the solute is chemical substance \(j\). The system is at equilibrium at temperature \(\mathrm{T}\). The chemical potential of water in the aqueous solution is related to the mole fraction \(\mathrm{x}_{1}\) of water using equation (a) which is based on Raoult’s Law for the solvent. \[\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{1}^{0}(\ell ; \mathrm{T})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right)+\int_{\mathrm{p}^{0}}^{\mathrm{p}} \mathrm{V}_{1}^{*}(\ell) \, \mathrm{dp}\]\[\text { By definition, at all } \mathrm{T} \text { and } \mathrm{p}, \operatorname{limit}\left(\mathrm{x}_{1} \rightarrow 1\right) \mathrm{f}_{\mathrm{1}}=1\]If ambient pressure is close to the standard pressure \(\mathrm{p}^{0}\), the chemical potential of solvent water in the aqueous solution is given by equation (c). \[\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{1}^{0}(\ell ; \mathrm{T})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right)\]For an ideal solution, \(\mathrm{f}_{1} = 1\). \[\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p} ; \mathrm{id})=\mu_{1}^{0}(\ell ; \mathrm{T})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right)\]But for a solution \(\mathrm{x}_{1} < 1.0\) and so \(\ln \left(x_{1}\right)<0\). In other words, by adding a solute to water (forming an ideal solution) we stabilise the solvent. We define a quantity \(\Delta(\ell \rightarrow a q) \mu_{1}(\mathrm{~T}, \mathrm{p})\) using equation (e) which measures the change in chemical potential of water when one mole of water is transferred from water(\(\ell\)) to an ideal aqueous solution. \[\Delta(\ell \rightarrow \mathrm{aq}) \mu_{1}(\mathrm{~T}, \mathrm{p})=\mu_{1}(\mathrm{aq})-\mu_{1}^{*}(\ell)=\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right)\]i.e. \(\Delta(\ell \rightarrow \mathrm{aq}) \mu_{1}(\mathrm{~T}, \mathrm{p} ; \mathrm{id})<0\)In the case of a real solution, the extent of stabilisation depends on whether \(\mathrm{f}_{1}\) is either larger or smaller than unity. [Note that \(\mathrm{f}_{1}\) cannot be negative]. This line of argument leads to an important theme in the description of the properties of aqueous solutions. We compare the chemical potentials of water in real and in the corresponding ideal solutions. The difference is the excess chemical potential, \(\mu_{1}^{E}(a q ; T ; p)\). \[\text { By definition, } \quad \mu_{1}^{\mathrm{E}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})-\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p} ; \mathrm{id})\]\[\text { Hence } \quad \mu_{1}^{\mathrm{E}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{f}_{\mathrm{l}}\right)\]If \(\mathrm{f}_{1} > 1.0\), then \(\mu_{1}^{\mathrm{E}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})>0\); if \(\mathrm{f}_{1} < 1.0\), then \(\mu_{1}^{\mathrm{E}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})<0\). In the latter case, interactions involving solute and solvent are responsible for the fact that the properties of a given solution are not ideal and the fact that these interactions stabilise the solvent relative to that for an ideal solution.This page titled 1.5.7: Chemical Potentials- Solutions- Raoult's Law is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.5.8: Chemical Potentials- Solutions- Osmotic Coefficient

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The chemical potential of (solvent) water in an aqueous solution can be related to the mole fraction composition of the solution. However, there is a possible disadvantage in an approach using the mole fraction scale to express the composition of a solution. We note that our interest is often in the properties of solutes in aqueous solutions, that the amount of solvent greatly exceeds the amount of solute in a solution, and that the sensitivity of equipment developed by chemists is sufficient to probe the properties of quite dilute solutions. Consequently the mole fraction scale for the solvent is not the most convenient method for expressing the composition of a given solution. Hence another equation relating \(\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\) to the composition of a solution finds favour. By definition, for a solution containing a single solute, chemical substance \(j\),, \[\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\]In terms of the standard chemical potential for water at temperature \(|mathrm{T}\), \[\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\]\(\mathrm{M}_{1}\) is the molar mass of water; \(\phi\) is the practical osmotic coefficient which is characteristic of the solute, molality \(\mathrm{m}_{j}\), temperature and pressure. By definition, \(\phi\) is unity for ideal solutions at all temperatures and pressures. \[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi=1.0 \text { at all } \mathrm{T} \text { and } \mathrm{p}\]Further for ideal solutions, the partial differentials \((\partial \phi / \partial \mathrm{T})_{\mathrm{p}}\), \(\left(\partial^{2} \phi / \partial T^{2}\right)_{p}\) and \((\partial \phi / \partial \mathrm{p})_{\mathrm{T}}\) are zero. \[\text { For an ideal solution, } \quad \mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p} ; \mathrm{id})=\mu_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})-\mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\]We rewrite equation (d) in the following form: \[\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p} ; \mathrm{id})-\mu_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})=-\mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\]Hence with an increase in molality of solute in an ideal aqueous solution, the solvent is stabilised, being at a lower chemical potential than that for pure water(\(\ell\)).We contrast the chemical potentials of the solvent in real and ideal solutions using an excess chemical potential, \(\mu_{1}^{\mathrm{E}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\); \[\begin{aligned} \mu_{1}^{\mathrm{E}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})-\mu_{1}(\mathrm{aq} ; \mathrm{id} ; \mathrm{T} ; \mathrm{p}) \\ &=(1-\phi) \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \end{aligned}\]The term \((1-\phi)\) is often encountered because it expresses succinctly the impact of a solute on the properties of a solvent. At a given molality (and fixed temperature and pressure), \(\phi\) is characteristic of the solute.Footnote Mole fractions of solvent \(\mathrm{x}_{1}\) and solute \(\mathrm{x}_{j}\) for aqueous solutions having gradually increasing molality of solute \(\mathrm{m}_{j}\). \(\begin{aligned} &{\left[\mathrm{J} \mathrm{mol}^{-1}\right]=} \\ &{\left[\mathrm{J} \mathrm{mol}^{-1}\right]- \,\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}] \,\left[\mathrm{kg} \mathrm{mol}^{-1}\right] \,\left[\mathrm{mol} \mathrm{kg}^{-1}\right]} \end{aligned}\) The definitions of ideal solutions expressed here and in terms of mole fraction of solvent are not in conflict. For an ideal solution these equations require that, \(-\mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}=\ln \left(\mathrm{x}_{1}\right)\) But \(\ln \left(\mathrm{x}_{1}\right)=\ln \left[\mathrm{M}_{1}^{-1} /\left(\mathrm{M}_{1}^{-1}+\mathrm{m}_{\mathrm{j}}\right)\right]=-\ln \left(1.0+\mathrm{m}_{\mathrm{j}} \, \mathrm{M}_{1}\right)\) Bearing in mind that \(\mathrm{M}_{1}=0.018 \mathrm{~kg} \mathrm{~mol}^{-1}\), for dilute solutions \(\ln \left(1.0+\mathrm{m}_{\mathrm{j}} \, \mathrm{M}_{1}\right)=\mathrm{m}_{\mathrm{j}} \, \mathrm{M}_{1}\).This page titled 1.5.8: Chemical Potentials- Solutions- Osmotic Coefficient is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.5.9: Chemical Potentials; Excess; Aqueous Solution

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A given aqueous solution, at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\) (\(\cong \mathrm{p}^{0}\)), contains a solute, chemical substance \(j\). If the thermodynamic properties of the solution are ideal, the chemical potential of the solute is given by equation (a). \[\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{id})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)\]For the corresponding real solution, \[\mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)\]Here \(\gamma_{j}\) is the activity coefficient. The excess chemical potential, \(\mu_{\mathrm{j}}^{\mathrm{E}}(\mathrm{aq})\) is given by equation (c). \[\mu_{\mathrm{j}}^{\mathrm{E}}(\mathrm{aq})=\mu_{\mathrm{j}}(\mathrm{aq})-\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{id})\]\[\text { Then, } \mu_{j}^{E}(a q)=R \, T \, \ln \left(\gamma_{j}\right)\]Often an excess chemical potential \(\mu_{\mathrm{j}}^{\mathrm{E}}(\mathrm{aq})\) is written in the form \(\mathrm{G}_{j}^{\mathrm{E}}\). The latter notation stems from the fact that chemical potentials are partial molar Gibbs energies. In the case of the solvent, water(\(\ell\)), the corresponding equations for the chemical potentials in solutions having either ideal or real thermodynamic properties are given by equations (e) and (f). \[\mu_{1}(\mathrm{aq} ; \mathrm{id})=\mu_{1}^{*}(\ell)-\mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\]\[\mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\ell)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\]\[\mu_{1}^{\mathrm{E}}(\mathrm{aq})=(1-\phi) \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\]This page titled 1.5.9: Chemical Potentials; Excess; Aqueous Solution is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.5.10: Chemical Potentials- Solutions- Henry's Law

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A given aqueous solutions at temperature \(\mathrm{T}\) contains a simple solute \(j\), molality \(\mathrm{m}_{j}\). Experiment shows that at equilibrium the partial pressure \(\mathrm{p}_{j}\) is close to a linear function of molality \(\mathrm{m}_{j}\), the constant of proportionality being the Henry’s Law constant for this particular solute in a defined solvent; equation(a). \[\mathrm{p}_{\mathrm{j}} \cong \mathrm{H}_{\mathrm{j}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)\]By definition \(\mathrm{m}^{0}=1.0 \mathrm{~mol} \mathrm{~kg}{ }^{-1}\). Experiment shows that as a given real solution becomes more dilute so the relationship given in (a) can be written as an equation. The relationship in (a) is rewritten as an equation to describe the properties of a solution having thermodynamic properties which are ideal. \[p_{j}(\mathrm{id})=\mathrm{H}_{\mathrm{j}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)\]In other words \(\mathrm{H}_{j}\) is the partial pressure of volatile solute in a solution having thermodynamic properties which are ideal and where the molality of the solute equals \(1.0 \mathrm{~mol kg}^{-1}\).For a real solution at equilibrium and at temperature \(\mathrm{T}\), the partial pressure \(\mathrm{p}_{\mathrm{j}}(\text { real })\) is related to molality mj using equation (c) where \(\gamma_{j}\) is the activity coefficient describing the properties of solute \(j\) in solution \[\mathrm{p}_{\mathrm{j}}(\text { real })=\mathrm{H}_{\mathrm{j}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right) \, \gamma_{\mathrm{j}}\]\[\text { By definition, at all T and } p \lim \operatorname{it}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{\mathrm{j}}=1\]As \(\mathrm{m}_{j}\) decreases so \(\mathrm{p}_{\mathrm{j}}(\text { real })\) approaches \(\mathrm{p}_{\mathrm{j}}(\mathrm{id})\).Henry’s Law forms the basis of equations which are used to related the chemical potential of solute \(j\), \(\mu_{j}\) to the composition of a solution.This page titled 1.5.10: Chemical Potentials- Solutions- Henry's Law is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.5.11: Chemical Potentials- Solutes

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A given aqueous solution is prepared using \(1 \mathrm{~kg}\) of water at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\). The molality of solute \(j\) is \(\mathrm{m}_{j}\). The chemical potential of solute \(j\), \(\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\) is related to \(\mathrm{m}_{j}\) using equation (a). \[\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)+\int_{\mathrm{p}^{\mathrm{a}}}^{\mathrm{p}} \mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq} ; \mathrm{T}) \, \mathrm{dp}\]\[\text { By definition, } \operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{\mathrm{j}}=1.0\]\(\mu_{j}^{0}(\mathrm{aq} ; \mathrm{T})\) is the chemical potential of solute \(j\) in an ideal solution (where \(\gamma_{j} =1\)) at temperature \(\mathrm{T}\) and standard pressure \(\mathrm{p}^{0}\left[=10^{5} \mathrm{~Pa}\right]\).For solutions at ambient pressure which is close to \(\mathrm{p}^{0}\), \(\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\) is related to \(\mathrm{m}_{j}\) using equation (c). \[\mu_{j}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)\]Henry’s law forms the basis of equations (a) and (c).This page titled 1.5.11: Chemical Potentials- Solutes is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.5.12: Chemical Potentials- Solute; Molality Scale

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A given aqueous solution comprises \(\mathrm{n}_{1}\) moles of solvent (e.g. water) and \(\mathrm{n}_{j}\) moles of solute (e.g. urea) at equilibrium, temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\). Thus the molality of solute \(j\) is given by equation (a). \[\mathrm{m}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} / \mathrm{w}_{1}=\mathrm{n}_{\mathrm{j}} / \mathrm{n}_{1} \, \mathrm{M}_{1}\]\[\mu_{j}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{\mathrm{j}}^{0}\left(\mathrm{aq} ; \mathrm{T} ; \mathrm{p}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)+\int_{\mathrm{p}^{0}}^{\mathrm{p}} \mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq}) \, \mathrm{dp}\]Activity coefficient \(\gamma_{j}\) takes account of the fact that the thermodynamic properties of real solutions are not ideal. An important consideration in understanding the factors which affect \(\gamma_{j}\) is the distance between solute molecules in solution. As we dilute the solution such that \(\mathrm{m}_{j}\) approaches zero so inter solute distances approache infinity; i.e. in the limit of infinite dilution or zero molality.If pressure is ambient and hence close to the standard pressure the integral term in equation (b) is negligibly small. \[\text { Hence, } \mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)\]At this stage we focus attention on the activity coefficient \(\gamma_{j}\). We start with equation (c) and by split the logarithm term. For solutions where the pressure \(\mathrm{p}\) is close to the standard pressure \(\mathrm{p}^{0}\) \[\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\gamma_{\mathrm{j}}\right)\]If the properties of the solution are ideal, equation (c) takes the following form. \[\mu_{j}(\mathrm{aq} ; \mathrm{id} ; \mathrm{T} ; \mathrm{p})=\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)\]If \(\mathrm{m}_{\mathrm{j}}<1.0 \mathrm{~mol} \mathrm{~kg}{ }^{-1}, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)\) is \(<0\). \(\mu_{j}(\mathrm{aq} ; \mathrm{id} ; \mathrm{T} ; \mathrm{p})<\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T})\). Solute-\(j\) is stabilised relative to solute \(j\) in the solution reference state, an ideal solution having unit molality. If \(\mathrm{m}_{\mathrm{j}}>1.0 \mathrm{~mol} \mathrm{~kg}\), \(\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)\) is \(>\) zero. Hence, \(\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{id} ; \mathrm{T} ; \mathrm{p})>\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T})\); solute-\(j\) is destabilised relative to solute \(j\) in the (ideal) solution reference state.We also compare the chemical potentials of solute \(j\) in real and ideal solutions at the same molality leading to the definition of an excess chemical potential for solute \(j\), \(\mu_{\mathrm{j}}^{\mathrm{E}}(\mathrm{aq})\). \[\mu_{j}^{\mathrm{E}}(\mathrm{aq})=\mu_{j}(\mathrm{aq})-\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{id})\]\[\text { Hence, } \mu_{\mathrm{j}}^{\mathrm{E}}(\mathrm{aq})=\mathrm{R} \, \mathrm{T} \, \ln \left(\gamma_{\mathrm{j}}\right)\]\[\text { where } \lim \operatorname{it}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{\mathrm{j}}=1 \text { and } \ln \left(\gamma_{\mathrm{j}}\right)=0 \text { at all T and } \mathrm{p} \text {. }\]Equation (g) highlights the role played by activity coefficient \(\gamma_{j}\); \(\gamma_{j}\) can be neither zero nor negative; the range for \(\gamma_{j}\) is from below to above unity. In contrast \(\ln \left(\gamma_{j}\right)\) can be zero (as in an ideal solution) and be either greater or less than zero.Activity coefficients are interesting quantities. For a given solute \(j\) at molality \(\mathrm{m}_{j}\) in an aqueous solution (at fixed temperature and pressure) \(\gamma_{j}\) describes the impact on the chemical potential \(\mu_{j}(\mathrm{aq})\) of solute - solute interactions. The basis of this conclusion follows from the definition given in equation (h). As a solution is diluted, so the mean distance of separation of solute molecules increases. In these terms a model for an ideal solution, molality mj is one in which each solute molecule contributes to the properties of a given solution independently of all other solutes in the system. In an operational sense, each solute molecule in unaware of the presence of other solute molecules in solution and in these terms the solute molecules are infinitely far apart.We emphasise the point that activity coefficient \(\gamma_{j}\) is an interesting and important quantity; \(\gamma_{j}\) describes the impact on chemical potential \(\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\) of solute - solute interactions. These interactions can be cohesive (i.e. attractive) such that \(\gamma_{j}<1, \ln \left(\gamma_{j}\right)<0\) and \(\mu_{j}(\text { aq } ; \text { real; } T ; p)<\mu_{j}(\text { aq;ideal; } T ; p)\), a stabilising influence. On the other hand, solute - solute interactions may be repulsive. In view of the fact that molecules have a real size, this contribution is always present. Consequently the latter (together with other forms of solute – solute repulsions) contribute to cases where \(\gamma_{j}>1.0\), \(\ln \left(\gamma_{j}\right)>0\) and \(\mu_{\mathrm{j}}(\text { aq; real; } \mathrm{T} ; \mathrm{p})>\mu_{\mathrm{j}}(\mathrm{aq} ; \text { ideal } ; \mathrm{T} ; \mathrm{p})\). In principle, activity coefficient \(\gamma_{j}\) contains an enormous amount of information. Many of the interesting properties of aqueous solutions are packed in the parameter \(\gamma_{j}\). Unfortunately only in rare instances is it possible to dissect a given \(\gamma_{j}\) into the several contributing interactions. A common though obviously dangerous procedure sets \(\gamma_{j}\) equal to unity, assuming that the properties of a given solute \(j\) are ideal. However in many cases we have no alternative but to make this assumption at least in initial stages of an analysis of experimental results.Equation (e) is satisfactory for very dilute solutions of neutral solutes. Indeed this equation has enormous technological significance. The task of producing a very pure liquid requires lowering the molalities of solutes in a solution. With decreasing molality of a given solute, the chemical potential of a solute decreases; i.e. the solute is stabilised. So as more solute, an impurity, is removed, the trace remaining is increasingly stabilised.We cannot put a number value to either \(\mu_{j}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\) or \(\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T})\). These quantities measure the contribution made by a solute \(j\) to the total energy of a solution. One contribution to, for example, \(\mu_{j}^{0}(\mathrm{aq} ; \mathrm{T})\) emerges from solvent-solute interactions. Interestingly, we can in general put a number value to the corresponding limiting partial molar volume, \(V_{j}^{\infty}(a q)\). The concept of infinite dilution is extremely important in a practical sense. Nevertheless, we enter a word of caution. Returning to equation (b) for the chemical potential, we note that \(\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \ln \left(\mathrm{m}_{\mathrm{j}}\right)\) tends to minus \(\infty\). \[\text { Hence (at all T and p) limit }\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mu_{\mathrm{j}}(\mathrm{aq})=-\infty\]The practical significance of equation (i) is that with increasing dilution so the chemical potential of a solute decreases - the solute is increasingly stabilised. That is why the challenge of removing the last traces of unwanted solute presents such a formidable task, particularly to those industries where very high solvent purity is essential; e.g. the pharmaceutical industry.Footnotes Activity coefficients have a ‘bad press’. They are not ‘loved’ except by a minority of chemists. Nevertheless these coefficients contain information concerning the way in which solute molecules ‘communicate’ to each other in solution. M. Spiro, Educ. Chem.,1966, 3,139. The concept of an activity coefficient for a solute tending to unity at infinite dilution was proposed by A.A. Noyes and W.C. Bray: J. Am. Chem. Soc., 1911, 33,1643. E. Wilhelm, Thermochim. Acta, 1987,119,17; Interactions in Ionic and Non-Ionic Hydrates, ed. H. Kleenberg, Springer –Verlag, Berlin, 1987,p.118. S. F. Sciamanna and J. M. Prausnitz, AIChE J., 1987, 33, 1315. Throughout this subject, it is good practice to examine equations describing the dependence of partial molar properties of solvent and solute in the limit that the composition of the solution tends to increasingly dilute solutions; see J. E. Garrod and T. M. Herringon, J. Chem. Educ., 1969, 46, 165.This page titled 1.5.12: Chemical Potentials- Solute; Molality Scale is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.5.13: Chemical Potentials- Solutes- Mole Fraction Scale

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For the most part we use either the molality scale or the concentration scale to express the composition of aqueous solutions. Nevertheless, the mole fraction scale is often used. Hence we express the chemical potential of solute \(j\), \(\mu_{j}\) as a function of mole fraction of solute \(j\), \(\mathrm{x}_{j} \left[=\mathrm{n}_{\mathrm{j}} /\left(\mathrm{n}_{1}+\mathrm{n}_{\mathrm{j}}\right)\right]\). Note that we are relating the property \(\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\) to the composition of the solution using a different method from that used where the composition is expressed in terms of the molality or concentration of a solutes. \end{aligned}\]\[\text { By definition, } \operatorname{limit}\left(\mathrm{x}_{\mathrm{j}} \rightarrow 0\right) \mathrm{f}_{\mathrm{j}}^{*}=1 \text { at all } \mathrm{T} \text { and } \mathrm{p} \text {. }\]\(f_{j}^{*}\) is the asymmetric solute activity coefficient on the mole fraction scale. The word 'asymmetric', although rarely used, emphasises the difference between \(f_{j}^{*}\) and the rational activity coefficients.For solutions at ambient pressure, the integral term in equation (a) is negligibly small. At pressure \(\mathrm{p}\) and temperature \(\mathrm{T}\), \[\mu_{j}(a q)=\mu_{j}^{0}(a q ; x-\text { scale })+R \, T \, \ln \left(x_{j} \, f_{j}^{*}\right)\]For an ideal solution, \(f_{j}^{*}\) is unity. The reference state for the solute is the solution where the mole fraction of solute-\(j\) is unity. This is clearly a hypothetical solution but we assume that the properties of the solute j in this solution can be obtained by extrapolating from the properties of solute-\(j\) at low mole fractions. For an ideal solution at ambient pressure and temperature, \[\mu_{j}(\mathrm{aq} ; \mathrm{id})=\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{x}-\text { scale })+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{\mathrm{j}}\right)\]\[\text { or, } \mu_{j}(\mathrm{aq} ; \mathrm{id})-\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{x}-\text { scale })=\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{\mathrm{j}}\right)\]Because, \(\mathrm{x}_{\mathrm{j}}<1.0, \quad \ln \left(\mathrm{x}_{\mathrm{j}}\right)<0\) Hence, \(\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{id})<\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{x}-\text { scale })\). The solute is at a lower chemical potential than in the solution reference state.Footnote The extrapolation defining the reference state as a solution wherein the mole fraction of solute is unity might seem strange. In fact such long extrapolations are common in chemical thermodynamics. For example at \(0.1 \mathrm{~MPa}\) and \(298.15 \mathrm{~K}\), liquid water is the stable phase. At \(0.1 \mathrm{~MPa}\) and \(273.15 \mathrm{~K}\) both water(\(\ell\)) and ice-1h are the stable phases. Nevertheless we know that, assuming water(\(g\)) is an ideal gas, the volume occupied by \(0.018 \mathrm{~kg}\) of water(\(g\)) at \(273 \mathrm{~K}\) and \(0.1 \mathrm{~MPa}\) equals \(22.4 \mathrm{~dm}^{3}\). The fact that this involves a long extrapolation into a state where water vapour is not the stable phase does not detract from the usefulness of the concept.This page titled 1.5.13: Chemical Potentials- Solutes- Mole Fraction Scale is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.5.14: Chemical Potentials; Solute; Concentration Scale

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Both molalities and mole fractions are based on the masses of solvent and solute in a solution. Hence neither the molality \(\mathrm{m}_{j}\) nor mole fraction \(\mathrm{x}_{j}\) of a non-reacting solute depend on temperature and pressure. In fact, when we differentiate the equations for \(\mu_{j}(\mathrm{aq} ; \mathrm{T})\) with respect to pressure we take advantage of the fact that \(\mathrm{m}_{j}\) and \(\mathrm{x}_{j}\) do not depend on pressure. In addition, molalities and mole fractions are precise; there is no ambiguity concerning the amount of solvent and solute in the solution.However, when we describe the meaning and significance of the activity coefficient \(\gamma_{j}\) and the meaning of the term 'infinite dilution' we refer to the distance between solute molecules. In fact, in reviewing the properties of solutions, chemists are often more interested in the distance between solute molecules than in their masses. [The same can be said about the interest shown by humans in the behaviour of other human beings!] Therefore, chemists often use concentrations to express the composition of solutions.The concentration of solute \(\mathrm{c}_{j}\) describes the amount of chemical substance \(j\) in a given volume of solution; \(\mathrm{c}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} / \mathrm{V}\). The common method for expressing \(\mathrm{c}_{j}\) uses the unit '\(\mathrm{mol} \mathrm{dm}\)'. By definition [at temperature \(\mathrm{T}\) and pressure \(p\left(\cong p^{0}\right)\)] \[\mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{c}-\mathrm{scale})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{c}_{\mathrm{j}} \, \mathrm{y}_{\mathrm{j}} / \mathrm{c}_{\mathrm{r}}\right)\]\(\mathrm{c}_{\mathrm{r}}\) is the reference concentration, \(1 \mathrm{~mol dm}^{-3}\); \(\mathrm{c}_{j}\) is expressed using the same unit; \(\mathrm{y}_{j}\) is the activity coefficient for the solute \(j\) on the concentration scale. \[\text { By definition, } \quad \lim \operatorname{it}\left(c_{j} \rightarrow 0\right) y_{j}=1 \quad \text { (at all } T \text { and } p \text { ) }\]\(\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{c}-\text { scale })\) is the chemical potential of solute \(j\) in an ideal \(\left(\mathrm{y}_{\mathrm{j}}=1.0\right)\) aqueous solution (at the same \(\mathrm{T}\) and \(\mathrm{p}\)) where the concentration of solute \(\mathrm{c}_{\mathrm{j}}=1.0 \mathrm{~mol} \mathrm{dm}^{-3}\)Footnotes Using the base SI units, concentration is given by the ratio \(\left(n_{j} / V\right)\) where \(\mathrm{V}\) is expressed using the unit \(\mathrm{m}^{3}\); \(\mathrm{n}_{j}\) is the amount of chemical substance \(j\), the unit being the mole. Nevertheless in the present context, general practice uses the reference concentration \(1 \mathrm{~mol dm}^{-3}\); \(\mathrm{c}_{j}\) is expressed using the unit, \(\mathrm{mol dm}^{-3}\). This practice emerges from the fact that for dilute aqueous solutions at ambient \(\mathrm{T}\) and \(\mathrm{p}\), unit concentration of solute, \(1 \mathrm{~mol dm}^{-3}\) is almost exactly unit molality, \(1 \mathrm{~mol kg}^{-1}\). For comments on standard states see E. M. Arnett and D. R. McKelvey, in Solute-Solvent Interactions, ed. J. F. Coetzee, M. Dekker, New York, 1969, chapter 6.This page titled 1.5.14: Chemical Potentials; Solute; Concentration Scale is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.5.15: Chemical Potentials- Solute- Concentration and Molality Scales

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For a given solution we can express the chemical potential of solute \(j\), \(\mu_{\mathrm{j}}(\mathrm{aq})\) in an aqueous solution at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\left(\approx \mathrm{p}^{0}\right)\) using two equations. Therefore, at fixed \(\mathrm{T}\) and \(\mathrm{p}\), \[\begin{aligned} &\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)= \\ &\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{c}-\text { scale })+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{c}_{\mathrm{j}} \, \mathrm{y}_{\mathrm{j}} / \mathrm{c}_{\mathrm{r}}\right) \end{aligned}\]Therefore, \[\ln \left(\mathrm{y}_{\mathrm{j}}\right)=\ln \left(\gamma_{\mathrm{j}}\right)+\ln \left(\mathrm{m}_{\mathrm{j}} \, \mathrm{c}_{\mathrm{r}} / \mathrm{m}^{0} \, \mathrm{c}_{\mathrm{j}}\right) +(1 / \mathrm{R} \, \mathrm{T}) \,\left[\mu_{\mathrm{j}}^{0}(\mathrm{aq})-\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{c}-\text { scale })\right]\]In the latter two equations the composition variables \(\mathrm{m}_{j}\) and \(\mathrm{c}_{j}\) are expressed in the units ‘\(\mathrm{mol kg}^{-1}\)’ and ‘\(\mathrm{mol dm}^{-3}\)’ respectively. The ratio ‘\(\mathrm{c}_{\mathrm{j}} / \mathrm{m}_{\mathrm{j}}\)’ equals the density expressed in the unit ‘\(\mathrm{kg dm}^{-3}\)’. For dilute solutions, \(\mathrm{c}_{\mathrm{j}} / \mathrm{m}_{\mathrm{j}}=\rho_{1}^{*}(\ell)\), the density of the pure solvent. \[\text { Also, } \mathrm{c}_{\mathrm{r}} / \mathrm{m}^{0}=\left[\mathrm{mol} \mathrm{d \textrm {dm } ^ { - 3 }}\right] /\left[\mathrm{mol} \mathrm{kg}^{-1}\right]=\left[\mathrm{kg} \mathrm{dm}^{-3}\right]\]For dilute aqueous solutions at ambient pressure and \(298.2 \mathrm{~K}\), \[\ln \left(\mathrm{m}_{\mathrm{j}} \, \mathrm{c}_{\mathrm{r}} / \mathrm{m}^{0} \, \mathrm{c}_{\mathrm{j}}\right)=-\ln (0.997)\]With reference to equation (b), with increasing dilution, \(\mathrm{y}_{\mathrm{j}} \rightarrow 1, \gamma_{\mathrm{j}} \rightarrow 1,\left(\mathrm{~m}_{\mathrm{j}} \, \mathrm{c}_{\mathrm{r}} / \mathrm{m}^{0} \, \mathrm{c}_{\mathrm{j}}\right) \rightarrow \mathrm{c}_{\mathrm{r}} / \mathrm{m}^{0} \, \rho_{1}^{*}(\ell)\) Hence, \[\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{c}-\mathrm{scale})-\mu_{\mathrm{j}}^{0}(\mathrm{aq})=\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{c}_{\mathrm{r}} / \mathrm{m}^{0} \, \rho_{1}^{*}(\ell)\right]\]We combine equations (b) and (e). \[\ln \left(\mathrm{y}_{\mathrm{j}}\right)=\ln \left(\gamma_{\mathrm{j}}\right)+\ln \left(\mathrm{m}_{\mathrm{j}} \, \mathrm{c}_{\mathrm{r}} / \mathrm{m}^{0} \, \mathrm{c}_{\mathrm{j}}\right)-\ln \left[\mathrm{c}_{\mathrm{r}} / \mathrm{m}^{0} \, \rho_{1}^{*}(\ell)\right]\]\[\ln \left(\mathrm{y}_{\mathrm{j}}\right)=\ln \left(\gamma_{\mathrm{j}}\right)+\ln \left(\mathrm{m}_{\mathrm{j}} \, \mathrm{c}_{\mathrm{r}} / \mathrm{m}^{0} \, \mathrm{c}_{\mathrm{j}}\right)-\ln \left[\mathrm{c}_{\mathrm{r}} / \mathrm{m}^{0} \, \rho_{1}^{*}(\ell)\right]\]Footnotes A given solution is prepared by adding \(\mathrm{n}_{j}\) moles of solute \(j\) to \(\mathrm{w}_{1} \mathrm{~kg}\) of solvent.Molality of solute \(\mathrm{j} / \mathrm{mol} \mathrm{kg}{ }^{-1}=\left(\mathrm{n}_{\mathrm{j}} / \mathrm{w}_{1}\right)\)Total mass of solution/kg \(=w_{1}+n_{j} \, M_{j}\) where molar mass of solute/kg \(\mathrm{mol}^{-1}=\mathrm{M}_{\mathrm{j}}\)Volume of solution/\(\mathrm{m}^{3} = \mathrm{V}\)Density of solution \(\rho / \mathrm{kg} \mathrm{m}^{-3}=\left[\frac{\mathrm{w}_{1}+\mathrm{n}_{\mathrm{j}} \, \mathrm{M}_{\mathrm{j}}}{\mathrm{V}}\right]\)By convention chemists express the composition of solutions in terms of (i) concentration using the unit ‘\(\mathrm{mol dm}^{-3}\)’ and (ii) molality using the unit, ‘\(\mathrm{mol kg}^{-1}\)’. These composition scales stem from the fact that at \(298.15 \mathrm{~K}\), \(1 \mathrm{~dm}^{3}\) of water has a mass of approx. \(1 \mathrm{~kg}\). So as we swap composition scales a conversion factor is often required .For dilute solutions \(w_{1}>n_{j} \, M_{j}\) and density of solution \(\rho\) equals the density of the pure solvent (at same temperature and pressure), i.e. density \(\rho=\rho 1(\ell) \mathrm{kg} \mathrm{m} \mathrm{m}^{-3}\) A typical conversion takes the following form for water at \(298.2 \mathrm{~K}\) and ambient pressure.\(\begin{aligned} \text { Density }=0.997 \mathrm{~g} \mathrm{~cm}^{-3} &=0.997\left(10^{-3} \mathrm{~kg}\right)\left(10^{-2} \mathrm{~m}^{-3}\right.\\ &=0.997 \mathrm{X} \mathrm{} 10^{3} \mathrm{~kg} \mathrm{~m}^{-3} \\ =& 997 \mathrm{~kg} \mathrm{~m}^{-3}=0.997 \mathrm{~kg} \mathrm{\textrm {dm } ^ { - 3 }} \end{aligned}\)\(\text { Then } \frac{\mathrm{c}_{\mathrm{j}} / \mathrm{mol} \mathrm{dm}^{-3}}{\mathrm{~m}_{\mathrm{j}} / \mathrm{mol} \mathrm{kg}^{-1}}=\frac{\mathrm{n}_{\mathrm{j}} / \mathrm{mol}}{\mathrm{V} / \mathrm{dm}^{3}} \, \frac{\mathrm{w}_{1} / \mathrm{kg}}{\mathrm{n}_{\mathrm{j}} / \mathrm{mol}}=\frac{\mathrm{w}_{1} / \mathrm{kg}}{\mathrm{V} / \mathrm{dm}^{3}}=\rho / \mathrm{kg} \mathrm{dm}^{-3}\) \(\begin{aligned} \ln \left(\mathrm{m}_{\mathrm{j}} \, \mathrm{c}_{\mathrm{r}} / \mathrm{m}^{0} \, \mathrm{c}_{\mathrm{j}}\right) &=\ln \left[\left(\mathrm{c}_{\mathrm{r}} / \mathrm{m}^{0}\right) /\left(\mathrm{c}_{\mathrm{j}} / \mathrm{m}_{\mathrm{j}}\right)\right] \\ =& \ln \left[\left(\mathrm{kg} \mathrm{d \textrm {m } ^ { - 3 } ) / \rho ]}=-\ln \left(\rho / \mathrm{kg} \mathrm{d \textrm {dm } ^ { - 3 } )}\right.\right.\right. \end{aligned}\)This page titled 1.5.15: Chemical Potentials- Solute- Concentration and Molality Scales is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.5.16: Chemical Potentials- Solute- Molality and Mole Fraction Scales

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.05%3A_Chemical_Potentials/1.5.16%3A_Chemical_Potentials-_Solute-_Molality_and_Mole_Fraction_Scales

The chemical potential of solute j in aqueous solution at temperature \(\mathrm{T}\) and at close to ambient pressure is related to the molality \(\mathrm{m}_{j}\) and mole fraction \(\mathrm{x}_{j}\). \[\begin{aligned} \mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right) \\ &=\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p} ; \mathrm{x}-\mathrm{scale})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{\mathrm{j}} \, \mathrm{f}_{\mathrm{j}}^{*}\right) \end{aligned}\]Therefore, \[\begin{aligned} \ln \left(\mathrm{f}_{\mathrm{j}}^{*}\right)=\ln \gamma_{\mathrm{j}} &+\ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{x}_{\mathrm{j}} \, \mathrm{m}^{0}\right) \\ &+(\mathrm{l} / \mathrm{R} \, \mathrm{T})\left[\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})-\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p} ; \mathrm{x} \text { - scale })\right] \end{aligned}\]For dilute solutions, \(\left(1 / \mathrm{M}_{1}\right)>\mathrm{m}_{\mathrm{j}}\). Hence \(\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0} \, \mathrm{x}_{\mathrm{j}}\right)\) equals \(\left(m^{0} \, M_{1}\right)^{-1}\), a dimension-less quantity. Therefore, \[\begin{aligned} \ln \left(f_{j}^{*}\right)=\ln \gamma_{j} &-\ln \left(m^{0} \, M_{1}\right) \\ &+(1 / R \, T)\left[\mu_{j}^{0}(a q ; T ; p)-\mu_{j}^{0}(a q ; T ; p ; x-s c a l e)\right] \end{aligned}\]It is unrealistic to expect that \(\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\) equals \(\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p} ; \mathrm{x}-\mathrm{scale})\) because the two reference states for solute-\(j\) are quite different. In general terms, \(\mathrm{f}_{\mathrm{j}}^{*}\) does not equal \(\gamma_{j}\) for the same solution. Nevertheless, both \(\mathrm{f}_{\mathrm{j}}^{*}\) and \(\gamma_{j}\) tend to the same limit, unity, as the solution approaches infinite dilution.Hence as \(\mathrm{n}_{j}\) tends to zero, \[\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})-\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p} ; \mathrm{x}-\mathrm{scale})=\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}^{0} \, \mathrm{M}_{\mathrm{l}}\right)\]For example, in the case of aqueous solutions at \(298.15 \mathrm{~K}\), \(\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}^{0} \, \mathrm{M}_{1}\right)\) equals \(\left(-9.96 \mathrm{~kJ} \mathrm{~mol}^{-1}\right)\) meaning that, with respect to the reference states for the two solutions, the chemical potential of solute \(j\) is higher on the mole fraction scale than on the molality scale. Combination of equations (b) and (d) yields an equation relating the two activity coefficients with the two terms describing the composition of the solution. \[\ln \left(f_{j}^{*}\right)=\ln \gamma_{j}+\ln \left(m_{j} \, M_{1} / x_{j}\right)\]The term ‘unitary’ is sometimes used to describe reference chemical potentials on the mole fraction scale, \(\mu_{\mathrm{j}}^{0}\left(\mathrm{aq} ; \mathrm{T} ; \mathrm{p} ; \mathrm{x}_{\mathrm{j}}=1\right)\). The term \(\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}^{0} \, \mathrm{M}_{1}\right)\) in equation (d) is called cratic because it refers to different amounts of solute and solvent which are mixed to form reference states for the solute on molality, mole fraction and concentration scales. The impression is sometimes given that standard states for solutes based on the mole fraction scale (sometimes identified as the unitary scale) are more fundamental but there is little experimental evidence to support this view.Footnotes For a solution prepared using \(\mathrm{w}_{1}\) kg of water and \(\mathrm{n}\) moles of solute \(j\),\(\begin{gathered} \mathrm{x}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} /\left[\left(\mathrm{w}_{1} / \mathrm{M}_{1}\right)+\mathrm{n}_{\mathrm{j}}\right] \text { and } \mathrm{m}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} / \mathrm{w}_{1} \, \\ \mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0} \, \mathrm{x}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} \,\left[\left(\mathrm{w}_{\mathrm{l}} / \mathrm{M}_{\mathrm{l}}\right)+\mathrm{n}_{\mathrm{j}}\right] / \mathrm{w}_{1} \, \mathrm{m}^{0} \, \mathrm{n}_{\mathrm{j}} \\ \mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0} \, \mathrm{x}_{\mathrm{j}}=\left[\left(1 / \mathrm{M}_{1}\right)+\mathrm{m}_{\mathrm{j}}\right] / \mathrm{m}^{0} \end{gathered}\) The terms 'unitary' and 'cratic' were suggested by R. W. Gurney, Ionic Processes in Solution, McGraw-Hill, New York, 1953.This page titled 1.5.16: Chemical Potentials- Solute- Molality and Mole Fraction Scales is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.5.17: Chemical Potentials- Solutions- Salts

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.05%3A_Chemical_Potentials/1.5.17%3A_Chemical_Potentials-_Solutions-_Salts

We consider a salt j having the following general formula; \[v_{+} \mathrm{M}^{\mathrm{z}+} v_{-} \mathrm{X}^{\mathrm{z}-} .\]Here \(ν_{+}\) and \(ν_{-}\) are the (integer) stoichiometric coefficients; \(\mathrm{z}\), and \(\mathrm{z}_{-}\) are the (integer) charge numbers. We assume the salt \(j\) is completely dissociated in aqueous solution. Hence the solution contains (apart from solvent) two chemical substances. With complete dissociation each mole of salt produces \(v\left(=v_{+}+v_{-}\right)\) moles of ions. The condition of electric neutrality is expressed by equation (b). \[\left|v_{+} \, z_{+}\right|=\left|v_{-} \, z_{-}\right|\]For a solution molality \(\mathrm{m}_{j}\), the molalities of cations and anions are \(ν_{+} \, \mathrm{m}_{j}\) and \(ν_{−} \, \mathrm{m}_{j}\) respectively. If the chemical potentials of cations and anions are \(\mu_{+}(\mathrm{aq})\) and \(\mu_{-}(\mathrm{aq})\) respectively, the chemical potential of salt \(j\) in aqueous solution (at molality, \(\mathrm{m}_{j}\) temperature \(\mathrm{T}\) and ambient pressure) is given by equation (c). \[\mu_{j}(a q)=v_{+} \, \mu_{+}(a q)+v_{-} \, \mu_{-}(a q)\]In an ideal solution (at the same \(\mathrm{T}\) and ambient pressure) where the molality of the salt \(j\) is \(1 \mathrm{~mol kg}^{–1}\).\[\mu_{\mathrm{j}}^{0}(\mathrm{aq})=\mathrm{v}_{+} \, \mu_{+}^{0}(\mathrm{aq})+\mathrm{v}_{-} \, \mu_{-}^{0}(\mathrm{aq})\]For each ionic substance \(\mathrm{i}\), the chemical potential (at the same \(\mathrm{T}\) and \(\mathrm{p}\)) is given by equation (e) where \(\gamma_{\mathrm{i}}\) is the single ionic activity coefficient. \[\mu_{\mathrm{i}}(\mathrm{aq})=\mu_{\mathrm{i}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{i}} \, \gamma_{\mathrm{i}} / \mathrm{m}^{0}\right)\]Hence for the salt with \(\mathrm{m}_{+}=v_{+} \, \mathrm{m}_{\mathrm{j}}\) and \(\mathrm{m}_{-}=v_{-} \, \mathrm{m}_{\mathrm{j}}\). \[\begin{aligned} \mu_{\mathrm{j}}(\mathrm{aq})=\left[\mathrm{v}_{+}\right.&\left.\, \mu_{+}^{0}(\mathrm{aq})+\mathrm{v}_{-}-\mu_{-}^{0}(\mathrm{aq})\right] \\ &+\mathrm{v}_{+} \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{v}_{+} \, \mathrm{m}_{\mathrm{j}} \, \gamma_{+} / \mathrm{m}^{0}\right) \\ &+\mathrm{v}_{-} \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{v}_{-} \, \mathrm{m}_{\mathrm{j}} \, \gamma_{-} / \mathrm{m}^{0}\right) \end{aligned}\]We draw together the logarithm terms. \[\begin{aligned} \mu_{j}(\mathrm{aq})=\left[v_{+}\right.&\left.\, \mu_{+}^{0}(\mathrm{aq})+\mathrm{v}_{-} \, \mu_{-}^{0}(\mathrm{aq})\right] \\ &+\mathrm{R} \, \mathrm{T} \, \ln \left[\left(\mathrm{v}_{+} \, \mathrm{m}_{\mathrm{j}} \, \gamma_{+} / \mathrm{m}^{0}\right)^{\mathrm{v}+} \,\left(\mathrm{v}_{-} \, \mathrm{m}_{\mathrm{j}} \, \gamma_{-} / \mathrm{m}^{0}\right)^{\mathrm{v}-}\right] \end{aligned}\]The latter far from elegant equation contains all the parameters we expect to be present. Here it is convenient to introduce a parameter \(\mathrm{Q}\). \[\mathrm{Q}^{v}=v_{+}^{v+} \, v_{-}^{v-}\]The (geometric) mean activity coefficient is defined by equation (i). \[\gamma_{\pm}^{v}=\gamma_{+}^{v+} \, \gamma_{-}^{v-}\]Hence equation (g) can be rewritten in the following form. \[\mu_{j}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+v \, \mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{Q} \, \mathrm{m}_{\mathrm{j}} \, \gamma_{\pm} / \mathrm{m}^{0}\right]\]The quantity \(\mathrm{Q}\) takes account of the stoichiometric composition of the salt. In preparing the salt solution we target the molality but this does not take account of how many moles of each ionic substance are produced by one mole of salt; the quantity \(ν\) only records how many moles of ionic substances are produced by each mole of salt. For the salt \(\mathrm{M}^{2+} 2 \mathrm{X}^{-}\) [e.g. \mathrm{Mg Br}_{2}\)] \(v_{+}=1\), \(v_{-}=2\) where \(\mathrm{Q}^{3} =1 ^{2}\). \(2^{2} = 4\). For this salt, \[\mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+3 \, \mathrm{R} \, \mathrm{T} \, \ln \left[4^{1 / 3} \, \mathrm{m}_{\mathrm{j}} \, \gamma_{\pm} / \mathrm{m}^{0}\right]\]The complex algebra associated with a thermodynamic description of salt solutions stems from a conflict of interests and practical chemistry. The ground rules in this subject are quite simple--- measurements are made on electrically neutral solutions; e.g. \(\mathrm{NaCl}(\mathrm{aq})\). In the latter case chemists often favour a description of this system in terms of an aqueous solution of two solutes, sodium ions and chloride ions. Rather than using an activity coefficient for the solute [e.g; \(\gamma(\mathrm{NaCl})\)], we define a mean activity coefficient \(\gamma_{\pm}\) which recognises the presence of two ionic substances. An indication of the presence of two solutes (i.e. \(\mathrm{Na}^{+}\) and \(\mathrm{Cl}^{-}\)) rather than one solute (e.g. \mathrm{NaCl}\)) is the stoichiometric parameter \(ν\) in the equation for the chemical potential of the solvent. This parameter is readily determined from the depression of solvent freezing points ( i.e. cryoscopy) and osmotic pressures. Both properties are directed at the properties of solvents. The cryoscopic technique is based on measurement of the temperature at which solvent in a solution is in equilibrium (at fixed pressure) with pure solid solvent. The osmotic pressure \(\pi\) characterises the equilibrium (a fixed temperature) between the solution at pressure \(\mathrm{p} + \pi\) and pure solvent at pressure \(\mathrm{p}\). Both techniques determine \(ν\) or, in colloquial terms, count solute particles. The molar mass of \(\mathrm{NaCl}\) and urea are roughly equal; the number of solute particles in \(\mathrm{NaCl}(\mathrm{aq})\) is twice that in urea(\(\mathrm{aq}\)) for the same mass of solute in \(1 \mathrm{~kg}\) of water. We confirm this observation by measuring the depressions of freezing points or the osmotic pressures of two solutions.Equation (j) signals an important challenge. If we could separate out ionic contributions to \(\mu_{\mathrm{j}}^{0}(\mathrm{aq})\) for salt \(j\), we could probe the contributions made by ion-water interactions, the hydration properties for a given ion \(\mathrm{i}\) at defined \(\mathrm{T}\) and \(\mathrm{p}\) to \(\mu_{j}^{0}(\mathrm{aq})\). In this exercise we might then extend the analysis to single ion enthalpies, \(\mathrm{H}_{\mathrm{i}}^{0}(\mathrm{aq})\), volumes \(\mathrm{V}_{\mathrm{i}}^{0}(\mathrm{aq})\), and entropies \(\mathrm{S}_{\mathrm{i}}^{0}(\mathrm{aq})\). Unfortunately the story is not simple. Indeed we cannot measure these chemical potentials and then obtain absolute estimates for the above derived properties.Footnotes R. H. Stokes and R. H. Robinson, J. Am. Chem.Soc.,1948,70,1870. R. G. Bates, Pure Appl. Chem.,1973,36,407. J. C. R. Reis, J. Chem. Soc. Faraday Trans.,1997,93,2171.This page titled 1.5.17: Chemical Potentials- Solutions- Salts is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.5.18: Chemical Potentials- Solutions- 1-1 Salts

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.05%3A_Chemical_Potentials/1.5.18%3A_Chemical_Potentials-_Solutions-_1-1_Salts

There is an important point to consider in the context of salt solutions. For a dilute aqueous solution containing sodium chloride, osmotic and colligative properties confirm that for each mole of sodium chloride the aqueous solution contains (almost exactly) two moles of solutes. These observations result in an added complexity in that chemists describe the solute, sodium chloride in two ways. In one description there is one solute - 'sodium chloride'. In another description there are two solutes sodium ions and chloride ions. The latter description is certainly attractive because we can ring the changes through a series of solutes; \(\mathrm{NaCl } \rightarrow \mathrm{ NaBr } \rightarrow \mathrm{ KBr } \rightarrow \mathrm{ KCl } \rightarrow\) etc. Here we change in stepwise fashion one chemical substance in the salt to produce a new solute. There is, however, one crucial condition. Aqueous solutions are electrically neutral although the solutions contain ions. Therefore, the total charge on all cations equals in magnitude the total charge on all anions in the same solution. There is, therefore, a major problem. We cannot examine the properties of aqueous solutions containing, for example, just cations. We can only examine the properties of electrically neutral solutions. How can we obtain the properties of ionic substance (e.g. \(\mathrm{Na}^{+}\)) in aqueous solutions at defined temperature and pressure? The frustrating answer is that we cannot measure the thermodynamic properties of single ions in solution. This realisation does not stop us speculating about such properties. In fact, a common procedure involves estimating the properties of single ions but then in the last stage of the analysis we pull the derived single ion properties together to describe the properties of a given salt solution.The chemical potential of each ionic substance i in solution is related to its molality mi using equation (a) for a solution at fixed temperature and fixed pressure. We assume that the latter is ambient pressure and therefore close to the standard pressure \(\mathrm{p}^{0}\). \[\mu_{i}(a q)=\mu_{i}^{0}(a q)+R \, T \, \ln \left(m_{i} \, \gamma_{i} / m^{0}\right)\]Here \(\mu_{\mathrm{i}}^{0}(\mathrm{aq})\) is the standard chemical potential of ion i in an aqueous solution where both the molality \(\mathrm{m}_{\mathrm{i}}\) and single ion activity coefficient \(\gamma_{\mathrm{i}\) are unity (at the same \(\mathrm{T}\) and \(\mathrm{p}\)). However, the terms \(\mu_{i}(\mathrm{aq})\), \(\mu_{\mathrm{i}}^{0}(\mathrm{aq})\) and \(\gamma_{\mathrm{i}}\) have no practical significance because, we cannot prepare a solution containing just one ionic chemical substance. The way forward involves using equation (a) for all ionic substances in the solution. In order to show how the argument develops we consider an aqueous solution containing a 1:1 salt (e.g. \(\mathrm{NaCl}\)) which we assert is fully dissociated into ions. We make two (extrathermodynamic) assumptions.\[\begin{aligned} &\mu_{j}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{m}_{\mathrm{j}}=1 \mathrm{~mol} \mathrm{~kg}^{-1}\right)= \\ &\mu_{+}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{m}_{+}=1 \mathrm{~mol} \mathrm{~kg}{ }^{-1}\right)+\mu_{-}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{m}_{-}=1 \mathrm{~mol} \mathrm{~kg}^{-1}\right) \end{aligned}\]As demanded by the analysis, such an ideal solution would be electrically neutral. But we have no information concerning the contributions which the ions make to the overall chemical potential, \(\mu_{\mathrm{j}}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{m}_{\mathrm{j}}=1 \mathrm{~mol} \mathrm{} \mathrm{kg}^{-1}\right)\). We anticipate that such contributions are characteristic of the ions. For a 1:1 salt combination of the three previous equations yields for a solution at fixed \(\mathrm{T}\) and \(\mathrm{p}\), equation (d). \[\begin{aligned} &\mu_{j}(\mathrm{aq})= \\ &\mu_{j}^{0}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{m}_{\mathrm{j}}=1 \mathrm{~mol} \mathrm{~kg} \mathrm{~kg}^{-1}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{+} \, \mathrm{m}_{-} \, \gamma_{+} \, \gamma_{-} / \mathrm{m}^{0} \, \mathrm{m}^{0}\right) \end{aligned}\]A (geometric) mean ionic activity coefficient \(\gamma_{\pm}\) is defined by equation (e). \[\gamma_{\pm}^{2}=\gamma_{+} \, \gamma_{-}\]Also for a 1: 1 salt \(\mathrm{m}_{\mathrm{j}}^{2}=\mathrm{m}_{+} \, \mathrm{m}_{-}\). Therefore (at fixed temperature and pressure), \[\mu_{j}(a q)=\mu_{j}^{0}(a q)+2 \, R \, T \, \ln \left(m_{j} \, \gamma_{\pm} / m^{0}\right)\]where \(\mu_{j}^{0}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{m}_{\mathrm{j}}=1 \mathrm{~mol} \mathrm{~kg}{ }^{-1} ; \mathrm{p} \equiv \mathrm{p}^{0}\right)\) \[\text { By definition limit }\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{\pm}=1.0 \text { at all } \mathrm{T} \text { and } \mathrm{p}\]The origin of the integer '2' in equation (f) is the stoichiometry of the salt; each mole of salt forms two moles of ions assuming complete dissociation. Hence \(\mu_{\mathrm{j}}^{0}(\mathrm{aq})\) ( ) is the chemical potential of salt \(j\) in an ideal solution at the same \(\mathrm{T}\) and \(\mathrm{p}\)_ (\(\cong \mathrm{p}^{0}\)) where the molality of the salt is \(1 \mathrm{~mol kg}^{-1}\). If the properties of the salt are ideal the chemical potential of the salt is given by equation (h). \[\mu_{j}(\mathrm{aq} ; \mathrm{id})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+2 \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)\]When \(\mathrm{m}_{\mathrm{j}}>\mathrm{m}^{0}\), the chemical potential of the salt in the ideal solution \(\mu_{j}(\mathrm{aq} ; \mathrm{id})>\mu_{\mathrm{j}}^{0}(\mathrm{aq})\); the salt is at a higher chemical potential than in the reference state. When \(\mathrm{m}_{\mathrm{j}}<1.0 \mathrm{~mol} \mathrm{} \mathrm{kg}^{-1}\), the chemical potential of the salt in the ideal solution is lower than in the reference solution where \(\mathrm{m}_{\mathrm{j}}=1.0 \mathrm{~mol} \mathrm{~kg}^{-1}\).Returning to the equation (f), there is merit in writing the equation in the following form. \[\mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+2 \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)+2 \, \mathrm{R} \, \mathrm{T} \, \ln \left(\gamma_{\pm}\right)\]\[\text { Or, } \quad \mu_{j}(\mathrm{aq})=\mu_{j}(\mathrm{aq} ; \mathrm{id})+2 \, \mathrm{R} \, \mathrm{T} \, \ln \left(\gamma_{\pm}\right)\]The difference \(\left[\mu_{j}(\mathrm{aq})-\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{id})\right]\) is a measure of the extent to which the chemical potential of a salt in a real salt solution differs from the chemical potential of the same salt in an ideal solution. For \(\mathrm{KCl}\)(\(\mathrm{aq}\); \(298.2 \mathrm{~K}\); \(0.1 \mathrm{~mol kg}^{-1}\)) the mean ionic activity coefficient \(\gamma_{\pm}\) equals \(0.769\); \(0.769 ; 2 \, \mathrm{R} \, \mathrm{T} \, \ln (0.769) =-1.30 \mathrm{~kJ} \mathrm{~mol}^{-1}\). In other words, \(\mathrm{KCl}\) in this solution is at a lower chemical potential than in the corresponding ideal solution. In fact, \(\gamma_{\pm}\) for most dilute aqueous salt solution is \(< 1.0\) at ambient \(\mathrm{T}\) and \(\mathrm{p}\), and so this pattern in chemical potentials is quite common. However, even though we know \(\gamma_{\pm}\) for these systems we are not in a position to comment on the single ion activity coefficients for the reasons discussed above.The difference described by equation (j) prompts the definition of an excess chemical potential, \(\mu_{\mathrm{j}}^{\mathrm{E}}(\mathrm{aq})\). \[\text { Thus } \quad \mu_{\mathrm{j}}^{\mathrm{E}}(\mathrm{aq})=2 \, \mathrm{R} \, \mathrm{T} \, \ln \left(\gamma_{\pm}\right)\]A key contribution to \(\mu_{j}^{0}(\mathrm{aq})\) emerges from cation-water and anion-water interactions, namely ionic hydration. In contrast \(\gamma_{\pm}\) is determined by ion-ion interactions in real solutions. For a solution at pressure \(\mathrm{p}\), equation (f) takes the following form. \[\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{p})=\mu_{\mathrm{j}}^{0}\left(\mathrm{aq} ; \mathrm{p}^{0}\right)+2 \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\pm} / \mathrm{m}^{0}\right)+\int_{\mathrm{p}^{0}}^{\mathrm{p}} \mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq}) \, \mathrm{dp}\]Footnotes For comparison of \(\gamma_{\pm}\) for dilute salt solutions in aqueous solution and in \(\mathrm{D}_{2}\mathrm{O}\) see O. D. Bonner, J. Chem. Thermodyn., 1971, 3,837; and references therein.This page titled 1.5.18: Chemical Potentials- Solutions- 1-1 Salts is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.5.19: Chemical Potentials- Solutions- Salt Hydrates in Aqueous Solution

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An aqueous solution is prepared using \(\mathrm{n}_{j}\) moles of salt \(\mathrm{MX}\) and \(\mathrm{n}_{1}\) moles of water. The properties of the system are accounted for using one of two possible Descriptions.The solute \(j\) comprises a 1:1 salt MX molality \(\mathrm{m}(\mathrm{MX})\left[=\mathrm{n}_{\mathrm{j}} / \mathrm{w}_{1}\right. \text { where } \mathrm{w}_{1} \text { is the mass of water} \right]\).The single ion chemical potentials, are defined in the following manner \[\begin{aligned} &\mu\left(\mathrm{M}^{+}\right)=\left[\partial \mathrm{G} / \partial \mathrm{n}\left(\mathrm{M}^{+}\right)\right]_{\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}\left(\mathrm{x}^{-}\right)} \\ &\mu\left(\mathrm{X}^{-}\right)=\left[\partial \mathrm{G} / \partial \mathrm{n}\left(\mathrm{X}^{-}\right)\right]_{\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}\left(\mathrm{M}^{+}\right)} \end{aligned}\]The total Gibbs energy (at fixed \(\mathrm{T}\) and \(\mathrm{p}\) where \(p \approx p^{0}\)) is given by equation (b). \[\begin{aligned} &\mathrm{G}(\mathrm{aq} ; \mathrm{I})=\mathrm{n}_{1} \, \mu_{1}^{\mathrm{eq}}(\mathrm{aq}) \\ &\quad+\mathrm{n}_{\mathrm{j}} \,\left\{\mu^{0}\left(\mathrm{M}^{+} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{m}\left(\mathrm{M}^{+}\right) \, \gamma_{+}(\mathrm{I}) / \mathrm{m}^{0}\right]\right\} \\ &\quad+\mathrm{n}_{\mathrm{j}} \,\left\{\mu^{0}\left(\mathrm{X}^{-} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{m}\left(\mathrm{X}^{-}\right) \, \gamma_{-}(\mathrm{I}) / \mathrm{m}^{0}\right]\right\} \end{aligned}\]According to this Description each mole of cations is hydrated by \(\mathrm{h}_{\mathrm{m}\left(\mathrm{H}_{2}\mathrm{O}\right)\) moles of water and each mole of anions is hydrated by \(\mathrm{h}_{\mathrm{x}\left(\mathrm{H}_{2}\mathrm{O}\right)\) moles of water.The single ion chemical potentials are defined as follows. \[\mu\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O}\right)=\left[\partial \mathrm{G} / \partial\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O}\right)\right]\]at constant \(\mathrm{T}\), \(\mathrm{p}\), \(\mathrm{n}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O}\right),\left[\mathrm{n}_{1}-\mathrm{n}_{\mathrm{j}} \,\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right)\right]\left(\mathrm{H}_{2} \mathrm{O}\right) \mu\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O}\right)=\left[\partial \mathrm{G} / \partial \mathrm{n}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O}\right)\right]\)\[\text { at constant } \mathrm{T}, \mathrm{p}, \mathrm{n}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O}\right),\left[\mathrm{n}_{1}-\mathrm{n}_{\mathrm{j}} \,\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right)\right]\left(\mathrm{H}_{2} \mathrm{O}\right)\]\[\mathrm{m}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O}\right)=\mathrm{n}_{\mathrm{j}} / \mathrm{M}_{1} \,\left[\mathrm{n}_{1}-\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right) \, \mathrm{n}_{\mathrm{j}}\right] ;\]\[\mathrm{m}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O}\right)=\mathrm{n}_{\mathrm{j}} / \mathrm{M}_{1} \,\left[\mathrm{n}_{1}-\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right) \, \mathrm{n}_{\mathrm{j}}\right] .\]The (equilibrium) Gibbs energy (at defined \(\mathrm{T}\) and \(\mathrm{p}\)) is given by the following equation. \[\begin{aligned} &\mathrm{G}(\mathrm{aq} ; \mathrm{II})=\left[\mathrm{n}_{1}-\mathrm{n}_{\mathrm{j}} \,\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right)\right] \, \mu_{1}(\mathrm{aq}) \\ &+\mathrm{n}_{\mathrm{j}} \,\left[\mu^{0} \,\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O}\right) \, \gamma_{+}(\mathrm{II}) / \mathrm{m}^{0}\right\}\right] \\ &+\mathrm{n}_{\mathrm{j}} \,\left[\mu^{0}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O}\right) \, \gamma_{-}(\mathrm{II}) / \mathrm{m}^{0}\right\}\right] \end{aligned}\]But the Gibbs energies defined by equations ( b) and (g) are identical (at equilibrium at defined \(\mathrm{T}\) and \(\mathrm{p}\)). After all, it is the same solution. Hence, (dividing by \(\mathrm{n}_{j}\)) \[\begin{aligned} &{\left[\mu^{0}\left(\mathrm{M}^{+} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{M}^{+} ; \mathrm{I}\right) \, \gamma_{+}(\mathrm{I}) / \mathrm{m}^{0}\right\}\right]} \\ &\quad+\left[\mu^{0}\left(\mathrm{X}^{-} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{X}^{-} ; \mathrm{I}\right) \, \gamma_{-}(\mathrm{I}) / \mathrm{m}^{0}\right\}\right]= \\ &\quad-\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right) \, \mu_{1}^{\mathrm{eq}}(\mathrm{aq})+ \\ &\quad\left[\mu^{0}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O}\right) \, \gamma_{+}(\mathrm{II}) / \mathrm{m}^{0}\right\}\right] \\ &+\left[\mu^{0}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O}\right) \, \gamma_{-}(\mathrm{II}) / \mathrm{m}^{0}\right\}\right] \\ &\text { Then, } \mu^{0}\left(\mathrm{M}^{+} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{M}^{+} ; \mathrm{I}\right) \, \gamma_{+}(\mathrm{I}) / \mathrm{m}^{0}\right\} \\ &+\mu^{0}\left(\mathrm{X}^{-} ; \mathrm{aq}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}(\mathrm{X} \, ; \mathrm{I}) \, \gamma_{-}(\mathrm{I}) / \mathrm{m}^{0}\right\} \\ &=-\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right) \,\left\{\mu_{1}^{*}(\ell)-2 \, \phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\right\} \\ &+\left\{\mu^{0}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{M}^{+} \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{II}\right) \, \gamma_{+}(\mathrm{II}) / \mathrm{m}^{0}\right\}\right] \\ &+\left[\mu^{0}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{X}^{-} \mathrm{h}_{\mathrm{X}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{II}\right) \, \gamma_{-}(\mathrm{II}) / \mathrm{m}^{0}\right\}\right] \end{aligned}\]We use the latter equation to explore what happens in the limit that \(\mathrm{n}_{j}\) approaches zero. Thus, \[\begin{aligned} &\operatorname{limit}\left(\mathrm{n}_{\mathrm{j}} \rightarrow 0\right) \gamma_{+}(\mathrm{I})=1 \quad \gamma_{-}(\mathrm{I})=1 \\ &\gamma_{+}(\mathrm{II})=1 \quad \gamma_{-}(\mathrm{II})=1 \\ &\mathrm{~m}_{\mathrm{j}}=0 \\ &\mathrm{~m}\left(\mathrm{M}^{+} \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{II}\right) \, \mathrm{m}\left(\mathrm{X}^{-} \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{II}\right) / \mathrm{m}\left(\mathrm{M}^{+} ; \mathrm{I}\right) \, \mathrm{m}\left(\mathrm{X}^{-} ; \mathrm{I}\right)=1.0 \end{aligned}\]\[\begin{gathered} \text { Hence, } \mu^{0}\left(\mathrm{M}^{+} ; \mathrm{aq}\right)+\mu^{0}\left(\mathrm{X}^{-} ; \mathrm{aq}\right)= \\ \mu^{0}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{aq}\right)+\mu^{0}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{aq}\right) \\ -\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right) \, \mu_{1}^{*}(\ell) \end{gathered}\]We obtain an equation linking the ionic chemical potentials. Thus, \[\ln \gamma_{+}(\mathrm{I})+\ln \gamma_{-}(\mathrm{I})=2 \, \phi \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right)+\ln \left\{\gamma_{+}(\mathrm{II})\right\}+\ln \left\{\gamma_{-} \text {(II) }\right\}\]\[ \begin{aligned} &\text { But } \ln \left\{\gamma_{+}(\mathrm{I})\right\}+\ln \left\{\gamma_{-}(\mathrm{I})\right\}=2 \,\left\{\ln \gamma_{\pm}(\mathrm{I})\right\} \\ &\text { Then, } 2 \, \ln \left\{\gamma_{\pm}(\mathrm{I})\right\}=2 \, \phi \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right)+2 \, \ln \left\{\gamma_{\pm} \text {(II) }\right\}\]We identify relationships between single ion activity coefficients in an extra-thermodynamic analysis. Thus from equation (l), \[\ln \left\{\gamma_{+} \text {(II) }\right\}=\ln \left\{\gamma_{+} \text {(I) }\right\}-\phi \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \, \mathrm{h}_{\mathrm{m}}\]\[\ln \left\{\gamma_{-}(\mathrm{II})\right\}=\ln \left\{\gamma_{-}(\mathrm{I})\right\}-\phi \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \, \mathrm{h}_{\mathrm{x}}\]It is noteworthy that in these terms the solution can be ideal using description I where \(\gamma_{\pm} = 1.0\) but non-ideal using description II. Nevertheless, these equations show how the activity coefficient of the hydrated ion (description II) is related to the activity coefficient of the simple ion (description I).This page titled 1.5.19: Chemical Potentials- Solutions- Salt Hydrates in Aqueous Solution is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.5.20: Chemical Potentials- Salt Solutions- Ion-Ion Interactions

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.05%3A_Chemical_Potentials/1.5.20%3A_Chemical_Potentials-_Salt_Solutions-_Ion-Ion_Interactions

For most dilute aqueous salt solutions (at ambient temperature and pressure), mean ionic activity coefficients \(\gamma_{\pm}\) are less than unity. Ion-ion interactions within a real solution lower chemical potentials below those of salts in the corresponding ideal solutions. A quantitative treatment of this stabilisation is enormously important. In fact for almost the whole of the 20th Century, scientists have offered theoretical bases for expressing \(\ln \left(\gamma_{\pm}\right)\) as a function of the composition of a salt solution, temperature, pressure and electrical permittivity of the solvent.In effect we offer as much information as demanded by the theory (e.g. molality of salt, nature of salt, permittivity of solvent, ion sizes, temperature, pressure, .....). We set the apparently simple task - please calculate the mean activity coefficient of the salt in this solution.Many models and treatments have been proposed. Most models start by considering a reference j-ion in an aqueous salt solution. In order to calculate the electric potential at the \(j\)-ion arising from all other ions in solution, we need to know the distribution of these ions about the \(j\)-ion. Unfortunately this distribution is unknown and so we need a model for this distribution. Further activity coefficients also reflect the impact of ions on water-water interactions in aqueous salt solutions.Footnotes H. S. Frank, Z. Phys. Chem, 1965,228,364.This page titled 1.5.20: Chemical Potentials- Salt Solutions- Ion-Ion Interactions is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.5.21: Chemical Potentials- Salt Solutions- Debye-Huckel Equation

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The chemical potential of salt j in an aqueous solution at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\) (which is close to the standard pressure \(\mathrm{p}^{0}\)) is related to the molality of salt \(\mathrm{m}_{j}\) using equation (a). \[\mu_{j}(a q)=\mu_{j}^{0}(a q)+v \, R \, T \, \ln \left(Q \, m_{j} \, \gamma_{\pm} / m^{0}\right)\]\[\text { Here } Q^{v}=v_{+}^{v(+)} \, v_{-}^{v(-)}\]In equation (b), \(ν_{+}\) and \(ν_{-}\) are the number of moles of cations and anions respectively produced on complete dissociation by one mole of salt; \(ν = ν_{+} + ν_{-}\). Here \(\gamma_{\pm}\) is the mean ionic activity coefficient where by definition, at all \(\mathrm{T}\) and \(\mathrm{p}\), \[\operatorname{limit}\left(m_{j} \rightarrow 0\right) \gamma_{\pm}=1\]If the thermodynamic properties of the solution are ideal than \(\gamma_{\pm}\) is unity. However the thermodynamic properties of salt solutions, even quite dilute solutions, are not ideal as a consequence of strong long-range charge-charge interactions between ions in solution. The challenge is therefore to come up with an equation for \(\gamma_{\pm}\) granted that the temperature, pressure and properties of the solvent and salt are known together with the composition of the solution. The first successful attempt to meet this challenge was published by Debye and Huckel in 1923 and 1924.In most published accounts, the CGS system of units is used. However here we use the SI system and trace the units through the treatment. The solvent is a dielectric (structureless) continuum characterised by its relative permittivity, \(\mathcal{\varepsilon}_{\mathrm{r}}\). The solute (salt) comprises ions characterised by their charge and radius; e.g. for ion-\(j\), charge \(\mathrm{z}_{j} \, e\) and radius \(\mathrm{r}_{j}\) such that for cations \(\mathrm{z}_{\mathrm{j}} \geq 1\) and for anions \(\mathrm{z}_{j} \leq -1\) where \(\mathrm{z}_{j}\) is an integer.The analysis combines two important physical chemical relationships; Boltzmann’s Law and Poisson’s Equation.We consider an aqueous salt solution containing \(\mathrm{i}\)-ionic substances, each substance having molality mi. The solution contains cations and anions. A KEY condition requires that the electric charge on the solution is zero. \[\text { Thus, } \sum_{j=1}^{j=i} m_{j} \, z_{j}=0\]Published accounts of the Debye-Huckel equation almost always use the concentration scale because the analysis concentrates on the distances between ions in solution rather than their mass. The concentration of \(j\) ions in a solution, volume \(\mathrm{V}\), is given by equation (e) \[\text { Thus, } \sum_{j=1}^{j=i} c_{j} \, z_{j}=0\]\[\mathrm{c}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} / \mathrm{V}\]Here \(\mathrm{n}_{j}\) is the amount of solute \(j\) (expressed using the unit, mole) and \(\mathrm{V}\) is the volume of solution.These equations describe the solution as seen from the standpoint of a chemist interested in the properties of a given solution. However the ‘view’ from the standpoint of, for example, a cation in the solution is quite different. The neutrality condition in equation (d) requires that the electric charge on the solution surrounding the cation \(j\) with charge \(+\left|z_{j} \, e\right|\) equals \(-\left|z_{j} \, e\right|\); i.e. equal in magnitude but opposite in sign. This is the electric charge on the rest of the solution and constitutes the ‘ion atmosphere’ of the \(j\) ion. Every ion in the solution has its own atmosphere having a charge equal in magnitude but opposite in sign. Moreover interaction between a j ion and its atmosphere stabilises the \(j\) ion in solution. The task of the theory is to obtain an equation for this stabilisation of the salt (i.e. the lowering of its chemical potential in solution). Intuitively we might conclude that this stabilisation is a function of the ionic strength of the salt solution and the dielectric properties of the solvent.We consider a reference \(j\) ion, radius \(\mathrm{r}_{j}\), in solution together with a small volume element, \(\mathrm{dV}\), a distance not more than say (\(50 \times \mathrm{r}_{j}\)) from the \(j\) ion. In terms of probabilities, if the \(j\) ion is a cation the probability of finding an anion in the reference volume is greater than finding a cation. Again with the \(j\) ion as reference, we identify a time averaged electric potential \(\psi_{j}\) at the volume element. The distribution of ions about the cation \(j\) is assumed to follow the Boltzmann distribution law. The time average number of cations \(\mathrm{dn}_{+}\) and anions \(\mathrm{dn}_{-}\) in the volume element is given by equation (g) where ion \(\mathrm{i}\) is, in turn, taken as a cation and then as an anion. \[\mathrm{dn}_{\mathrm{i}}=\mathrm{p}_{\mathrm{i}} \, \exp \left(-\frac{\mathrm{z}_{\mathrm{i}} \, \mathrm{e} \, \boldsymbol{\psi}_{\mathrm{j}}}{\mathrm{k} \, \mathrm{T}}\right) \, \mathrm{dV}\]Here \(\mathrm{p}_{\mathrm{i}}\) is the number of \(\mathrm{i}\) ions in unit volume of the solution. Each i ion has electric charge \(z_{i} \, e\). Hence the electric charge on the volume \(\mathrm{dV}\) is obtained by summing over the charge on the time average number of all ions. The charge density \(\rho_{j}\) is given by equation (h), where the subscript \(j\) on \(\rho_{j}\) stresses that the charge is described with respect to the charge on the \(j\) ion. \[\rho_{\mathrm{j}}=\sum \mathrm{p}_{\mathrm{i}} \, \mathrm{z}_{\mathrm{i}} \, \mathrm{e} \, \exp \left(-\frac{\mathrm{z}_{\mathrm{i}} \, \mathrm{e} \, \psi_{\mathrm{j}}}{\mathrm{k} \, \mathrm{T}}\right)\]The subscript \(j\) on \(\rho_{j}\) and \(\psi_{j}\) identifies the impact of ion \(j\) on the composition and electric potential of the reference volume \(\mathrm{dV}\) distance \(\mathrm{r}\) from the \(j\) ion. At this point some simplification is welcomed. We expand the exponential in equation (h). \[\text { Hence, } \quad \rho_{j}= \sum \mathrm{p}_{\mathrm{i}} \, \mathrm{z}_{\mathrm{i}} \, \mathrm{e}-\sum \mathrm{p}_{\mathrm{i}} \, \mathrm{z}_{\mathrm{i}} \, \mathrm{e} \,\left(\frac{\mathrm{z}_{\mathrm{i}} \, \mathrm{e} \, \psi_{\mathrm{j}}}{\mathrm{k} \, \mathrm{T}}\right)+\sum \mathrm{p}_{\mathrm{i}} \, \mathrm{z}_{\mathrm{i}} \, \mathrm{e} \,(1 / 2) \,\left(\frac{\mathrm{z}_{\mathrm{i}} \, \mathrm{e} \, \psi_{\mathrm{j}}}{\mathrm{k} \, \mathrm{T}}\right)^{2}-\ldots \ldots\]The solution as a whole has zero electric charge. \[\text { Hence } \quad \sum \mathrm{p}_{\mathrm{i}} \, \mathrm{z}_{\mathrm{i}} \, \mathrm{e}=0\]\[\text { Also for dilute solutions, }\left(\frac{\mathrm{z}_{\mathrm{i}} \, \mathrm{e} \, \psi_{\mathrm{j}}}{\mathrm{k} \, \mathrm{T}}\right) \ll<1\]Hence the third and all subsequent terms in equation (i) are negligibly small. \[\text { Therefore } \rho_{j}=-\sum \frac{p_{i} \,\left(z_{i} \, e\right)^{2} \, \psi_{j}}{k \, T}\]The approximation leading to equation (l) is welcome for an important reason. Equation (l) satisfies a key condition which requires a linear interdependence between \(\rho_{j}\) and \(\psi_{j}\).Equation (l) relates charge density \(\rho_{j}\) and electric potential \(\psi_{j}\). These two properties are also related by Poisson’s theorem: \[\nabla^{2} \psi_{j}=-\rho_{j} / \varepsilon_{0} \, \varepsilon_{\mathrm{r}}\]Here \(\varepsilon_{0}\) is the permittivity of free space; \(\varepsilon_{\mathrm{r}\) is the relative permittivity of the solvent; \(\rho_{j}\) is the charge density per unit volume.In the case considered here, the electric charges (ions) are spherically distributed about the reference \(j\) ion. Then Poisson’s equation takes the following form. \[\left[\frac{1}{r^{2}}\right] \, \frac{d}{d r}\left(r^{2} \, \frac{d \psi_{j}}{d r}\right)=-\frac{\rho_{j}}{\varepsilon_{0} \, \varepsilon_{r}}\]Combination of equations (l) and (n) yields the key equation (o). \[\left[\frac{1}{r^{2}}\right] \, \frac{d}{d r}\left(r^{2} \, \frac{d \psi_{j}}{d r}\right)=\frac{e^{2}}{\varepsilon_{0} \, \varepsilon_{r} \, k \, T} \, \sum p_{i} \, z_{i}^{2} \, \psi_{j}\]\[\text { Or, } \quad\left[\frac{1}{\mathrm{r}^{2}}\right] \, \frac{\mathrm{d}}{\mathrm{dr}}\left(\mathrm{r}^{2} \, \frac{\mathrm{d} \psi_{\mathrm{j}}}{\mathrm{dr}}\right)=\kappa^{2} \, \psi_{\mathrm{j}}\]\[\text { where } \kappa^{2}=\frac{e^{2}}{\varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{k} \, \mathrm{T}} \sum \mathrm{p}_{\mathrm{i}} \, \mathrm{z}_{\mathrm{i}}^{2}\]Property \(\kappa\) has the unit ‘reciprocal distance’. Equation (p) is a second order differential equation having the general solution given by equation (p). \[\psi_{j}=A_{1} \, \exp (-\kappa \, r) / r+A_{2} \, \exp (\kappa \, r) / r\]However \(\operatorname{limit}(r \rightarrow \infty) \exp (\kappa \, r) / r\) is very large where \(\psi_{j}\) is zero. Hence \(\mathrm{A}_{2}\) must be zero. \[\text { Therefore, } \Psi_{j}=\mathrm{A}_{1} \, \exp (-\kappa \, \mathrm{r}) / \mathrm{r}\]We combine equations (l) and (s). \[\rho_{j}=-A_{1} \, \frac{\exp (-\kappa \, r)}{r} \, \sum \frac{p_{i} \,\left(z_{i} \, e\right)^{2}}{k \, T}\]Using the definition of \(\kappa^{2}\) in equation (q), \[\rho_{j}=-A_{1} \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \kappa^{2} \, \frac{\exp (-\kappa \, r)}{r}\]At this point, a geometric condition is taken into account. Charge density \(\rho_{j}\) describes the electrical properties of the solution ‘outside ‘ the \(j\) ion. No other ions can approach the \(j\) ion closer than a ‘distance of closest approach’ \(\mathrm{a}_{j}\). The total charge on the solution ‘outside’ the \(j\) ion equals in magnitude but opposite in sign that on the \(j\) ion. Hence, \[4 \, \pi \, \int_{a(j)}^{\infty} \rho_{j} \, r^{2} \, d r=-z_{j} \, e\]\[\text { Or, } 4 \, \pi \, \int_{a(j)}^{\infty}\left[-A_{1} \, \varepsilon_{0} \, \varepsilon_{r} \, K^{2} \, \frac{\exp (-K \, r)}{r}\right] \, r^{2} \, d r=-z_{j} \, e\]This integration yields an equation for \(\mathrm{A}_{1}\). \[A_{1}=\frac{\left(z_{j} \, e\right) \, \exp \left(\kappa \, a_{j}\right)}{4 \, \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \,\left(1+\kappa \, a_{j}\right)}\]\[\text { Hence } \psi_{\mathrm{j}}=\frac{\left(\mathrm{z}_{\mathrm{j}} \, \mathrm{e}\right) \, \exp \left(\kappa \, \mathrm{a}_{\mathrm{j}}\right)}{4 \, \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \,\left(1+\kappa \, \mathrm{a}_{\mathrm{j}}\right)} \, \frac{\exp (-\kappa \, \mathrm{r})}{\mathrm{r}}\]We recall that \(\psi_{j}\) is the electric potential at distance \(\mathrm{r}\) from the \(j\) ion. In the event that the solution contains just the \(j\) ion (i.e. an isolated \(j\) ion) with charge \(z_{j} \, e\), the electric potential \(\psi(\text { iso })\), distance \(\mathrm{r}\) from the \(j\) ion, is given by equation (z). \[\psi_{j}(i S 0)=\frac{z_{j} \, e}{4 \, \pi \, \varepsilon_{0} \, \varepsilon_{r} \, r}\]The electric potential \(psi_{j}\) given by equation (y) is the sum of \(\psi(\text { iso })\) and the electric potential produced by all other ions in solution \(\psi(\text { rest })\). \[\text { Then } \quad \psi_{j}=\psi_{j}(\text { iso })+\psi_{j}(\text { rest })\]\[\text { Hence, } \quad \psi_{\mathrm{j}}(\text { rest })=\frac{\mathrm{z}_{\mathrm{j}} \, \mathrm{e}}{4 \, \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{r}}\left[\frac{\exp \left(\kappa \, \mathrm{a}_{\mathrm{j}}\right) \, \exp (-\kappa \, \mathrm{r})}{1+\kappa \, \mathrm{a}_{\mathrm{j}}}-1\right]\]Equation (zb) is valid for all values of \(\mathrm{r}\), including for \(\mathrm{r}=\mathrm{a}_{\mathrm{j}}\). Then from equation (zb), \(\psi_{\mathrm{j}}(\text { rest })\) at \(\mathrm{r} = \mathrm{a}_{j}\) is given by equation (zc). \[\psi_{j}(\text { rest })=-\frac{z_{j} \, e}{4 \, \pi \, \varepsilon_{0} \, \varepsilon_{r}} \, \frac{\kappa}{1+\kappa \, a_{j}}\]We imagine that the \(j\) ion is isolated in solution and that the electrical interaction with all other \(i\) ions is then switched on at fixed \(\mathrm{T}\) and \(\mathrm{p}\). The change in chemical potential of single \(j\) ion is given by equation (zd), \[\Delta \mu_{j}(\mathrm{elec})=-\frac{\mathrm{z}_{\mathrm{j}}^{2} \, \mathrm{e}^{2}}{4 \, \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}}} \, \frac{\kappa}{1+\kappa \, \mathrm{a}_{\mathrm{j}}}\]For one mole of \(j\) ions, \(\Delta \mu_{\mathrm{j}}(\mathrm{elec})\) is given by equation (ze) where an additional factor of ‘2’ is introduced into the denominator. Otherwise each ion would be counted twice; i.e. once as the \(j\) ion and once in solution around the \(j\) ion. \[\Delta \mu_{\mathrm{j}}(\mathrm{elec} ; \text { one mole })=-\frac{\mathrm{z}_{\mathrm{j}}^{2} \, \mathrm{e}^{2} \, \mathrm{N}_{\mathrm{A}}}{8 \, \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}}} \, \frac{\kappa}{1+\kappa \, \mathrm{a}_{\mathrm{j}}}\]The chemical potential of single ion \(j\) in an aqueous solution, \(\mu_{\mathrm{j}}(\mathrm{aq})\) is related to molality \(\mathrm{m}_{j}\) and single ion activity coefficient \(\gamma_{j}\) using equation (zf). \[\mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\gamma_{\mathrm{j}}\right)\]Comparison of equations (ze) and (zf) yields equation (zg). \[\ln \left(\gamma_{\mathrm{j}}\right)=-\frac{\mathrm{z}_{\mathrm{j}}^{2} \, \mathrm{e}^{2}}{8 \, \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{k} \, \mathrm{T}} \, \frac{\kappa}{1+\kappa \, \mathrm{a}_{\mathrm{j}}}\]The mean ionic activity coefficient \(\gamma_{\pm}\) for the salt in solution is given by equation (zh); i.e. for a simple salt where each mole of salt contains \(ν_{+}\) moles of cations and \(ν_{-}\) moles of anions. \[\gamma_{\pm}=\left(\gamma_{+}^{v+} \, \gamma_{-}^{v-}\right)^{1 / v}\]\[\text { Or, } \quad\left(v_{+}+v_{-}\right) \, \ln \left(\gamma_{\pm}\right)=v_{+} \, \ln \left(\gamma_{+}\right)+v_{-} \, \ln \left(\gamma_{-}\right)\]We envisage closest approaches only between differently charged ions. Then for a given salt, \(\mathrm{a}_{+}=\mathrm{a}_{-}=\mathrm{a}_{\mathrm{j}}\). Hence from equation (zg), \[\ln \left(\gamma_{\pm}\right)=-\frac{\mathrm{e}^{2}}{8 \, \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{k} \, \mathrm{T}} \,\left(\frac{\kappa}{1+\kappa \, \mathrm{a}_{\mathrm{j}}}\right) \,\left(\frac{\mathrm{v}_{+} \, \mathrm{z}_{+}^{2}+\mathrm{v}_{-} \, \mathrm{z}_{-}^{2}}{\mathrm{v}_{+}+\mathrm{v}_{-}}\right)\]But the salt is overall electrically neutral. \[\text { Or, } \quad \mathrm{V}_{+} \, \mathrm{Z}_{+}=-\mathrm{V}_{-} \, \mathrm{Z}_{-}\]\[\text { Whence, } \quad v_{+}=-v_{-} \, z_{-} / z_{+}\]\[\text { So }, \frac{\mathrm{v}_{+} \, \mathrm{z}_{+}^{2}+\mathrm{v}_{-} \, \mathrm{z}_{-}^{2}}{\mathrm{v}_{+}+\mathrm{v}_{-}}=-\mathrm{z}_{+} \, \mathrm{z}_{-}\]Hence we arrive at an equation for the mean ionic activity coefficient, \(\gamma_{\pm}\). \[\ln \left(\gamma_{\pm}\right)=\frac{z_{+} \, z_{-} \, e^{2}}{8 \, \pi \, \varepsilon_{0} \, \varepsilon_{s} \, k \, T} \,\left(\frac{K}{1+\kappa \, a_{j}}\right)\]At this point we return to equation (q) and recall that \(\mathrm{p}_{\mathrm{i}\) is the number of ions in unit volume of solution. If the concentration of \(\mathrm{i}\) ions equals \(\mathrm{c}_{\mathrm{i}}\), (with \(\mathrm{N}_{\mathrm{A}} =\) the Avogadro constant), \[\text { then } \mathrm{p}_{\mathrm{i}}=\mathrm{N}_{\mathrm{A}} \, \mathrm{c}_{\mathrm{i}}\]\[\text { Therefore }, \quad \kappa^{2}=\frac{e^{2} \,\left(N_{A}\right)^{2}}{\varepsilon_{0} \, \varepsilon_{r} \, R \, T} \, \sum c_{i} \, z_{i}^{2}\]The convention is to express concentrations using the unit, \(\mathrm{mol dm}^{-3}\) for which we use the symbol, \(\mathrm{c}^{\prime}\). \[\text { Hence } \quad \kappa^{2}=\frac{\mathrm{e}^{2} \,\left(\mathrm{N}_{\mathrm{A}}\right)^{2}}{10^{3} \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{R} \, \mathrm{T}} \, \sum \mathrm{c}_{\mathrm{i}}^{\prime} \, \mathrm{z}_{\mathrm{i}}^{2}\]For dilute solutions, the following approximation is valid where \[\text { ionic strength } \mathrm{I}=(1 / 2) \, \sum \mathrm{m}_{\mathrm{i}} \, \mathrm{z}_{\mathrm{i}}^{2}\]\[\kappa^{2}=\frac{2 \, \mathrm{e}^{2} \,\left(\mathrm{N}_{\mathrm{A}}\right)^{2} \, \rho_{1}^{*}(\ell)}{\varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{R} \, \mathrm{T}} \, \mathrm{I}\]From equations (zn) and (zs), \[\ln \left(\gamma_{\pm}\right)=\frac{\mathrm{z}_{+} \, \mathrm{z}_{-} \, \mathrm{e}^{2}}{8 \, \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{k} \, \mathrm{T}} \,\left[\frac{2 \, \mathrm{e}^{2}\left(\mathrm{~N}_{\mathrm{A}}\right)^{2} \, \rho_{1}^{*}(\ell)}{\varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{R} \, \mathrm{T}}\right]^{1 / 2} \, \frac{(\mathrm{I})^{1 / 2}}{1+\kappa \, \mathrm{a}_{\mathrm{j}}}\]For very dilute solutions, the Debye Huckel Limiting Law (DHLL) is used where it is assumed that \(1+\kappa \, \mathrm{a}_{\mathrm{j}}=1.0\). Hence, \[\ln \left(\gamma_{\pm}\right)=\frac{\mathrm{e}^{3} \,\left[2 \, \mathrm{N}_{\mathrm{A}} \, \rho_{1}^{*}(\ell)\right]^{1 / 2}}{8 \, \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{k} \, \mathrm{T}} \,\left[\frac{\mathrm{N}_{\mathrm{A}}}{\varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{R} \, \mathrm{T}}\right]^{1 / 2} \, \mathrm{z}_{+} \, \mathrm{z}_{-} \,(\mathrm{I})^{1 / 2}\]Equation (zu) may be written in the following form. \[\ln \left(\gamma_{\pm}\right)=\frac{\mathrm{e}^{3} \,\left[2 \, \mathrm{N}_{\mathrm{A}} \, \rho_{1}^{*}(\ell)\right]^{1 / 2}}{8 \, \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{k} \, \mathrm{T}} \,\left[\frac{\mathrm{N}_{\mathrm{A}}}{\varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{R} \, \mathrm{T}}\right]^{1 / 2} \, \mathrm{z}_{+} \, \mathrm{z}_{-} \,(\mathrm{I})^{1 / 2}\]For aqueous solutions at ambient pressure and \(298.15 \mathrm{~K}\), \(\rho_{1}^{*}(\ell)=997.047 \mathrm{~kg} \mathrm{~m}^{-3}\) and \(\varepsilon_{\mathrm{r}}=78.36\). \[\text { Hence } \ln \left(\gamma_{\pm}\right)=(1.1749) \, \mathrm{z}_{+} \, \mathrm{z}_{-} \,\left(\mathrm{I} / \mathrm{mol} \mathrm{kg}^{-1}\right)^{1 / 2}\]We note that with \(Z_{+} \, Z_{-}=-\left|Z_{+} \, Z_{-}\right|\), \(\ln \left(\gamma_{\pm}\right)<0\).In other words \(\ln \left(\gamma_{\pm}\right)\) is a linear function of the square root of the ionic strength I. Most authors choose to write equation (zx) using logarithms to base 10. \[\text { Then } \log \left(\gamma_{\pm}\right)=\left|\mathrm{z}_{+} \, \mathrm{z}_{-}\right| \, \mathrm{A}_{\gamma} \,\left(\mathrm{I} / \mathrm{mol} \mathrm{kg}^{-1}\right)^{1 / 2}\]Here \(\mathrm{A}_{\gamma}=0.510\). Certainly the latter constant is readily remembered as ‘one-half’. Slight disagreements between published estimates of \(\mathrm{A}_{\gamma}\) are a result of different estimates of \(\varepsilon_{\mathrm{r}}\) and \(\rho_{1}^{*}(\ell)\). Harned and Owen [1d] published a useful Table for \(\mathrm{A}_{\gamma}\) as a function of temperature for aqueous solutions.The full equation for \(\ln \left(\gamma_{\pm}\right)\) following on from equation (zt) takes the following form. \[\ln \left(\gamma_{\pm}\right)=\frac{-\left|z_{+} \, z_{-}\right| \, S_{\gamma} \,\left(\mathrm{I} / \mathrm{mol} \mathrm{kg}^{-1}\right)^{1 / 2}}{\left.1+\beta \, a_{j} \,(\mathrm{I} / \mathrm{mol} \mathrm{kg})^{-1}\right)^{1 / 2}}\]For aqueous solutions at ambient pressure and \(298.15 \mathrm{~K}\), \(\mathrm{S}_{\gamma}=1.175\) and \(\beta=3.285 \mathrm{~nm}^{-1}\). Adam suggests that aj can be treated as a variable in fitting the measured dependence of \(\ln \left(\gamma_{\pm}\right)\) on ionic strength for a given salt.Footnotes P. Debye and E. Huckel, Physik. Z.,1923, 24,185,334;1924,25,97. For accounts of the theory see-- In equation (g), dni describes the number of ions in volume \(\mathrm{dV}\); \(\mathrm{p}_{\mathrm{i}}\) describes the number of ions in unit volume of solution. [In other words the units used to express \(\mathrm{p}_{\mathrm{i}}\) and \(\mathrm{dn}_{\mathrm{i}}\) differ. \[\begin{gathered} \frac{\mathrm{z}_{\mathrm{i}} \, \mathrm{e} \, \psi_{\mathrm{j}}}{\mathrm{k} \, \mathrm{T}}=\frac{ \,[\mathrm{C}] \,[\mathrm{V}]}{\left[\mathrm{J} \mathrm{K}^{-1}\right] \,[\mathrm{K}]}=\frac{ \,[\mathrm{A} \mathrm{s}] \,\left[\mathrm{J} \mathrm{A}^{-1} \mathrm{~s}^{-1}\right]}{\left[\mathrm{J} \mathrm{K}^{-1}\right] \,[\mathrm{K}]}= \\ \mathrm{p}_{\mathrm{i}} \, \exp \left(-\frac{\mathrm{z}_{1} \, \mathrm{e} \, \psi_{\mathrm{j}}}{\mathrm{k} \, \mathrm{T}}\right) \, \mathrm{dV}=\left[\frac{1}{\mathrm{~m}^{3}}\right] \, \,\left[\mathrm{m}^{3}\right]= \\ \mathrm{dn}_{\mathrm{i}}= \quad \mathrm{p}_{\mathrm{i}}=\left[\mathrm{m}^{-3}\right] \end{gathered}\] \(\frac{\mathrm{z}_{\mathrm{i}} \, \mathrm{e} \, \psi}{\mathrm{k} \, \mathrm{T}}=\frac{ \,[\mathrm{C}] \,[\mathrm{V}]}{\left[\mathrm{J} \mathrm{K}^{-1}\right] \,[\mathrm{K}]}=\frac{[\mathrm{A} \mathrm{s}] \,\left[\mathrm{J} \mathrm{A}^{-1} \mathrm{~s}^{-1}\right]}{[\mathrm{J}]}=\) Then \(\rho_{\mathrm{j}}=\left[\frac{1}{\mathrm{~m}^{3}}\right] \, \,[\mathrm{C}]=\left[\frac{\mathrm{C}}{\mathrm{m}^{3}}\right]\); i.e. charge per unit volume \(\exp (x)=1+\frac{x}{1 !}+\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}+\ldots \ldots\) \[\rho_{\mathrm{j}}=\frac{\left[\mathrm{m}^{-3}\right] \,^{2} \,[\mathrm{C}]^{2} \,[\mathrm{V}]}{\left[\mathrm{J} \mathrm{K}^{-1}\right] \,[\mathrm{K}]}=\frac{\left[\mathrm{m}^{-3}\right] \,[\mathrm{A} \mathrm{s}]^{2} \,\left[\mathrm{J} \mathrm{A}^{-1} \mathrm{~s}^{-1}\right]}{[\mathrm{J}]}=\left[\mathrm{C} \mathrm{m}^{-3}\right]\]i.e. charge per unit volume \(\nabla^{2} \psi_{\mathrm{j}}=\frac{1}{\left[\mathrm{~m}^{2}\right]} \,[\mathrm{V}]=\left[\frac{\mathrm{V}}{\mathrm{m}^{2}}\right]\) \[\rho_{\mathrm{j}} / \varepsilon_{0} \, \varepsilon_{\mathrm{r}}=\frac{\left[\mathrm{C} \mathrm{m}^{-3}\right]}{\left[\mathrm{Fm}^{-1}\right] \,}=\frac{\left[\mathrm{As} \mathrm{} \mathrm{m}^{-3}\right]}{\left[\mathrm{AsV^{-1 } \mathrm { m } ^ { - 1 } ]}\right.}=\left[\frac{\mathrm{V}}{\mathrm{m}^{2}}\right]\] \(\left[\frac{1}{r^{2}}\right] \, \frac{d}{d r}\left(r^{2} \, \frac{d \psi_{j}}{d r}\right)=\left[\frac{1}{m^{2}}\right] \, \frac{1}{[m]} \,\left[m^{2}\right] \, \frac{\left[\mathrm{J} \mathrm{A}^{-1} \mathrm{~s}^{-1}\right]}{[m]}=\left[\frac{\mathrm{J} \mathrm{A}^{-1} \mathrm{~s}^{-1}}{\mathrm{~m}^{2}}\right]\)\[\frac{\rho}{\varepsilon_{0} \, \varepsilon_{\mathrm{r}}}=\left[\frac{\mathrm{C}}{\mathrm{m}^{3}}\right] \, \frac{1}{\left[\mathrm{Fm}^{-1}\right]} \, \frac{1}{}=\left[\frac{\mathrm{As}}{\mathrm{m}^{3}}\right] \,\left[\frac{1}{\mathrm{As} \mathrm{V}^{-1} \mathrm{~m}^{-1}}\right]=\left[\frac{\mathrm{V}}{\mathrm{m}^{2}}\right]=\left[\frac{\mathrm{J} \mathrm{A}^{-1} \mathrm{~s}^{-1}}{\mathrm{~m}^{2}}\right]\] \[\begin{aligned} &\kappa^{2}=\frac{[\mathrm{C}]^{2}}{\left[\mathrm{Fm}^{-1}\right] \, \,\left[\mathrm{J} \mathrm{K}^{-1}\right] \,[\mathrm{K}]} \,\left[\frac{1}{\mathrm{~m}^{3}}\right] \,^{2}\\ &\kappa=\left[\mathrm{m}^{-1}\right] \end{aligned}\] \[\begin{aligned} &\frac{\mathrm{d}}{\mathrm{dr}}\left(\mathrm{r}^{2} \, \frac{\mathrm{d} \psi_{\mathrm{j}}}{\mathrm{dr}}\right)=2 \, \mathrm{r} \, \frac{\mathrm{d} \psi_{\mathrm{j}}}{\mathrm{dr}}+\mathrm{r}^{2} \, \frac{\mathrm{d}^{2} \psi_{\mathrm{j}}}{\mathrm{dr}^{2}}=\kappa^{2} \, \psi_{\mathrm{j}} \\ &\text { Then, } \mathrm{r}^{2} \, \frac{\mathrm{d}^{2} \psi_{\mathrm{j}}}{\mathrm{dr}}+2 \, \mathrm{r} \, \frac{\mathrm{d} \psi_{\mathrm{j}}}{\mathrm{dr}}-\mathrm{\kappa}^{2} \, \psi_{\mathrm{j}}=0 \end{aligned}\] \(\psi_{j}=[\mathrm{V}]\) and \(\psi_{\mathrm{j}}=\mathrm{A}_{1} \, \exp \left([\mathrm{m}]^{-1} \,[\mathrm{m}]\right) /[\mathrm{m}] \quad \mathrm{A}_{1}=[\mathrm{Vm}]\) \(\rho_{\mathrm{j}}=\left[\frac{\mathrm{C}}{\mathrm{m}^{3}}\right]\) \[\mathrm{A}_{1} \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \kappa^{2} \, \frac{\exp (-\mathrm{K} \, \mathrm{r})}{\mathrm{r}}=[\mathrm{V} \mathrm{m}] \,\left[\mathrm{Fm} \mathrm{m}^{-1}\right] \, \,[\mathrm{m}]^{-2} \, \frac{}{[\mathrm{m}]}\] \(\psi_{j}(\text { iso })=\frac{ \,[\mathrm{C}]}{ \, \,\left[\mathrm{Fm}^{-1}\right] \, \,[\mathrm{m}]}=\frac{[\mathrm{As}]}{\left[\mathrm{As} \mathrm{V}^{-1}\right]}=[\mathrm{V}]\) \[\begin{aligned} &\Delta \mu_{j}(\mathrm{elec} ; \text { one mole })=\frac{^{2} \,[\mathrm{C}]^{2} \,[\mathrm{mol}]^{-1}}{ \,\left[\mathrm{F} \mathrm{m}^{-1}\right] \,} \,\left[\frac{[\mathrm{m}]^{-1}}{1+[\mathrm{m}]^{-1} \,[\mathrm{m}]}\right] \\ &=\frac{[\mathrm{A} \mathrm{s}]^{2} \,\left[\mathrm{mol}^{-1}\right] \,[\mathrm{m}]^{-1}}{\left[\mathrm{~A} \mathrm{~s} \mathrm{} \mathrm{V}^{-1} \mathrm{~m}^{-1}\right]}=\frac{[\mathrm{A} \mathrm{s}] \,[\mathrm{mol}]^{-1}}{\left[\mathrm{~J}^{-1} \mathrm{~A} \mathrm{~s}\right]}=\left[\mathrm{J} \mathrm{mol}^{-1}\right] \end{aligned}\] \[\begin{aligned} &{ \frac{\mathrm{v}_{+} \, \mathrm{z}_{+}^{2}+\mathrm{v}_{-} \, \mathrm{z}_{-}^{2}}{\mathrm{v}_{+}+\mathrm{v}_{-}}=\frac{-\left(\mathrm{v}_{-} \, \mathrm{z}_{-} \, \mathrm{z}_{+}\right)+\left(\mathrm{v}_{-} \, \mathrm{z}_{-}^{2}\right)}{-\left(\mathrm{v}_{-} \, \mathrm{z}_{-} / \mathrm{z}_{+}\right)+\mathrm{v}_{-}}} \\ &=-\mathrm{z}_{+} \, \mathrm{z}_{-}\left[\frac{\mathrm{v}_{-}-\left(\mathrm{v}_{-} \, \mathrm{z}_{-} / \mathrm{z}_{+}\right)}{-\left(\mathrm{v}_{-} \, \mathrm{z}_{-} / \mathrm{z}_{+}\right)+\mathrm{v}_{-}}\right]=-\mathrm{z}_{+} \, \mathrm{z}_{-} \end{aligned}\] \[\begin{aligned} &\ln \left(\gamma_{\pm}\right)=\frac{ \,[\mathrm{C}]^{2}}{ \,\left[\mathrm{F} \mathrm{m}^{-1}\right] \, \,\left[\mathrm{J} \mathrm{K}^{-1}\right] \,[\mathrm{K}]} \,\left[\frac{[\mathrm{m}]^{-1}}{1+[\mathrm{m}]^{-1} \,[\mathrm{m}]}\right]\\ &=\frac{[\mathrm{A} \mathrm{s}]^{2}}{\left[\mathrm{As} \mathrm{} \mathrm{V}^{-1}\right]} \, \frac{1}{[\mathrm{~J}]}=\frac{[\mathrm{As}]}{\left[\mathrm{J}^{-1} \mathrm{As}\right] \,[\mathrm{J}]}= \end{aligned}\] \(\mathrm{N}_{\mathrm{A}} \, \mathrm{c}_{\mathrm{i}}=\left[\mathrm{mol}^{-1}\right] \,\left[\mathrm{mol} \mathrm{m}^{-3}\right]=\left[\mathrm{m}^{-3}\right]\) \[\begin{aligned} &\kappa^{2}=\frac{[\mathrm{C}]^{2} \,\left[\mathrm{mol}^{-1}\right]^{2}}{\left[\mathrm{~F} \mathrm{~m}^{-1}\right] \, \,\left[\mathrm{J} \mathrm{mol}^{-1} \mathrm{~K}^{-1}\right] \,[\mathrm{K}]} \,\left[\mathrm{mol} \mathrm{m}^{-3}\right] \\ &=\frac{[\mathrm{A} \mathrm{s}]^{2} \,[\mathrm{m}]^{-2}}{\left[\mathrm{~A} \mathrm{~s} \mathrm{} \mathrm{V}^{-1}\right] \,[\mathrm{J}]}=\frac{[\mathrm{As}] \,[\mathrm{m}]^{-2}}{\left[\mathrm{~J}^{-1} \mathrm{~A} \mathrm{~s}\right] \,[\mathrm{J}]}=[\mathrm{m}]^{-2} \end{aligned}\] \[\begin{aligned} &\frac{\mathrm{z}_{+} \, \mathrm{z}_{-} \, \mathrm{e}^{2}}{8 \, \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{k} \, \mathrm{T}}=\frac{ \,[\mathrm{C}]^{2}}{ \, \,\left[\mathrm{Fm}^{-1}\right] \,\left[\mathrm{JK}^{-1}\right] \,[\mathrm{K}]}\\ &=\frac{\left[\mathrm{A}^{2} \mathrm{~s}^{2}\right]}{\left[\mathrm{A} \mathrm{s} \mathrm{} \mathrm{J}^{-1} \mathrm{As} \mathrm{} \mathrm{m}^{-1}\right] \,[\mathrm{J}]}=[\mathrm{m}]\\ &\left[\frac{2 \, \mathrm{e}^{2} \,\left(\mathrm{N}_{\mathrm{A}}\right)^{2} \, \rho_{1}^{*}(\ell)}{\varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{R} \, \mathrm{T}}\right]^{1 / 2}=\left[\frac{ \,[\mathrm{C}]^{2} \,\left[\mathrm{mol}^{-1}\right]^{2} \,\left[\mathrm{kg} \mathrm{m}^{-3}\right]}{\left[\mathrm{Fm}^{-1}\right] \, \,\left[\mathrm{J} \mathrm{mol}^{-1} \mathrm{~K}^{-1}\right] \,[\mathrm{K}]}\right]^{1 / 2}\\ &=\left[\frac{\left[\mathrm{A}^{2} \mathrm{~s}^{2}\right] \,\left[\mathrm{mol}^{-1}\right]^{2} \,\left[\mathrm{kg} \mathrm{m}^{-3}\right]}{\left[\mathrm{As} \mathrm{J} \mathrm{A} \mathrm{s} \mathrm{}{ }^{-1}\right] \,\left[\mathrm{J} \mathrm{mol}^{-1}\right]}\right]^{1 / 2}=\left[\left[\mathrm{mol}^{-1}\right] \,[\mathrm{kg}] \,\left[\mathrm{m}^{-2}\right]\right]^{1 / 2}\\ &=\frac{\left[\mathrm{m}^{-1}\right]}{\left[\mathrm{mol} \mathrm{kg}^{-1}\right]^{1 / 2}}\\ &\kappa \, \mathrm{a}_{\mathrm{j}}=\left[\mathrm{m}^{-1}\right] \,[\mathrm{m}]=\\ &\frac{\mathrm{z}_{+} \, \mathrm{z}_{-} \, \mathrm{e}^{2}}{8 \, \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{k} \, \mathrm{T}} \,\left[\frac{2 \, \mathrm{e}^{2}\left(\mathrm{~N}_{\mathrm{A}}\right)^{2} \, \rho_{1}^{*}(\ell)}{\varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{R} \, \mathrm{T}}\right]^{1 / 2} \, \frac{(\mathrm{I})^{1 / 2}}{1+\kappa \, \mathrm{a}_{\mathrm{j}}}\\ &=[\mathrm{m}] \, \frac{\left[\mathrm{m}^{-1}\right]}{\left[\mathrm{mol} \mathrm{kg}^{-1}\right]^{1 / 2}} \, \frac{\left[\mathrm{mol} \mathrm{kg}^{-1}\right]^{1 / 2}}{}= \end{aligned}\]As required \(\ln \left(\gamma_{\pm}\right)=\) J. S. Winn, Physical Chemistry, Harper Collins, New York, 1995,page 315. M. L. McGlashan, Chemical Thermodynamics, Academic Press, 1979, p 304. Using N. K. Adam, Physical Chemistry, Oxford, 1956, page 395.This page titled 1.5.21: Chemical Potentials- Salt Solutions- Debye-Huckel Equation is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.6.1: Composition- Mole Fraction- Molality- Concentration

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.06%3A_Composition/1.6.01%3A_Composition-_Mole_Fraction-_Molality-_Concentration

A solution comprises at least two different chemical substances where at least one substance is in vast molar excess. The term ‘solution’ is used to describe both solids and liquids. Nevertheless the term ‘solution’ in the absence of the word ‘solid’ refers to a liquid. Chemists are particularly expert at identifying the number and chemical formulae of chemical substances present in a given closed system. Here we explore how the chemical composition of a given system is expressed. We consider a simple system prepared using water(\(\ell\)) and urea(s) at ambient temperature and pressure. We designate water as chemical substance 1 and urea as chemical substance \(j\), so that the closed system contains an aqueous solution. The amounts of the two substances are given by \(\mathrm{n}_{j} \left(=\mathrm{w}_{1} / \mathrm{M}_{1}\right)\) and \(\mathrm{n}_{\mathrm{j}}\left(=\mathrm{w}_{\mathrm{j}} / \mathrm{M}_{\mathrm{j}}\right)\) where \(\mathrm{w}_{1}\) and \(\mathrm{w}_{j}\) are masses; \(\mathrm{M}_{1}\) and \(\mathrm{M}_{j}\) are the molar masses of the two chemical substances. In these terms, \(\mathrm{n}_{1}\) and \(\mathrm{n}_{j}\) are extensive variables. \[\text { Mass of solution, } \mathrm{w}=\mathrm{n}_{1} \, \mathrm{M}_{1}+\mathrm{n}_{\mathrm{j}} \, \mathrm{M}_{\mathrm{j}}\]\[\text { Mass of solvent, } \mathrm{w}_{1}=\mathrm{n}_{1} \, \mathrm{M}_{1}\]For water, \(\mathrm{M}_{1} = 0.018 \mathrm{~kg mol}^{-1}\). However in reviewing the properties of solutions, chemists prefer intensive composition variables.The mole fractions of the two substances \(\mathrm{x}_{1}\) and \(\mathrm{x}_{j}\) are given by the following two equations: \[\mathrm{x}_{1}=\mathrm{n}_{\mathrm{l}} /\left(\mathrm{n}_{1}+\mathrm{n}_{\mathrm{j}}\right) \quad \mathrm{x}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} /\left(\mathrm{n}_{1}+\mathrm{n}_{\mathrm{j}}\right)\]Here \(x_{1}+x_{j}=1.0\). In general terms for a system comprising \(\mathrm{i}\)-chemical substances, the mole fraction of substance \(\mathrm{k}\) is given by equation (d). \[\mathbf{x}_{\mathrm{k}}=\mathrm{n}_{\mathrm{k}} / \sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{n}_{\mathrm{j}}\]Hence \(\sum_{j=1}^{j=i} x_{j}=1.0\). The advantage of the dimensionless mole fraction scale is that in the absence of chemical reaction, the mole fraction \(\mathrm{x}_{\mathrm{k}}\) of chemical substance \(\mathrm{k}\) is independent of both temperature and pressure.For liquid systems, the chemical substance in vast excess is called the solvent whereas the other substances are called solutes. In the urea + water system, water is the solvent if \(\mathrm{n}_{1}>>\mathrm{n}_{\mathrm{j}}\). We as observers of the properties of this system draw a distinction between the two substances, identifying urea as the solute. The molality of solute \(\mathrm{m}_{j}\) is given by the amount of solute in \(1 \mathrm{~kg}\) of solvent. \[\mathrm{m}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} / \mathrm{w}_{1}\]Because the molality of solute \(j\) is defined in terms of masses of solute and solvent, \(\mathrm{m}_{j}\) is independent of temperature and pressure. Hence for precise characterization of the properties of solutes the molality scale is preferred.The concentration of chemical substance \(j\) in a system volume \(\mathrm{V}\), \[c_{j}=n_{j} / V\]The latter statement is quite general. However chemists interested in the properties of solutions normally use the term ‘concentration’ with reference to the properties of a solute, substance \(\mathrm{j}\). Several problems are associated with equation (f). The major problem is that the volume of a solution depends on both temperature and pressure so both these intensive variables should be stated when \(\mathrm{c}_{j}\) is quoted. A further problem emerges when there is a need to specify the precise composition of a given solution. Many chemists prepare a solution by dissolving a known mass of solute \(j\) in small volume of solvent. The volume of the solution is then ‘made up to the mark’ for a given volume (e.g. \(250 \mathrm{~cm}^{3}\)). But often chemists do not record precisely how much solvent is used. In these terms we see why the molality is often the preferred composition scale for solutions because the amounts of solvent and solute are precisely defined. However when thinking about the properties of solutions, chemists consider the distance between solute molecules rather than their masses. An interesting calculation offers insight into the dependence of intermolecular separation for solutes as a function of solute concentration, \(\mathrm{c}_{j}\)_. As the concentration of the solute (e.g. urea) in an aqueous solution increases so the mean distance between the solute molecules decreases. In the event that the solute is a 1:1 salt (e.g. potassium bromide, \(\mathrm{KBr}\)), the calculation takes account of the fact that each mole of salt produces, with complete dissociation two moles of solute ions. We gain insight into the problem by considering a solution of \(\mathrm{KBr}(\mathrm{aq}, 1 \mathrm{~mol dm}^{-3}\)). The calculated distance between ion centres is \(0.94 \mathrm{~nm}\). The radii of these ions are approx. \(0.15 \mathrm{~nm}\). The diameter of a water molecule is around \(0.4 \mathrm{~nm}\).. So there are relatively few water molecules between the ions at this concentration.These calculations are important because they indicate how solute-solute distances change on increasing the concentration of solute. Chemists often want to know how solute - solute molecular interactions affect the properties of solutions. Certainly the distance between solute molecules is a key consideration in reviewing the properties of solutes in aqueous solutions. The task of understanding the properties of aqueous solutions is usually divided into two parts. For the first part we use the term hydration to describe solute - solvent interactions. We imagine a molecule of solute, chemical substance \(j\), in an infinite expanse of solvent and direct attention to the organisation of solvent molecules surrounding each solute molecule, the cosphere. The term hydration number often refers to the number of water molecules contiguous to each solute molecule but the term 'hydration shell' often extends to include solvent molecules outside the immediate sheath.With increase in solute concentration the mean separation between solute molecules decreases. In responding to the task of understanding the properties of real solutions we define the properties of ideal solutions. In the case of salt solutions, strong and long - range ion - ion interactions contribute to marked deviations from the properties of ideal solutions.Footnotes \(\mathrm{m}_{\mathrm{j}}=\left[\mathrm{mol} \mathrm{} \mathrm{kg}^{-1}\right]\) R. A. Robinson and R. H. Stokes, Electrolyte Solutions, Butterworths, London, 2nd edn. revised, 1965, page 15. Consider a solution in which each solute molecule is placed at the centre of a cube, edge d metres. Then distance between solute molecules\(/\mathrm{m} = \mathrm{d}\) Volume of one \(text{cube}/\mathrm{m}^{3} = \mathrm{d}^{3}\); Volume of \(\mathrm{n}_{j}\) moles of cubes \(/\mathrm{m}^{3} = \mathrm{n}_{j} \,s \mathrm{N}_{\mathrm{A} \,s \mathrm{d}^{3}\) where \mathrm{N}_{\mathrm{A}}\) is the Avogadro constant. Thus \(V(s \ln )=n_{j} \, N_{A} \, d^{3} \text { or } d=\left(c_{j} \, N_{A}\right)^{-1 / 3}\) where \(\mathrm{c}_{j}\) is expressed in \(\mathrm{mol m}^{-3}\). \(\mathrm{d}=\left\{\left[\mathrm{mol} \mathrm{m}^{-3}\right] \,\left[\mathrm{mol}^{-1}\right]\right\}^{-1 / 3}=[\mathrm{m}]\) If \(\mathrm{c}_{j}\) is expressed, as is conventional, in \(\mathrm{mol dm}^{-3}\), \(\mathrm{d}=\left(10^{3} \, \mathrm{c}_{\mathrm{j}} \, \mathrm{N}_{\mathrm{A}}\right)^{-1 / 3}\). \[\begin{array}{lll} \mathrm{c}_{\mathrm{j}} / \mathrm{mol} \mathrm{dm}^{-3} & \text { Single Solute } & 1: 1 \text { salt } \\ & \mathrm{d} / \mathrm{nm} & \mathrm{d} / \mathrm{nm} \\ 10^{-4} & 25.5 & 20.2 \\ 10^{-3} & 11.8 & 9.4 \\ 10^{-2} & 5.5 & 4.4 \\ 10^{-1} & 2.6 & 2.0 \\ 1 & 1.2 & 0.94 \\ 5 & 0.69 & 0.55 \end{array}\] N. E. Dorsey, Properties of Ordinary Water Substance, Reinhold, New York, 1940. The estimate quoted in the main text is based on the estimates given in Table 15 of this fascinating monograph. The latter offers information concerning, for example, the load which ice will support. Apparently, ice having a thickness of 20 cm will support a battery of artillery with carriages and horses (see p. 458). Molalities are based on mass, and concentrations on distances. R. W. Gurney, Ionic Processes in Solution, McGraw-Hill, New York, 1953. H. L. Friedman and C. V. Krishnan,in Water - A Comprehensive Treatise, ed. F. Franks, Plenum Press, New York, 1973, Vol. 3, Chapter 1. J.-Y. Huot and C. Jolicoeur, in The Chemical Physics of Solvation, Part (a) Theory of Solvation,ed. J.Ulstrup, Elsevier, New York, 1985. One interesting feature is common throughout aqueous chemistry. If a given water molecule is strongly hydrogen bonded to four other water molecules, that water molecule exists in a local state which has low density (high volume). In other words, strong cohesion implies low density; a pattern contrary to that encountered in nearly all natural systems (and in human activities); A. Ben-Naim, Chem. Phys. Lett., 1972,13, 406. [Exceptions to this statement are found in the structures of high pressure ice polymorphs - but then there are always exceptions to general statements.] The seminal paper in this subject, aqueous chemistry, is probably that written by J. D. Bernal and R. H. Fowler, J. Chem. Phys., 1933,1, 515. This paper is remarkable in that despite the subjects covered (i.e. ice, liquid water and aqueous solutions), the term 'hydrogen-bond' is not used. The authors refer to the tendency of water to group in 'tetrahedral coordination’.This page titled 1.6.1: Composition- Mole Fraction- Molality- Concentration is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.6.2: Composition- Scale Conversions- Molality

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.06%3A_Composition/1.6.02%3A_Composition-_Scale_Conversions-_Molality

For a solution in a single solvent, chemical substance 1, containing solute \(j\), \[\text { Molaity, } \mathrm{m}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} / \mathrm{w}_{1}\]Here \(\mathrm{n}_{j}\) is the amount of solute and \(\mathrm{w}_{1}\) is the mass of solventFor the same system, the amount of solvent, \[\mathrm{n}_{1}=\mathrm{n}_{\mathrm{j}} / \mathrm{m}_{\mathrm{j}} \, \mathrm{M}_{1}\]But mole fraction, \(\mathrm{x}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} /\left(\mathrm{n}_{\mathrm{1}}+\mathrm{n}_{\mathrm{j}}\right)\) \[\text { Thus } \quad x_{j}=\frac{m_{j} \, n_{1} \, M_{1}}{n_{1}+m_{j} \, n_{1} \, M_{1}} \quad \text { or } \quad x_{j}=\frac{m_{j} \, M_{1}}{1+m_{j} \, M_{1}}\]For dilute solutions, \(1>>m_{j} \, M_{1}\). \[\text { Then } x_{j}=m_{j} \, M_{1}\]For water(\(\ell\)), \(\mathrm{M}_{1}=0.018 \mathrm{~kg} \mathrm{~mol}^{-1}\). From equation (c) \(x_{j}+x_{j} \, m_{j} \, M_{1}=m_{j} \, M_{1}\) \[\text { Then } \quad x_{j}=m_{j} \, M_{1} \,\left(1-x_{j}\right) \quad \text { or } \quad m_{j}=x_{j} /\left[M_{1} \,\left(1-x_{j}\right)\right]\]For dilute solutions \(1-x_{j} \approx 1.0\). and we recover equation (d). In short, equations (c) and (e) provide exact conversions between \(\mathrm{m}_{j}\) and \(\mathrm{x}_{j}\) whereas equation (d) is only valid for dilute solutions.We consider a solution having volume \(\mathrm{V}(\mathrm{s} \ln )\). \(\text { Mass of solution }=\rho(s \ln ) \, V(s \ln )\) where (at defined \(\mathrm{T}\) and \(\mathrm{p}\)), density \(=\rho(\mathrm{sln})\). If amount of substance \(j\) in this solution is \(\mathrm{n}_{j}\) mol then mass of solute \(\mathrm{w}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} \, \mathrm{M}_{\mathrm{j}}\), where \(\mathrm{M}_{j} =\) molar mass of solute.Mass of solvent in system \(=\rho(s \ln ) \, V(s \ln )-n_{j} \, M_{j}\) \[\text { Hence molality } \quad m_{j}=\frac{n_{j}}{\rho(s \ln ) \, V(s \ln )-n_{j} \, M_{j}}\]\[\text { and concentration } \quad c_{j}=\frac{n_{j}}{V(s \ln )}\]From (f) and (g) \(\mathrm{m}_{\mathrm{j}}=\frac{\mathrm{n}_{\mathrm{j}}}{\frac{\rho(\mathrm{s} \ln ) \, \mathrm{n}_{\mathrm{j}}}{\mathrm{c}_{\mathrm{j}}}-\mathrm{n}_{\mathrm{j}} \, \mathrm{M}_{\mathrm{j}}}=\frac{1}{\frac{\rho(\mathrm{s} \ln )}{\mathrm{c}_{\mathrm{j}}}-\mathrm{M}_{\mathrm{j}}}\) \[\text { Or, } m_{j}=\frac{c_{j}}{\rho(s \ln )-M_{j} \, c_{j}}\]\[\text { Or, } \quad c_{j}=\frac{m_{j} \, \rho(s \ln )}{1+m_{j} \, M_{j}}\]For dilute solutions \(\rho(\mathrm{s} \ln )>>\mathrm{M}_{\mathrm{j}} \, \mathrm{c}_{\mathrm{j}}\). \[\text { Then } c_{j}(s \ln )=m_{j} \, \rho(s \ln )\]Therefore the exact conversion is given by equations (h) and (i) which reduce to equation (j) for dilute solutions. An elegant conversion is possible between \(\mathrm{m}_{j}\) and \(\mathrm{c}_{j}\) scales. The volume of a simple solution is given by equation (k). \[\mathrm{V}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\]Here \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)\) is the apparent molar volume of solute \(j\). \[\text { Or, } \quad \mathrm{V}(\mathrm{aq}) / \mathrm{n}_{\mathrm{j}}=\left(\mathrm{n}_{1} / \mathrm{n}_{\mathrm{j}}\right) \,\left[\mathrm{M}_{1} / \rho_{1}^{*}(\ell)\right]+\phi\left(\mathrm{V}_{\mathrm{j}}\right)\]\[\text { Or, } \quad 1 / \mathrm{c}_{\mathrm{j}}=\left[1 / \mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]+\phi\left(\mathrm{V}_{\mathrm{j}}\right)\]\[\text { Then, } 1 / \mathrm{m}_{\mathrm{j}}=\left[\rho_{1}^{*}(\ell) / \mathrm{c}_{\mathrm{j}}\right]-\left[\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \rho_{1}^{*}(\ell)\right]\] Amount of solvent, molar mass \(\mathrm{M}_{1}\), \(\mathrm{n}_{1}=\mathrm{w}_{1} / \mathrm{M}_{1}\) Then, \(\mathrm{m}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} / \mathrm{n}_{1} \, \mathrm{M}_{1}\) Units \(\mathrm{m}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} / \mathrm{n}_{1} \, \mathrm{M}_{1}=[\mathrm{mol}] /[\mathrm{mol}] \,\left[\mathrm{kg} \mathrm{mol}^{-1}\right]=\left[\mathrm{mol} \mathrm{kg}^{-1}\right]\) \(c_{j}=\left[\mathrm{mol} \mathrm{m}^{-3}\right]=\left[\mathrm{mol} \mathrm{kg}^{-1}\right] \,\left[\mathrm{kg} \mathrm{m}^{-3}\right]\)This page titled 1.6.2: Composition- Scale Conversions- Molality is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.6.3: Composition- Scale Conversion- Solvent Mixtures

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.06%3A_Composition/1.6.03%3A_Composition-_Scale_Conversion-_Solvent_Mixtures

A given mixed solvent is prepared (at fixed \(\mathrm{T}\) and \(\mathrm{p}\)) by mixing \(\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell) \mathrm{m}^{3}\) of liquid 1 and \(\mathrm{n}_{2} \, \mathrm{V}_{2}^{*}(\ell) \mathrm{m}^{3}\) of liquid 2. We will assume that the thermodynamic properties of the mixture are ideal. \[\text { Then volume } \mathrm{V}=\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{n}_{2} \, \mathrm{V}_{2}^{*}(\ell)\]Then volume% of liquid 2 in the mixture is given by equation (b). \[\mathrm{V}_{2} \%=\left[10^{2} \, \mathrm{n}_{2} \, \mathrm{V}_{2}^{*}(\ell)\right] /\left[\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{n}_{2} \, \mathrm{V}_{2}^{*}(\ell)\right]\]The mass of a given mixed solvent system equals \(\mathrm{w}_{\mathrm{s}}\). Further mass% of liquid 2 is \(\mathrm{w}_{2}\)%. \[\text { Thus } \mathrm{w}_{2} \%=\mathrm{w}_{2} \, 10^{2} /\left(\mathrm{w}_{1}+\mathrm{w}_{2}\right)\]\[\text { Mole fraction } \mathrm{x}_{2}=\left(\mathrm{w}_{2} \% / \mathrm{M}_{2}\right) /\left[\frac{\left(10^{2}-\mathrm{w}_{2} \%\right)}{\mathrm{M}_{1}}+\frac{\mathrm{w}_{2} \%}{\mathrm{M}_{2}}\right]\]\[\text { Also } \left.\mathrm{V}_{2} \%(\mathrm{mix} ; \text { id })=\left[10^{2} \, \mathrm{w}_{2} / \rho_{2}^{*}(\ell)\right] / \frac{\mathrm{w}_{1}}{\rho_{1}^{*}(\ell)}+\frac{\mathrm{w}_{2}}{\rho_{2}^{*}(\ell)}\right]\]If \(\left(\mathrm{w}_{1}+\mathrm{w}_{2}\right)=100 \mathrm{~kg}\), \[\mathrm{V}_{2} \%(\operatorname{mix} ; \mathrm{id})=\left[10^{2} \, \mathrm{w}_{2} \% / \rho_{2}^{*}(\ell)\right] /\left[\frac{\left(10^{2}-\mathrm{w}_{2} \%\right)}{\rho_{1}^{*}(\ell)}+\frac{\mathrm{w}_{2} \%}{\rho_{2}^{*}(\ell)}\right]\]A given solvent mixture has mass \(10^{2} \mathrm{~kg}\) is prepared using \(\mathrm{w}_{2} \mathrm{~kg}\left[=\mathrm{w}_{2} \% \right]\) of liquid 2; nj moles of solute are dissolved in this mixture. \[\text { Molality } \mathrm{m}_{\mathrm{j}} / \mathrm{mol} \mathrm{} \mathrm{kg}^{-1}=\mathrm{n}_{\mathrm{j}} / 10^{2}\]\[\text { Mole fraction, } \mathrm{x}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} /\left\{\left[\left(10^{2}-\mathrm{w}_{2} \%\right) / \mathrm{M}_{1}\right]+\left[\mathrm{w}_{2} \% / \mathrm{M}_{2}\right]+\mathrm{n}_{\mathrm{j}}\right\}\]For dilute solutions, \(\mathrm{n}_{\mathrm{j}}<<\left(\mathrm{n}_{1}+\mathrm{n}_{2}\right)\) \[\text { Then, } \quad \mathrm{x}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} /\left\{\left[\left(10^{2}-\mathrm{w}_{2} \%\right) / \mathrm{M}_{1}\right]+\left[\mathrm{w}_{2} \% / \mathrm{M}_{2}\right]\right\}\]\[\text { Or, } \left.\mathrm{x}_{\mathrm{j}}=10^{2} \, \mathrm{m}_{\mathrm{j}} /\left\{\left[10^{2}-\mathrm{w}_{2} \%\right] / \mathrm{M}_{1}\right]+\left[\mathrm{w}_{2} \% / \mathrm{M}_{2}\right]\right\}\]A given solution is prepared (at fixed \(\mathrm{T}\) and \(\mathrm{p}\)) using \(\mathrm{n}_{1}\) moles of liquid 1, \(\mathrm{n}_{2}\) moles of liquid 2 and \(\mathrm{n}_{j}\) moles of a simple solute (e.g. urea) where \(n_{j}<<\left(n_{1}+n_{2}\right)\). \[\text { Mass of mixed solvent } \mathrm{w}_{\mathrm{s}}=\mathrm{n}_{1} \, \mathrm{M}_{1}+\mathrm{n}_{2} \, \mathrm{M}_{2}\]\[\text { Mass of system, } w=n_{j} \, M_{j}+n_{1} \, M_{1}+n_{2} \, M_{2}\]\[\text { Molality of solute } \mathrm{m}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} /\left[\mathrm{n}_{1} \, \mathrm{M}_{1}+\mathrm{n}_{2} \, \mathrm{M}_{2}\right]=\mathrm{n}_{\mathrm{j}} /\left[\mathrm{w}_{1}+\mathrm{w}_{2}\right]\]\[\text { Or, } \quad \mathrm{m}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} / \mathrm{w}_{\mathrm{s}}\]Density of solution \(= \rho\) Mass of solution \(= \mathrm{w}\) \[\text { Volume of solution } \mathrm{V}=\left[\mathrm{n}_{\mathrm{j}} \, \mathrm{M}_{\mathrm{j}}+\mathrm{n}_{1} \, \mathrm{M}_{1}+\mathrm{n}_{2} \, \mathrm{M}_{2}\right] / \rho\]\[\text { Concentration of solute } j, \mathrm{c}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} \, \rho /\left[\mathrm{n}_{\mathrm{j}} \, \mathrm{M}_{\mathrm{j}}+\mathrm{n}_{1} \, \mathrm{M}_{1}+\mathrm{n}_{2} \, \mathrm{M}_{2}\right]\]\[\text { For dilute solutions, } \mathrm{n}_{\mathrm{j}} \, \mathrm{M}_{\mathrm{j}} \ll\left[\mathrm{n}_{1} \, \mathrm{M}_{1}+\mathrm{n}_{2} \, \mathrm{M}_{2}\right]\]\[\text { Then, } \mathrm{c}_{\mathrm{j}} \cong \mathrm{n}_{\mathrm{j}} \, \rho /\left[\mathrm{n}_{1} \, \mathrm{M}_{1}+\mathrm{n}_{2} \, \mathrm{M}_{2}\right]\]If the solution is dilute, the density of the solution is approx. equal to density of the solvent \(\rho_{s}\) at the same \(\mathrm{T}\) and \(\mathrm{p}\). \[\text { Hence } \mathrm{c}_{\mathrm{j}} \cong \mathrm{n}_{\mathrm{j}} \, \rho_{\mathrm{s}} /\left[\mathrm{n}_{1} \, \mathrm{M}_{1}+\mathrm{n}_{2} \, \mathrm{M}_{2}\right]\]\[\text { Molality of solute } \mathrm{m}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} /\left[\mathrm{n}_{1} \, \mathrm{M}_{1}+\mathrm{n}_{2} \, \mathrm{M}_{2}\right]\]Then \(c_{j} \cong m_{j} \, p_{s}\)This page titled 1.6.3: Composition- Scale Conversion- Solvent Mixtures is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.7.1: Compressions and Expansions- Liquids

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The isothermal compressions of solutions and liquids have been extensively studied and the subject has a remarkable history. The term compression, symbol \(\mathrm{K}\) describes the sensitivity of the volume of a system to an isothermal change in pressure, \((\partial V / \partial p)\). Reference is usually made to the voyage made by HMS Challenger and the report of experiments undertaken by Tait into the compression of water. Kell summarises various equations which have been proposed describing the isothermal dependence of the molar volume of water on pressure; see also references.The dependence of the volume of water(\(\ell\)) at low pressures and at a given temperature on pressure can be represented by equation (a) where \(\mathrm{A}\) and \(\mathrm{B}\) are constants. \[[\mathrm{V}(\text { ref })-\mathrm{V}] / \mathrm{V}(\text { ref }) \, \mathrm{p}=\mathrm{A} /(\mathrm{B}+\mathrm{p})\]Here \(\mathrm{V}(\text{ref})\) is the volume ‘at zero pressure’, usually ambient pressure (i.e. approx \(105 \mathrm{~N m}^{-2}\)). This equation often called the Tait equation has the form shown in equation (b). \[-\left(1 / \mathrm{V}^{0}\right) \,(\partial \mathrm{V} / \partial \mathrm{p})=\mathrm{A} /(\mathrm{B}+\mathrm{p})\]\[\text { Alternatively } \mathrm{V}=\mathrm{V}^{0}\{1-\mathrm{A} \, \ln [(\mathrm{B}+\mathrm{p}) / \mathrm{B}]\}\]The challenge of measuring the isothermal compression of liquids has been taken up by many investigators; e.g. references. The isothermal compressions of a liquid \(\mathrm{K}_{\mathrm{T}}\) is defined by equation (d). \[\mathrm{K}_{\mathrm{T}}=-(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{T}}\]The isothermal compressibility is given by equation (e). \[\kappa_{\mathrm{T}}=-\mathrm{V}^{-1} \,(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{T}}\]For all thermodynamic equilibrium states, both \(\mathrm{K}_{\mathrm{T}}\) and \(\kappa_{\mathrm{T}}\) are positive variables. A related variable is the isochoric thermal pressure coefficient, \((\partial p / \partial T)_{v}\).We develop the story in the context of systems containing two liquid components. For a closed system containing \(\mathrm{n}_{1}\) and \(\mathrm{n}_{2}\) moles of chemical substances 1 and 2, the Gibbs energy is a dependent variable and the variables \(\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{2}\right]\) are the independent variables. Temperature \(\mathrm{T}\) is the thermal potential; pressure \(\mathrm{p}\) is the mechanical variable. The number of thermodynamic variables necessary to define the system is established using the Gibbs Phase Rule. For a closed system (at defined \(\mathrm{T}\) and \(\mathrm{p}\)) at thermodynamic equilibrium the composition/organisation is represented by \(\xi^{e q}\). The affinity for spontaneous change is zero consistent with the Gibbs energy being a minimum; equation (f). \[\mathrm{A}=-(\partial \mathrm{G} / \partial \xi)_{\mathrm{T}, \mathrm{p}}^{\mathrm{eq}}=0\]The Gibbs energy, volume and entropy of a solution at equilibrium are state variables. We contrast these properties with those properties which are associated with a process (pathway). Thus we contrast the state variable V with an unspecified compression of a solution. We need to define the path followed by the system when the pressure is changed. The Gibbs energy of a closed system at thermodynamic equilibrium (where the affinity for spontaneous change is zero and where the molecular composition/organisation is characterised by \(\xi^{e q}\)) is described by equation (g). \[\mathrm{G}=\mathrm{G}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{2}, \mathrm{~A}=0\right]\]The same state is characterised by the equilibrium volume and equilibrium entropy by equations (h) and (i) respectively. \[\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{2}, \mathrm{~A}=0\right]\]\[\mathrm{S}=\mathrm{S}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{2}, \mathrm{~A}=0\right]\]We use two intensive variables, \(\mathrm{T}\) and \(\mathrm{p}\), in the definition of extensive variables \(\mathrm{G}\), \(\mathrm{V}\) and \(\mathrm{S}\). When the pressure is increased by finite increments from \(\mathrm{p}\) to (\(\mathrm{p} + \Delta \mathrm{p}\)), the volume changes in finite increments from \(\mathrm{V}\) to (\(\mathrm{V} + \Delta \mathrm{V}\)). For an important pathway, the temperature is constant. However to satisfy the condition that the affinity for spontaneous change \(\mathrm{A}\) is zero, the molecular organisation/composition \(\xi\) changes. The volume at pressure (\(\mathrm{p} + \Delta \mathrm{p}\)) is defined using equation (j). \[\mathrm{V}=\mathrm{V}\left[\mathrm{T},(\mathrm{p}+\Delta \mathrm{p}), \mathrm{n}_{1}, \mathrm{n}_{2}, \mathrm{~A}=0\right]\]In principle we plot the volume as a function of pressure at constant temperature, \(\mathrm{n}_{1}\), \(\mathrm{n}_{2}\), and at ‘\(\mathrm{A} = 0\)’. The gradient of the plot defined by equation (h) yields the equilibrium isothermal compression, \(\mathrm{K}_{\mathrm{T}}(\mathrm{A}=0)\); equation (k) \[\mathrm{K}_{\mathrm{T}}(\mathrm{A}=0)=-(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{T}, \mathrm{A}=0}\]\(\mathrm{K}_{\mathrm{T}}(\mathrm{A} = 0)\) characterises the state defined by the set of variables, \(\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{2}, \mathrm{~A}=0\right]\).We turn our attention to another property starting with a system having a volume defined by equation (h). The system is perturbed by a change in pressure from \(\mathrm{p}\) to (\(\mathrm{p} + \Delta \mathrm{p}\)) in an equilibrium displacement. However on this occasion we require that the entropy of the system remains constant at a value defined by equation (i). In principle we plot the volume \(\mathrm{V}\) as a function of pressure at constant \(\mathrm{n}_{1}\), \(\mathrm{n}_{2}\), at ‘\(\mathrm{A}=0\)’ and at a constant entropy defined by equation (i). The gradient of the plot at the point where the volume is defined by equation (g) yields the equilibrium isentropic compression \(\mathrm{K}_{\mathrm{S}} (\mathrm{A}=0)\); equation (l) where isentropic = adiabatic and ‘at equilibrium’. \[\mathrm{K}_{\mathrm{S}}(\mathrm{A}=0)=-(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{S}, \mathrm{A}=0}\]The equilibrium state characterised by \(\mathrm{K}_{\mathrm{S}}(\mathrm{A}=0)\) is defined by the variables \(\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{2}, \mathrm{~A}=0\right]\). In other words an isentropic volumetric property describes a solution defined in part by the intensive variables \(\mathrm{T}\) and \(\mathrm{p}\). Significantly the condition on the partial derivative in equation (l) is an extensive variable, entropy. For a stable phase \(\mathrm{K}_{\mathrm{S}}\) is positive.The arguments outlined above are repeated with respect to both isobaric equilibrium expansions \(\mathrm{E}_{\mathrm{p}}(\mathrm{A}=0)\) and isentropic equilibrium expansions, \(\mathrm{E}_{\mathrm{S}}(\mathrm{A}=0)\); equations (m) and (n). \[\mathrm{E}_{\mathrm{p}}(\mathrm{A}=0)=-(\partial \mathrm{V} / \partial \mathrm{T})_{\mathrm{p}, \mathrm{A}=0}\]\[\mathrm{E}_{\mathrm{S}}(\mathrm{A}=0)=-(\partial \mathrm{V} / \partial \mathrm{T})_{\mathrm{S}, \mathrm{A}=0}\]The (equilibrium) volume intensive isothermal \(\kappa_{\mathrm{T}}\) and isentropic \(\kappa_{\mathrm{S}}\) compressibilities are defined by equations (o) and (p) . \[\kappa_{\mathrm{T}}=-(1 / \mathrm{V}) \,(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{T}}=\mathrm{K}_{\mathrm{T}} \, \mathrm{V}^{-1}\]\[\kappa_{\mathrm{s}}=-(1 / \mathrm{V}) \,(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{s}}=\mathrm{K}_{\mathrm{s}} \, \mathrm{V}^{-1}\]In 1914 Tyrer reported isentropic and isothermal compressibilities for many liquids. Equations (q) and (r) define two (equilibrium) expansibilities, isentropic and isobaric, volume intensive properties. \[\alpha_{\mathrm{s}}=(1 / \mathrm{V}) \,(\partial \mathrm{V} / \partial \mathrm{T})_{\mathrm{s}}=\mathrm{E}_{\mathrm{S}} \, \mathrm{V}^{-1}\]\[\alpha_{p}=(1 / V) \,(\partial \mathrm{V} / \partial \mathrm{T})_{p}=\mathrm{E}_{\mathrm{p}} \, \mathrm{V}^{-1}\]Rowlinson and Swinton state that the property \(\alpha_{\mathrm{S}}\) is ‘of little importance’. The isobaric heat capacity per unit volume \(\sigma\) is the ratio \(\left[\mathrm{C}_{\mathrm{p}} / \mathrm{V}\right]\). A property of some importance is the difference between compressibilities, \(\delta\); equation (s). \[\delta=\kappa_{\mathrm{T}}-\kappa_{\mathrm{S}}=\mathrm{T} \,\left[\alpha_{\mathrm{p}}\right]^{2} \, \mathrm{V} / \mathrm{C}_{\mathrm{p}}=\mathrm{T} \,\left[\alpha_{p}\right]^{2} / \sigma\]The property \(\sigma\) is given different symbols and names; e.g. volumetric specific heat. Here we identify \(\sigma\) as the thermal (or, heat) capacitance. The property \(\varepsilon\) is the difference between isobaric and isentropic expansibilities; equation (t). \[\varepsilon=\alpha_{p}-\alpha_{s}=\kappa_{T} \, \sigma / T \, \alpha_{p}\]The Newton–Laplace equation is the starting point for the determination of isentropic compressibilities of liquids using sound speeds and densities; equation (u). \[u^{2}=\left(\kappa_{\mathrm{s}} \, \rho\right)^{-1}\]The isentropic condition on \(\kappa_{\mathrm{S}}\) means that as a sound wave passes through a liquid the pressure and temperature fluctuate within each microscopic volume but the entropy remains constant.Footnotes P. G. Tait, ‘Voyage of HMS Challenger’ (Physics and Chemistry), 1888, Volume II, Part IV, 76pp. P. G. Tait, ‘Scientific Papers’, The University Press, Cambridge, 1898, Volume I, p.261. See also N. E. Dorsey, Properties of Ordinary Water Substance, Reinhold, New York , 1940, pp. 207-253. G.S Kell, Water A Comprehensive Treatise, ed. F Franks, Plenum Press, New York, 17972, Volume 1, pp. 382-383. J. H. Dymond and R. Malhotra, Int. J. Thermophys., 1988, 9,941. A. T. J. Hayward, Brit.J. Appl. Phys., 1967, 18,965. G. A. Neece and D. R. Squire, J.Phys.Chem.,1968,72,128. J. H. Hildebrand, Phys.Rev.,1929,34,649. D. Tyrer, J. Chem. Soc., 1914,105,2534. H. E. Eduljee, D. M. Newitt and K. E. Weale, J.Chem.Soc.,1951,3086. L. A. K. Staveley, W. I. Tupman, and K. R. Hart, Trans. Faraday Soc.,1955,51,323. D. N. Newitt and K.Weale, J.Chem. Soc.,1951,3092. \(\mathrm{K}_{\mathrm{T}}=\left[\mathrm{m}^{3}\right] /\left[\mathrm{N} \mathrm{m}^{-2}\right]=\left[\mathrm{m}^{3} \mathrm{~Pa}^{-1}\right]\) \(\mathrm{K}_{\mathrm{T}}=\frac{1}{\left[\mathrm{~m}^{3}\right]} \, \frac{\left[\mathrm{m}^{3}\right]}{[\mathrm{Pa}]}=\left[\mathrm{Pa}^{-1}\right]\) \((\partial \mathrm{p} / \partial \mathrm{T})_{\mathrm{V}}=\left[\mathrm{Pa} \mathrm{K}{ }^{-1}\right]\) Phase Rule; \(\mathrm{P} = 1\); \(\mathrm{C} = 2\). Hence \(\mathrm{F} = 3\). Then we define \(\mathrm{T}\), \(\mathrm{p}\) and mole fraction composition. J. S. Rowlinson and F. L. Swinton, Liquids and Liquid Mixtures, Butterworths, London, 3rd edn., 1982, pp 16-17.This page titled 1.7.1: Compressions and Expansions- Liquids is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.7.2: Compressibilities (Isothermal) and Chemical Potentials- Liquids

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The (equilibrium) isothermal compressibility of a closed system containing a condensed phase is given by equation (a). \[\kappa_{\mathrm{T}}=-\frac{1}{\mathrm{~V}} \,\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{T}}\]\[\text { Or, } \quad \kappa_{\mathrm{T}}=-\left(\frac{\partial \ln (\mathrm{V})}{\partial \mathrm{p}}\right)_{\mathrm{T}}\]Here we assume that over a range of pressures of interest here , \(\kappa_{\mathrm{T}}\) is independent of pressure. \[\text { Hence at fixed temperature, } \int_{p=0}^{p} d \ln (V)=-K_{T} \, \int_{p=0}^{p} d p\]We define a property \(V(p=0)\), the volume of the system under consideration extrapolated to zero pressure at fixed temperature. \[\text { Therefore } \ln [\mathrm{V}(\mathrm{p}) / \mathrm{V}(\mathrm{p}=0)]=-\kappa_{\mathrm{T}} \, \mathrm{p}\]\[\text { Or, } \mathrm{V}(\mathrm{T}, \mathrm{p})=\mathrm{V}(\mathrm{T}, \mathrm{p}=0) \, \exp \left(-\mathrm{K}_{\mathrm{T}} \, \mathrm{p}\right)\]For systems at ordinary pressures, \(\kappa_{\mathrm{T}} \, \mathrm{P}<<1\). \[\text { Hence } \mathrm{V}(\mathrm{T}, \mathrm{p})=\mathrm{V}(\mathrm{T}, \mathrm{p}=0) \,\left[1-\kappa_{\mathrm{T}} \, \mathrm{p}\right]\]For example, in the case of a pure liquid , chemical substance 1 [e.g. water] \[\mathrm{V}_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})=\mathrm{V}_{1}^{*}(\ell ; \mathrm{T}, \mathrm{p}=0) \,\left[1-\kappa_{\mathrm{T} 1}^{*}(\ell) \, \mathrm{p}\right]\]\[\text { But for water }(\ell),\left\lfloor\partial \mu_{1}^{*}(\ell) / \partial \mathrm{p}\right\rfloor=\mathrm{V}_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})\]\[\text { Hence } \quad \left[\frac{\partial \mu_{1}^{*}(\ell)}{\partial \mathrm{p}}\right]=\mathrm{V}_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p}=0) \,\left[1-\kappa_{\mathrm{T1}}^{*}(\ell) \, \mathrm{p}\right]\]Or, following integration between limits ‘\(\mathrm{p}=0\)’ and \(\mathrm{p}\), \[\begin{aligned} &\mu_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})=\mu_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p}=0) \\ &\quad+\mathrm{p} \, \mathrm{V}_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p}=0) \,\left[1-(1 / 2) \, \kappa_{\mathrm{T} 1}^{*}(\ell) \, \mathrm{p}\right] \end{aligned}\]The latter equation relates the chemical potential of a liquid at pressure p to the isothermal compressibility of the liquid.Footnote With \(\exp (x)=1+x+\left(x^{2} / 2 !\right)+\left(x^{3} / 3 !\right)+\ldots \ldots\) At small \(\mathrm{x}\), \(\exp (x) \approx 1+x\) I. Prigogine and R. Defay, Chemical Thermodynamics, transl D. H. Everett, Longmans Green, London, 1953.This page titled 1.7.2: Compressibilities (Isothermal) and Chemical Potentials- Liquids is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.7.3: Compressions- Isentropic- Solutions- General Comments

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At fixed \(\mathrm{T}\) and \(\mathrm{p}\), the equilibrium state for an aqueous solution is a minimum in Gibbs energy, \(\mathrm{G}^{\text{eq}}\). The first derivative of \(\mathrm{G}^{\text{eq}}\) with respect to temperature at constant pressure yields the equilibrium enthalpy \(\mathrm{H}^{\text{eq}}\). The first derivative of \(\mathrm{H}^{\text{eq}}\) with respect to temperature also at constant pressure yields the equilibrium isobaric heat capacity \({\mathrm{C}_{\mathrm{p}}}^{\text{eq}}\). Alternatively we can track the pressure derivatives of \(\mathrm{G}^{\text{eq}}\). The first derivative of \(\mathrm{G}^{\text{eq}}\) with respect to pressure at fixed temperature is the equilibrium volume \(\mathrm{V}^{\text{eq}}\). The first derivative of \(\mathrm{V}^{\text{eq}}\) with respect to pressure at fixed temperature yields the equilibrium isothermal compression \(\mathrm{K}_{\mathrm{T}}^{\mathrm{eq}}\), the ratio \(\mathrm{K}_{\mathrm{T}}^{\mathrm{eq}} / \mathrm{V}^{\mathrm{eq}}\) yielding the equilibrium isothermal compressibility \(\kappa_{\mathrm{T}}^{\mathrm{eq}}\). Concentrating attention on equilibrium properties of aqueous solutions, an extensive literature concerns \(\mathrm{V}(\mathrm{aq})\) in terms of the corresponding densities, \(\rho(\mathrm{aq})\). An extensive literature describes isobaric heat capacities \(\mathrm{C}_{\mathrm{p}}(\mathrm{aq}\), effectively the second derivative of \(\mathrm{G}(\mathrm{aq}\). Rather less literature describes \(\kappa_{\mathrm{T}}(\mathrm{aq}\), a second derivative of \(\mathrm{G}(\mathrm{aq})\) with respect to pressure. However an extensive literature reports isentropic compressibilities, \(\kappa_{\mathrm{S}(\mathrm{aq})\); equation (a). \[\kappa_{\mathrm{S}}=-(1 / \mathrm{V}) \,(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{S}}=\mathrm{K}_{\mathrm{S}} \, \mathrm{V}^{-1}\]This perhaps surprising observation is accounted for by the fact that speeds of sound (at low frequency, e.g. \(1 \mathrm{~MHz}\)) in aqueous solutions are conveniently and precisely measured using either the ‘sing-around; or ‘pulse-echo-overlap’ methods {for a summary of the ‘History of Sound’ see reference 3.) Then using the Newton-LaPlace equation \(\kappa_{\mathrm{S}}(\mathrm{aq})\) is obtained; equation (b). \[\mathrm{u}^{2}=\left(\kappa_{\mathrm{s}} \, \rho\right)^{-1}\]The speed of sound at zero frequency is a thermodynamically defined property. The isentropic compressibility of water(\(\ell\)) at ambient \(\mathrm{T}\) and \(\mathrm{p}\) can be calculated using either the speed of sound \(\kappa_{\mathrm{s}}^{*}(\ell ; \text { acoustic })\) or using \(\kappa_{\mathrm{T}}^{*}(\ell)\), \(\alpha_{\mathrm{P}}^{*}(\ell)\) and \({\sigma}^{*}(\ell)\) to yield \(\kappa_{\mathrm{s}}^{*}(\ell ; \text { thermodynamic })\). The two estimates agree lending support to the practice of calculating isentropic compressibilities of solutions using the Newton-Laplace equation. We equate the isentropic condition with adiabatic, provided that the compression is reversible.An important quantity is the difference \(\delta\) between compressibilities; equation (c). \[\delta=\kappa_{\mathrm{T}}-\kappa_{\mathrm{S}}=\mathrm{T} \,\left(\alpha_{\mathrm{p}}\right)^{2} / \sigma\]The property \(\sigma\) is given a number of different names but here we use the term, heat (or, thermal ) capacitance. The ratio of isothermal to isentropic compressions equals the ratio of isobaric to isochoric heat capacities. \[\mathrm{K}_{\mathrm{T}} / \mathrm{K}_{\mathrm{S}}=\mathrm{C}_{\mathrm{p}} / \mathrm{C}_{\mathrm{V}}\]Interest in the isentropic compressibilities of solutions was stimulated by Gucker and co-workers and, in particular, by Harned and Owen. The latter authors defined a property of the solute, here called \(\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \text { def }\right)\) using equation (e) where the composition of a given aqueous solution is expressed using concentration \(\mathrm{c}_{j}\). \[\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \operatorname{def}\right) \equiv\left[\kappa_{\mathrm{S}}(\mathrm{aq})-\kappa_{\mathrm{Sl}}^{*}(\ell)\right] \,\left[\mathrm{c}_{\mathrm{j}}\right]^{-1}+\kappa_{\mathrm{S} 1}^{*}(\ell) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\]Also \[\begin{aligned} &\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \operatorname{def}\right) \equiv \\ &\quad\left[\kappa_{\mathrm{S}}(\mathrm{aq}) \, \rho_{1}^{*}(\ell)-\kappa_{\mathrm{Sl}}^{*}(\ell) \, \rho(\mathrm{aq})\right] \,\left[\mathrm{c}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1}+\kappa_{\mathrm{S} 1}^{*}(\ell) \, \mathrm{M}_{\mathrm{j}} \,\left[\rho_{1}^{*}(\ell)\right]^{-1} \end{aligned}\]Similar equations relate \(\phi\left(K_{\mathrm{Sj}} ; \text { def }\right)\) to the molality of the solute, \(\mathrm{m}_{j}\). \[\phi\left(\mathrm{K}_{\mathrm{sj}} ; \mathrm{def}\right) \equiv\left[\kappa_{\mathrm{s}}(\mathrm{aq})-\kappa_{\mathrm{s} 1}^{*}(\ell)\right] \,\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1}+\kappa_{\mathrm{s}}(\mathrm{aq}) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\]\[\begin{aligned} &\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \operatorname{def}\right) \equiv \\ &{\left[\kappa_{\mathrm{S}}(\mathrm{aq}) \, \rho_{1}^{*}(\ell)-\kappa_{\mathrm{Sl}}^{*}(\ell) \, \rho(\mathrm{aq})\right] \,\left[\mathrm{m}_{\mathrm{j}} \, \rho(\mathrm{aq}) \, \rho_{1}^{*}(\ell)\right]^{-1}} \\ &+\kappa_{\mathrm{S}}(\mathrm{aq}) \, \mathrm{M}_{\mathrm{j}} \,[\rho(\mathrm{aq})]^{-1} \end{aligned}\]The latter four equations are stated by analogy with those relating \(\phi\left(\mathrm{K}_{\mathrm{Tj}}\right)\) to the composition of a solution. In these terms equations (e) to (h) are said to describe the same property of a given solute. A crucial feature of equations (e) - (h) is the equivalence symbol (i.e.. \(\equiv\)). In this sense Harned and Owen defined an apparent isentropic compression of solute-\(j\) in terms of the quantities on the r.h.s. of equation (a). They recognised that \(\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \text { def }\right)\) does not have thermodynamic basis. The target quantity is the apparent molar isentropic compression defined by equation (i) which, however, is not a description of an isentropic process as its name might suggest. \[\phi\left(\mathrm{K}_{\mathrm{sj}} ; \mathrm{def}\right)=\left(1 / \mathrm{n}_{\mathrm{j}}\right) \, \mathrm{K}_{\mathrm{s}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})-\left(\mathrm{n}_{1} / \mathrm{n}_{\mathrm{j}}\right) \, \mathrm{K}_{\mathrm{S} 1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})\]In fact \(\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \operatorname{def}\right)\) is a measure of the change in the isentropic compression of a solvent when solute \(j\) is added under isothermal-isobaric conditions. The equivalence symbol in equations (e) - (h) is important. In fact reservations are often expressed especially when estimates of \(\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \text { def }\right)\) are discussed, particularly the dependence of \(\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \text { def }\right)\) on solution composition. Franks and co-workers recognised that the lack of isobaric heat capacity data forces the adoption of an approach in which \(\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \text { def }\right)\) is often effectively assumed equal to \(\phi\left(\mathrm{K}_{\mathrm{Tj}^{\mathrm{j}}}\right)\). Owen and Simons comment that overlooking the difference between \(\kappa_{\mathrm{S}}(\mathrm{aq})\) and \(\kappa_{\mathrm{T}}(\mathrm{aq})\) causes errors of approximately 7.5% in estimates of \(\phi\left(\mathrm{K}_{\mathrm{Tj}}\right)^{\infty}\) for \(\mathrm{NaCl}(\mathrm{aq})\) and \(\mathrm{KCl}(\mathrm{aq})\) at \(298 \mathrm{~K}\).In terms of the development of the theory, a problem is encountered with the differential dependence of the molar volume of the solvent on pressure at constant entropy of the solution. The task is to describe how the molar volume of the solvent would depend on pressure if it were held at the same entropy of the solution.Footnotes R. Garnsey, R. J. Boe, R. Mahoney and T. A. Litovitz, J. Chem. Phys., 1969, 50, 5222. E. P. Papadakis, J.Acoust. Soc. Am.,1972,52,843. R Taton Science in the Nineteenth Century, trans., A J Pomerans, Basic Books, New York, 1965,chapter 3. G. Horvath-Szabo, H. Hoiland and E. Hogseth, Rev. Sci.Instrum.,1994,65,1644. G. Douheret, M. I. Davis, J. C. R. Reis and M. J. Blandamer, Chem. Phys. Phys Chem., 2001, 2, 148. J. S. Rowlinson and F. L. Swinton, Liquids and Liquid Mixtures, Butterworths, London, 3rd. edn., 1982, pp.16-17. J. O. Hirschfelder, C. F. Curtiss and R. B. Bird, Molecular Theory of Gases and Liquids, Wiley, New York, 1954, chapters 5 and 11. We use several calculus operations. Thus, \((\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{T}}=-(\partial \mathrm{T} / \partial \mathrm{p})_{\mathrm{V}} \,(\partial \mathrm{V} / \partial \mathrm{T})_{\mathrm{p}}=-(\partial \mathrm{T} / \partial \mathrm{p})_{\mathrm{V}} \,(\partial \mathrm{V} / \partial \mathrm{S})_{\mathrm{p}} \,(\partial \mathrm{S} / \partial \mathrm{T})_{\mathrm{p}}\) And, \((\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{s}}=-(\partial \mathrm{S} / \partial \mathrm{p})_{\mathrm{V}} \,(\partial \mathrm{V} / \partial \mathrm{S})_{\mathrm{p}}=-(\partial \mathrm{T} / \partial \mathrm{p})_{\mathrm{V}} \,(\partial \mathrm{V} / \partial \mathrm{S})_{\mathrm{p}} \,(\partial \mathrm{S} / \partial \mathrm{T})_{\mathrm{V}}\) Then, \((\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{T}} /(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{S}}=(\partial \mathrm{S} / \partial \mathrm{T})_{\mathrm{p}} /(\partial \mathrm{S} / \partial \mathrm{T})_{\mathrm{V}}\) Further \(\mathrm{H}=\mathrm{G}+\mathrm{T} \, \mathrm{S}\) and \(\mathrm{S}=-(\partial \mathrm{G} / \partial \mathrm{T})_{\mathrm{p}}\) Hence, \(\mathrm{H}=\mathrm{G}-\mathrm{T} \,(\partial \mathrm{G} / \partial \mathrm{T})_{\mathrm{p}}\) Then \((\partial \mathrm{H} / \partial \mathrm{T})_{\mathrm{p}}=\mathrm{C}_{\mathrm{p}}=-\mathrm{T} \,\left(\partial^{2} \mathrm{G} / \partial \mathrm{T}^{2}\right)_{\mathrm{p}}=\mathrm{T} \,(\partial \mathrm{S} / \partial \mathrm{T})_{\mathrm{p}}\) Similarly \((\partial \mathrm{U} / \partial \mathrm{T})_{\mathrm{V}}=\mathrm{C}_{\mathrm{V}}=\mathrm{T} \,(\partial \mathrm{S} / \partial \mathrm{T})_{\mathrm{V}}\) Therefore, \(\mathrm{K}_{\mathrm{T}} / \mathrm{K}_{\mathrm{S}}=\mathrm{C}_{\mathrm{p}} / \mathrm{C}_{\mathrm{V}}\) F. T. Gucker, Chem. Rev.,1933,13,127. F. T. Gucker, F. W. Lamb, G. A. Marsh and R. M. Haag, J. Amer. Chem. Soc., 1950, 72, 310; and references therein. H.S. Harned and B. B. Owen, The Physical Chemistry of Electrolytic Solutions, Reinhold, New York, 3rd. edn., 1958, section 8.7. M. J. Blandamer, J. Chem. Soc. Faraday Trans., 1998, 94,1057. M. J. Blandamer, M. I. Davis, G. Douheret, and J. C. R. Reis, Chem.Revs.,2001,30,8. F. Franks, J. R. Ravenhill and D. S. Reid, J. Solution Chem., 1972, 1, 3. B. B. Owen and H. L. Simons, J. Phys. Chem., 1957, 61, 479.This page titled 1.7.3: Compressions- Isentropic- Solutions- General Comments is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.7.4: Compressibilities- Isentropic- Related Properties

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A given closed system at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\) contains chemical substances 1 and \(j\). The system at specified \(\mathrm{T}\) and \(\mathrm{p}\) is at equilibrium where the affinity for spontaneous change is zero. We describe the volume and the entropy of the system using the following two equations. \[\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}, \mathrm{A}=0\right]\]\[\mathrm{S}=\mathrm{S}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}, \mathrm{A}=0\right]\]The system is perturbed by a change in pressure. We envisage two possible paths tracked by the system accompanying a change in volume. In the first case the temperature is constant along the path for which ‘\(\mathrm{A}=0\)’. The isothermal equilibrium dependence of volume on pressure, namely the equilibrium isothermal compression \(\mathrm{K}_{\mathrm{T}}(\mathrm{A}=0)\), is defined by equation (c). \[K_{T}(A=0)=-\left(\frac{\partial V}{\partial p}\right)_{T, A=0}\]In the second case the entropy remains constant along the path travelled by the system where ‘\(\mathrm{A}=0\)’. The differential equilibrium isentropic compression is given by equation (d); isentropic = adiabatic + equilibrium \[\mathrm{K}_{\mathrm{s}}(\mathrm{A}=0)=-\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{S}, \mathrm{A}=0}\]For all stable phases the volume of a given system decreases with increase in pressure at fixed temperature. The minus signs in equations (c) and (d) mean that compressions are positive variables. Neither \(\mathrm{K}_{\mathrm{T}}(\mathrm{A}=0)\) or \(\mathrm{K}_{\mathrm{S}}(\mathrm{A}=0)\) are strong functions of state because both variables describe pathways between states. The partial differentials in equations (c) and (d) differ in an important respect. The isothermal condition refers to an intensive variable whereas the isentropic condition refers to an extensive variable. The two properties \(\mathrm{K}_{\mathrm{T}}(\mathrm{A}=0)\) and \(\mathrm{K}_{\mathrm{S}}(\mathrm{A}=0)\) are related using a calculus operation. \[\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{S}, \mathrm{A}=0}=\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{A}=0}-\left(\frac{\partial \mathrm{S}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{A}=0} \,\left(\frac{\partial \mathrm{T}}{\partial \mathrm{S}}\right)_{\mathrm{p}, \mathrm{A}=0} \,\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{P}, \mathrm{A}=0}\]Hence, \[\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{S}, \mathrm{A}=0}=\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{A}=0}+\left[\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \mathrm{A}=0}\right]^{2} \, \frac{\mathrm{T}}{\mathrm{C}_{\mathrm{p}}(\mathrm{A}=0)}\]But the (equilibrium) isobaric expansibility, \[\alpha_{p}(A=0)=\frac{1}{V} \,\left(\frac{\partial V}{\partial T}\right)_{p, A=0}\]\[\operatorname{Then}\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{S}, \mathrm{A}=0}=\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{A}=0}+\left[\alpha_{\mathrm{p}}(\mathrm{A}=0)\right]^{2} \, \frac{\mathrm{V}^{2} \, \mathrm{T}}{\mathrm{C}_{\mathrm{p}}(\mathrm{A}=0)}\]By definition, the equilibrium isobaric heat capacity per unit volume {also called heat capacitance}, \[\sigma(A=0)=C_{p}(A=0) / V\]In terms of compressions, \[\mathrm{K}_{\mathrm{S}}(\mathrm{A}=0)=\mathrm{K}_{\mathrm{T}}(\mathrm{A}=0)-\left[\alpha_{\mathrm{p}}(\mathrm{A}=0)\right]^{2} \, \frac{\mathrm{V} \, \mathrm{T}}{\sigma(\mathrm{A}=0)}\]Three terms in equation (j), \(\mathrm{K}_{\mathrm{S}}(\mathrm{A}=0)\), \(\mathrm{K}_{\mathrm{T}}(\mathrm{A}=0)\) and \(\mathrm{V}\), are volume extensive variables. However it is convenient to rewrite these equations using volume intensive variables. Two equations define the isentropic equilibrium compressibility \(\kappa_{\mathrm{S}}(\mathrm{A}=0)\) and isothermal equilibrium compressibility \(\kappa_{\mathrm{T}}(\mathrm{A}=0)\) of a given system. \[\kappa_{\mathrm{T}}(\mathrm{A}=0)=-\frac{1}{\mathrm{~V}} \,\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{A}=0}=\frac{\mathrm{K}_{\mathrm{T}}(\mathrm{A}=0)}{\mathrm{V}}\]\[\kappa_{\mathrm{S}}(\mathrm{A}=0)=-\frac{1}{\mathrm{~V}} \,\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{s}, \mathrm{A}=0}=\frac{\mathrm{K}_{\mathrm{S}}(\mathrm{A}=0)}{\mathrm{V}}\]\[\text { Therefore } \kappa_{\mathrm{S}}(\mathrm{A}=0)=\kappa_{\mathrm{T}}(\mathrm{A}=0)-\left[\alpha_{\mathrm{p}}(\mathrm{A}=0)\right]^{2} \, \frac{\mathrm{T}}{\sigma(\mathrm{A}=0)}\]\[\text { By definition, } \delta=\kappa_{\mathrm{T}}-\kappa_{\mathrm{S}}\]\[\text { Then } \delta(A=0)=\left[\alpha_{p}(A=0)\right]^{2} \, \frac{T}{\sigma(A=0)}\]Footnotes From a Maxwell relationship for the condition at ‘\(\mathrm{A}=0\)’; i.e. at equilibrium, \(\partial^{2} \mathrm{G} / \partial \mathrm{T} \, \partial \mathrm{p}=\partial^{2} \mathrm{G} / \partial \mathrm{p} \, \partial \mathrm{T}\). Then, \(\mathrm{E}_{\mathrm{p}}=(\partial \mathrm{V} / \partial \mathrm{T})_{\mathrm{p}}=-(\partial \mathrm{S} / \partial \mathrm{p})_{\mathrm{T}}\) From the Gibbs - Helmholtz equation, we combine the equations, \(\mathrm{H}=\mathrm{G}+\mathrm{T} \, \mathrm{S}\) and \(\mathrm{S}=-(\partial \mathrm{G} / \partial \mathrm{T})_{\mathrm{p}}\). Hence, \(\mathrm{H}=\mathrm{G}-\mathrm{T} \,(\partial \mathrm{G} / \partial \mathrm{T})_{\mathrm{p}}\) Then, \((\partial \mathrm{H} / \partial \mathrm{T})_{\mathrm{p}}=\mathrm{C}_{\mathrm{p}}=-\mathrm{T} \,\left(\partial^{2} \mathrm{G} / \partial \mathrm{T}^{2}\right)_{\mathrm{p}}=\mathrm{T} \,(\partial \mathrm{S} / \partial \mathrm{T})_{\mathrm{p}}\) \(\kappa_{\mathrm{S}}(\mathrm{A}=0)=[\mathrm{Pa}]^{-1} \quad \kappa_{\mathrm{T}}(\mathrm{A}=0)=[\mathrm{Pa}]^{-1}\) \(\begin{aligned} &{\left[\alpha_{p}(\mathrm{~A}=0)\right]^{2} \, \frac{\mathrm{T}}{\sigma(\mathrm{A}=0)}=\left[\mathrm{K}^{-1}\right]^{2} \,[\mathrm{K}] \,\left[\mathrm{J} \mathrm{K}{ }^{-1} \mathrm{~m}^{-3}\right]^{-1}=\left[\mathrm{N} \mathrm{m}^{-2}\right]^{-1}=\mathrm{Pa}^{-1}} \\ &\sigma(\mathrm{A}=0)=\mathrm{C}_{\mathrm{p}}(\mathrm{A}=0) / \mathrm{V}=\left[\mathrm{J} \mathrm{K}{ }^{-1}\right] \,[\mathrm{m}]^{-3} \end{aligned}\) M. J. Blandamer, M. I. Davis, G. Douheret and J. C. R. Reis, Chem. Soc Rev., 2001, 30,8.This page titled 1.7.4: Compressibilities- Isentropic- Related Properties is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.7.5: Compressions- Isentropic- Solutions- Partial and Apparent Molar

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.07%3A_Compressions/1.7.05%3A_Compressions-_Isentropic-_Solutions-_Partial_and_Apparent_Molar

Isentropic properties of aqueous solutions are defined in a manner analogous to that used to define isothermal compressions and isothermal compressibilities. The assertion is made that a system (e.g. an aqueous solution) can be perturbed along a pathway where the affinity for spontaneous change is zero by a small change in pressure \(\delta \mathrm{p}\), to a neighbouring state having the same entropy. The (equilibrium) isentropic compression is defined by equation (a). \[\mathrm{K}_{\mathrm{S}}(\mathrm{aq})=-[\partial \mathrm{V}(\mathrm{aq}) / \partial \mathrm{p}]_{\mathrm{S}(\mathrm{aq}), A=0}\]The constraint on this partial differential refers to 'at constant \(\mathrm{S}(\mathrm{aq})\)'. The definition of \(\mathrm{K}_{\mathrm{S}}(\mathrm{aq})\) uses non-Gibbsian independent variables. In other words, isentropic parameters do not arise naturally from the formalism which expresses the Gibbs energy in terms of independent variables in the case of, for example, a simple solution, \(\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{\mathrm{l}}, \mathrm{n}_{\mathrm{j}}\right]\). The isothermal compression of a solution \(\mathrm{K}_{\mathrm{T}}(\mathrm{aq})\) and partial molar isothermal compressions of both solvent \(\mathrm{K}_{\mathrm{T} 1}(\mathrm{aq})\) and solute \(\mathrm{K}_{\mathrm{T} j}(\mathrm{aq})\) are defined using Gibbsian independent variables. Unfortunately the corresponding equations cannot be simply carried over to the isentropic property \(\mathrm{K}_{\mathrm{S}}(\mathrm{aq})\). The volume of a solution is expressed in terms of the amounts of solvent \(\mathrm{n}_{1}\) and solute \(\mathrm{n}_{j}\). \[\mathrm{V}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{V}_{1}(\mathrm{aq})+\mathrm{n}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}(\mathrm{aq})\]The latter equation is differentiated with respect to pressure at constant entropy of the solution \(\mathrm{S}(\mathrm{aq})\). The latter condition includes the condition that the system remains at equilibrium where the affinity for spontaneous change is zero. We emphasize a point. The entropy which remains constant is that of the solution. \[\mathrm{K}_{\mathrm{s}}(\mathrm{aq})=-\mathrm{n}_{1} \,\left[\partial \mathrm{V}_{1}(\mathrm{aq}) / \partial \mathrm{p}\right]_{\mathrm{S}(\mathrm{aq}) ; \mathrm{A}=0}-\mathrm{n}_{\mathrm{j}} \,\left[\partial \mathrm{V}_{\mathrm{j}}(\mathrm{aq}) / \partial \mathrm{p}\right]_{\mathrm{S}(\mathrm{aq}) ; \mathrm{A}=0}\]\(\mathrm{K}_{\mathrm{S}}(\mathrm{aq})\) is an extensive property of the aqueous solution. \(\mathrm{K}_{\mathrm{S}}(\mathrm{aq})\) may also be re-expressed using Euler’s theorem as a function of the composition of the solution. \[\mathrm{K}_{\mathrm{s}}(\mathrm{aq})=\mathrm{n}_{1} \,\left[\partial \mathrm{K}_{\mathrm{s}}(\mathrm{aq}) / \partial \mathrm{n}_{1}\right]_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{j})}+\mathrm{n}_{\mathrm{j}} \,\left[\partial \mathrm{K}_{\mathrm{s}}(\mathrm{aq}) / \partial \mathrm{n}_{\mathrm{j}}\right]_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{l})}\]Because \(\mathrm{K}_{\mathrm{S}}(\mathrm{aq})\) is defined using non-Gibbsian independent variables, two important inequalities follow. \[-\left[\partial \mathrm{V}_{1}(\mathrm{aq}) / \partial \mathrm{p}\right]_{\mathrm{S}(\mathrm{aq})} \neq\left[\partial \mathrm{K}_{\mathrm{s}}(\mathrm{aq}) / \partial \mathrm{n}_{1}\right]_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{j})}\]\[-\left[\partial \mathrm{V}_{\mathrm{j}}(\mathrm{aq}) / \partial \mathrm{p}\right]_{\mathrm{S}(\mathrm{aq})} \neq\left[\partial \mathrm{K}_{\mathrm{S}}(\mathrm{aq}) / \partial \mathrm{n}_{\mathrm{j}}\right]_{\mathrm{T}, \mathrm{p}, \mathrm{n}}\]\(\left[\partial \mathrm{K}_{\mathrm{s}}(\mathrm{aq}) / \partial \mathrm{n}_{1}\right]_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{j})}\) and \(\left[\partial \mathrm{K}_{\mathrm{S}}(\mathrm{aq}) / \partial \mathrm{n}_{\mathrm{j}}\right]_{\mathrm{T}, \mathrm{p}, \mathrm{n}}\) are respectively the partial molar properties of the solvent and solute. Because partial molar properties should describe the effects of a change in composition on the properties of a solution, we write equation (d) for an aqueous solution in the following form. \[\mathrm{K}_{\mathrm{S}}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{K}_{\mathrm{S} 1}(\mathrm{aq} ; \text { def })+\mathrm{n}_{\mathrm{j}} \, \mathrm{K}_{\mathrm{Sj}}(\mathrm{aq} ; \text { def })\]\[\text { Hence, } \quad \mathrm{K}_{\mathrm{sj}}(\mathrm{aq} ; \text { def }) \neq-\left[\partial \mathrm{V}_{\mathrm{j}}(\mathrm{aq}) / \partial \mathrm{p}\right]_{\mathrm{S}(\mathrm{aq})}\]In view of the latter inequality \(\mathrm{K}_{\mathrm{Sj}}(\mathrm{aq} ; \mathrm{def})\) is a non-Lewisian partial molar property. We could define a molar isentropic compression of solute \(j\) as (minus) the isentropic differential dependence of partial molar volume on pressure. This alternative definition is consistent with equation (g) expressing a summation rule analogous to that used for partial molar properties. However some other thermodynamic relationships involving partial molar properties would not be valid in this case. Therefore, \(-\left[\partial V_{j}(a q) / \partial p\right]_{S(a q)}\) is a semi-partial molar property. A similar problem is encountered in defining an apparent molar compression for solute \(j\), \(\phi\left(\mathrm{K}_{\mathrm{Sj}_{\mathrm{j}}}\right)\) in a solution where the solute has apparent molar volume \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)\); cf. equation (h). We might assert that \(\phi\left(\mathrm{K}_{\mathrm{Sj}_{\mathrm{j}}}\right)\) is related to the isentropic differential dependence of \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)\) on pressure, \(-\left[\phi\left(\mathrm{V}_{\mathrm{j}}\right) / \partial \mathrm{p}\right]_{\mathrm{S}(\mathrm{aq})}\). Alternatively, using as a guide the apparent molar properties \(\phi\left(\mathrm{E}_{\mathrm{pj}}\right)\) and \(\phi\left(\mathrm{K}_{\mathrm{Tj}^{\mathrm{j}}}\right)\), we could define \(\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \mathrm{def}\right)\) using equation (i). \[\mathrm{K}_{\mathrm{S}}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{K}_{\mathrm{S} 1}^{*}(\mathrm{l})+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{K}_{\mathrm{S} \mathrm{j}} ; \text { def }\right)\]\(\mathrm{K}_{\mathrm{Sj}}(\mathrm{aq} ; \mathrm{def})\) as given by equation (d) and \(\phi\left(K_{S j} ; \text { def }\right)\) are linked; equation (j). \[\mathrm{K}_{\mathrm{Sj}}(\mathrm{aq} ; \operatorname{def})=\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \operatorname{def}\right)+\mathrm{n}_{\mathrm{j}} \,\left[\partial \phi\left(\mathrm{K}_{\mathrm{Sj}} ; \text { def }\right) / \partial \mathrm{n}_{\mathrm{j}}\right]_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{l})}\]Equation (j) is of the general form encountered for other apparent and partial molar properties. This form is also valid in the case of partial and apparent molar isobaric expansions, isothermal compressions and isobaric heat capacities. On the other hand, the semi-partial molar isentropic compression defined by \(-\left[\partial \mathrm{V}_{\mathrm{j}}(\mathrm{aq}) / \partial \mathrm{p}\right]_{\mathrm{s}(\mathrm{aq})}\) and the semi-apparent molar isentropic compression defined by \(-\left[\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right) / \partial \mathrm{p}\right]_{\mathrm{S}(\mathrm{aq})}\) are related. The isentropic pressure dependence of \(\mathrm{V}_{\mathrm{j}}(\mathrm{aq})\) is given by equation (k). \[\begin{aligned} &-\left[\partial \mathrm{V}_{\mathrm{j}}(\mathrm{aq}) / \partial \mathrm{p}\right]_{\mathrm{S}(\mathrm{aq})}= \\ &-\left[\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right) / \partial \mathrm{p}\right]_{\mathrm{S}(\mathrm{aq})}-\mathrm{n}_{\mathrm{j}} \,\left\{\partial\left[\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right) / \partial \mathrm{n}_{\mathrm{j}}\right]_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{l})} / \partial \mathrm{p}\right\}_{\mathrm{s}(\mathrm{aq})} \end{aligned}\]However, \[\left\{\partial\left[\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right) / \partial \mathrm{n}_{\mathrm{j}}\right]_{\mathrm{T}, \mathrm{p}, \mathrm{n}} / \partial \mathrm{p}\right\}_{\mathrm{S}(\mathrm{aq})} \neq\left\{\partial\left[\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right) / \partial \mathrm{p}\right]_{\mathrm{S}(\mathrm{aq})} / \partial \mathrm{n}_{\mathrm{j}}\right\}_{\mathrm{T}, \mathrm{p}, \mathrm{n}}\]Hence, the analogue of equation (j) does not hold for these 'semi' properties. The inequalities (e) and (f) highlight the essence of non-Lewisian properties. Their origin is a combination of properties defined in terms of Gibbsian and non-Gibbsian independent variables as in equations (e) and (f). This combination is also the reason for the inequality (l). We stress that the isentropic condition in equations (e) and (f) refers to the entropy \(\mathrm{S}(\mathrm{aq})\) of the solution defined as is the volume \(\mathrm{V}(\mathrm{aq})\) by the Gibbsian independent variables \(\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}\right]\). But this is not the entropy \(\mathrm{S}_{1}^{*}(\ell)\) of the pure solvent having volume \(\mathrm{V}_{1}^{*}(\ell)\). \(\mathrm{S}(\mathrm{aq})\) at fixed composition is not simply related to \(\mathrm{S}_{1}^{*}(\ell)\) as, for example, linear functions of temperature and pressure.The isentropic condition is involved in the definitions of isentropic compression, \(\mathrm{K}_{\mathrm{S} 1}^{*}(\ell)\) and isentropic compressibility \(\kappa_{\mathrm{S} 1}^{*}(\ell)\) of the solvent. \[\mathrm{K}_{\mathrm{S} 1}^{*}(\ell)=-\left[\partial \mathrm{V}_{1}^{*}(\ell) / \partial \mathrm{p}\right] \text { at constant } \mathrm{S}_{1}^{*}(\ell)\]\[\begin{aligned} &\kappa_{\mathrm{Sl}}^{*}(\ell)=\mathrm{K}_{\mathrm{Sl}}^{*}(\ell) / \mathrm{V}_{1}^{*}(\ell)\\ &=-\left[\partial \mathrm{V}_{1}^{*}(\ell) / \partial \mathrm{p}\right] / \mathrm{V}_{1}^{*}(\ell) \text { at constant } \mathrm{S}_{1}^{*}(\ell) \end{aligned}\]The different isentropic conditions in equation (a) and in equations (m) and (n) signal a complexity in the isentropic differentiation of equation (o) with respect to pressure. \[\mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\mathrm{M}_{1}^{-1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\]Footnotes J. C. R. Reis, M. J. Blandamer, M. I. Davis and G. Douheret, Chem. Phys. Phys. Chem., 2001, 3,1465. J. C. R. Reis, J. Chem. Soc. Faraday Trans.,2,1982, 78,1565. M. J. Blandamer, M. I. Davis, G. Douheret and J. C. R. Reis, Chem. Soc. Rev.,2001,30,8. J. C. R. Reis, J. Chem. Soc. Faraday Trans.,1998,94,2385. M. J. Blandamer, J. Chem. Soc. Faraday Trans.,1998,94,1057. M. J. Blandamer, Chem. Soc. Rev.,1998,27,73.This page titled 1.7.5: Compressions- Isentropic- Solutions- Partial and Apparent Molar is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.7.6: Compressions- Isentropic- Neutral Solutes

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Granted that \(\phi\left(K_{\mathrm{Sj}} ; \text { def }\right)\) has been measured for solutions containing neutral solutes (at defined \(\mathrm{T}\) and \(\mathrm{p}\)), interesting patterns emerge for the dependences of \(\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \operatorname{def}\right)\) on molality \(\mathrm{m}_{j}\) and on solute \(j\). Further these dependences are readily extrapolated (geometrically) to infinite dilution to yield estimates of \(\phi\left(K_{\mathrm{S}_{j}} ; \operatorname{def}\right)^{\infty}\). These comments apply to solutions of neutral solutes in both aqueous and non-aqueous solutions; e.g. solutions in propylene carbonate and aqueous solutions of carbohydrates.For dilute solutions of neutral solutes \(\phi\left(K_{\mathrm{Sj}} ; \text { def }\right)\) is often approximately a linear function of the molality \(\mathrm{m}_{j}\). \[\text { Thus } \phi\left(\mathrm{K}_{\mathrm{S}_{\mathrm{j}}} ; \text { def }\right)=\phi\left(\mathrm{K}_{\mathrm{S}_{\mathrm{j}}} ; \text { def }\right)^{\infty}+\mathrm{b}_{\mathrm{KS}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)\]For aqueous solutions containing ureas, acetamides and \(\alpha,\omega\)-alkanediols, the slope \(b_{\mathrm{KS}}\) is positive. For dextrose(aq), sucrose(aq), urea(aq) and thiourea(aq) φ(; ) K def Sj ∞ is negative. In contrast φ(; ) K def Sj ∞ is positive for dioxan(aq) and acetamide(aq). In other words \(\phi\left(K_{\mathrm{S}_{j}} ; \operatorname{def}\right)^{\infty}\) is characteristic of the solute. Group additivity schemes are discussed for \(\phi\left(K_{\mathrm{S}_{j}} ; \operatorname{def}\right)^{\infty}\) with respect to glycylpeptides(aq), amino acids(aq) and alcohols. With increase in temperature \(\phi\left(K_{\mathrm{S}_{j}} ; \operatorname{def}\right)^{\infty}\) for amino acids(aq) and glycyl dipeptides(aq) increases. Particularly interesting in terms of solute-water interactions is the study reported by Galema et al who comment on the calculation of \(\phi\left(K_{\mathrm{Sj}} ; \text { def }\right)\) for solute-\(j\) using equation (b). \[\mathrm{K}_{\mathrm{sj}}(\mathrm{aq} ; \operatorname{def})=\phi\left(\mathrm{K}_{\mathrm{sj}} ; \operatorname{def}\right)+\mathrm{m}_{\mathrm{j}} \,\left[\partial \phi\left(\mathrm{K}_{\mathrm{s} j} ; \operatorname{def}\right) / \partial \mathrm{m}_{\mathrm{j}}\right]_{\mathrm{T}, \mathrm{p}}\]This study confirmed the importance of the stereochemistry of carbohydrates on their hydration. A clear contrast is drawn between those solutes where the hydrophilic groups match and mismatch into the three dimensionally hydrogen - bonded structure of liquid water. With increase in solute concentration, the dependence of \(\mathrm{K}_{\mathrm{Sj}}(\mathrm{aq} ; \text { def })\) on composition is non-linear. For amines(aq) \(\mathrm{K}_{\mathrm{Sj}}(\mathrm{aq} ; \text { def })\) passes through minima.Chalikian discusses the isentropic compression of a wide range of solutes with reference to group contributions, the discussion being extended to proteins and oligopeptides.Footnotes H. Høiland, J. Solution Chem., 1977, 6, 291. P. J. Bernal and W. A. Van Hook, J. Chem. Thermodyn., 1986,18,955. A. Lo Surdo, C. Shin and F. J. Millero, J. Chem. Eng. Data, 1978, 23, 197. F. Franks, J. R. Ravenhill and D. S. Reid, J Solution Chem.,1972, 1,3. M. Iqbal and R. E. Verrall, J.Phys.Chem.,1987,91,967. D. P. Kharakoz, J.Phys.Chem.,1991,95,5634. F. J. Millero, A. Lo Surdo and C. Shin, J.Phys.Chem.,1978,82,784. T. V. Chalikian, A. P. Sarvazyan, T. Funck, C. A. Cain and K. J. Breslauer, J. Phys. Chem., 1994, 98, 321. M. Kikuchi, M. Sakurai and K. Nitta, J. Chem. Eng. Data, 1995,40,935 M. Sakurai, K. Nakamura, K. Nitta and N. Takenaka, J. Chem. Eng. Data, 1995,40,301. T. Nakajima, T. Komatsu and T. Nakagawa, Bull. Chem. Soc Jpn., 1975,48,788. G. R. Hedwig, H. Hoiland and E. Hogseth, J. Solution Chem.,1996,25,1041. G. R. Hedwig, J. D. Hastie and H. Hoiland, J. Solution Chem.,1996,25,615. S. A. Galema and H. Høiland, J. Phys. Chem., 1991, 95, 5321. S. A. Galema, J. B. F. N. Engberts, H. Hoiland and G. M. Forland, J. Phys. Chem., 1993, 97, 6885. M. Kaulgud and K. J. Patil, J. Phys. Chem., 1974,78,714. T. V. Chalikian, J. Phys. Chem.B,2001,105,12566. N Taulier and T.V. Chalikian, Biochem. Biophys. Acta, 2002,1595,48. A. W.Hakin, H. Hoiland and G. R. Hedwig, Phys. Chem. Chem. Phys.,2000,2,4850.This page titled 1.7.6: Compressions- Isentropic- Neutral Solutes is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.7.7: Compressions- Isentropic- Salt Solutions

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An extensive literature describes the isentropic compressibilities of salt solutions prompted by earlier studies by Passynski described by Owen.The isentropic compression of a given aqueous salt solution \(\mathrm{K}_{\mathrm{s}}(\mathrm{aq})\) is determined using the Newton-Laplace Equation in conjunction with speeds of sound and densities. An apparent molar compression of salt \(j \phi\left(K_{s} ; \text { def }\right)\) is calculated using equation (a). \[\mathrm{K}_{\mathrm{S}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{K}_{\mathrm{S} 1}^{*}(\ell)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{K}_{\mathrm{s}} ; \operatorname{def}\right)\]Here \(\mathrm{K}_{\mathrm{S} 1}^{*}(\ell)\) is the isentropic compression of the solvent at the same \(\mathrm{T}\) and \(\mathrm{p}\). For salt solutions, particularly aqueous salt solutions, the dependence of \(\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \text { def }\right)\) on the molality of the salt is generally examined in the light of equations describing the role of ion-ion interactions [3; see also reference 4]. For dilute salt solutions, equation (b) forms the basis for examining the dependence of \(\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \text { def }\right)\) on \(\left(\mathrm{m}_{\mathrm{j}}\right)^{1 / 2}\) where \(\mathrm{m}_{j}\) is the molality of the salt-\(j\). \[\text { Then, } \phi\left(\mathrm{K}_{\mathrm{Sj}} ; \text { def }\right)=\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \text { def }\right)^{\infty}+\mathrm{S}_{\mathrm{KS}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}\]The form of the equation (b) has all the hallmarks of a pattern required by the \(\mathrm{DHLL}\). In practice \(\mathrm{S}_{\mathrm{KS}}\) cannot be calculated because the required isentropic dependence of the relative permittivity of the solvent on pressure is generally not known. However a plot is obtained using equation (b) yielding an estimate of \(\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \operatorname{def}\right)^{\infty}\).For a large range of 1:1 salts \(\phi\left(K_{S_{j}} ; \operatorname{def}\right)^{\infty}\) is negative, a pattern attributed to electrostriction of neighbouring solvent molecules by electric charges on the ions. \(\phi\left(\mathrm{K}_{\mathrm{S}_{j}} ; \text { def }\right)^{\infty}\) is more negative for solutions in \(\mathrm{D}_{2}\mathrm{O}\) than in \(\mathrm{H}_{2}\mathrm{O}\) as a consequence of more intense electrostriction in \(\mathrm{D}_{2}\mathrm{O}\). Further on the basis of the Desnoyers-Philip Equation, the difference \(\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \mathrm{def}\right)^{\infty}-\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)^{\infty}\) is small but not negligible, amounting to approx. 10%. For alkylammonium ions in aqueous solutions \(\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \operatorname{def}\right)^{\infty}\) decreases with increase in the hydrophobic character, matching a general increase in \(\phi\left(V_{j}\right)^{\infty}\). Group and ionic contributions to \(\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \operatorname{def}\right)^{\infty}\) have been estimated. Indeed \(\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \text { def }\right)\) is approximately a linear function of \(\left(m_{j} / m^{0}\right)^{1 / 2}\) for a wide range of aqueous and non-aqueous salt solutions; e.g. salts in \(\mathrm{DMSO}\) and in propylene carbonate. \(\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \text { def }\right)\) for copper(I) and sodium perchlorates in cyanobenzene, pyridine and cyanomethane show almost no dependence on salt molality.A problem is further complicated by the fact that the \(\mathrm{DHLL}\) for \(\phi\left(K_{\mathrm{S} j} ; \text { def }\right)\) is itself a complicated function of salt molality, \(\mathrm{S}_{\mathrm{KS}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}\) being however the leading term. A problem is encountered with the differential dependence of the molar volume of the solvent \(\mathrm{V}_{1}^{*}(\ell)\) on pressure at constant \(\mathrm{S}(\mathrm{s} \ln )\) describing how the volume of the solution would depend on pressure if it were held at the same entropy of the solution. Thus \[\begin{aligned} -\left[\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right]_{\mathrm{s}(\mathrm{aq})} &=\left[\kappa_{\mathrm{s}}(\mathrm{aq})-\kappa_{\mathrm{s} 1}^{*}(\ell)\right] \,\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1}+\kappa_{\mathrm{s}}(\mathrm{aq}) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \\ &+\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \, \mathrm{T} \, \alpha_{1}^{*}(\ell) \,\left\{\left[\alpha_{\mathrm{p}}(\mathrm{aq}) / \sigma(\mathrm{aq})\right]-\left[\alpha_{\mathrm{p} 1}^{*}(\ell) / \sigma_{1}^{*}(\ell)\right]\right\} \end{aligned}\]By definition, \[\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \operatorname{def}\right)=\left[\kappa_{\mathrm{s}}(\mathrm{aq})-\kappa_{\mathrm{S} 1}^{*}(\ell)\right] \,\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1}+\kappa_{\mathrm{s}}(\mathrm{aq}) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\]Then, \[\begin{aligned} &-\left[\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right]_{\mathrm{S}(\mathrm{aq})}=\phi\left(\mathrm{K}_{\mathrm{sj}} ; \mathrm{def}\right) \\ &\quad+\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \, \mathrm{T} \, \alpha_{1}^{*}(\ell) \,\left\{\left[\alpha_{\mathrm{p}}(\mathrm{aq}) / \sigma(\mathrm{aq})\right]-\left[\alpha_{\mathrm{p} 1}^{*}(\ell) / \sigma_{1}^{*}(\ell)\right]\right\} \end{aligned}\]Consequently the difference between \(-\left[\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right]_{\mathrm{s}(\mathrm{aq})}\) and \(\phi\left(\mathrm{K}_{\mathrm{sj}} ; \mathrm{def}\right)\) is determined by the property \(\Delta \phi\), defined in equation (f). \[\Delta \phi=\left\{\left[\alpha_{\mathrm{p}}(\mathrm{aq}) / \sigma(\mathrm{aq})\right]-\left[\alpha_{\mathrm{p} 1}^{*}(\ell) / \sigma_{1}^{*}(\ell)\right]\right\}\]However \(\Delta \phi / \mathrm{m}_{\mathrm{j}}\) is indeterminate at infinite dilution . But using L’Hospital‘s rule, \[\operatorname{Limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \Delta \phi / \mathrm{m}_{\mathrm{j}}= \left[\left[\rho_{1}^{*}(\ell) \, \alpha_{p 1}^{*}(\ell) / \sigma_{1}^{*}(\ell)\right] \,\left\{\frac{\phi\left(E_{p j}\right)^{\infty}}{\alpha_{p 1}^{*}(\ell)}\right]-\left[\frac{\phi\left(C_{p j}\right)^{\infty}}{\sigma_{1}^{*}(\ell)}\right]\right\}\]Despite the thermodynamic polish given to the analysis of isentropic compressions, the problem of contrasting conditions ‘at constant \(\mathrm{S}(\mathrm{aq})\)’ and ‘at constant \(\mathrm{S}_{1}^{*}(\ell)\)’ underlies the analysis.Footnotes A. Passynski, Acta Physicochim. URSS, 1938,8,385. B. B. Owen and P. L. Kronick, J. Am. Chem.Soc.,1961,65,84. F. J. Millero, in Activity Coefficients of Electrolyte Solutions, ed. R. M. Pytkowicz, CRC Press, Boca Raton, Fl,1979 , chapter 13. F. J. Millero, F. Vinokurova, M. Fernandez and J. P. Hershey, J. Solution Chem.,1987,16,269. J. G. Mathieson and B. E. Conway, J. Chem. Soc Faraday Trans.1, 1974, 70,752. B. B. Owen and H. L. Simons, J. Am. Chem. Soc.,1957,61,479 Transition metal chlorides(aq); A. Lo Surdo and F. J. Millero, J. Phys. Chem.,1980,84,710. Nitroammino cobalt(III) complexes(aq); F. Kawaizumi, K. Matsumoto and H. Nomura, J. Phys. Chem., 1983, 87,3161. Sodium nitrate(aq) and sodium thiosulfate(aq); N. Rohman, and S. Mahiuddin, J. Chem. Soc. Faraday Trans.,1997,93,2053. Bipyridine and phenanthroline complexes of Fe(II), Cu(II), Ni(II) and Cu(II) chlorides(aq); F. Kawaizumi, H. Nomura and F. Nakao, J. Solution Chem.,1987, 16,133 R. Buwalda, J. B. F. N. Engberts, H. Høiland and M. J. Blandamer, J. Phys. Org. Chem., 1998, 11, 59. E. Ayranci and B. E. Conway, J. Chem. Soc. Faraday Trans.1, 1983,79,1357. G. Peron, G. Trudeau and J.E.Desnoyers, Can J.Chem.,1987,65,1402. J. I. Lankford, W. T. Holladay and C. M. Criss, J. Solution Chem.,1984,13,699. J. I. Lankford and C. M. Criss, J. Solution Chem.,1987,16,753. D. S. Gill, P. Singh, J. Singh, P. Singh, G. Senanayake and G. T. Hefter, J. Chem. Soc., Faraday Trans., 1995, 91, 2789. J. C. R. Reis and M. A. P. Segurado, Phys.Chem.Chem.Phys.,1999,1,1501. M. J. Blandamer, J. Chem. Soc. Faraday Trans.,1998,94,1057.This page titled 1.7.7: Compressions- Isentropic- Salt Solutions is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.7.8: Compresssions- Isentropic- Aqueous Solution

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A given aqueous solution is prepared using \(\mathrm{n}_{1}\) moles of water and \(\mathrm{n}_{j}\) moles of solute \(j\). The thermodynamic properties of this solution are ideal. \[\text { Then, } \quad V_{m}(a q ; \text { id })=x_{1} \, V_{1}^{*}(\ell)+x_{j} \, \phi\left(V_{j}\right)^{\infty}\]\[\text { Here } \phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}=\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq})=\operatorname{limit}\left(\mathrm{n}_{\mathrm{j}} \rightarrow 0\right)\left(\frac{\partial \mathrm{V}(\mathrm{aq} ; \mathrm{id})}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}}\]The molar entropy of the ideal solution is given by equation (c). \[\mathrm{S}_{\mathrm{m}}(\mathrm{aq} ; \mathrm{id})=\mathrm{x}_{1} \, \mathrm{S}_{1}^{*}(\ell)-\mathrm{x}_{1} \, \mathrm{R} \, \ln \left(\mathrm{x}_{1}\right)+\mathrm{x}_{\mathrm{j}} \, \mathrm{S}_{\mathrm{j}}(\mathrm{aq} ; \mathrm{id})\]\(\mathrm{S}_{\mathrm{j}}(\mathrm{aq} ; \mathrm{id})\) is the partial molar entropy of solute \(j\) at the same \(\mathrm{T}\) and \(\mathrm{p}\). The solution is perturbed by a change in pressure and displaced to a neighbouring state having the same entropy, \(\mathrm{S}_{\mathrm{m}}(\mathrm{aq} ; \mathrm{id})\). \[\mathrm{K}_{\mathrm{Sm}}(\mathrm{aq} ; \mathrm{id})=-\left(\frac{\partial \mathrm{V}_{\mathrm{m}}(\mathrm{aq} ; \mathrm{id})}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{m} ; \mathrm{aq} ; \mathrm{jd}), \mathrm{x}(\mathrm{j})}\]From equation (a), \[\mathrm{K}_{\mathrm{Sm}(\mathrm{a} ; ; \mathrm{dd})}=-\mathrm{x}_{1} \,\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{m} ; \mathrm{aq} ; \mathrm{id}), \mathrm{x}(\mathrm{j})}-\mathrm{x}_{\mathrm{j}} \,\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{m} ; \mathrm{aq} ; ; \mathrm{dd}), \mathrm{x}(\mathrm{j})}\]On these partial differentials, the isentropic condition is not the most convenient because it refers to the entropy of an ideal solution. Using the technique adopted for liquid mixtures, we obtain in equation (f), an expression for the unconventional isentropic compression of the solvent . \[\begin{aligned} &\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{m} ; \mathrm{aq} ; \mathrm{dd}), \mathrm{x}(\mathrm{j})}= \\ &-\mathrm{K}_{\mathrm{S} 1}^{*}(\ell)-\mathrm{T} \,\left[\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)\right]^{2} / \mathrm{C}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{T} \, \mathrm{E}_{\mathrm{p} 1}^{*}(\ell) \, \mathrm{E}_{\mathrm{pm}}(\mathrm{aq} ; \mathrm{id}) / \mathrm{C}_{\mathrm{pm}}(\mathrm{aq} ; \mathrm{id}) \end{aligned}\]Or, \[\begin{aligned} &\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{m} ; \mathrm{qq} ; \mathrm{id}), \mathrm{x}(\mathrm{j})}= \\ &-\mathrm{K}_{\mathrm{S} 1}^{*}(\ell)-\mathrm{T} \, \mathrm{E}_{\mathrm{p} 1}^{*}(\ell) \,\left[\mathrm{E}_{\mathrm{p} 1}^{*}(\ell) / \mathrm{C}_{\mathrm{p} 1}^{*}(\ell)-\mathrm{E}_{\mathrm{pm}}(\mathrm{aq} ; \mathrm{id}) / \mathrm{C}_{\mathrm{pm}}(\mathrm{aq} ; \mathrm{id})\right] \end{aligned}\]Except for the different ideal reference state, equation (g) for the solvent is formally identical to the corresponding equation for liquid mixtures. However, in this case we need to follow a different approach for chemical substance \(j\). The appropriate choice for isentropic conditions on solute properties is the entropy of the pure solvent at same \(\mathrm{T}\) and \(\mathrm{p}\). Hence, \[\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}}{\partial \mathrm{p}}\right)_{\mathrm{s}(\mathrm{m} ; \mathrm{aq} ; \mathrm{id}), \mathrm{x}(\mathrm{j})}=\]Or, \[\begin{aligned} &\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{m} ; \mathrm{aq} ; \mathrm{id}), \mathrm{x}(\mathrm{j})}=\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}}{\partial \mathrm{p}}\right)_{\mathrm{s}_{1}^{*}(\ell)} \\ &-\mathrm{T} \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right)^{\infty} \, \frac{\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)}{\mathrm{C}_{\mathrm{pl}}^{*}(\ell)}+\mathrm{T} \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right)^{\infty} \, \frac{\mathrm{E}_{\mathrm{pm}}(\mathrm{aq} ; \mathrm{id})}{\mathrm{C}_{\mathrm{pm}}(\mathrm{aq} ; \mathrm{id})} \end{aligned}\]Or, \[\begin{aligned} &\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{m} ; \mathrm{aq} ; \mathrm{id}), \times(\mathrm{j})}=\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}}{\partial \mathrm{p}}\right)_{\mathrm{s}_{1}^{*}(\ell)} \\ &-\mathrm{T} \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right)^{\infty} \,\left[\frac{\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}-\frac{\mathrm{E}_{\mathrm{pm}}(\mathrm{aq} ; \mathrm{id})}{\mathrm{C}_{\mathrm{pm}}(\mathrm{aq} ; \mathrm{id})}\right] \end{aligned}\]The isentropic pressure dependences of apparent and partial molar volumes are complicated functions. We are interested in obtaining an expression for \(\mathrm{K}_{\mathrm{Sm}}(\mathrm{aq} ; \mathrm{id})\) in terms of the limiting apparent or partial molar isentropic compression of solute \(j\), \(\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)^{\infty}\). We use the following expression. \[\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}}{\partial \mathrm{p}}\right)_{\mathrm{s}_{1}^{*}(\ell)}=-\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)^{\infty}-\mathrm{T} \,\left(\frac{\left[\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}\right) \,\left(\frac{\phi\left(\mathrm{E}_{\mathrm{pj}}\right)^{\infty}}{\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)}-\frac{\phi\left(\mathrm{C}_{\mathrm{pj}}\right)^{\infty}}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}\right)\]We combine the results in equations (e), (g), (j) and (k) to obtain an equation for \(\mathrm{K}_{\mathrm{Sm}}(\mathrm{aq} ; \mathrm{id})\); equation (l). \[\begin{aligned} &\mathrm{K}_{\mathrm{Sm}}(\mathrm{aq} ; \mathrm{id})= \\ &\mathrm{x}_{1} \, \mathrm{K}_{\mathrm{s} 1}^{*}(\ell)+\mathrm{x}_{1} \, \mathrm{T} \, \mathrm{E}_{\mathrm{p} 1}^{*}(\ell) \,\left[\frac{\mathrm{E}_{\mathrm{pl}}^{*}(\ell)}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}-\frac{\mathrm{E}_{\mathrm{pm}}(\mathrm{aq} ; \mathrm{id})}{\mathrm{C}_{\mathrm{pm}}(\mathrm{aq} ; \mathrm{id})}\right] \\ &+\mathrm{x}_{\mathrm{j}} \, \phi\left(\mathrm{K}_{\mathrm{sj}}\right)^{\infty}+\mathrm{x}_{\mathrm{j}} \, \mathrm{T} \,\left[\frac{\left[\mathrm{E}_{\mathrm{pl}}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{pl}}^{*}(\ell)}\right] \,\left[\frac{\phi\left(\mathrm{E}_{\mathrm{pj}}\right)^{\infty}}{\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)}-\frac{\phi\left(\mathrm{C}_{\mathrm{pj}}\right)^{\infty}}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}\right] \\ &+\mathrm{x}_{\mathrm{j}} \, \mathrm{T} \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right)^{\infty} \,\left[\frac{\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}-\frac{\mathrm{E}_{\mathrm{pm}}(\mathrm{aq} ; \mathrm{id})}{\mathrm{C}_{\mathrm{pm}}(\mathrm{aq} ; \mathrm{id})}\right] \end{aligned}\]Finally, by noting that \(\mathrm{E}_{\mathrm{pm}}(\mathrm{aq} ; \mathrm{id})=\mathrm{x}_{1} \, \mathrm{E}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{x}_{\mathrm{j}} \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right)^{\infty}\) after slight simplification we arrive at an expression for \(\mathrm{K}_{\mathrm{Sm}}(\mathrm{aq} ; 1 \mathrm{~d})\); equation (m). \[\begin{aligned} &\mathrm{K}_{\mathrm{Sm}}(\mathrm{aq} ; \mathrm{id})=\mathrm{x}_{1} \,\left\{\mathrm{K}_{\mathrm{s} 1}^{*}(\ell)+\mathrm{T} \,\left[\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)\right]^{2} / \mathrm{C}_{\mathrm{p} 1}^{*}(\ell)\right\} \\ &+\mathrm{x}_{\mathrm{j}} \,\left\{\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)^{\infty}+\mathrm{T} \,\left(\frac{\left[\mathrm{E}_{\mathrm{pl}}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}\right) \,\left(\frac{2 \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right)^{\infty}}{\mathrm{E}_{\mathrm{pl} 1}^{*}(\ell)}-\frac{\phi\left(\mathrm{C}_{\mathrm{pj}}\right)^{\infty}}{\mathrm{C}_{\mathrm{pl} 1}^{*}(\ell)}\right)\right. \\ &-\mathrm{T} \, \frac{\left[\mathrm{E}_{\mathrm{pm}}(\mathrm{aq} ; \mathrm{id})\right]^{2}}{\mathrm{C}_{\mathrm{pm}}(\mathrm{aq} ; \mathrm{id})} \end{aligned}\]The complexity of equation (m) for solutions can be attributed to a combination of the non-Gibbsian character of \(\mathrm{K}_{\mathrm{Sm}}(\mathrm{aq} ; 1 \mathrm{~d})\) with the non-Lewisian character of \(\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)^{\infty}\). Clearly \(\mathrm{K}_{\mathrm{Sm}}(\mathrm{aq} ; 1 \mathrm{~d})\) and \(\mathrm{K}_{\mathrm{mix}}(\mathrm{aq} ; 1 \mathrm{~d})\) are not equal because the reference states for chemical substance \(j\) differ. We are interested in the apparent molar isentropic compression of solute \(j\) in ideal aqueous solutions \(\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)(\mathrm{aq} ; \mathrm{id})\), which is defined in equation (n) and expressed by equation (o). \[\mathrm{K}_{\mathrm{Sm}}(\mathrm{aq} ; \mathrm{id})=\mathrm{x}_{1} \, \mathrm{K}_{\mathrm{S} I}^{*}(\ell)+\mathrm{x}_{\mathrm{j}} \, \phi\left(\mathrm{K}_{\mathrm{Sj}}\right)(\mathrm{aq} ; \mathrm{id})\]\[\begin{aligned} &\phi\left(\mathrm{K}_{\mathrm{sj}}\right)(\mathrm{aq} ; \mathrm{id})=\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)^{\infty} \\ &+\mathrm{T} \,\left(\frac{\left[\mathrm{E}_{\mathrm{pl} 1}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}\right) \,\left[\frac{2 \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right)^{\infty}}{\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)}-\frac{\phi\left(\mathrm{C}_{\mathrm{pj}}\right)^{\infty}}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}\right] \\ &+\mathrm{T} \,\left(\frac{\mathrm{x}_{1} \,\left[\mathrm{E}_{\mathrm{pl}}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{pl} 1}^{*}(\ell)}-\frac{\left[\mathrm{E}_{\mathrm{pm}}(\mathrm{aq} ; \mathrm{id})\right]^{2}}{\mathrm{C}_{\mathrm{pm}}(\mathrm{aq} ; \mathrm{id})}\right) \, \frac{1}{\mathrm{x}_{\mathrm{j}}} \end{aligned}\]Limiting values for \(\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)(\mathrm{aq} ; \mathrm{id})\) are interesting. For the ideal solution at \(\mathrm{x}_{j} = 0\), which is the same state as the real solution at infinite dilution, we naturally obtain \(\phi\left(\mathrm{K}_{\mathrm{Sj}_{j}}\right)^{\infty}\) although using equation (o) for this purpose requires solving an indeterminate form. For the ideal solution at \(\mathrm{x}_{j} = 0\) we obtain equation (p), which yields equation (q) after major reorganisation. \[\begin{aligned} &\operatorname{limit}\left(\mathrm{x}_{\mathrm{j}} \rightarrow 1\right) \phi\left(\mathrm{K}_{\mathrm{sj}}\right)(\mathrm{aq} ; \mathrm{id})=\phi\left(\mathrm{K}_{\mathrm{s} j}\right)^{\infty} \\ &+\mathrm{T} \,\left(\frac{\left[\mathrm{E}_{\mathrm{pl}}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}\right) \,\left[\frac{2 \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right)^{\infty}}{\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)}-\frac{\phi\left(\mathrm{C}_{\mathrm{pj}}\right)^{\infty}}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}\right]-\mathrm{T} \,\left(\frac{\left[\phi\left(\mathrm{E}_{\mathrm{pj}}\right)^{\infty}\right]^{2}}{\phi\left(\mathrm{C}_{\mathrm{pj}}\right)^{\infty}}\right) \end{aligned}\]\[\begin{aligned} &\operatorname{limit}\left(\mathrm{x}_{\mathrm{j}} \rightarrow 1\right) \phi\left(\mathrm{K}_{\mathrm{s} j}\right)(\mathrm{aq} ; \mathrm{id})=\phi\left(\mathrm{K}_{\mathrm{sj}}\right)^{\infty} \\ &-\mathrm{T} \,\left(\frac{\left[\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)\right]^{2}}{\phi\left(\mathrm{C}_{\mathrm{pj}}\right)^{\infty}}\right) \,\left[\frac{\phi\left(\mathrm{E}_{\mathrm{p} j}\right)^{\infty}}{\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)}-\frac{\phi\left(\mathrm{C}_{\mathrm{pj}}\right)^{\infty}}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}\right] \end{aligned}\]The latter equation expresses the molar isentropic compression of solute \(j\) in a standard state of unit mole fraction in terms of properties for the pure solvent and for the solute at infinite dilution. The ideal aqueous solution may be described as a non-ideal liquid mixture. An excess property is defined by equation (r). \[\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)^{\mathrm{E}}=\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)(\mathrm{aq} ; \mathrm{id})-\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)(\text { mix} ; \mathrm{id})\]\[\text { Or, } \phi\left(K_{S j}\right)^{E}=\left[K_{S m}(\mathrm{aq} ; i d)-K_{S m}(\operatorname{mix} ; i \mathrm{~d})\right] / x_{j}\]A working equation for \(\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)^{\mathrm{E}}\) can be generated from equation (o). After little reorganisation, we obtain equation (t). \[\begin{aligned} &\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)^{\mathrm{E}}=\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)^{\infty}-\mathrm{K}_{\mathrm{Sj}}^{*}(\ell) \\ &+\mathrm{T} \,\left[\frac{\left[\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}\right] \,\left[\frac{2 \, \phi\left(\mathrm{E}_{\mathrm{pj}}^{\infty}\right)}{\mathrm{E}_{\mathrm{pl} 1}^{*}(\ell)}-\frac{\phi\left(\mathrm{C}_{\mathrm{pj}}\right)^{\infty}}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}\right]-\mathrm{T} \, \frac{\left[\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)} \\ &-\mathrm{T} \,\left[\frac{\left[\mathrm{E}_{\mathrm{pm}}(\mathrm{aq} ; \mathrm{id})\right]^{2}}{\mathrm{C}_{\mathrm{pm}}(\mathrm{aq} ; \mathrm{id})}-\frac{\left[\mathrm{E}_{\mathrm{pm}}(\mathrm{mix} ; \mathrm{id})\right]^{2}}{\mathrm{C}_{\mathrm{pm}}(\mathrm{mix} ; \mathrm{id})}\right] \, \frac{1}{\mathrm{x}_{\mathrm{j}}} \end{aligned}\]Interestingly, the first four terms on the right end side of equation (t) express the difference \(\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)^{\infty}-\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}^{*}(\ell)\). For solution chemists the important reference state is at infinite dilution. The limiting excess property \(\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)^{\mathrm{E}, \infty}\) is given by equation (u). \[\begin{aligned} &\phi\left(\mathrm{K}_{\mathrm{S} \mathrm{j}}\right)^{\mathrm{E}, \infty}=\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)^{\infty}-\mathrm{K}_{\mathrm{S} 1}^{*}(\ell) \\ &-\mathrm{T} \, \mathrm{C}_{\mathrm{p} 1}^{*}(\ell) \,\left[\frac{\mathrm{E}_{\mathrm{pj}}^{*}(\ell)}{\mathrm{C}_{\mathrm{p} j}^{*}(\ell)}-\frac{\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}\right]^{2} \end{aligned}\]Estimates of \(\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)^{\infty}\) using \(\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)^{\mathrm{E}, \infty}\) data for binary liquid mixtures often neglect the last term in equation (u).Equation (u) works in two ways. A solution chemist will estimate \(\phi\left(K_{S j}\right)^{\infty}\) from data reporting \(\phi\left(K_{\mathrm{Sj}}\right)^{\mathrm{E}, \infty}\). A chemist interested in the properties of liquid mixtures will estimate \(\phi\left(K_{\mathrm{Sj}}\right)^{\mathrm{E}, \infty}\) from data reporting \(\phi\left(K_{S j}\right)^{\infty}\).Footnotes J. C. R. Reis, J. Chem. Soc., Faraday Trans.2. 1982, 78, 1595. M. J. Blandamer, J. Chem. Soc., Faraday Trans., 1998, 94, 1057. J. C. R. Reis, J. Chem. Soc., Faraday Trans., 1998, 94, 2395. M. I. Davis, G. Douheret, J. C. R. Reis and M. J. Blandamer, Phys. Chem. Chem. Phys., 2001,3,4555.This page titled 1.7.8: Compresssions- Isentropic- Aqueous Solution is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.7.9: Compressions- Isentropic and Isothermal- Solutions- Approximate Limiting Estimates

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.07%3A_Compressions/1.7.09%3A_Compressions-_Isentropic_and_Isothermal-_Solutions-_Approximate_Limiting_Estimates

The Newton Laplace Equation relates the speed of sound \(\mathrm{u}\) in an aqueous solution, density \(\rho(\mathrm{aq})\) and isentropic compressibility \(\kappa_{\mathrm{S}}(\mathrm{aq})\); equation (a). \[\mathrm{u}^{2}=\left[\kappa_{\mathrm{s}}(\mathrm{aq}) \, \rho(\mathrm{aq})\right]^{-1}\]The differential dependence of sound velocity \(\mathrm{u}\) on \(\kappa_{\mathrm{S}}(\mathrm{aq})\) and \(\rho(\mathrm{aq})\) is given by equation (b). \[\begin{aligned} &2 \, u(a q) \, d u(a q)= \\ &\quad-\frac{1}{\left[\kappa_{\mathrm{s}}(a q)\right]^{2} \, \rho(a q)} \, d \kappa_{s}(a q)-\frac{1}{\left.\kappa_{s}(a q)\right] \,[\rho(a q)]^{2}} \, d \rho(a q) \end{aligned}\]We divide equation (b) by equation (a). \[2 \, \frac{\mathrm{du}(\mathrm{aq})}{\mathrm{u}(\mathrm{aq})}=-\frac{\mathrm{d} \kappa_{\mathrm{s}}(\mathrm{aq})}{\kappa_{\mathrm{s}}(\mathrm{aq})}-\frac{\mathrm{d} \rho(\mathrm{aq})}{\rho(\mathrm{aq})}\]We explore three approaches based on equation (c)Two extra-thermodynamic assumptions are made.\[\text { Thus } \quad \mathrm{u}(\mathrm{aq})=\mathrm{u}_{1}^{*}(\ell)+\mathrm{A}_{\mathrm{u}} \, \mathrm{c}_{\mathrm{j}}\] \[\text { By definition, } \quad \mathrm{du}(\mathrm{aq})=\mathrm{u}(\mathrm{aq})-\mathrm{u}_{1}^{*}(\ell)=\mathrm{A}_{\mathrm{u}} \, \mathrm{c}_{\mathrm{j}}\]\[\text { Thus } \quad \rho(\mathrm{aq})=\rho_{1}^{*}(\ell)+\mathrm{A}_{\rho} \, \mathrm{c}_{\mathrm{j}}\] Hence from equations (c)-(f), \[2 \, \frac{\mathrm{A}_{\mathrm{u}} \, \mathrm{c}_{\mathrm{j}}}{\mathrm{u}(\mathrm{aq})}=-\frac{\mathrm{d} \kappa_{\mathrm{S}}(\mathrm{aq})}{\mathrm{K}_{\mathrm{s}}(\mathrm{aq})}-\frac{\mathrm{A}_{\mathrm{\rho}} \, \mathrm{c}_{\mathrm{j}}}{\rho(\mathrm{aq})}\] \[\frac{\mathrm{d} \kappa_{\mathrm{S}}(\mathrm{aq})}{\kappa_{\mathrm{S}}(\mathrm{aq})}=-2 \, \frac{\mathrm{A}_{\mathrm{u}} \, \mathrm{c}_{\mathrm{j}}}{\mathrm{u}(\mathrm{aq})}-\frac{\mathrm{A}_{\rho} \, \mathrm{c}_{\mathrm{j}}}{\rho(\mathrm{aq})}\]In principle the change in \(\kappa_{\mathrm{S}}(\mathrm{aq})\) resulting from addition of a solute \(j\) to form a solution concentration \(\mathrm{c}_{j}\) can be obtained from the experimentally determined parameters \(\mathrm{A}_{\rho}\) and \(\mathrm{A}_{\mathrm{u}}\).Another approach expresses the two dependences using a general polynomial in \(\mathrm{c}_{j}\). \[\text { By definition, } \quad \mathrm{A}_{\mathrm{u}}^{\infty}=\operatorname{limit}\left(\mathrm{c}_{\mathrm{j}} \rightarrow 0\right)\left(\frac{\partial \mathrm{u}(\mathrm{aq})}{\partial \mathrm{c}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}}\]\[\text { and } \mathrm{A}_{\rho}^{\infty}=\operatorname{limit}\left(\mathrm{c}_{\mathrm{j}} \rightarrow 0\right)\left(\frac{\partial \rho(\mathrm{aq})}{\partial \mathrm{c}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}}\]The assumption is made that both \(\mathrm{A}_{\mathrm{u}}^{\infty}\) and \(\mathrm{A}_{\rho}^{\infty}\) are finite. \[\text { Similarly } \operatorname{limit}\left(\mathrm{c}_{\mathrm{j}} \rightarrow 0\right)\left(\frac{\kappa_{\mathrm{s}}(\mathrm{aq})-\kappa_{\mathrm{S}}^{*}(\ell)}{\mathrm{c}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}}=\left(\frac{\partial \kappa_{\mathrm{S}}(\mathrm{aq})}{\partial \mathrm{c}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}}^{\infty}\]The procedures described above are incorporated into the following equation for \(\phi\left(\mathrm{K}_{\mathrm{sj}} ; \mathrm{def}\right)\). \[\text { Thus } \phi\left(\mathrm{K}_{\mathrm{s} j} ; \text { def }\right)=\left[\mathrm{c}_{\mathrm{j}}\right]^{-1} \,\left[\kappa_{\mathrm{s}}(\mathrm{aq})-\kappa_{\mathrm{s} 1}^{*}(\ell)\right]+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \kappa_{\mathrm{s} 1}^{*}(\ell)\]Hence using equation (h) with \(\mathrm{d}_{\mathrm{s}}(\mathrm{aq})=\kappa_{\mathrm{s}}(\mathrm{aq})-\kappa_{\mathrm{s} 1}^{*}(\ell)\) \[\begin{aligned} &\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \text { def }\right)= \\ &\qquad\left[\kappa_{\mathrm{S}}(\mathrm{aq}) / \mathrm{c}_{\mathrm{j}}\right] \,\left[-\frac{2 \, \mathrm{A}_{\mathrm{u}} \, \mathrm{c}_{\mathrm{j}}}{\mathrm{u}(\mathrm{aq})}-\frac{\mathrm{A}_{\rho} \, \mathrm{c}_{\mathrm{j}}}{\rho(\mathrm{aq})}\right]+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \mathrm{K}_{\mathrm{S} 1}^{\mathrm{*}}(\ell) \end{aligned}\]If we assume that \(\kappa_{\mathrm{S}}(\mathrm{aq})\) is close to \(\kappa_{\mathrm{S} 1}^{*}(\ell)\), then \[\phi\left(\mathrm{K}_{\mathrm{Sj}_{j}} ; \operatorname{def}\right)=\kappa_{\mathrm{s}}(\mathrm{aq}) \,\left[-\frac{2 \, \mathrm{A}_{\mathrm{u}}}{\mathrm{u}(\mathrm{aq})}-\frac{\mathrm{A}_{\rho}}{\rho(\mathrm{aq})}+\phi\left(\mathrm{V}_{\mathrm{j}}\right)\right]\]Equation (n) is complicated in the sense that the properties \(\kappa_{\mathrm{S}}(\mathrm{aq})\), \(\mathrm{u}(\mathrm{aq})\), \(\rho(\mathrm{aq})\) and \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)\) depend on concentration \(\mathrm{c}_{j}\). With respect to \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)\), the following equation is exact. \[\phi\left(\mathrm{V}_{\mathrm{j}}\right)=\left[\mathrm{c}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\rho_{1}^{*}(\ell)-\rho(\mathrm{aq})\right]+\mathrm{M}_{\mathrm{j}} / \rho_{1}^{*}(\ell)\]\[\text { Using equation }(f), \phi\left(V_{j}\right)=-\frac{A_{\rho}}{\rho_{1}^{*}(\ell)}+\frac{M_{j}}{\rho_{1}^{*}(\ell)}\]\[\text { Or, }-\frac{A_{\rho}}{\rho_{1}^{*}(\ell)}=\phi\left(V_{j}\right)-\frac{M_{j}}{\rho_{1}^{*}(\ell)}\]Equation (q) is multiplied by the ratio, \(\rho_{1}^{*}(\ell) / \rho(\mathrm{aq})\). \[\text { Thus }-\frac{\mathrm{A}_{\rho}}{\rho(\mathrm{aq})}=\frac{\rho_{1}^{*}(\ell)}{\rho(\mathrm{aq})} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)-\frac{\mathrm{M}_{\mathrm{j}}}{\rho(\mathrm{aq})}\]Combination of equations (n) and (r) yields equation (s). \[\begin{aligned} &\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \mathrm{def}\right)= \\ &\kappa_{\mathrm{s}} \,\left[-\frac{2 \, \mathrm{A}_{\mathrm{u}}}{\mathrm{u}(\mathrm{aq})}+\frac{\rho_{1}^{*}(\ell)}{\rho(\mathrm{aq})} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)-\frac{\mathrm{M}_{\mathrm{j}}}{\rho(\mathrm{aq})}+\phi\left(\mathrm{V}_{\mathrm{j}}\right)\right] \end{aligned}\]The argument is advanced that \(\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \mathrm{def}\right)\) can be meaningfully extrapolated to infinite dilution. \[\operatorname{limit}\left(c_{j} \rightarrow 0\right) \phi\left(K_{\mathrm{Sj}_{j}} ; \operatorname{def}\right)=\phi\left(\mathrm{K}_{\mathrm{sj}} ; \operatorname{def}\right)^{\infty}\]In the same limit \(\rho_{1}^{*}(\ell) / \rho(\mathrm{aq})=1.0\) and \(\mathrm{K}_{\mathrm{S}}(\mathrm{aq})=\mathrm{K}_{\mathrm{S}}^{*}(\ell)\). \[\phi\left(\mathrm{K}_{\mathrm{S}_{j}} ; \operatorname{def}\right)^{\infty}=\kappa_{\mathrm{s} 1}^{*}(\ell) \,\left[2 \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}-\frac{\mathrm{M}_{\mathrm{j}}}{\rho_{1}^{*}(\ell)}-\frac{2 \, \mathrm{A}_{\mathrm{u}}}{\mathrm{u}_{1}^{*}(\ell)}\right]\]\[\text { But from equation }(\mathrm{d}), \mathrm{A}_{\mathrm{u}}=\left[\mathrm{u}(\mathrm{aq})-\mathrm{u}_{1}^{*}(\ell)\right] / \mathrm{c}_{\mathrm{j}}\]\[\phi\left(\mathrm{K}_{\mathrm{sj}} ; \operatorname{def}\right)^{\infty}=\kappa_{\mathrm{sl}}^{*}(\ell) \,\left[2 \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}-2 \, \mathrm{U}-\frac{\mathrm{M}_{\mathrm{j}}}{\rho_{1}^{*}(\ell)}\right]\]where (cf. equation (v)), \[\mathrm{U}=\left[\mathrm{u}(\mathrm{aq})-\mathrm{u}_{1}^{*}(\ell)\right] /\left[\mathrm{u}_{1}^{*}(\ell) \, \mathrm{c}_{\mathrm{j}}\right]\]The symbol \(\mathrm{U}\) identifies the relative molar increment of the speed of sound. Equation (w) shows \(\phi\left(K_{S_{j}} ; \operatorname{def}\right)^{\infty}\) is obtained from \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}\) and the speed of sound in a solution concentration \(\mathrm{c}_{j}\). \[\text { In this approach we assume that }\left(\frac{\partial \mathrm{u}}{\partial \mathrm{c}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}}=\frac{\mathrm{u}(\mathrm{aq})-\mathrm{u}_{1}^{*}(\ell)}{\mathrm{c}_{\mathrm{j}}}\]\[\text { Then, } U=\frac{1}{\mathrm{u}_{1}^{*}(\ell)} \,\left(\frac{\mathrm{du}(\mathrm{aq})}{\mathrm{dc}_{\mathrm{j}}}\right)\]However \(\left(\frac{\mathrm{du}(\mathrm{aq})}{\mathrm{dc}}\right)\) and similarly \(\left(\frac{\mathrm{du}(\mathrm{aq})}{\mathrm{dm}_{\mathrm{j}}}\right)\) are obtained using experimental results for real concentrations. Hence the estimated \(\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \operatorname{def}\right)^{\infty}\) is likely to be poor.The apparent molar isothermal compression of solute \(j\) is related to the concentration \(\mathrm{c}_{j}\) using the following exact equation. \[\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\left[\mathrm{c}_{\mathrm{j}}\right]^{-1} \,\left[\kappa_{\mathrm{T}}(\mathrm{aq})-\kappa_{\mathrm{T} 1}^{*}(\ell)\right]+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \kappa_{\mathrm{T} 1}^{*}(\ell)\]\[\text { By definition. } \quad \delta(a q)=\kappa_{\mathrm{T}}(\mathrm{aq})-\kappa_{\mathrm{S}}(\mathrm{aq})\]\[\text { and } \delta_{1}^{*}=\kappa_{\mathrm{T} 1}^{*}(\ell)-\kappa_{\mathrm{S} 1}^{*}(\ell)\]\[\text { For an aqueous solution, } \kappa_{\mathrm{T}}(\mathrm{aq})=\delta(\mathrm{aq})+\kappa_{\mathrm{s}}(\mathrm{aq})\]According to the Newton-Laplace Equation. \[[u(\mathrm{aq})]^{2}=\left[\kappa_{\mathrm{s}}(\mathrm{aq}) \, \rho(\mathrm{aq})\right]^{-1}\]\[\text { From equation }(\mathrm{zd}), \kappa_{\mathrm{T}}(\mathrm{aq})=\delta(\mathrm{aq})+\left\{[\mathrm{u}(\mathrm{aq})]^{2} \, \rho(\mathrm{aq})\right\}^{-1}\]At this stage, assumptions are made concerning the dependences of \(\kappa_{\mathrm{T}}(\mathrm{aq})\) and \(\delta(\mathrm{aq})\) on concentration \(\mathrm{c}_{j}\). \[\text { Thus } \quad \kappa_{\mathrm{T}}(\mathrm{aq})=\kappa_{\mathrm{T} 1}^{*}(\ell)+\mathrm{A}_{\mathrm{KT}} \, \mathrm{c}_{\mathrm{j}}\]\[\text { and } \quad \delta(\mathrm{aq})=\delta_{1}^{*}(\ell)+\mathrm{A}_{\delta} \, \mathrm{c}_{\mathrm{j}}\]Using equations (d), (f) and (zf), \[\begin{aligned} \kappa_{\mathrm{Tl}}^{*}(\ell)+\mathrm{A}_{\kappa \mathrm{T}} \, \mathrm{c}_{\mathrm{j}}=& \delta_{1}^{*}(\ell)+\mathrm{A}_{\delta} \, \mathrm{c}_{\mathrm{j}} \\ &+\frac{1}{\left\{\mathrm{u}_{1}^{*}(\ell)+\mathrm{A}_{\mathrm{u}} \, \mathrm{c}_{\mathrm{j}}\right\}^{2} \,\left\{\rho_{1}^{*}(\ell)+\mathrm{A}_{\rho} \, \mathrm{c}_{\mathrm{j}}\right\}} \end{aligned}\]Or, \[\begin{aligned} &\kappa_{\mathrm{T} 1}^{*}(\ell)+\mathrm{A}_{\kappa \mathrm{T}} \, \mathrm{c}_{\mathrm{j}}=\delta_{1}^{*}(\ell)+\mathrm{A}_{\delta} \, \mathrm{c}_{\mathrm{j}} \\ &+\frac{1}{\left[\mathrm{u}_{1}^{*}(\ell)\right]^{2} \,\left\{1+\mathrm{A}_{\mathrm{u}} \, \mathrm{c}_{\mathrm{j}} / \mathrm{u}_{1}^{*}(\ell)\right\}^{2} \, \rho_{1}^{*}(\ell) \,\left\{1+\mathrm{A}_{\rho} \, \mathrm{c}_{\mathrm{j}} / \rho_{1}^{*}(\ell)\right\}} \end{aligned}\]Assuming \(\mathrm{A}_{\mathrm{u}} \, \mathrm{c}_{\mathrm{j}} / \mathrm{u}_{1}^{*}(\ell)<<1\) and \(A_{\rho} \, c_{j} / \rho_{1}^{*}(\ell)<<1\), \[\begin{aligned} &\kappa_{\mathrm{T} 1}^{*}(\ell)+\mathrm{A}_{\mathrm{kT}} \, \mathrm{c}_{\mathrm{j}}=\delta_{1}^{*}(\ell)+\mathrm{A}_{\delta} \, \mathrm{c}_{\mathrm{j}} \\ &+\frac{1}{\left[\mathrm{u}_{1}^{*}(\ell)\right]^{2} \, \rho_{1}^{*}(\ell)} \,\left[1-\frac{2 \, \mathrm{A}_{\mathrm{u}} \, \mathrm{c}_{\mathrm{j}}}{\mathrm{u}_{1}^{*}(\ell)}\right] \,\left[1-\frac{\mathrm{A}_{\rho} \, \mathrm{c}_{\mathrm{j}}}{\rho_{1}^{*}(\ell)}\right] \end{aligned}\]\[\text { We assume that }\left[\frac{2 \, \mathrm{A}_{\mathrm{u}} \, \mathrm{c}_{\mathrm{j}}}{\mathrm{u}_{1}^{*}(\ell)}\right] \,\left[\frac{\mathrm{A}_{\rho} \, \mathrm{c}_{\mathrm{j}}}{\rho_{1}^{*}(\ell)}\right]<<1\]\[\begin{aligned} &\text { Therefore, } \\ &\kappa_{\mathrm{T} 1}^{*}(\ell)+\mathrm{A}_{\mathrm{KT}} \, \mathrm{c}_{\mathrm{j}}=\delta_{1}^{*}(\ell)+\mathrm{A}_{\delta} \, \mathrm{c}_{\mathrm{j}} \\ &+\frac{1}{\left[\mathrm{u}_{1}^{*}(\ell)\right]^{2} \, \rho_{1}^{*}(\ell)} \,\left[1-\frac{2 \, \mathrm{A}_{\mathrm{u}} \, \mathrm{c}_{\mathrm{j}}}{\mathrm{u}_{1}^{*}(\ell)}-\frac{\mathrm{A}_{\rho} \, \mathrm{c}_{\mathrm{j}}}{\rho_{1}^{*}(\ell)}\right] \end{aligned}\]\[\text { But } \kappa_{\mathrm{s} 1}^{*}(\ell)=\left\{\left[u_{1}^{*}(\ell)\right]^{2} \, \rho_{1}^{*}(\ell)\right\}^{-1}\]\[\text { and } \kappa_{\mathrm{T} 1}^{*}(\ell)=\delta_{1}^{*}(\ell)+\kappa_{\mathrm{S} 1}^{*}(\ell)\]Then, \[\begin{aligned} &\delta_{1}^{*}(\ell)+\kappa_{\mathrm{S} 1}^{*}(\ell)+\mathrm{A}_{\mathrm{KT}} \, \mathrm{c}_{\mathrm{j}}=\delta_{1}^{*}(\ell)+\mathrm{A}_{\delta} \, \mathrm{c}_{\mathrm{j}} \\ &+\kappa_{\mathrm{Sl}}^{*}(\ell) \,\left[1-\frac{2 \, \mathrm{A}_{\mathrm{u}} \, \mathrm{c}_{\mathrm{j}}}{\mathrm{u}_{1}^{*}(\ell)}-\frac{\mathrm{A}_{\rho} \, \mathrm{c}_{\mathrm{j}}}{\rho_{1}^{*}(\ell)}\right] \end{aligned}\]\[\text { Or } \mathrm{A}_{\mathrm{K}}=\mathrm{A}_{\delta}-\kappa_{\mathrm{S} 1}^{*}(\ell) \,\left[\frac{2 \, \mathrm{A}_{\mathrm{u}}}{\mathrm{u}_{1}^{*}(\ell)}+\frac{\mathrm{A}_{\rho}}{\rho_{1}^{*}(\ell)}\right]\]From equations (za) and (zg), \[\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\mathrm{A}_{\mathrm{KT}}+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \kappa_{\mathrm{Tl}}^{*}(\ell)\]Equations (zq) and (zr) yield equation (as), \[\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\mathrm{A}_{\delta}-\kappa_{\mathrm{S} 1}^{*}(\ell) \,\left[\frac{2 \, \mathrm{A}_{\mathrm{u}}}{\mathrm{u}_{1}^{*}(\ell)}+\frac{\mathrm{A}_{\rho}}{\rho_{1}^{*}(\ell)}\right]+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \kappa_{\mathrm{T} 1}^{*}(\ell)\]Or, using equation (q) \[\begin{aligned} \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\mathrm{A}_{\delta}-\mathrm{K}_{\mathrm{S} 1}^{*}(\ell) \,[&\left.\frac{2 \, \mathrm{A}_{\mathrm{u}}}{\mathrm{u}_{1}^{*}(\ell)}+\frac{\mathrm{M}_{\mathrm{j}}}{\rho_{1}^{*}(\ell)}-\phi\left(\mathrm{V}_{\mathrm{j}}\right)\right] \\ &+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \kappa_{\mathrm{T} 1}^{*}(\ell) \end{aligned}\]Using equation (zc), \[\begin{aligned} \phi\left(\mathrm{K}_{\mathrm{Tj}_{\mathrm{j}}}\right)=& \mathrm{A}_{\delta}-\kappa_{\mathrm{S} 1}^{*}(\ell) \,\left[\frac{2 \, \mathrm{A}_{\mathrm{u}}}{\mathrm{u}_{1}^{*}(\ell)}+\frac{\mathrm{M}_{\mathrm{j}}}{\rho_{1}^{*}(\ell)}-\phi\left(\mathrm{V}_{\mathrm{j}}\right)\right] \\ &+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \delta_{1}^{*}(\ell)+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \kappa_{\mathrm{S} 1}^{*}(\ell) \end{aligned}\]Or, \[\begin{aligned} &\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\mathrm{A}_{\delta}+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \delta_{1}^{*}(\ell) \\ &+\kappa_{\mathrm{Sl}}^{*}(\ell) \,\left[2 \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)-\frac{\mathrm{M}_{\mathrm{j}}}{\left.\rho_{1}^{*} \ell\right)}-\frac{2 \, \mathrm{A}_{u}}{\left.\mathrm{u}_{1}^{*} \ell\right)}\right] \end{aligned}\]The latter is the Owen-Simons Equation which takes the following form in the limit of infinite dilution. \[\begin{aligned} \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)^{\infty}=\left[\mathrm{A}_{\delta}\right.&\left.+\phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty} \, \delta_{1}^{*}(\ell)\right] \\ &+\kappa_{\mathrm{Sl}}^{*}(\ell) \,\left[2 \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}-\frac{\mathrm{M}_{\mathrm{j}}}{\rho_{1}^{*}(\ell)}-\frac{2 \, \mathrm{A}_{\mathrm{u}}}{\mathrm{u}_{1}^{*}(\ell)}\right] \end{aligned}\]The term \(\left[\mathrm{A}_{\delta}+\phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty} \, \delta_{1}^{*}(\ell)\right]\) is not negligibly small. Using equation (u), equation (zw) takes the following form, \[\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)^{\infty}=\left[\mathrm{A}_{\delta}+\phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty} \, \delta_{1}^{*}(\ell)\right]+\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)^{\infty}\]Clearly the approximation which sets \(\phi\left(\mathrm{K}_{\mathrm{Tj}}\right)^{\infty}\) equal to \(\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)^{\infty}\) is poor although often made. In fact Hedwig and Hoiland show that for N-acetylamino acids in aqueous solution at \(298.15 \mathrm{~K} \mathrm{} \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)^{\infty}\) and \(\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)^{\infty}\) can have different signs, offering convincing evidence that the assumption is untenable.Footnotes \(\begin{aligned} &A_{u}=\left[\frac{m}{s}\right] \,\left[\frac{m^{3}}{m o l}\right]=\left[m^{4} \mathrm{~s}^{-1} \mathrm{~mol}^{-1}\right] \\ &A_{\rho}=\left[\frac{k g}{m^{3}}\right] \,\left[\frac{m^{3}}{m o l}\right]=\left[k g \mathrm{~mol}^{-1}\right] \end{aligned}\) \(\begin{aligned} &2 \, \frac{\mathrm{A}_{\mathrm{u}}}{\mathrm{u}} \, \mathrm{K}_{\mathrm{S}}(\mathrm{aq})= \, \frac{1}{\left[\mathrm{~m} \mathrm{~s}^{-1}\right]} \,\left[\mathrm{m}^{4} \mathrm{~s}^{-1} \mathrm{~mol}^{-1}\right] \, \frac{1}{\left[\mathrm{~N} \mathrm{~m}^{-2}\right]}=\frac{\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]}{\left[\mathrm{N} \mathrm{m}^{-2}\right]} \\ &\frac{\mathrm{A}_{\rho}}{\rho} \, \kappa_{\mathrm{S}}(\mathrm{aq})=\frac{\left[\mathrm{kg} \mathrm{m}^{-3}\right]}{\left[\mathrm{mol} \mathrm{m}^{-3}\right]} \, \frac{1}{\left.\mathrm{~kg} \mathrm{~m}^{-3}\right]} \, \frac{1}{\left[\mathrm{~N} \mathrm{~m}^{-2}\right]}=\frac{\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]}{\left[\mathrm{Nm}^{-2}\right]} \\ &\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \kappa_{\mathrm{s}}(\mathrm{aq})=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right] \, \frac{1}{\left[\mathrm{~N} \mathrm{~m}^{-2}\right]}=\frac{\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]}{\left[\mathrm{N} \mathrm{m}^{-2}\right]} \end{aligned}\) S. Barnatt, J. Chem. Phys.,1952,20,278. B. B. Owen and H. L. Simons, J. Phys.Chem.,1957,61,479. H. S. Harned and B. B. Owen, The Physical Chemistry of Electrolytic Solutions, Reinhold, New York, 1958, 3rd. edn., section 8.7. D. P. Kharakov, J. Phys.Chem.,1991,95,5634. T. V. Chalikian, A. P .Sarvazyan, T. Funck, C. A.Cain, and K. J. Breslauer, J. Phys.Chem.,1994,98,321. T. V. Chalikian, A.P.Sarvazyan and K. J. Breslauer, Biophys. Chem.,1994,51,89. P. Bernal and J. McCluan, J Solution Chem.,2001,30,119. G. R. Hedwig and H. Hoiland, Phys. Chem. Chem. Phys.,2004,6,2440.This page titled 1.7.9: Compressions- Isentropic and Isothermal- Solutions- Approximate Limiting Estimates is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.7.10: Compressions- Desnoyers - Philip Equation

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.07%3A_Compressions/1.7.10%3A_Compressions-_Desnoyers_-_Philip_Equation

In terms of isentropic and isothermal compressibilities the Desnoyers-Philip Equation is important. A key equation expresses the difference between two apparent properties, \(\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{J}}}\right)^{\infty}\) and \(\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \operatorname{def}\right)^{\infty}\). We develop the proof in the general case starting from equation (a). \[\delta=\kappa_{\mathrm{T}}-\kappa_{\mathrm{S}}=\mathrm{T} \,\left(\alpha_{\mathrm{p}}\right)^{2} / \sigma\]\[\text { Hence, for an aqueous solution, } \delta(\mathrm{aq})=T \,\left[\alpha_{\mathrm{p}}(\mathrm{aq})\right]^{2} / \sigma(\mathrm{aq})\]\[\text { For water }(\ell) \text { at the same } \mathrm{T} \text { and } \mathrm{p}, \delta_{1}^{*}(\ell)=\mathrm{T} \,\left[\alpha_{\mathrm{p} 1}^{*}(\ell)\right]^{2} / \sigma_{1}^{*}(\ell)\]We formulate an equation for the difference, \(\delta(\mathrm{aq})-\delta_{1}^{*}(\ell)\) \[\delta(\mathrm{aq})-\delta_{1}^{*}(\ell)=\mathrm{T} \,\left[\alpha_{\mathrm{p}}(\mathrm{aq})\right]^{2} / \sigma(\mathrm{aq})-\mathrm{T} \,\left[\alpha_{\mathrm{p} 1}^{*}(\ell)\right]^{2} / \sigma_{1}^{*}(\ell)\]We add and subtract the same term. With some slight reorganisation, \[\begin{aligned} \delta(\mathrm{aq})-\delta_{1}^{*}(\ell)=& \mathrm{T} \,\left[\alpha_{\mathrm{p}}(\mathrm{aq})\right]^{2} / \sigma(\mathrm{aq})-\mathrm{T} \,\left[\alpha_{\mathrm{p} 1}^{*}(\ell)\right]^{2} / \sigma(\mathrm{aq}) \\ &-\mathrm{T} \,\left[\alpha_{\mathrm{p} 1}^{*}(\ell)\right]^{2} / \sigma_{1}^{*}(\ell)+\mathrm{T} \,\left[\alpha_{\mathrm{p} 1}^{*}(\ell)\right]^{2} / \sigma(\mathrm{aq}) \end{aligned}\]Or, \[\begin{aligned} \delta(\mathrm{aq})-\delta_{1}^{*}(\ell)=& \mathrm{T} \,[\sigma(\mathrm{aq})]^{-1} \,\left\{\left[\alpha_{\mathrm{p}}(\mathrm{aq})\right]^{2}-\left[\alpha_{\mathrm{p} 1}^{*}(\ell)\right]^{2}\right\} \\ &-\mathrm{T} \,\left[\alpha_{\mathrm{p} 1}^{*}(\ell)\right]^{2} \,\left[\frac{1}{\sigma_{1}^{*}(\ell)}-\frac{1}{\sigma(\mathrm{aq})}\right] \end{aligned}\]We identify the term \(\left\{\left[\alpha_{p}(a q)\right]^{2}-\left[\alpha_{p 1}^{*}(\ell)\right]^{2}\right\}\) as ‘a square minus a square’. \[\begin{aligned} \delta(\mathrm{aq})-\delta_{1}^{*}(\ell)=& \mathrm{T} \,[\sigma(\mathrm{aq})]^{-1} \,\left\{\alpha_{\mathrm{p}}(\mathrm{aq})+\alpha_{\mathrm{p} 1}^{*}(\ell)\right\} \,\left\{\alpha_{\mathrm{p}}(\mathrm{aq})-\alpha_{\mathrm{p} 1}^{*}(\ell)\right\} \\ &-\mathrm{T} \, \frac{\left[\alpha_{\mathrm{p} 1}^{*}(\ell)\right]^{2}}{\sigma(\mathrm{aq}) \, \sigma_{1}^{*}(\ell)} \,\left[\sigma(\mathrm{aq})-\sigma_{1}^{*}(\ell)\right] \end{aligned}\]We use equations (b) for \(\delta(\mathrm{aq})\) and (c) for \(\delta_{1}^{*}(\ell)\) to remove explicit reference to temperature in equation (g). \[\begin{aligned} &\delta(\mathrm{aq})-\delta_{1}^{*}(\ell)= \\ &\delta(\mathrm{aq}) \,\left[\alpha_{\mathrm{p}}(\mathrm{aq})\right]^{-2} \,\left\{\alpha_{\mathrm{p}}(\mathrm{aq})+\alpha_{\mathrm{p} 1}^{*}(\ell)\right\} \,\left\{\alpha_{\mathrm{p}}(\mathrm{aq})-\alpha_{\mathrm{pl}}^{*}(\ell)\right\} \\ &-\delta_{1}^{*}(\ell) \,[\sigma(\mathrm{aq})]^{-1} \,\left[\sigma(\mathrm{aq})-\sigma_{1}^{*}(\ell)\right] \end{aligned}\]\[\text { But } \phi\left(\mathrm{K}_{\mathrm{Tj}_{\mathrm{j}}}\right)-\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \operatorname{def}\right)=\left(\mathrm{c}_{\mathrm{j}}\right)^{-1} \,\left[\delta(\mathrm{aq})-\delta_{1}^{*}(\ell)\right]+\delta_{1}^{*}(\ell) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\]We insert equation (h) for the difference \(\delta(\mathrm{aq})-\delta_{1}^{*}(\ell)\) into equation (i). \[\begin{aligned} &\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)-\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \mathrm{def}\right)= \\ &\frac{\delta(\mathrm{aq}) \,\left[\alpha_{\mathrm{p}}(\mathrm{aq})+\alpha_{\mathrm{p} 1}^{*}(\ell)\right]}{\left[\alpha_{\mathrm{p}}(\mathrm{aq})\right]^{2}} \, \frac{\left[\alpha_{\mathrm{p}}(\mathrm{aq})-\alpha_{\mathrm{p} 1}^{*}(\ell)\right]}{\mathrm{c}_{\mathrm{j}}} \\ &-\frac{\delta_{1}^{*}(\ell)}{\sigma(\mathrm{aq})} \, \frac{\left[\sigma(\mathrm{aq})-\sigma_{1}^{*}(\ell)\right]}{\mathrm{c}_{\mathrm{j}}}+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \delta_{1}^{*}(\ell) \end{aligned}\]\[\text { But, } \phi\left(E_{p j}\right)=\left[c_{j}\right]^{-1} \,\left[\alpha_{p}(a q)-\alpha_{p l}^{*}(\ell)\right]+\alpha_{p 1}^{*}(\ell) \, \phi\left(V_{j}\right)\]We identify the difference \(\left[\alpha_{p}(a q)-\alpha_{p 1}^{*}(\ell)\right]\). \[\text { Then } \phi\left(E_{p j}\right)-\alpha_{p 1}^{*}(\ell) \, \phi\left(V_{j}\right)=\left[c_{j}\right]^{-1} \,\left[\alpha_{p}(a q)-\alpha_{p 1}^{*}(\ell)\right]\]\[\text { Similarly } \phi\left(\mathrm{C}_{\mathrm{pj}}\right)-\sigma_{1}^{*}(\ell) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)=\left[\mathrm{c}_{\mathrm{j}}\right]^{-1} \,\left[\sigma(\mathrm{aq})-\sigma_{1}^{*}(\ell)\right]\]Then from equations (k), (l), (m) and (n), \[\begin{gathered} \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)-\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \operatorname{def}\right)=\frac{\delta(\mathrm{aq}) \,\left[\alpha_{\mathrm{p}}(\mathrm{aq})+\alpha_{\mathrm{p} 1}^{*}(\ell)\right]}{\left[\alpha_{\mathrm{p}}(\mathrm{aq})\right]^{2}} \,\left[\phi\left(\mathrm{E}_{\mathrm{pj}}\right)-\alpha_{\mathrm{p} 1}^{*}(\ell) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\right] \\ -\frac{\delta_{1}^{*}(\ell)}{\sigma(\mathrm{aq})} \,\left[\phi\left(\mathrm{C}_{\mathrm{pj}}\right)-\sigma_{1}^{*}(\ell) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\right]+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \delta_{1}^{*}(\ell) \end{gathered}\]We collect the \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)\) terms. \[\begin{gathered} \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)-\phi\left(\mathrm{K}_{\mathrm{sj}} ; \mathrm{def}\right)= \\ \frac{\delta(\mathrm{aq})}{\alpha_{\mathrm{p}}(\mathrm{aq})} \,\left\{1+\frac{\alpha_{\mathrm{p} 1}^{*}(\ell)}{\alpha_{\mathrm{p}}(\mathrm{aq})}\right\} \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right)-\frac{\delta_{1}^{*}(\ell)}{\sigma(\mathrm{aq})} \, \phi\left(\mathrm{C}_{\mathrm{pj}}\right) \\ +\left\{-\frac{\delta(\mathrm{aq}) \, \alpha_{\mathrm{p} 1}^{*}(\ell)}{\alpha_{\mathrm{p}}(\mathrm{aq})}-\frac{\delta(\mathrm{aq}) \,\left[\alpha_{\mathrm{p} 1}^{*}(\ell)\right]^{2}}{\left[\alpha_{\mathrm{p}}(\mathrm{aq})\right]^{2}}+\frac{\delta_{1}^{*}(\ell) \, \sigma_{1}^{*}(\ell)}{\sigma(\mathrm{aq})}+\delta_{1}^{*}(\ell)\right\} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \end{gathered}\]We note that in the second {----} bracket, the product term of \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)\). By using equations (b) for \(\delta(\mathrm{aq})\) and (c) for \(\delta_{1}^{*}(\ell)\) the second and third terms are together equal to zero.Hence \[\begin{aligned} &\phi\left(\mathrm{K}_{\mathrm{Tj}}\right)-\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \mathrm{def}\right)= \\ &\qquad \frac{\delta(\mathrm{aq})}{\alpha_{\mathrm{p}}(\mathrm{aq})} \,\left\{1+\frac{\alpha_{\mathrm{p} 1}^{*}(\ell)}{\alpha_{\mathrm{p}}(\mathrm{aq})}\right\} \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right)-\frac{\delta_{1}^{*}(\ell)}{\sigma(\mathrm{aq})} \, \phi\left(\mathrm{C}_{\mathrm{pj}}\right) \\ &+\left\{\delta_{1}^{*}(\ell)-\frac{\delta(\mathrm{aq}) \, \alpha_{\mathrm{p} 1}^{*}(\ell)}{\alpha_{\mathrm{p}}(\mathrm{aq})}\right\} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \end{aligned}\]The latter is the full Desnoyers–Philip equation. But \[\begin{gathered} \operatorname{limit}\left(c_{j} \rightarrow 0\right) \alpha_{p}(a q)=\alpha_{p l}^{*}(\ell) \\ \phi\left(E_{p j}\right)=\phi\left(E_{p j}\right)^{\infty}, \\ \phi\left(C_{p j}\right)=\phi\left(C_{p j}\right)^{\infty}, \delta(a q)=\delta_{1}^{*}(\ell) \\ \text { and } \sigma(a q)=\sigma_{1}^{*}(\ell) \end{gathered}\]\[\text { Then } \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)^{\infty}-\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \text { def }\right)^{\infty}=\delta_{1}^{*}(\ell) \,\left\{\frac{2 \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right)^{\infty}}{\alpha_{\mathrm{p} 1}^{*}(\ell)}-\frac{\phi\left(\mathrm{C}_{\mathrm{pj}}\right)^{\infty}}{\sigma_{1}^{*}(\ell)}\right\}\]Footnotes J. E. Desnoyers and P. R. Philip, Can. J. Chem, 1972, 50,1094. M. J. Blandamer, M. I. Davis, G. Douheret and J. C. R. Reis, Chem. Soc. Rev., 2001, 30, 8.This page titled 1.7.10: Compressions- Desnoyers - Philip Equation is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.7.11: Compression- Isentropic- Apparent Molar Volume

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.07%3A_Compressions/1.7.11%3A_Compression-_Isentropic-_Apparent_Molar_Volume

A given liquid system is prepared using \(\mathrm{n}_{1}\) moles of water, molar mass \(\mathrm{M}_{1}\), and \(\mathrm{n}_{j}\) moles of substance \(j\). The closed system is at equilibrium, at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\). The volume of the system is given by equation (a). \[\mathrm{V}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{V}_{\mathrm{1}}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\]Here \(V_{1}^{*}(\ell)\) is the molar volume of pure water and \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)\) is the apparent molar volume of substance \(j\) in the system; \(\mathrm{V}(\mathrm{aq})\) and \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)\) depend on the composition of the system, but \(\mathrm{V}_{1}^{*}(\ell)\) does not.The solution is perturbed to a local equilibrium state by a change in pressure along a path for which the entropy remains constant at \(\mathrm{S}(\mathrm{aq})\). At a specified molality \(\mathrm{m}_{j the change in volume is characterised by the isentropic compressibility, \(\mathrm{K}_{\mathrm{s}}(\mathrm{aq})\) defined in equation (b). \[\kappa_{\mathrm{s}}(\mathrm{aq})=-\frac{1}{\mathrm{~V}(\mathrm{aq})} \,\left(\frac{\partial \mathrm{V}(\mathrm{aq})}{\partial \mathrm{p}}\right)_{\mathrm{s}(\mathrm{aq}) ; \mathrm{m}(\mathrm{j})}\]Hence, \[\mathrm{V}(\mathrm{aq}) \, \mathrm{K}_{\mathrm{s}}(\mathrm{aq})=-\mathrm{n}_{1} \,\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{aq}) ; \mathrm{m}(\mathrm{j})}-\mathrm{n}_{\mathrm{j}} \,\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{aq}) ; \mathrm{m}(\mathrm{j})}\]The isentropic condition on the first partial differential in equation (c) refers to the entropy of an aqueous solution at molality, \(\mathrm{m}_{j}\). There is interest in relating this partial differential to the isentropic compressibility of the pure liquid substance 1 at the same \(\mathrm{T}\) and \(\mathrm{p}\), which is defined in equation (d). \[\kappa_{\mathrm{s} 1}^{*}(\ell)=-\frac{1}{\mathrm{~V}_{1}^{*}(\ell)} \,\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{s}^{*}(\ell)}\]For substance 1 the different isentropic conditions are related by equation (e). \[\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{s}^{*}(\mathrm{aq}) \mathrm{m}(\mathrm{j})}=\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{s}^{*}(\ell)}+\left(\frac{\partial \mathrm{S}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{aq}) / \mathrm{m}(\mathrm{j})} \,\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{S}_{1}^{*}(\mathrm{l})}\right)_{\mathrm{p}^{*}}\]In the latter equation we identify \(\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{s}^{*}(\ell)}\) and \(\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{S}_{1}^{*}}\right)_{\mathrm{p}^{*}}\) with, respectively, \(-\mathrm{V}_{1}^{*}(\ell) \, \kappa_{\mathrm{S} 1}^{*}(\ell)\) and \(\mathrm{T} \, \alpha_{\mathrm{p} 1}^{*}(\ell) / \sigma_{1}^{*}(\ell)\), which are thermodynamic properties of water (\(\ell\)). Here \(\sigma_{1}^{*}(\ell)\) is the heat capacitance (or heat capacity per unit volume) of water (\(\ell\)) Using the same calculus operation, the remaining partial differential is related to an isothermal property in equation (f). \[\left(\frac{\partial \mathrm{S}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{s}^{*}(\mathrm{aq}) ; \mathrm{m}(\mathrm{j})}=\left(\frac{\partial \mathrm{S}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{T}}+\left(\frac{\partial \mathrm{T}}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{aq}) ; \mathrm{m}(\mathrm{j})} \,\left(\frac{\partial \mathrm{S}_{1}^{*}(\ell)}{\partial \mathrm{T}}\right)_{\mathrm{p}^{*}}\]Since \(\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{T}}=-\mathrm{V}_{1}^{*}(\ell) \, \alpha_{\mathrm{pl}}^{*}(\ell),\left(\frac{\partial \mathrm{T}}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{aq})) \mathrm{m}(\mathrm{j})}=\mathrm{T} \, \frac{\alpha_{\mathrm{p}}(\mathrm{aq})}{\sigma(\mathrm{aq})}\), and \(\left(\frac{\partial \mathrm{S}_{1}^{*}(\ell)}{\partial \mathrm{T}}\right)_{\mathrm{p}^{*}}=\frac{\mathrm{V}_{1}^{*}(\ell) \, \sigma_{1}^{*}(\ell)}{\mathrm{T}}\), we combine these results with equation (f) to express equation (e) as equation (g). \[\begin{aligned} &\frac{1}{\mathrm{~V}_{1}^{*}(\ell)} \,\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{aq}) ; \mathrm{m}(\mathrm{j})}= \\ &-\kappa_{\mathrm{S} 1}^{*}(\ell)-\mathrm{T} \, \frac{\left[\alpha_{\mathrm{p} 1}^{*}(\ell)\right]^{2}}{\sigma_{1}^{*}(\ell)}+\mathrm{T} \, \frac{\alpha_{\mathrm{p} 1}^{*}(\ell) \, \alpha_{\mathrm{p}}(\mathrm{aq})}{\sigma(\mathrm{aq})} \end{aligned}\]We return to equation (c). Using equation (a) for \(\mathrm{V}(\mathrm{aq})\), equation (c) yields equation (h). \[\begin{aligned} &-\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right)_{\mathrm{s}(\mathrm{aq}) ; \mathrm{m}(\mathrm{j})}= \\ &{\left[\left(\frac{\mathrm{n}_{1}}{\mathrm{n}_{\mathrm{j}}}\right) \, \mathrm{V}_{1}^{*}(\ell)+\phi\left(\mathrm{V}_{\mathrm{j}}\right)\right] \, \mathrm{K}_{\mathrm{s}}(\mathrm{aq})+\left(\frac{\mathrm{n}_{1}}{\mathrm{n}_{\mathrm{j}}}\right) \,\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{s}(\mathrm{aq}) ; \mathrm{m}(\mathrm{j})}} \end{aligned}\]We note that \(\frac{\mathrm{n}_{1}}{\mathrm{n}_{\mathrm{j}}}=\frac{1}{\mathrm{~m}_{\mathrm{j}} \, \mathrm{M}_{1}}\) And that density \(\rho_{1}^{*}(\ell)=\frac{\mathrm{M}_{1}}{\mathrm{~V}_{1}^{*}(\ell)}\). Then combining equations (g) and (h) leads to equation (i) after slight simplification. \[\begin{aligned} &-\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right)_{\mathrm{s}(\mathrm{aq}) ; \mathrm{m}(\mathrm{j})}= \\ &{\left[\kappa_{\mathrm{s}}(\mathrm{aq})-\kappa_{\mathrm{s} 1}^{*}(\ell)\right] \,\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1}+\kappa_{\mathrm{s}}(\mathrm{aq}) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)} \\ &+\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \, \mathrm{T} \, \alpha_{\mathrm{p} 1}^{*}(\ell) \,\left[\frac{\alpha_{\mathrm{p}}(\mathrm{aq})}{\sigma(\mathrm{aq})}-\frac{\alpha_{\mathrm{p} 1}^{*}(\ell)}{\sigma_{1}^{*}(\ell)}\right] \end{aligned}\]An equivalent derivation of equation (i) has been given.Footnotes M. J. Blandamer, J. Chem. Soc., Faraday Trans., 1998, 94, 1057.This page titled 1.7.11: Compression- Isentropic- Apparent Molar Volume is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.7.12: Compressions- Isentropic- Binary Liquid Mixtures

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.07%3A_Compressions/1.7.12%3A_Compressions-_Isentropic-_Binary_Liquid_Mixtures

A given liquid mixture is prepared using \(\mathrm{n}_{1}\) moles of water and \(\mathrm{n}_{j}\) moles of liquid substance \(j\). The closed system is at equilibrium, at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\). The volume of the system is defined by the following equation. \[\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}\right]\]Similarly the entropy of the system is defined by equation (b). \[\mathrm{S}=\mathrm{S}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}\right]\]The volume of the system is given by equation (c). \[\mathrm{V}=\mathrm{n}_{1} \, \mathrm{V}_{1}+\mathrm{n}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}\]Here \(\mathrm{V}_{1}\) and \(\mathrm{V}_{j}\) are the partial molar volumes of the two substances in the system; \(\mathrm{V}\), \(\mathrm{V}_{1}\) and \(\mathrm{V}_{j}\) depend on the composition of the system. Similarly the entropy of the system is given by equation (d). \[\mathrm{S}=\mathrm{n}_{1} \, \mathrm{S}_{1}+\mathrm{n}_{\mathrm{j}} \, \mathrm{S}_{\mathrm{j}}\]The molar volume of the system \(\mathrm{V}_{\mathrm{m}}\) is given by the ratio \(\mathrm{V} /\left(\mathrm{n}_{1}+\mathrm{n}_{\mathrm{j}}\right)\). \[\text { Hence, } \quad \mathrm{V}_{\mathrm{m}}=\mathrm{x}_{1} \, \mathrm{V}_{1}+\mathrm{x}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}\]\[\text { Similarly } \quad S_{m}=x_{1} \, S_{1}+x_{j} \, S_{j}\]The system under examination is a binary liquid mixture such that the thermodynamic properties of the mixture are ideal. \[\mathrm{V}_{\mathrm{m}}(\operatorname{mix} ; \mathrm{id})=\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{x}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}^{*}(\ell)\]\[\mathrm{S}_{\mathrm{m}}(\mathrm{mix} ; \mathrm{id})=\mathrm{x}_{1} \, \mathrm{S}_{1}^{*}(\ell)+\mathrm{x}_{\mathrm{j}} \, \mathrm{S}_{\mathrm{j}}^{*}(\ell)+\Delta_{\operatorname{mix}} \mathrm{S}_{\mathrm{m}}(\operatorname{mix} ; \mathrm{id})\]In other words the two reference states are the pure substances at the initially fixed \(\mathrm{T}\) and \(\mathrm{p}\). In equation (h) the term \(\Delta_{\operatorname{mix}} \mathrm{S}_{\mathrm{m}}(\mathrm{mix} ; \mathrm{id})\) describes the ideal molar entropy of mixing, which is a function of composition only. The liquid mixture is perturbed to a local equilibrium state by a change in pressure along a path for which the entropy remains at that given by equation (h). At a specified mole fraction \(\mathrm{x}_{j}\) the change in volume is characterised by the isentropic compression, \(\mathrm{K}_{\mathrm{Sm}}(\mathrm{mix} ; \mathrm{id})\) which is \(\mathrm{K}_{\mathrm{m}}\left(\text { at constant } \mathrm{S}_{\mathrm{m}}(\mathrm{mix} ; \mathrm{id})\right)\) defined in equation (i). \[\mathrm{K}_{\mathrm{Sm}}\left(\operatorname{mix} ; \mathrm{x}_{\mathrm{j}} ; \mathrm{id}\right)=-\left(\frac{\partial \mathrm{V}_{\mathrm{m}}(\mathrm{mix} ; \mathrm{id})}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{m} ; \operatorname{mix} ; \mathrm{id}), \mathrm{x}(\mathrm{j})}\]Hence, \[\begin{aligned} &\mathrm{K}_{\mathrm{Sm}}\left(\mathrm{mix} ; \mathrm{x}_{\mathrm{j}} ; \mathrm{id}\right)= \\ &\quad-\mathrm{x}_{1} \,\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{m} ; \mathrm{mix} ; \mathrm{dd}), x(\mathrm{j})}-\mathrm{x}_{\mathrm{j}} \,\left(\frac{\partial \mathrm{V}_{2}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{m} ; \mathrm{mix} ; \mathrm{id}), x(j)} \end{aligned}\]We note that the isentropic condition on the partial differentials in equation (j) refers to the entropy of an ideal mixture at mole fraction \(\mathrm{x}_{j}\). There is merit in relating these partial differentials to the isentropic compressions of the pure liquid substances 1 and \(j\) at the same \(\mathrm{T}\) and \(\mathrm{p}\), which are defined in equations (k) and (l). \[\mathrm{K}_{\mathrm{si}}^{*}(\ell)=-\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{s}^{*}}\]\[\mathrm{K}_{\mathrm{Sj}}^{*}(\ell)=-\left(\frac{\partial \mathrm{V}_{\mathrm{j}}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{s}(j)^{*}}\]The required relationship is obtained using a calculus operation. For substance 1 the different isentropic conditions are related by equation (m). \[\begin{aligned} &\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{m} ; \operatorname{mix} ; \mathrm{jd}) \times(\mathrm{j})}= \\ &\left.\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{S}^{*}}+\left(\frac{\partial \mathrm{S}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{m} ; \operatorname{mix} ; ; \mathrm{d}) \times(j)} \,\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{S}_{1}^{*}(\ell)}\right)_{\mathrm{p}}\right)^{*} \end{aligned}\]In the latter equation we identify \(\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{s}^{*}} \text { and }\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{S}_{1}^{*}(\ell)}\right)_{\mathrm{p}}\) with, respectively, \(-\mathrm{K}_{\mathrm{Sl}}^{*}(\ell)\) and \(\mathrm{T} \, \frac{\mathrm{E}_{\mathrm{pl}}^{*}(\ell)}{\mathrm{C}_{\mathrm{pl}}^{*}(\ell)}\) which are thermodynamic properties of water(\(\ell\)). Using the same calculus operation, the remaining partial differential is related to an isothermal property in equation (n). \[\begin{aligned} &\left(\frac{\partial \mathrm{S}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{m} ; \mathrm{mix} \text {;id }) \times(\mathrm{j})}= \\ &\left(\frac{\partial \mathrm{S}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{T}}+\left(\frac{\partial \mathrm{T}}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{m} ; \operatorname{mix} ; \mathrm{id}) \times(\mathrm{j})} \,\left(\frac{\partial \mathrm{S}_{1}^{*}(\ell)}{\partial \mathrm{T}}\right)_{\mathrm{p}} \end{aligned}\]But \(\left(\frac{\partial \mathrm{S}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{T}}=-\mathrm{E}_{\mathrm{pl}}^{*}(\ell)\), \(\left(\frac{\partial T}{\partial p}\right)_{\mathrm{S}(\mathrm{m} ; \operatorname{mix} ; \mathrm{jd}) \mathrm{x}(\mathrm{j})}=\mathrm{T} \, \frac{\mathrm{E}_{\mathrm{pm}}(\mathrm{A} ; \mathrm{mix} ; \mathrm{id})}{\mathrm{C}_{\mathrm{pm}}(\mathrm{A} ; \mathrm{mix} ; \mathrm{id})}\), and \(\left(\frac{\partial \mathrm{S}_{1}^{*}(\ell)}{\partial \mathrm{T}}\right)_{\mathrm{p}}=\frac{\mathrm{C}_{\mathrm{pm}}^{*}(\ell)}{\mathrm{T}}\),We combine these results with equation (n) to re-express equation (m) as equation (o). \[\begin{aligned} &\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{m} ; \mathrm{mix} ; \mathrm{id}) \mathrm{x}(\mathrm{j})}= \\ &-\mathrm{K}_{\mathrm{s} 1}^{*}(\ell)-\mathrm{T} \, \frac{\left[\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}+\mathrm{T} \, \mathrm{E}_{\mathrm{p} 1}^{*}(\ell) \, \frac{\mathrm{E}_{\mathrm{pm}}(\mathrm{mix} ; \mathrm{id})}{\mathrm{C}_{\mathrm{pm}}(\mathrm{mix} ; \mathrm{id})} \end{aligned}\]Similarly, for the substance j we obtain equation (p). \[\begin{aligned} &\left(\frac{\partial \mathrm{V}_{\mathrm{j}}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{S}(\mathrm{m} ; \operatorname{mix} ; \mathrm{dd}) \times(\mathrm{j})}= \\ &-\mathrm{K}_{\mathrm{Sj}}^{*}(\ell)-\mathrm{T} \, \frac{\left[\mathrm{E}_{\mathrm{pj}}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{pj}}^{*}(\ell)}+\mathrm{T} \, \mathrm{E}_{\mathrm{pj}}^{*}(\ell) \, \frac{\mathrm{E}_{\mathrm{pm}}(\mathrm{mix} ; \mathrm{id})}{\mathrm{C}_{\mathrm{pm}}(\mathrm{mix} ; \mathrm{id})} \end{aligned}\]Equations (o) and (p) can be used to recast equation (j) in the form of equation (q). \[\begin{aligned} &\mathrm{K}_{\mathrm{Sm}}(\mathrm{mix} ; \mathrm{id})=\mathrm{x}_{1} \, \mathrm{K}_{\mathrm{Sl} 1}^{*}(\ell)+\mathrm{x}_{\mathrm{j}} \, \mathrm{K}_{\mathrm{Sj}}^{*}(\ell)+ \\ &\mathrm{T} \,\left\{\mathrm{x}_{1} \, \frac{\left[\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}+\mathrm{x}_{\mathrm{j}} \, \frac{\left[\mathrm{E}_{\mathrm{pj}}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{pj}}^{*}(\ell)}\right. \\ &\left.-\left[\mathrm{x}_{1} \, \mathrm{E}_{\mathrm{pl}}^{*}(\ell)+\mathrm{x}_{\mathrm{j}} \, \mathrm{E}_{\mathrm{pj}}^{*}(\ell)\right] \, \frac{\mathrm{E}_{\mathrm{pm}}(\mathrm{mix} ; \mathrm{id})}{\mathrm{C}_{\mathrm{pm}}(\mathrm{mix} ; \mathrm{id})}\right\} \end{aligned}\]\(\mathrm{K}_{\mathrm{Sm}}(\mathrm{A} ; \text { mix; id })\) is the ideal molar isentropic compression, which is commonly denoted by id \(\mathrm{K}_{\mathrm{Sm}}^{\mathrm{id}}\). The sum, \(\left[x_{1} \, E_{p 1}^{*}(\ell)+x_{j} \, E_{p j}^{*}(\ell)\right]\) is the ideal molar isobaric thermal expansion \(E_{p m}^{\text {id }}\) for the binary liquid mixture, here denoted as \(\mathrm{E}_{\mathrm{pm}}(\operatorname{mix} ; \mathrm{id})\), and analogously for the ideal molar isobaric heat capacity \(\mathrm{C}_{\mathrm{pm}}^{\mathrm{id}}= \mathrm{C}_{\mathrm{pm}}(\operatorname{mix} ; \mathrm{id})\). Thus, \[\begin{aligned} &\mathrm{K}_{\mathrm{Sm}}(\mathrm{mix} ; \mathrm{id})=\mathrm{x}_{1} \, \mathrm{K}_{\mathrm{s} 1}^{*}(\ell)+\mathrm{x}_{\mathrm{j}} \, \mathrm{K}_{\mathrm{Sj}}^{*}(\ell)+ \\ &\mathrm{T} \,\left\{\mathrm{x}_{1} \, \frac{\left[\mathrm{E}_{\mathrm{pl}}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{pl}}^{*}(\ell)}+\mathrm{x}_{\mathrm{j}} \, \frac{\left[\mathrm{E}_{\mathrm{pj}}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{pj}}^{*}(\ell)}-\frac{\left[\mathrm{E}_{\mathrm{pm}}(\mathrm{mix} ; \mathrm{id})\right]^{2}}{\mathrm{C}_{\mathrm{pm}}(\mathrm{mix} ; \mathrm{id})}\right\} \end{aligned}\]The last term of equation (r) expresses a mixing property. In general terms, \[\mathrm{K}_{\mathrm{Sm}}(\operatorname{mix} ; \text { id })=\mathrm{x}_{1} \, \mathrm{K}_{\mathrm{sl}}^{*}(\ell)+\mathrm{x}_{\mathrm{j}} \, \mathrm{K}_{\mathrm{Sj}}^{*}(\ell)+\Delta_{\text {mix }} \mathrm{K}_{\mathrm{Sm}}(\operatorname{mix} ; \mathrm{id})\]From equations (r) and (s) we obtain the following expression. \[\begin{aligned} &\Delta_{\text {mix }} \mathrm{K}_{\mathrm{Sm}}(\operatorname{mix} ; \mathrm{id})= \\ &\mathrm{T} \,\left\{\mathrm{x}_{1} \, \frac{\left[\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}+\mathrm{x}_{\mathrm{j}} \, \frac{\left[\mathrm{E}_{\mathrm{pj}}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{pj}}^{*}(\ell)}-\frac{\left[\mathrm{E}_{\mathrm{pm}}(\mathrm{mix} ; \mathrm{id})\right]^{2}}{\mathrm{C}_{\mathrm{pm}}(\operatorname{mix} ; \mathrm{id})}\right\} \\ &\Delta_{\text {mix }} \mathrm{K}_{\mathrm{Sm}}(\mathrm{A} ; \operatorname{mix} ; \mathrm{id})=\mathrm{T}\left\{\mathrm{x}_{1}\left[\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)\right]^{2} / \mathrm{C}_{\mathrm{p} 1}^{*}(\ell)\right. \\ &\left.+\mathrm{x}_{\mathrm{j}}\left[\mathrm{E}_{\mathrm{pj}}^{*}(\ell)\right]^{2} / \mathrm{C}_{\mathrm{pj}}^{*}(\ell)-\left[\mathrm{E}_{\mathrm{pm}}(\text { mix } ; \mathrm{id})\right]^{2} / \mathrm{C}_{\mathrm{pm}}(\operatorname{mix} ; \mathrm{id})\right\} \end{aligned}\]This is an important, albeit frequently neglected term, in the calculation of isentropic compressions of thermodynamically ideal liquid mixtures. In general ideal molar mixing values are non-zero for non-Gibbsian properties, the origin of which has been discussed.We recall the definition of an apparent molar property. Hence, \[\mathrm{K}_{\mathrm{Sm}}(\operatorname{mix} ; \mathrm{id})=\mathrm{x}_{1} \, \mathrm{K}_{\mathrm{S} 1}^{*}(\ell)+\mathrm{x}_{\mathrm{j}} \, \phi\left(\mathrm{K}_{\mathrm{S} j}\right)(\mathrm{mix} ; \mathrm{id})\]Here \(\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)(\operatorname{mix} ; \mathrm{id})\) is the apparent molar isentropic compression of chemical substance \(j\) in the ideal liquid mixture. Combination of equations (r) and (u) yields equation (v). \[\begin{aligned} &\phi\left(\mathrm{K}_{\mathrm{S}_{\mathrm{j}}}\right)(\operatorname{mix} ; \mathrm{id})= \\ &\mathrm{K}_{\mathrm{Sj}}^{*}(\ell)+\mathrm{T} \,\left\{\mathrm{x}_{1} \, \frac{\left[\mathrm{E}_{\mathrm{pl} 1}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{pl}}^{*}(\ell)}+\mathrm{x}_{\mathrm{j}} \, \frac{\left[\mathrm{E}_{\mathrm{pj}}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{pj}}^{*}(\ell)}-\frac{\left[\mathrm{E}_{\mathrm{pm}}(\mathrm{mix} ; \mathrm{id})\right]^{2}}{\mathrm{C}_{\mathrm{pm}}(\mathrm{mix} ; \mathrm{id})}\right\} / \mathrm{x}_{\mathrm{j}} \end{aligned}\]The limiting values for \(\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)\) at \(\mathrm{x}_{j} = 1\) and \(\mathrm{x}_{j} = 0\) are of particular interest. For the pure liquid substance \(j\), equation (w) is readily obtained. \[\operatorname{limit}\left(\mathrm{x}_{\mathrm{j}} \rightarrow 1\right) \phi\left(\mathrm{K}_{\mathrm{Sj}}\right)(\operatorname{mix} ; \mathrm{id})=\mathrm{K}_{\mathrm{Sj}}^{*}(\ell)\]The latter is the expected property. However, the infinite dilution limit of equation (v) is not immediately obvious. In fact, both the numerator and denominator in the last term approach zero as we approach infinite dilution (\(\mathrm{x}_{j} = 0\)). What emerges is equation (x), which is an example of the unusual formalism for non-Lewisian properties. \[\begin{aligned} &\operatorname{limit}\left(\mathrm{x}_{\mathrm{j}} \rightarrow 0\right) \phi\left(\mathrm{K}_{\mathrm{Sj}}\right)(\operatorname{mix} ; \mathrm{id})= \\ &\mathrm{K}_{\mathrm{Sj}}^{*}(\ell)+\mathrm{T} \, \mathrm{C}_{\mathrm{pj}}^{*}(\ell) \,\left[\frac{\mathrm{E}_{\mathrm{pj}}^{*}(\ell)}{\mathrm{C}_{\mathrm{pj}}^{*}(\ell)}-\frac{\mathrm{E}_{\mathrm{pl}}^{*}(\ell)}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}\right]^{2} \end{aligned}\]Thus for ideal liquid mixtures \(\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)^{\infty}\) is given by a combination of properties for both pure liquid components. In other words, the chemical nature of component 1 affects the non-Lewisian properties of its mate j in the ideal mixture. This is in contrast with apparent molar Lewisian properties, such as \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)\), for which the values in ideal mixtures are the same as in the pure liquid state of substance \(j\).An extensive literature describes isentropic compressions of binary liquid mixturesFootnotes G. Douhéret, M. I. Davis, J. C. R. Reis and M. J. Blandamer, ChemPhysChem, 2001, 2, 148. M. I. Davis, G. Douhéret, J. C. R. Reis and M. J. Blandamer, Phys. Chem. Chem. Phys., 2001, 3, 4555. It is instructive to show how the limiting value for \(\phi\left(\mathrm{K}_{\mathrm{Sj}}\right)(\operatorname{mix} ; \mathrm{id})\) at \(\mathrm{x}_{j} = 0\) (and hence \(\mathrm{x}_{1} = 1\)) is obtained from equation (v). In this limit the last term (without \(\mathrm{T}\)) of equation (v) becomes, \[\begin{aligned} &\operatorname{limit}\left(\mathrm{x}_{\mathrm{j}} \rightarrow 0\right)\left\{\mathrm{x}_{1} \, \frac{\left[\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}+\mathrm{x}_{\mathrm{j}} \, \frac{\left[\mathrm{E}_{\mathrm{pj}}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{pj}}^{*}(\ell)}-\frac{\left[\mathrm{E}_{\mathrm{pm}}(\operatorname{mix} ; \mathrm{id})\right]^{2}}{\mathrm{C}_{\mathrm{pm}}(\operatorname{mix} ; \mathrm{id})}\right\} / 0\\ &=0 / 0 \end{aligned}\]We apply L'Hospital's rule which asserts that this limit is equal to the ratio of limits (bb) and (cc). \[\begin{aligned} &\operatorname{limit}\left(\mathrm{x}_{\mathrm{j}} \rightarrow 0\right) \\ &\mathrm{d}\left\{\mathrm{x}_{1} \, \frac{\left[\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}+\mathrm{x}_{\mathrm{j}} \, \frac{\left[\mathrm{E}_{\mathrm{pj}}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{pj}}^{*}(\ell)}-\frac{\left[\mathrm{x}_{1} \, \mathrm{E}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{x}_{\mathrm{j}} \, \mathrm{E}_{\mathrm{pj}}^{*}(\ell)\right]^{2}}{\mathrm{x}_{1} \, \mathrm{C}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{x}_{\mathrm{j}} \, \mathrm{C}_{\mathrm{pj}}^{*}(\ell)}\right\} / \mathrm{dx} \mathrm{x}_{\mathrm{j}} \end{aligned}\]\[\operatorname{limit}\left(\mathrm{x}_{\mathrm{j}} \rightarrow 0\right) \mathrm{dx}_{\mathrm{j}} / \mathrm{dx}_{\mathrm{j}}\]The latter limit is unity. The former is obtained from the following differential. \[\begin{aligned} &\mathrm{d}\left\{\mathrm{x}_{1} \, \frac{\left[\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}+\mathrm{x}_{\mathrm{j}} \, \frac{\left[\mathrm{E}_{\mathrm{pj}}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{pj}}^{*}(\ell)}-\frac{\left[\mathrm{x}_{1} \, \mathrm{E}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{x}_{\mathrm{j}} \, \mathrm{E}_{\mathrm{pj}}^{*}(\ell)\right]^{2}}{\mathrm{x}_{1} \, \mathrm{C}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{x}_{\mathrm{j}} \, \mathrm{C}_{\mathrm{pj}}^{*}(\ell)}\right\} / \mathrm{dx} \mathrm{x}_{\mathrm{j}} \\ &=-\frac{\left[\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}+\frac{\left[\mathrm{E}_{\mathrm{pj}}^{*}(\ell)\right]^{2}}{\mathrm{C}_{\mathrm{pj}}^{*}(\ell)} \\ &-\left\{2 \, \frac{\left[\mathrm{x}_{1} \, \mathrm{E}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{x}_{\mathrm{j}} \, \mathrm{E}_{\mathrm{pj}}^{*}(\ell)\right]}{\left[\mathrm{x}_{1} \, \mathrm{C}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{x}_{\mathrm{j}} \, \mathrm{C}_{\mathrm{pj}}^{*}(\ell)\right]}\right\} \,\left[-\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{E}_{\mathrm{pj}}^{*}(\ell)\right] \\ &+\left\{\frac{\left[\mathrm{x}_{1} \, \mathrm{E}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{x}_{\mathrm{j}} \, \mathrm{E}_{\mathrm{pj}}^{*}(\ell)\right]^{2}}{\left[\mathrm{x}_{1} \, \mathrm{C}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{x}_{\mathrm{j}} \, \mathrm{C}_{\mathrm{pj}}^{*}(\ell)\right]^{2}}\right\} \,\left[-\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{C}_{\mathrm{pj}}^{*}(\ell)\right] \end{aligned}\]The limiting value of this differential is, \[\operatorname{limit}\left(x_{1} \rightarrow 0\right) \frac{\left.\mathrm{d}_{\{} \ldots\right\}}{\mathrm{dx}_{\mathrm{j}}}=\mathrm{C}_{\mathrm{pj}}^{*}(\ell) \,\left[\frac{\mathrm{E}_{\mathrm{pj}}^{*}(\ell)}{\mathrm{C}_{\mathrm{p} j}^{*}(\ell)}-\frac{\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)}{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)}\right]^{2}\]This is the value for the limit of equation (aa), which was used to obtain equation (x) above. J. C. R. Reis, M. J. Blandamer, M. I. Davis and G. Douhéret, Phys. Chem. Chem. Phys., 2001, 3, 1465.This page titled 1.7.12: Compressions- Isentropic- Binary Liquid Mixtures is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.7.13: Compressions- Isothermal- Equilibrium and Frozen

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.07%3A_Compressions/1.7.13%3A_Compressions-_Isothermal-_Equilibrium_and_Frozen

The volume of a given closed system is defined by the set of independent variables \(\mathrm{T}\), \(\mathrm{p}\) and composition \(\xi\); \(\mathrm{V}=\mathrm{V}[\mathrm{T}, \mathrm{p}, \xi]\). We assert that in this state the affinity for spontaneous chemical reaction is \(\mathrm{A}\). The system is perturbed by a change in pressure such that the system can track one of two pathways; (i) at constant \(\mathrm{A}\) or (ii) at constant \(\xi\). The differential dependences of volume on pressure are related using equation (a). \[\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{A}}=\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \xi}-\left(\frac{\partial \mathrm{A}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \xi} \,\left(\frac{\partial \xi}{\partial \mathrm{A}}\right)_{\mathrm{T}, \mathrm{p}} \,\left(\frac{\partial \mathrm{V}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}\]The two differentials expressing the dependence of volume on pressure define the equilibrium isothermal compression and the frozen isothermal compression respectively. For a system at equilibrium, (i.e. minimum in \(\mathrm{G}\) at fixed \(\mathrm{T}\) and \(\mathrm{p}\)) following perturbation by a change in pressure, \[\mathrm{K}_{\mathrm{T}}(\mathrm{A}=0)=-\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{A}=0} \quad \mathrm{~K}_{\mathrm{T}}\left(\xi^{\mathrm{eq}}\right)=-\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \xi^{\mathrm{a}}}\]The negative signs recognise that for all thermodynamically stable systems, the volume decreases with increase in pressure. Nevertheless there is merit in thinking of compression (and compressibility) as a positive feature of a system. Both \(\mathrm{K}_{\mathrm{T}}\left(\xi^{\mathrm{eq}}\right)\) and \(\mathrm{K}_{\mathrm{T}}(\mathrm{A}=0)\) are extensive variables characterising two possible pathways. From equation (a), \[\mathrm{K}_{\mathrm{T}}(\mathrm{A}=0)=\mathrm{K}_{\mathrm{T}}\left(\xi^{\mathrm{eq}}\right)-\left[\left(\frac{\partial \mathrm{V}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}\right]^{2} \,\left(\frac{\partial \xi}{\partial \mathrm{A}}\right)_{\mathrm{T}, \mathrm{p}}\]But at equilibrium, \((\partial \mathrm{A} / \partial \xi)_{\mathrm{T}, \mathrm{p}}<0\). Hence, irrespective of the sign of \((\partial \mathrm{V} / \partial \xi)_{\mathrm{T}, \mathrm{p}}\), \(\mathrm{K}_{\mathrm{T}}(\mathrm{A}=0)>\mathrm{K}_{\mathrm{T}}\left(\xi^{\mathrm{eq}}\right)\). Equation (c) is rewritten in terms of compressibilities. \[\kappa_{\mathrm{T}}(\mathrm{A}=0)=-(1 / \mathrm{V}) \,(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{T}, \mathrm{A}=0} \quad \kappa_{\mathrm{T}}\left(\xi^{\mathrm{eq}}\right)=-(1 / \mathrm{V}) \,(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{T}, \xi^{\text {eq }}}\]\[\kappa_{\mathrm{T}}(\mathrm{A}=0)=\kappa_{\mathrm{T}}\left(\xi^{\mathrm{eq}}\right)-(1 / \mathrm{V}) \,\left[(\partial \mathrm{V} / \partial \xi)_{\mathrm{T}, \mathrm{p}}\right]^{2} \,(\partial \xi / \partial \mathrm{A})_{\mathrm{T}, \mathrm{p}}\]Because \((\partial \mathrm{A} / \partial \xi)_{\mathrm{T}, \mathrm{p}}<0\), for all stable systems, \(\kappa_{\mathrm{T}}(\mathrm{A}=0)>\kappa_{\mathrm{T}}\left(\xi^{\mathrm{eq}}\right)\). According therefore to equation (e) the volume decrease accompanying a given change in pressure is more dramatic under condition that \(\mathrm{A} = 0\) than under the condition where \(\xi\) remains constant at \(\xi^{\mathrm{eq}}\). Both \(\kappa_{\mathrm{T}}(\mathrm{A}=0)\) and \(\kappa_{\mathrm{T}}\left(\xi^{\mathrm{eq}}\right)\) are volume intensive properties of a solution.Footnotes The contrast between the two conditions is familiar to anyone who has dived into a swimming pool and “got it wrong”. Hitting the wall of water is similar to the conditions for\(\mathrm{K}_{\mathrm{T}}(\xi)\) whereas for a good dive the conditions resemble \(\mathrm{K}_{\mathrm{T}}(\mathrm{A}=0)\); the water molecules move apart to allow a smooth entry into the water. Consider \(\left(\frac{\partial^{2} \mathrm{G}}{\partial \mathrm{p} \, \mathrm{d} \xi}\right)=\left(\frac{\partial^{2} \mathrm{G}}{\partial \xi \, \mathrm{dp}}\right)\). Then, \(-\left(\frac{\partial \mathrm{A}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \xi}=\left(\frac{\partial \mathrm{V}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}\) \(\mathrm{K}_{\mathrm{T}}=\left[\mathrm{m}^{3} \mathrm{~Pa}^{-1}\right] ; \kappa_{\mathrm{T}}=\left[\mathrm{Pa}^{-1}\right]\) Equation (e) forms the basis of the pressure-jump fast reaction technique. A rapid change in pressure produces a “frozen” system which relaxes to the equilibrium state at a rate characteristic of the system. For information concerning \(\mathrm{D}_{2}\mathrm{O}(\ell)\), see R. A. Fine and F. J. Millero, J.Chem.Phys.,1975,63,89.This page titled 1.7.13: Compressions- Isothermal- Equilibrium and Frozen is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.