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1.7.14: Compressions- Ratio- Isentropic and Isothermal

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/1.7.14%3A_Compressions-_Ratio-_Isentropic_and_Isothermal

Using a calculus operation, we obtain equations relating isothermal and isentropic dependencies of volume on pressure.Thus,\[\begin{aligned} (\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{T}} &=-(\partial \mathrm{T} / \partial \mathrm{p})_{\mathrm{V}} \,(\partial \mathrm{V} / \partial \mathrm{T})_{\mathrm{p}} \\ &=-(\partial \mathrm{T} / \partial \mathrm{p})_{\mathrm{V}} \,(\partial \mathrm{V} / \partial \mathrm{S})_{\mathrm{p}} \,(\partial \mathrm{S} / \partial \mathrm{T})_{\mathrm{p}} \end{aligned}\]and,\[\begin{aligned} (\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{s}}=&-(\partial \mathrm{S} / \partial \mathrm{p})_{\mathrm{v}} \,(\partial \mathrm{V} / \partial \mathrm{S})_{\mathrm{p}} \\ &=-(\partial \mathrm{T} / \partial \mathrm{p})_{\mathrm{V}} \,(\partial \mathrm{V} / \partial \mathrm{S})_{\mathrm{p}} \,(\partial \mathrm{S} / \partial \mathrm{T})_{\mathrm{V}} \end{aligned}\]Then,\[(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{T}} /(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{s}}=(\partial \mathrm{S} / \partial \mathrm{T})_{\mathrm{p}} /(\partial \mathrm{S} / \partial \mathrm{T})_{\mathrm{V}}\]The Gibbs-Helmholtz equation requires that\[\mathrm{H}=\mathrm{G}-\mathrm{T} \,(\partial \mathrm{G} / \partial \mathrm{T})_{\mathrm{p}}\]Also\[(\partial \mathrm{H} / \partial \mathrm{T})_{\mathrm{p}}=\mathrm{C}_{\mathrm{p}}=-\mathrm{T} \,\left(\partial^{2} \mathrm{G} / \partial \mathrm{T}^{2}\right)_{\mathrm{p}}=\mathrm{T} \,(\partial \mathrm{S} / \partial \mathrm{T})_{\mathrm{p}}\]Similarly,\[(\partial \mathrm{U} / \partial \mathrm{T})_{\mathrm{V}}=\mathrm{C}_{\mathrm{V}}=\mathrm{T} \,(\partial \mathrm{S} / \partial \mathrm{T})_{\mathrm{V}}\]Hence,\[\mathrm{K}_{\mathrm{T}} / \mathrm{K}_{\mathrm{s}}=\mathrm{C}_{\mathrm{p}} / \mathrm{C}_{\mathrm{V}}\]This page titled 1.7.14: Compressions- Ratio- Isentropic and Isothermal is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.7.15: Compression- Isentropic and Isothermal- Solutions- 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/1.7.15%3A_Compression-_Isentropic_and_Isothermal-_Solutions-_Limiting_Estimates

The density of an aqueous solutions at defined \(\mathrm{T}\) and \(\mathrm{p}\) and solute molality \(\mathrm{m}_{j}\) yields the apparent molar volume of solute \(j\), \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)\). The dependence of \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)\) on \(\mathrm{m}_{j}\) can be extrapolated to yield the limiting (infinite dilution) property \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}\). The isothermal dependence of densities on pressure can be expressed in terms of an analogous infinite dilution apparent molar isothermal compression, \(\phi\left(\mathrm{K}_{\mathrm{Tj}}\right)^{\infty}\). Similarly the isentropic compressibilities of solutions are characterised by \(\phi\left(\mathrm{K}_{\mathrm{S} j} ; \operatorname{def}\right)^{\infty}\) which is accessible via the density of a solution and the speed of sound in the solution. Nevertheless the isothermal property \(\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{J}}}\right)^{\infty}\) presents fewer conceptual problems in terms of understanding the properties of solutes and solvents which control volumetric properties. The challenge is to use \(\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \operatorname{def}\right)^{\infty}\) in order to obtain \(\phi\left(\mathrm{K}_{\mathrm{Tj}}\right)^{\infty}\). The linking relationship is the Desnoyers-Philip equation. The apparent molar isothermal compression for solute \(j \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)\) is related to the concentration \(\mathrm{c}_{j}\) of solute using equation (a) where \(\phi\left(V_{j}\right)\) is the apparent molar volume of the solute. \[\phi\left(K_{\mathrm{T}_{\mathrm{j}}}\right)=\left[\kappa_{\mathrm{T}}(\mathrm{aq})-\kappa_{\mathrm{Tl}}^{*}(\ell)\right] \,\left(\mathrm{c}_{\mathrm{j}}\right)^{-1}+\kappa_{\mathrm{Tl}}^{*}(\ell) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\]The corresponding isentropic compression for solute \(j\), \(\phi\left(K_{\mathrm{Sj}} ; \text { def }\right)\) is related to the concentration \(\mathrm{c}_{j}\) using equation (b). \[\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \operatorname{def}\right) \equiv\left[\kappa_{\mathrm{S}}(\mathrm{aq})-\kappa_{\mathrm{S} 1}^{*}(\ell)\right] \,\left(\mathrm{c}_{\mathrm{j}}\right)^{-1}+\kappa_{\mathrm{S} 1}^{*}(\ell) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\][We replace the symbol ≡ by the symbol = in the following account.] \[\text { By definition } \delta(\mathrm{aq})=\kappa_{\mathrm{T}}(\mathrm{aq})-\kappa_{\mathrm{S}}(\mathrm{aq})\]\[\text { And } \delta_{1}^{*}(\ell)=\kappa_{\mathrm{T} 1}^{*}(\ell)-\kappa_{\mathrm{S} 1}^{*}(\ell)\]Hence \(\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)\) and \(\phi\left(K_{\mathrm{Sj}} ; \operatorname{def}\right)\) are related by equation (e). \[\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)-\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \operatorname{def}\right)=\left(\mathrm{c}_{\mathrm{j}}\right)^{-1} \,\left[\delta(\mathrm{aq})-\delta_{1}^{*}(\ell)\right]+\delta_{1}^{*}(\ell) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\]The difference \(\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)-\phi\left(\mathrm{K}_{\mathrm{S}_{\mathrm{j}}} ; \mathrm{def}\right)\) depends on the concentration of the solute \(\mathrm{c}_{j}\). Further \(\delta(\mathrm{aq})-\delta_{1}^{*}(\ell)\) is not zero. In fact, \[\begin{aligned} \delta(\mathrm{aq})-& \delta_{1}^{*}(\ell)=\\ &\left\{\mathrm{T} \,\left[\alpha_{\mathrm{p}}(\mathrm{aq})\right]^{2} / \sigma(\mathrm{aq})-\left\{\mathrm{T} \,\left[\alpha_{\mathrm{p} 1}^{*}(\ell)\right]^{2} / \sigma_{1}^{*}(\ell)\right\}\right. \end{aligned}\]Using the technique of adding and subtracting the same quantity, equation (f) is re-expressed as follows. \[\begin{aligned} &\delta(\mathrm{aq})-\delta_{1}^{*}(\ell)= \\ &\begin{aligned} \left\{\delta(\mathrm{aq}) /\left[\alpha_{\mathrm{p}}(\mathrm{aq})\right]^{2}\right\} \,\left[\, \alpha_{\mathrm{p}}(\mathrm{aq})+\alpha_{\mathrm{p} 1}^{*}(\ell)\right] \,\left[\alpha_{\mathrm{p}}(\mathrm{aq})-\alpha_{\mathrm{p} 1}^{*}(\ell)\right] \\ &-\left[\delta_{1}^{*}(\ell) / \sigma(\mathrm{aq})\right] \,\left[\sigma(\mathrm{aq})-\sigma_{1}^{*}(\ell)\right] \end{aligned} \end{aligned}\]The difference \(\left[\alpha_{\mathrm{p}}(\mathrm{aq})-\alpha_{\mathrm{p} 1}^{*}(\ell)\right]\) is related to \(\phi\left(\mathrm{E}_{\mathrm{pj}}\right)\) using equation (h). \[\phi\left(E_{p j}\right)=\left[\alpha_{p}(a q)-\alpha_{p 1}^{*}(\ell)\right] \,\left(c_{j}\right)^{-1}+\alpha_{p 1}^{*}(\ell) \, \phi\left(V_{j}\right)\]Similarly, \(\left[\sigma(\mathrm{aq})-\sigma_{1}^{*}(\ell)\right]\) is related to \(\phi\left(C_{\mathrm{p} j}\right)\) using equation (i). \[\phi\left(C_{p j}\right)=\left[\sigma(a q)-\sigma_{1}^{*}(\ell)\right] \,\left(c_{j}\right)^{-1}+\sigma_{1}^{*}(\ell) \, \phi\left(V_{j}\right)\]Using equations (g) - (i), we express equation (e) as follows. \[\begin{aligned} \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)-\phi\left(\mathrm{K}_{\mathrm{sj}} ; \mathrm{def}\right)=& \\ {\left[\delta(\mathrm{aq}) / \alpha_{\mathrm{p}}(\mathrm{aq})\right] \,\left\{1+\left[\alpha_{\mathrm{p} 1}^{*}(\ell) / \alpha_{\mathrm{p}}(\mathrm{aq})\right]\right\} \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right) } \\ &-\left[\delta_{1}^{*}(\ell) / \sigma(\mathrm{aq})\right] \, \phi\left(\mathrm{C}_{\mathrm{pj}}\right)+\left\{\delta_{1}^{*}(\ell)\right.\\ &\left.-\left[\delta(\mathrm{aq}) \, \alpha_{\mathrm{p} 1}^{*}(\ell) / \alpha_{\mathrm{p}}(\mathrm{aq})\right]\right\} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \end{aligned}\]Equation (j) was obtained by Desnoyers and Philip who showed that if \(\phi\left(K_{T_{j}}\right)^{\infty}\) and \(\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \mathrm{def}\right)^{\infty}\) are the limiting (infinite dilution) apparent molar properties, the difference is given by equation (k). \[\begin{aligned} \phi\left(\mathrm{K}_{\mathrm{Tj}}\right)^{\infty}-\phi\left(\mathrm{K}_{\mathrm{Sj}} ;\right.&\operatorname{def})^{\infty}=\\ \delta_{1}^{*}(\ell) \,\left\{\left[2 \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right)^{\infty} / \alpha_{\mathrm{pl} 1}^{*}(\ell)\right]-\left[\phi\left(\mathrm{C}_{\mathrm{pj}}\right)^{\infty} / \sigma_{1}^{*}(\ell)\right]\right\} \end{aligned}\]Using equation (b), \(\phi\left(K_{\mathrm{Sj}} ; \text { def }\right)\) is plotted as a function of cj across a set of different solutions having different entropies. \(\operatorname{Limit}\left(\mathrm{c}_{\mathrm{j}} \rightarrow 0\right) \phi\left(\mathrm{K}_{\mathrm{Sj}} ; \text { def }\right)\) defines \(\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \operatorname{def}\right)^{\infty}\). Granted two limiting quantities, \(\phi\left(E_{p j}\right)^{\infty}\) and \(\phi\left(C_{p j}\right)^{\infty}\) are available for the solution at the same \(\mathrm{T}\) and \(\mathrm{p}\), equation (k) is used to calculate \(\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{J}}}\right)^{\infty}\) using \(\phi\left(K_{\mathrm{S}_{\mathrm{j}}} ; \text { def }\right)^{\infty}\).An alternative form of equation (j) refers to a solution, molality mj. \[\begin{aligned} &\phi\left(\mathrm{K}_{\mathrm{Tj}}\right)-\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \text { def }\right)= \\ &\quad \delta_{1}^{*}(\ell) \,\left\{\left[2 \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right) / \alpha_{\mathrm{p} 1}^{*}(\ell)\right]-\left[\phi\left(\mathrm{C}_{\mathrm{pj}}\right) / \sigma_{1}^{*}(\ell)\right]\right. \\ &\quad+\left[\rho_{1}^{*}(\ell) \, \mathrm{m}_{\mathrm{j}} \,\left[\phi\left(\mathrm{E}_{\mathrm{pj}}\right)\right]^{2} /\left[\alpha_{\mathrm{p} 1}^{*}(\ell)\right]^{2}\right\} \,\left\{1+\left[\rho_{1}^{*}(\ell) \, \mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{C}_{\mathrm{pj}}\right) / \sigma_{1}^{*}(\ell)\right]\right\}^{-1} \end{aligned}\]The fact that \(\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)^{\infty}\) can be obtained from \(\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \text { def }\right)^{\infty}\) indicates the importance of the Desnoyers-Philip equation. Bernal and Van Hook used the Desnoyers-Philip equation to calculate \(\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)^{\infty}\) for glucose(aq), sucrose(aq) and fructose(aq) at \(348 \mathrm{~K}\). Similarly Hedwig et. al. used the Desnoyers –Philip equation to obtain estimates of \(\phi\left(K_{\mathrm{Tj}}\right)^{\infty}\) for glycyl dipeptides (aq) at \(298 \mathrm{~K}\).Footnotes J. E. Desnoyers and P. R. Philip, Can. J.Chem.,1972, 50,1095. J. C. R. Reis, J. Chem. Soc. Faraday Trans.,1998,94,2385. P. J. Bernal and W. A. Hook, J.Chem.Thermodyn.,1986,18,955. G. R. Hedwig, J. D. Hastie and H. Hoiland, J. Solution Chem.,1996, 25, 615.This page titled 1.7.15: Compression- Isentropic and Isothermal- Solutions- Limiting Estimates is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.7.16: Compressions- Isentropic and Isothermal- 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/1.7.16%3A_Compressions-_Isentropic_and_Isothermal-_Apparent_Molar_Volume

A given solution is perturbed by a change in pressure to a neighbouring state at constant affinity, \(\mathrm{A}\). \[(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{s}}=(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{T}}-(\partial \mathrm{S} / \partial \mathrm{p})_{\mathrm{T}} \,(\partial \mathrm{T} / \partial \mathrm{S})_{\mathrm{p}} \,(\partial \mathrm{V} / \partial \mathrm{T})_{\mathrm{p}}\]But for the pure solvent (at constant affinity \(\mathrm{A}\)) S * 1 * S1 K (l) = −(∂V (l)/ ∂p) and T * 1 * T1 K (l) = −(∂V (l)/ ∂p) (b)We confine attention to perturbation at ‘\(\mathrm{A} = 0\)’; i.e. an equilibrium process. [Note the change in sign.] \[\mathrm{K}_{\mathrm{s} 1}^{*}(\ell)=\mathrm{K}_{\mathrm{T} 1}^{*}(\ell)+\left(\partial \mathrm{S}_{1}^{*}(\ell) / \partial \mathrm{p}\right)_{\mathrm{T}} \,\left(\partial \mathrm{T} / \partial \mathrm{S}_{1}^{*}(\ell)\right)_{\mathrm{p}} \,\left(\partial \mathrm{V}_{1}^{*}(\ell) / \partial \mathrm{T}\right)_{\mathrm{p}}\]From a Maxwell Equation, \[\left(\partial \mathrm{S}_{1}^{*}(\ell) / \partial \mathrm{p}\right)_{\mathrm{T}}=-\left(\partial \mathrm{V}_{1}^{*}(\ell) / \partial \mathrm{T}\right)_{\mathrm{p}}\]From the Gibbs-Helmholtz Equation, \[\left(\partial \mathrm{S}_{1}^{*}(\ell) / \partial \mathrm{T}\right)_{\mathrm{p}}=\mathrm{C}_{\mathrm{p} 1}^{*}(\ell) / \mathrm{T}\]\(C_{p 1}^{*}(\ell)\) is the molar ( equilibrium) isobaric heat capacity of the solvent at defined \(\mathrm{T}\) and \(\mathrm{p}\). From equation (c), [Note change of sign.] \[\mathrm{K}_{\mathrm{S} 1}^{*}(\ell)=\mathrm{K}_{\mathrm{T} 1}^{*}(\ell)-\left[\left(\partial \mathrm{V}_{1}^{*}(\ell) / \partial \mathrm{T}\right)_{\mathrm{p}}\right]^{2} \, \mathrm{T} / \mathrm{C}_{\mathrm{p} 1}^{*}(\ell)\]But \[\alpha_{1}^{*}(\ell) \, V_{1}^{*}(\ell)=\left(\partial V_{1}^{*}(\ell) / \partial T\right)_{p}\]\[\mathrm{K}_{\mathrm{S} 1}^{*}(\ell)=\mathrm{K}_{\mathrm{T} 1}^{*}(\ell)-\left[\alpha_{1}^{*}(\ell) \, \mathrm{V}_{1}^{*}(\ell)\right]^{2} \, \mathrm{T} / \mathrm{C}_{\mathrm{p} 1}^{*}(\ell)\]But the ratio of isobaric heat capacity of the solvent to its molar volume, \[\mathrm{C}_{\mathrm{p} 1}^{*}(\ell) / \mathrm{V}_{1}^{*}(\ell)=\sigma_{1}^{*}(\ell) .\]\[\left.\mathrm{K}_{\mathrm{s} 1}^{*}(\ell)=\mathrm{K}_{\mathrm{T} 1}^{*}(\ell)-\left[\alpha_{1}^{*}(\ell)\right]^{2} \, \mathrm{V}_{1}^{*}(\ell)\right] \, \mathrm{T} / \sigma_{1}^{*}(\ell)\]But \[\kappa_{\mathrm{s} 1}^{*}(\ell)=\left[\mathrm{V}_{1}^{*}(\ell)\right]^{-1} \, \mathrm{K}_{\mathrm{S} 1}^{*}(\ell) \text { and } \kappa_{\mathrm{T} 1}^{*}(\mathrm{l})=\left[\mathrm{V}_{1}^{*}(\ell)\right]^{-1} \, \mathrm{K}_{\mathrm{T} 1}^{*}(\ell)\]\[\kappa_{\mathrm{S} 1}^{*}(\ell)=\kappa_{\mathrm{T} 1}^{*}(\ell)-\left[\alpha_{1}^{*}(\ell)\right]^{2} \, \mathrm{T} / \sigma_{1}^{*}(\ell)\]Similarly for an aqueous solution, molality \(\mathrm{m}_{j}\), \[\kappa_{\mathrm{S}}(\mathrm{aq})=\kappa_{\mathrm{T}}(\mathrm{aq})-[\alpha(\mathrm{aq})]^{2} \, \mathrm{T} / \sigma(\mathrm{aq})\]Also \[\mathrm{K}_{\mathrm{T}}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{K}_{\mathrm{T} 1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)\]\[\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\left(\mathrm{n}_{\mathrm{j}}\right)^{-1} \,\left\{\mathrm{K}_{\mathrm{T}}(\mathrm{aq})-\mathrm{n}_{1} \, \mathrm{K}_{\mathrm{T} 1}^{*}(\ell)\right]\]We convert from compressions to compressibilities. \[\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\left(\mathrm{n}_{\mathrm{j}}\right)^{-1} \,\left\{\mathrm{V}(\mathrm{aq}) \, \kappa_{\mathrm{T}}(\mathrm{aq})-\mathrm{n}_{1} \, \mathrm{V}_{\mathrm{T} 1}^{*}(\ell) \, \kappa_{\mathrm{T} 1}^{*}(\ell)\right]\]But we know \(\kappa_{\mathrm{T}}(\mathrm{aq})\) in terms of ) \(\kappa_{\mathrm{S}}(\mathrm{aq})\) (aq [see equation (m)] and \(\kappa_{\mathrm{T} 1}^{*}(\ell)\) in terms of \(\kappa_{\mathrm{s} 1}^{*}(\ell)\). Then [NB change of sign] \[\begin{aligned} \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\left[\frac{\mathrm{V}(\mathrm{aq})}{\mathrm{n}_{\mathrm{j}}}\right] \,\left[\mathrm{K}_{\mathrm{S}}(\mathrm{aq})+\frac{\{\alpha(\mathrm{aq})\}^{2} \, \mathrm{T}}{\sigma(\mathrm{aq})}\right] \\ &-\left[\frac{\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)}{\mathrm{n}_{\mathrm{j}}}\right] \,\left[\kappa_{\mathrm{Sl}}^{*}(\ell)+\frac{\left\{\alpha_{1}^{*}(\ell)\right\}^{2} \, \mathrm{T}}{\sigma_{1}^{*}(\ell)}\right] \end{aligned}\]We introduce densities into equation (q). For a solution having mass \(\mathrm{w}\), \[\begin{aligned} \mathrm{V}(\mathrm{aq}) / \mathrm{n}_{\mathrm{j}} &=\left[1 / \mathrm{n}_{\mathrm{j}}\right] \,[\mathrm{w} / \rho(\mathrm{aq})]=\left[1 / \mathrm{n}_{\mathrm{j}} \, \rho(\mathrm{aq})\right] \,\left[\mathrm{n}_{\mathrm{j}} \, \mathrm{M}_{\mathrm{j}}+\mathrm{n}_{1} \, \mathrm{M}_{1}\right] \\ &=\left[1 / \mathrm{m}_{\mathrm{j}} \, \rho(\mathrm{aq})\right]+\left[\mathrm{M}_{\mathrm{j}} / \rho(\mathrm{aq})\right] \end{aligned}\]Also, \[\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell) / \mathrm{n}_{\mathrm{j}}=\left[\mathrm{n}_{1} / \mathrm{n}_{\mathrm{j}}\right] \,\left[\mathrm{M}_{1} / \rho_{1}^{*}(\ell)\right]=\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1}\]From equations (q), (r) and (s), \[\begin{aligned} \phi\left(\mathrm{K}_{\mathrm{Tj}}\right)=\left[\frac{1}{\mathrm{~m}_{\mathrm{j}} \, \rho(\mathrm{aq})}+\frac{\mathrm{M}_{\mathrm{j}}}{\rho(\mathrm{aq})}\right] \,\left[\kappa_{\mathrm{s}}(\mathrm{aq})+\frac{\{\alpha(\mathrm{aq})\}^{2} \, \mathrm{T}}{\sigma(\mathrm{aq})}\right] \\ &-\left[\frac{1}{\mathrm{~m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)}\right] \,\left[\kappa_{\mathrm{s} 1}^{*}(\ell)+\frac{\left\{\alpha_{1}^{*}(\ell)\right\}^{2} \, \mathrm{T}}{\sigma_{1}^{*}(\ell)}\right] \end{aligned}\]We factor out the six terms . The order in which we write these terms anticipates the next but one step. \[\begin{aligned} \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\left[\frac{\kappa_{\mathrm{s}}(\mathrm{aq})}{\mathrm{m}_{\mathrm{j}} \, \rho(\mathrm{aq})}\right]-\left[\frac{\kappa_{\mathrm{Sl}}^{*}(\ell)}{\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)}\right]+\left[\frac{\mathrm{M}_{\mathrm{j}} \, \kappa_{\mathrm{s}}(\mathrm{aq})}{\rho(\mathrm{aq})}\right] \\ &+\left[\frac{\{\alpha(\mathrm{aq})\}^{2} \, \mathrm{T}}{\mathrm{m}_{\mathrm{j}} \, \rho(\mathrm{aq}) \, \sigma(\mathrm{aq})}\right] \\ &-\left[\frac{\left\{\alpha_{1}^{*}(\ell)\right\}^{2} \, \mathrm{T}}{\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell) \, \sigma_{1}^{*}(\ell)}\right]+\left[\frac{\mathrm{M}_{\mathrm{j}} \,\{\alpha(\mathrm{aq})\}^{2} \, \mathrm{T}}{\rho(\mathrm{aq}) \, \sigma(\mathrm{aq})}\right] \end{aligned}\]Hence, \[\begin{gathered} \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\left\{\mathrm{m}_{\mathrm{j}} \, \rho(\mathrm{aq}) \, \rho_{1}^{*}(\ell)\right\}^{-1} \,\left[\left\{\mathrm{K}_{\mathrm{s}}(\mathrm{aq}) \, \rho_{1}^{*}(\ell)\right\}-\left\{\kappa_{\mathrm{s} 1}^{*}(\ell) \, \rho(\mathrm{aq})\right\}\right] \\ +\left\{\kappa_{\mathrm{S}}(\mathrm{aq}) \, \mathrm{M}_{\mathrm{j}} / \rho(\mathrm{aq})+\mathrm{A}+\mathrm{B}\right. \end{gathered}\]where, \[\mathrm{A}=\left[\frac{\mathrm{T}}{\mathrm{m}_{\mathrm{j}} \, \rho(\mathrm{aq}) \, \rho_{1}^{*}(\ell)}\right] \,\left[\left(\frac{\{\alpha(\mathrm{aq})\}^{2} \, \rho_{1}^{*}(\ell)}{\sigma(\mathrm{aq})}\right)-\left(\frac{\left\{\alpha_{1}^{*}(\ell) \, \rho(\mathrm{aq})\right.}{\sigma_{1}^{*}(\ell)}\right)\right]\]and \[\mathrm{B}=\mathrm{M}_{\mathrm{j}} \,\{\alpha(\mathrm{aq})\}^{2} \, \mathrm{T} / \rho(\mathrm{aq}) \, \sigma(\mathrm{aq})\]With reference to solutions we compare the isentropic and isothermal dependences of \(\phi\left(V_{j}\right)\) on pressure. \[\left[\frac{\partial \phi\left(V_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right]_{\mathrm{S}(\mathrm{aq})}=\left[\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right]_{\mathrm{T}}-\left[\frac{\partial \mathrm{S}(\mathrm{aq})}{\partial \mathrm{p}}\right]_{\mathrm{T}} \,\left[\frac{\partial \mathrm{T}}{\partial \mathrm{S}(\mathrm{aq})}\right]_{\mathrm{p}} \,\left[\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{T}}\right]_{\mathrm{p}}\]Noting signs [cf. equations (d) and (e)] and the definition of \(\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)\), \[\left[\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right]_{\mathrm{S}(\mathrm{aq})}=-\phi\left(\mathrm{K}_{\mathrm{Tj}_{\mathrm{j}}}\right)-\left[-\left[\frac{\partial \mathrm{V}(\mathrm{aq})}{\partial \mathrm{T}}\right]_{\mathrm{p}}\right] \,\left[\frac{\mathrm{T}}{\mathrm{C}_{\mathrm{p}}(\mathrm{aq})}\right] \,\left[\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{T}}\right]_{\mathrm{p}}\]\[\left[\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right]_{\mathrm{S}(\mathrm{aq})}=\phi\left(\mathrm{K}_{\mathrm{Tj}}\right)-\left[\frac{\partial \mathrm{V}(\mathrm{aq})}{\partial \mathrm{T}}\right]_{\mathrm{p}} \,\left[\frac{\mathrm{T}}{\mathrm{C}_{\mathrm{p}}(\mathrm{aq})}\right] \,\left[\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{T}}\right]_{\mathrm{p}}\]We turn our attention to \(\left[\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right) / \partial \mathrm{T}\right]_{\mathrm{p}}\). We recall that \[\mathrm{V}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\]\[\phi\left(\mathrm{V}_{\mathrm{j}}\right)=\left(1 / \mathrm{n}_{\mathrm{j}}\right) \, \mathrm{V}(\mathrm{aq})-\left[\left(\mathrm{n}_{1} / \mathrm{n}_{\mathrm{j}}\right) \, \mathrm{V}_{1}^{*}(\ell)\right]\]Hence \[\left[\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{T}}\right]_{\mathrm{p}}=\frac{1}{\mathrm{n}_{\mathrm{j}}} \,\left[\frac{\partial \mathrm{V}(\mathrm{aq})}{\partial \mathrm{T}}\right]_{\mathrm{p}}-\frac{\mathrm{n}_{1}}{\mathrm{n}_{\mathrm{j}}} \,\left[\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{T}}\right]_{\mathrm{p}}\]We combine equations (za) and (zd). Hence \[\begin{aligned} &-\left[\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right]_{\mathrm{S}(\mathrm{aq})}= \\ &\phi\left(\mathrm{K}_{\left.\mathrm{T}_{\mathrm{j}}\right)}\right. \\ &-\left[\left[\frac{\partial \mathrm{V}(\mathrm{aq})}{\partial \mathrm{T}}\right]_{\mathrm{p}}\right]_{\mathrm{T}} \,\left[\frac{\mathrm{T}}{\mathrm{C}_{\mathrm{p}}(\mathrm{aq})}\right] \,\left[\frac{1}{\mathrm{n}_{\mathrm{j}}} \,\left[\frac{\partial \mathrm{V}(\mathrm{aq})}{\partial \mathrm{T}}\right]_{\mathrm{p}}-\frac{\mathrm{n}_{1}}{\mathrm{n}_{\mathrm{j}}} \,\left[\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{T}}\right]_{\mathrm{p}}\right] \end{aligned}\]Or, \[\begin{aligned} -\left[\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}]_{\mathrm{S}(\mathrm{aq})}}\right]=\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right) \\ &-\left[\frac{\mathrm{T}}{\mathrm{C}_{\mathrm{p}}(\mathrm{aq})}\right] \, \frac{1}{\mathrm{n}_{\mathrm{j}}} \,\left[\frac{\partial \mathrm{V}(\mathrm{aq})}{\partial \mathrm{T}}\right]_{\mathrm{p}}^{2} \\ &+\left[\frac{\mathrm{T}}{\mathrm{C}_{\mathrm{p}}(\mathrm{aq})}\right] \,\left[\frac{\mathrm{n}_{1}}{\mathrm{n}_{\mathrm{j}}} \,\left[\frac{\partial \mathrm{V}(\mathrm{aq})}{\partial \mathrm{T}}\right]_{\mathrm{p}}\left[\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{T}}\right]_{\mathrm{p}}\right] \end{aligned}\]But \[\alpha(\mathrm{aq}) \, \mathrm{V}(\mathrm{aq})=[\partial \mathrm{V}(\mathrm{aq}) / \partial \mathrm{T}]_{\mathrm{p}}\]And \[\alpha_{1}^{*}(\ell) \, V_{1}^{*}(\ell)=\left[\partial V_{1}^{*}(\ell) / \partial \mathrm{T}\right]_{\mathrm{p}}\]We introduce the latter two equations into equation (zf). \[\begin{aligned} {\left[\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right]_{\mathrm{S}(\mathrm{aq})} } &=\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right) \\ &-\left[\frac{\mathrm{T}}{\mathrm{C}_{\mathrm{p}}(\mathrm{aq})}\right] \, \frac{1}{\mathrm{n}_{\mathrm{j}}} \,[\mathrm{V}(\mathrm{aq}) \, \alpha(\mathrm{aq})]^{2} \\ &+\left[\frac{\mathrm{T}}{\mathrm{C}_{\mathrm{p}}(\mathrm{aq})}\right] \,\left[\frac{\mathrm{n}_{1}}{\mathrm{n}_{\mathrm{j}}}\right] \,\left[\alpha(\mathrm{aq}) \, \mathrm{V}(\mathrm{aq}) \, \alpha_{1}^{*}(\ell) \, \mathrm{V}_{1}^{*}(\ell)\right] \end{aligned}\]But \(\sigma(\mathrm{aq})=\mathrm{C}_{\mathrm{p}}(\mathrm{aq}) / \mathrm{V}(\mathrm{aq})\) and \(\sigma_{1}^{*}(\ell)=\mathrm{C}_{\mathrm{p} 1}^{*}(\ell) / \mathrm{V}_{1}^{*}(\ell)\). Also \(\mathrm{M}_{1} / \mathrm{V}_{1}^{*}(\ell)=\rho_{1}^{*}(\ell)\). Hence, \[\begin{aligned} -\left[\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right]_{\mathrm{S}(\mathrm{aq})} &=\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right) \\ -& {\left[\frac{\mathrm{T}}{\sigma(\mathrm{aq})}\right] \, \frac{1}{\mathrm{n}_{\mathrm{j}}} \, \mathrm{V}(\mathrm{aq}) \,[\alpha(\mathrm{aq})]^{2} } \\ &+\left[\frac{\mathrm{T}}{\sigma(\mathrm{aq})}\right] \,\left[\frac{\mathrm{n}_{1}}{\mathrm{n}_{\mathrm{j}}}\right] \,\left[\alpha(\mathrm{aq}) \, \alpha_{1}^{*}(\ell) \, \mathrm{M}_{1} / \rho_{1}^{*}(\ell)\right] \end{aligned}\]Also, \(\mathrm{n}_{\mathrm{j}} / \mathrm{n}_{1} \, \mathrm{M}_{1}=\mathrm{m}_{\mathrm{j}}\). And \(\mathrm{V}(\mathrm{aq}) / \mathrm{n}_{\mathrm{j}}=\left[1 / \mathrm{m}_{\mathrm{j}} \, \rho(\mathrm{aq})\right]+\left[\mathrm{M}_{\mathrm{j}} / \rho(\mathrm{aq})\right]\)Hence we obtain a relation between the two compressions of the apparent molar volumes \[\begin{aligned} -\left[\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right]_{\mathrm{S}(\mathrm{aq})} &=\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right) \\ -\left[\frac{\mathrm{T}}{\sigma(\mathrm{aq})}\right] \,\left[\frac{1}{\mathrm{~m}_{\mathrm{j}} \, \rho(\mathrm{aq})}+\frac{\mathrm{M}_{\mathrm{j}}}{\rho(\mathrm{aq})}\right] \,[\alpha(\mathrm{aq})]^{2} \\ &+\left[\frac{\mathrm{T}}{\sigma(\mathrm{aq})}\right] \,\left[\frac{1}{\mathrm{~m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)}\right] \, \alpha(\mathrm{aq}) \, \alpha_{1}^{*}(\ell) \end{aligned}\]Footnotes Unit check on equation (l). \(\left.\left[\mathrm{N} \mathrm{m}^{-2}\right]^{-1}=\left[\mathrm{N} \mathrm{m}^{-2}\right]^{-1}-\left\{\left[\mathrm{K}^{-1}\right]^{2} \,[\mathrm{K}]\right\} /\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]^{-1}\right\}\) \(\left[\mathrm{N} \mathrm{m}^{-2}\right]^{-1}=\left[\mathrm{N} \mathrm{m}^{-2}\right]^{-1}-\left[\mathrm{J} \mathrm{m}^{-3}\right]^{-1}\) But \(\left[\mathrm{N} \mathrm{m}^{-2}\right]=\left[\mathrm{J} \mathrm{m}^{-3}\right]\) From \(\begin{aligned} &V(a q)=\frac{w}{\rho(a q)}=\frac{n_{1} \, M_{1}}{\rho(a q)}+\frac{n_{j} \, M_{j}}{\rho(a q)} \\ &\frac{V(a q)}{n_{j}}=\frac{n_{1} \, M_{1}}{n_{j} \, \rho(a q)}+\frac{M_{j}}{\rho(a q)} \\ &\frac{V(a q)}{n_{j}}=\frac{1}{m_{j} \, \rho(a q)}+\frac{M_{j}}{\rho(a q)} \end{aligned}\)This page titled 1.7.16: Compressions- Isentropic and Isothermal- 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.

1.7.17: Compressions- Isothermal- Solutes- Partial Molar Compressions

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/1.7.17%3A_Compressions-_Isothermal-_Solutes-_Partial_Molar_Compressions

A given aqueous solution at temperature \(\mathrm{T}\) and near ambient pressure \(\mathrm{p}\) contains a solute \(j\) at molality \(\mathrm{m}_{j}\). The chemical potential \(\mu_{j}(\mathrm{aq})\) is related to the molality \(\mathrm{m}_{j}\) using equation (a). \[\mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\gamma_{\mathrm{j}}\right)\]Then \[\mathrm{V}_{\mathrm{j}}(\mathrm{aq})=\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \,\left[\partial \ln \left(\gamma_{\mathrm{j}}\right) / \partial \mathrm{p}\right]_{\mathrm{T}}\]By definition, the partial molar isothermal compression of solute \(j\), \[K_{T_{j}}(a q)=-\left(\frac{\partial V_{j}(a q)}{\partial p}\right)_{T}\]Then \[\mathrm{K}_{\mathrm{Tj}}(\mathrm{aq})=\mathrm{K}_{\mathrm{TJ}}^{\infty}(\mathrm{aq})-\mathrm{R} \, \mathrm{T} \,\left[\partial^{2} \ln \left(\gamma_{\mathrm{j}}\right) / \partial \mathrm{p}^{2}\right]_{\mathrm{T}}\]Thus by definition, \[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{K}_{\mathrm{T}_{\mathrm{j}}}(\mathrm{aq})=\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}^{\infty}(\mathrm{aq})\]Hence the difference between \(\mathrm{K}_{\mathrm{Tj}_{\mathrm{j}}}(\mathrm{aq})\) and \(\mathrm{K}_{\mathrm{T}_{j}}^{\infty}(\mathrm{aq})\) depends on the second differential of \(\ln \left(\gamma_{j}\right)\) with respect to pressure.Footnotes The formal definition of \(\mathrm{K}_{\mathrm{Tj}}(\mathrm{aq})\) is given by equation (a). \[\mathrm{K}_{\mathrm{Tj}}(\mathrm{aq})=\left(\frac{\partial \mathrm{K}_{\mathrm{T}}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{i} \neq \mathrm{j})}\]However, \[\mathrm{K}_{\mathrm{T}}=-(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{T}, \mathrm{n}(\mathrm{j})}\]Then, \[\mathrm{K}_{\mathrm{Tj}}(\mathrm{aq})=-\left(\frac{\partial\left(\partial \mathrm{V} / \partial \mathrm{n}_{\mathrm{j}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{i} \neq \mathrm{j})}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{n}(\mathrm{j})}\]Or, \[\mathrm{K}_{\mathrm{Tj}}(\mathrm{aq})=-\left(\frac{\partial \mathrm{V}_{\mathrm{j}}(\mathrm{aq})}{\partial \mathrm{p}}\right)_{\mathrm{T}}\]In other words, equation (c) shows that \(\mathrm{K}_{\mathrm{Tj}}(\mathrm{aq})\) is a Lewisian partial molar propertyThis page titled 1.7.17: Compressions- Isothermal- Solutes- Partial Molar Compressions is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.7.18: Compressions- Isothermal- Apparent Molar Compression

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/1.7.18%3A_Compressions-_Isothermal-_Apparent_Molar_Compression

The volume of a given solution prepared at fixed temperature and fixed pressure using \(\mathrm{n}_{1}\) moles of water and \(\mathrm{n}_{j}\) moles of solute \(j\) is given by equation (a). \[\mathrm{V}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{V}_{1}(\mathrm{aq})+\mathrm{n}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}(\mathrm{aq})\]If the solution is prepared using \(1 \mathrm{~kg}\) of water(\(\ell\)), \[\mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\mathrm{M}_{1}^{-1} \, \mathrm{V}_{1}(\mathrm{aq})+\mathrm{m}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}(\mathrm{aq})\]\(\mathrm{M}_{1}\) is the molar mass of the solvent, water(\(\ell\)); \(\mathrm{V}_{1}(\mathrm{aq})\) and \(\mathrm{V}_{j}(\mathrm{aq})\) are the partial molar volumes of water and solute \(j\) respectively in the solution. As we change \(\mathrm{m}_{j}\) (for a fixed mass of solvent) so both \(\mathrm{V}_{1}(\mathrm{aq})\) and \(\mathrm{V}_{j}(\mathrm{aq})\) change. An important procedure rewrites equation (b) in the following form where \(\mathrm{V}_{1}^{*}(\ell)\) is the molar volume of pure solvent at the same \(\mathrm{T}\) and \(\mathrm{p}\). Thus, \[\mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\mathrm{M}_{1}^{-1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\]\(\phi\left(\mathrm{V}_{\mathrm{j}}\right)\) is the apparent molar volume of the solute \(j\). The system, an aqueous solution, is displaced by a change in pressure (at fixed \(\mathrm{T}\)) along a path where the affinity for spontaneous change is zero. In other words the system is subjected to an equilibrium displacement. The isothermal differential dependence of volume \(\mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)\) is given by equation (d). \[\left(\frac{\partial \mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right.}{\partial \mathrm{p}}\right)_{\mathrm{T}}=\mathrm{M}_{1}^{-1} \,\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{T}}+\mathrm{m}_{\mathrm{j}} \,\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right)_{\mathrm{T}}\]By definition, the apparent molar (isothermal) compression of the solute, \[\phi\left(\mathrm{K}_{\mathrm{Tj}}\right)=-\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right)_{\mathrm{T}}\]Similarly for equilibrium molar compression of the pure solvent, \[\mathrm{K}_{\mathrm{T} 1}^{*}(\ell)=-\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{T}}\]By definition, \[\mathrm{K}_{\mathrm{T}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=-\left(\frac{\partial \mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)}{\partial \mathrm{p}}\right)_{\mathrm{T}}\]Hence, \[\mathrm{K}_{\mathrm{T}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\mathrm{M}_{1}^{-1} \, \mathrm{K}_{\mathrm{T} 1}^{*}(\ell)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{K}_{\mathrm{Tj}}\right)\]Moreover recalling that \(\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi\left(\mathrm{V}_{\mathrm{j}}\right)=\phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}=\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq})\), \[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)^{\infty}=\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}^{\infty}(\mathrm{aq})\]These equations combined with those yielding \(\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)\) from measured \(\kappa_{\mathrm{T} 1}^{*}(\ell)\) and \(\kappa_{\mathrm{T}}(\mathrm{aq})\) signal an attractive approach to the study of solvent-solute interactions via \(\mathrm{K}_{\mathrm{T} j}^{\infty}(\mathrm{aq})\). In this context Gurney identified a cosphere of solvent around a solute molecule where the organization differs from that in the bulk solvent at some distance from a given solute molecule \(j\). For example, the limiting partial molar volume of solute \(j\) can be understood as the sum of two terms, \(\mathrm{V}\)(intrinsic) and \(\mathrm{V}\)(cosphere). Then \(\mathrm{V}\)(cosphere) is an indicator of the role of solvent-solute interaction, hydration in aqueous solution. Thus, \[\left.V_{j}^{\infty}(a q)=V_{j} \text { (int rinsic }\right)+V_{j}(\cos p h e r e)\]Hence, \[\mathrm{K}_{\mathrm{T}_{j}}^{\infty}(\mathrm{aq})=\mathrm{K}_{\mathrm{T}_{\mathrm{j}}} \text { (int rinsic) }+\mathrm{K}_{\mathrm{T}_{j}}(\cos \text {phere})\]The argument is advanced that \(\mathrm{K}_{\mathrm{T}j}\)(intrinsic) for simple ions such as halide ions and alkali metal ions is zero. \(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}^{\infty}(\mathrm{aq})\) is an indicator of the hydration of a given solute in aqueous solution. \(\mathrm{K}_{\mathrm{T}_{j}}^{\infty}(\mathrm{aq})\) is obtained from the dependence of \(\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)\) on, for example, concentration \(\mathrm{c}_{j}\) using equation(l) for neutral solutes and equation (m) for salts, the latter being based on the DHLL. \[\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}^{\infty}(\mathrm{aq})+\mathrm{a}_{\mathrm{KT}} \,\left(\mathrm{c}_{\mathrm{j}} / \mathrm{c}_{\mathrm{r}}\right)\]\[\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}^{\infty}(\mathrm{aq})+\mathrm{b}_{\mathrm{KT}} \,\left(\mathrm{c}_{\mathrm{j}} / \mathrm{c}_{\mathrm{r}}\right)^{1 / 2}\]One might have expected an extensive scientific literature reporting \(\mathrm{K}_{\mathrm{T}_{j}}^{\infty}(\mathrm{aq})\) for a wide range of solutes. Unfortunately measurement of isothermal compressions of liquids is difficult at least to the precision required for the estimation of \(\mathrm{K}_{\mathrm{Tj}}^{\infty}(\mathrm{aq})\). Indeed direct measurement of the volume change of a liquid when compressed at constant temperature is difficult because the isothermal condition is difficult to satisfy. Two procedures have been adopted to over come this problem. In both cases isentropic compressibilities calculated from densities and speeds of sound have been used.In one set of procedures, isentropic compressibilities, densities and isobaric heat capacities are used to calculate isothermal compressions for a given solution, molality \(\mathrm{m}_{j}\). For example, Bernal and Van Hook use the Desnoyers-Philip Equation to evaluate \(\phi\left(K_{T_{j}}\right)^{\infty}\) for glucose, sucrose and fructose in aqueous solutions at \(348 \mathrm{~K}\). An alternative procedure equates \(\phi\left(\mathrm{K}_{\mathrm{Tj}}\right)^{\infty}\) with the experimentally accessible limiting apparent isentropic compression, ∞ φ(K ) Sj . In another approach, the starting point is equation (a) which is differentiated with respect to pressure at constant temperature to yield equation (n). \[\mathrm{K}_{\mathrm{T}}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{K}_{\mathrm{Tl}}(\mathrm{aq})+\mathrm{n}_{\mathrm{j}} \, \mathrm{K}_{\mathrm{T}_{\mathrm{j}}}(\mathrm{aq})\]Equation (n) is divided by volume \(\mathrm{V}(\mathrm{aq})\). Hence \[\begin{aligned} \mathrm{K}_{\mathrm{T}}(\mathrm{aq}) / \mathrm{V}(\mathrm{aq})=\left[\mathrm{n}_{1} \, \mathrm{V}_{1}(\mathrm{aq}) / \mathrm{V}(\mathrm{aq})\right] \,\left[1 / \mathrm{V}_{1}(\mathrm{aq})\right] \, \mathrm{K}_{1}(\mathrm{aq}) \\ &+\left[\mathrm{n}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}(\mathrm{aq}) / \mathrm{V}(\mathrm{aq})\right] \,\left[1 / \mathrm{V}_{\mathrm{j}}(\mathrm{aq})\right] \, \mathrm{K}_{\mathrm{j}}(\mathrm{aq}) \end{aligned}\]We use \(\phi_{1}\left[=\mathrm{n}_{1} \, \mathrm{V}_{1}(\mathrm{aq}) / \mathrm{V}(\mathrm{aq})\right] \text { and } \phi_{\mathrm{j}}\left[=\mathrm{n}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}(\mathrm{aq}) / \mathrm{V}(\mathrm{aq})\right]\) to express volume fractions. \[\kappa_{\mathrm{T}}(\mathrm{aq})=\phi_{1} \,\left[1 / \mathrm{V}_{1}(\mathrm{aq})\right] \, \mathrm{K}_{\mathrm{T} 1}(\mathrm{aq})+\phi_{\mathrm{j}} \,\left[1 / \mathrm{V}_{\mathrm{j}}(\mathrm{aq})\right] \, \mathrm{K}_{\mathrm{Tj}}(\mathrm{aq})\]The latter equation is not tremendously helpful. Although \(\kappa_{\mathrm{T}}(\mathrm{aq})\) can be measured, the right hand side involves six terms about which we have no information ‘a priori’ and which depend on the composition of the solution.Footnotes F. T. Gucker, Chem. Rev.,1933,14,127. F. T. Gucker, J. Am. Chem. Soc.,1933, 55,2709. Units; \(\phi\left(\mathrm{K}_{\mathrm{Tj}}\right)=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1} \mathrm{~Pa}^{-1}\right] ; \mathrm{K}_{\mathrm{T} 1}^{*}(\ell)=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1} \mathrm{~Pa}^{-1}\right]\); \(\mathrm{K}_{\mathrm{T}}(\mathrm{aq})=\left[\mathrm{m}^{3} \mathrm{~Pa}^{-1}\right]\)For the solution \(\kappa_{\mathrm{T}}(\mathrm{aq})=\mathrm{K}_{\mathrm{T}}(\mathrm{aq}) / \mathrm{V}(\mathrm{aq})=\left[\mathrm{Pa}^{-1}\right]\)For the solvent \(\kappa_{\mathrm{T} 1}^{*}(\ell)=\mathrm{K}_{\mathrm{T} 1}^{*}(\ell) / \mathrm{V}_{1}^{*}(\ell)=\left[\mathrm{Pa}^{-1}\right]\)Isothermal compressions have units of ‘volume per unit of pressure’ whereas compressibilities have units of ‘reciprocal pressure’. \(\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)\) is an apparent molar isothermal compression on the grounds that the units of this quantity are \(\left[\mathrm{m}^{3} \mathrm{~mol}^{-1} \mathrm{~Pa}^{-1}\right]\). Some reports use the term ‘apparent molar isothermal compressibility’ which should be avoided because in the present context this term corresponds to a different property; see J. C. R. Reis, J. Chem. Soc. Faraday Trans.,1998,94,2385. R.W. Gurney, Ionic Processes in Solution, McGraw-Hill, New York, 1953. P. D. Bernal and W. A. Van Hook, J. Chem. Thermodyn., 1986,18,955.This page titled 1.7.18: Compressions- Isothermal- Apparent Molar Compression is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.7.19: Compressions- Isothermal- Solutions- Apparent Molar- Determination

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/1.7.19%3A_Compressions-_Isothermal-_Solutions-_Apparent_Molar-_Determination

A given solution (at fixed \(\mathrm{T}\) and \(\mathrm{p}\)) is prepared using \(1 \mathrm{~kg}\) of solvent water and \(\mathrm{m}_{j}\) moles of solute \(j\). The compression of this solution \(\mathrm{K}_{\mathrm{T}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)\) is given by equations (a) and (b). \[\mathrm{K}_{\mathrm{T}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\mathrm{M}_{1}^{-1} \, \mathrm{K}_{\mathrm{T} 1}^{*}(\ell)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)\]\[\mathrm{K}_{\mathrm{T}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\mathrm{M}_{1}^{-1} \, \mathrm{K}_{\mathrm{T} 1}(\mathrm{aq})+\mathrm{m}_{\mathrm{j}} \, \mathrm{K}_{\mathrm{Tj}}(\mathrm{aq})\]where, \[\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}(\mathrm{aq})=-\left(\frac{\partial \mathrm{V}_{\mathrm{j}}}{\partial \mathrm{p}}\right)_{\mathrm{T}}\]and \[\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=-\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right)_{\mathrm{T}}\]Both \(\mathrm{K}_{\mathrm{Tj}}(\mathrm{aq})\) and \(\phi\left(\mathrm{K}_{\mathrm{Tj}^{\mathrm{j}}}\right)\) are Lewisian variables. With reference to partial molar volumes, \[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi\left(\mathrm{V}_{\mathrm{j}}\right)=\phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}=\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq})\]Hence \[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)^{\infty}=\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}^{\infty}(\mathrm{aq})\]\(\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)^{\infty}\) is the limiting (infinite dilution) apparent molar compression of solute--\(j\). For a given solution \(\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)\) is calculated using one of the following equations together with the isothermal compressions of solution and solvent.\[\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\kappa_{\mathrm{T}}(\mathrm{aq})-\kappa_{\mathrm{T} 1}^{*}(\ell)\right]+\kappa_{\mathrm{T}}(\mathrm{aq}) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\]\[\begin{gathered} \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\left[\mathrm{m}_{\mathrm{j}} \, \rho(\mathrm{aq}) \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\kappa_{\mathrm{T}}(\mathrm{aq}) \, \rho_{1}^{*}(\ell)-\kappa_{\mathrm{T} 1}^{*}(\ell) \, \rho(\mathrm{aq})\right] \\ +\kappa_{\mathrm{T}}(\mathrm{aq}) \, \mathrm{M}_{\mathrm{j}} \,[\rho(\mathrm{aq})]^{-1} \end{gathered}\]\[\begin{gathered} \phi\left(\mathrm{K}_{\mathrm{TJ}}\right)=\left[\mathrm{c}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\kappa_{\mathrm{T}}(\mathrm{aq}) \, \rho_{1}^{*}(\ell)-\kappa_{\mathrm{T} 1}^{*}(\ell) \, \rho(\mathrm{aq})\right] \\ +\kappa_{\mathrm{T} 1}^{*}(\ell) \, \mathrm{M}_{\mathrm{j}} / \rho_{1}^{*}(\ell) \end{gathered}\]Also \[\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\left[\mathrm{c}_{\mathrm{j}}\right]^{-1} \,\left[\kappa_{\mathrm{T}}(\mathrm{aq})-\kappa_{\mathrm{Tl}}^{*}(\ell)\right]+\kappa_{\mathrm{T} 1}^{*}(\ell) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\]The latter four equations are thermodynamically correct, no assumption being made in their derivation.In 1933, Gucker reviewed the direct determination of compressibilities of solutions leading to apparent molar compressions of solutes in aqueous solution calculated using equation (k). \[\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\frac{\kappa_{\mathrm{T}}(\mathrm{aq})}{\rho(\mathrm{aq})} \,\left[\frac{1}{\mathrm{~m}_{\mathrm{j}}}+\mathrm{M}_{\mathrm{j}}\right]-\frac{\kappa_{1}^{*}(\ell)}{\rho_{1}^{*}(\ell)} \, \frac{1}{\mathrm{~m}_{\mathrm{j}}}\]Compressibilties of solutions were directly determined by measuring the sensitivity of \(\mathrm{V}(\mathrm{aq})\) to an increase in pressure. Gucker showed that for aqueous salt solutions, \(\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)\) is negative and a linear function of \(\left(\mathrm{c}_{\mathrm{j}}\right)^{1 / 2}\). Moreover the limiting value, \(\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)^{\infty}\) is an additive property of \(\phi\left(\mathrm{K}_{\mathrm{T}}-1 \mathrm{ion}\right)^{\infty}\)A useful approximation is that for dilute solutions at constant \(\mathrm{T}\) and \(\mathrm{p}\) containing a neutral solute \(j\), \(\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)\) is linear function of molaity \(\mathrm{m}_{j}\). \[\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\mathrm{e} \, \mathrm{m}_{\mathrm{j}}+\mathrm{f}\]Hence, \[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{K}_{\mathrm{T}_{\mathrm{j}}}(\mathrm{aq})=\mathrm{K}_{\mathrm{Tj}}^{\infty}(\mathrm{aq})=\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)^{\infty}\]We identify ‘f’ in equation (l) as the limiting isothermal apparent molar isothermal compression of solute \(j\) in solution (at equilibrium).Footnotes \(\phi\left(\mathrm{K}_{\mathrm{Tj}}\right)=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1} \mathrm{~Pa}^{-1}\right] ; \mathrm{K}_{\mathrm{Tl}}^{*}=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1} \mathrm{~Pa}^{-1}\right]\); \(\mathrm{K}_{\mathrm{T}}(\mathrm{aq})=\left[\mathrm{m}^{3} \mathrm{~Pa}^{-1}\right]\)For the solution, \(\kappa_{\mathrm{T}}=\mathrm{K}_{\mathrm{T}}(\mathrm{aq}) / \mathrm{V}(\mathrm{aq})=\left[\mathrm{Pa}^{-1}\right]\)For the solvent, \(\kappa_{\mathrm{T} 1}^{*}(\ell)=\mathrm{K}_{\mathrm{T} 1}^{*}(\ell) / \mathrm{V}_{1}^{*}(\ell)=\left[\mathrm{Pa}^{-1}\right]\) Isothermal compressions have units ‘volume per unit of pressure’ whereas compressibilities have units of ‘reciprocal pressure’. \(\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)\) is an apparent molar isothermal compression on the grounds that the units of this property are \(\left[\mathrm{m}^{3} \mathrm{~mol}^{-1} \mathrm{~Pa}^{-1}\right]\). Some reports use the term ‘apparent molar isothermal compressibility’. For an aqueous solution at fixed temperature and pressure prepared using \(\mathrm{n}_{1}\) moles of water and \(\mathrm{n}_{j}\) moles of solute \(j\), \[\mathrm{V}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\]Hence with respect to an equilibrium displacement (i.e. at \(\mathrm{A} = 0\)) at defined temperature, \[(\partial \mathrm{V}(\mathrm{aq}) / \partial \mathrm{p})_{\mathrm{T}}=\mathrm{n}_{1} \,\left(\partial \mathrm{V}_{1}^{*}(\ell) / \partial \mathrm{p}\right)_{\mathrm{T}}+\mathrm{n}_{\mathrm{j}} \,\left(\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right) / \partial \mathrm{p}\right)_{\mathrm{T}}\]Hence \[(\partial \mathrm{V}(\mathrm{aq}) / \partial \rho)_{\mathrm{T}}=\mathrm{n}_{1} \,\left(\partial \mathrm{V}_{1}^{*}(\ell) / \partial \mathrm{p}\right)_{\mathrm{T}}-\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)\]For the solution the (equilibrium) isothermal compressibility, \[\kappa_{\mathrm{T}}(\mathrm{aq})=-\frac{1}{\mathrm{~V}(\mathrm{aq})} \,\left(\frac{\partial \mathrm{V}(\mathrm{aq})}{\partial \mathrm{p}}\right)_{\mathrm{T}}\]Similarly for the pure solvent, \[\kappa_{\mathrm{T} 1}^{*}(\ell)=-\frac{1}{\mathrm{~V}_{1}^{*}(\ell)} \,\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{T}}\]Hence from equation (c), \[\mathrm{V}(\mathrm{aq}) \, \mathrm{K}_{\mathrm{T}}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell) \, \mathrm{K}_{\mathrm{T} 1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{K}_{\mathrm{Tj}}\right)\]We use equation (as) for \(\mathrm{V}(\mathrm{aq})\) in conjunction with equation (f). \[\begin{aligned} \kappa_{\mathrm{T}}(\mathrm{aq}) \,\left[\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\right]=\\ \mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell) \, \mathrm{K}_{\mathrm{T} 1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right) \end{aligned}\]\[\begin{aligned} &\phi\left(\mathrm{K}_{\mathrm{T} \mathrm{j}}\right)= \\ &{\left[\kappa_{\mathrm{T}}(\mathrm{aq}) \, \mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)\right] / \mathrm{n}_{\mathrm{j}}-\left[\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell) \, \mathrm{K}_{\mathrm{T} 1}^{*}(\ell) / \mathrm{n}_{\mathrm{j}}\right]+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \kappa_{\mathrm{T}}(\mathrm{aq})} \end{aligned}\]But, \(\mathrm{V}_{1}^{*}(\ell)=\mathrm{M}_{1} / \rho_{1}^{*}(\ell)\) Then, \[\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\kappa_{\mathrm{T}}(\mathrm{aq}) \, \frac{\mathrm{n}_{1} \, \mathrm{M}_{1}}{\rho_{1}^{*}(\ell) \, \mathrm{n}_{\mathrm{j}}}-\frac{\mathrm{n}_{1} \, \mathrm{M}_{1}}{\rho_{1}^{*}(\ell) \, \mathrm{n}_{\mathrm{j}}} \, \mathrm{K}_{\mathrm{T} 1}^{*}(\ell)+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \mathrm{K}_{\mathrm{T}}(\mathrm{aq})\]Molality \(\begin{aligned} &\mathrm{m}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} / \mathrm{n}_{\mathrm{l}} \, \mathrm{M}_{\mathrm{l}} \\ &\phi\left(\mathrm{K}_{\mathrm{Tj}}\right)=\frac{\kappa_{\mathrm{T}}(\mathrm{aq})}{\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)}-\frac{\kappa_{\mathrm{T} 1}^{*}(\ell)}{\rho_{1}^{*}(\ell) \, \mathrm{m}_{\mathrm{j}}}+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \kappa_{\mathrm{T}}(\mathrm{aq}) \end{aligned}\)Or, \[\phi\left(\mathrm{K}_{\mathrm{TJ}}\right)=\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\kappa_{\mathrm{T}}(\mathrm{aq})-\kappa_{\mathrm{T} 1}^{*}(\ell)\right]+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \kappa_{\mathrm{T}}(\mathrm{aq})\]Using again equation (a) to substitute for \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)\), \[\begin{aligned} \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \, & {\left[\kappa_{\mathrm{T}}(\mathrm{aq})-\kappa_{\mathrm{T} 1}^{*}(\ell)\right] } \\ &+\kappa_{\mathrm{T}}(\mathrm{aq}) \,\left[\frac{\mathrm{V}(\mathrm{aq})-\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)}{\mathrm{n}_{\mathrm{j}}}\right] \end{aligned}\]With \(\mathrm{c}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} / \mathrm{V}(\mathrm{aq})\), \[\begin{aligned} \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\kappa_{\mathrm{T}}(\mathrm{aq})-\kappa_{\mathrm{T} 1}^{*}(\ell)\right] \\ &+\kappa_{\mathrm{T}}(\mathrm{aq}) \,\left[\frac{1}{\mathrm{c}_{\mathrm{j}}}-\frac{\mathrm{n}_{1} \, \mathrm{M}_{1}}{\rho_{1}^{*}(\ell) \, \mathrm{n}_{\mathrm{j}}}\right] \end{aligned}\]But \(\frac{1}{c_{j}}=\frac{M_{j}}{\rho(a q)}+\frac{1}{m_{j} \, \rho(a q)}\) Hence, \[\begin{aligned} \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\kappa_{\mathrm{T}}(\mathrm{aq})-\kappa_{\mathrm{Tl}}^{*}(\ell)\right] \\ &+\kappa_{\mathrm{T}}(\mathrm{aq}) \,\left[\frac{\mathrm{M}_{\mathrm{j}}}{\rho(\mathrm{aq})}+\frac{1}{\mathrm{~m}_{\mathrm{j}} \, \rho(\mathrm{aq})}-\frac{\mathrm{n}_{1} \, \mathrm{M}_{\mathrm{1}}}{\rho_{1}^{*}(\ell) \, \mathrm{n}_{\mathrm{j}}}\right] \end{aligned}\]\[\begin{gathered} \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\left[\mathrm{m}_{\mathrm{j}} \, \rho(\mathrm{aq}) \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\kappa_{\mathrm{T}}(\mathrm{aq}) \, \rho_{1}^{*}(\ell)-\kappa_{\mathrm{T} 1}^{*}(\ell) \, \rho(\mathrm{aq})\right] \\ +\frac{\kappa_{\mathrm{T}}(\mathrm{aq}) \, \mathrm{M}_{\mathrm{j}}}{\rho(\mathrm{aq})} \end{gathered}\]We start with equation (f). \[\mathrm{V}(\mathrm{aq}) \, \mathrm{K}_{\mathrm{T}}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell) \, \kappa_{\mathrm{T} 1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)\]Mass of solution, \[V(a q) \, \rho(a q)=n_{1} \, M_{1}+n_{j} \, M_{j}\]Or, \[\mathrm{n}_{1}=\left[\mathrm{V}(\mathrm{aq}) \, \rho(\mathrm{aq})-\mathrm{n}_{\mathrm{j}} \, \mathrm{M}_{\mathrm{j}}\right] / \mathrm{M}_{1}\]We combine equations (f) and (q). \[\begin{aligned} &\mathrm{V}(\mathrm{aq}) \, \kappa_{\mathrm{T}}(\mathrm{aq})= \\ &\quad \mathrm{V}_{1}^{*}(\ell) \, \kappa_{\mathrm{T} 1}^{*}(\ell) \,\left[\frac{\mathrm{V}(\mathrm{aq}) \, \rho(\mathrm{aq})-\mathrm{n}_{\mathrm{j}} \, \mathrm{M}_{\mathrm{j}}}{\mathrm{M}_{1}}\right]+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right) \end{aligned}\]With \(c_{j}=n_{j} / V(a q)\) \[\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\frac{\kappa_{\mathrm{T}}(\mathrm{aq})}{\mathrm{c}_{\mathrm{j}}}-\frac{\mathrm{V}_{1}^{*}(\ell) \, \kappa_{\mathrm{T} 1}^{*}(\ell) \, \rho(\mathrm{aq})}{\mathrm{c}_{\mathrm{j}} \, \mathrm{M}_{1}}+\frac{\mathrm{V}_{1}^{*}(\ell) \, \kappa_{\mathrm{T} 1}^{*}(\ell) \, \mathrm{M}_{\mathrm{j}}}{\mathrm{M}_{1}}\]Hence, \[\begin{array}{r} \phi\left(\mathrm{K}_{\mathrm{Tj}}\right)=\left[\mathrm{c}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\kappa_{\mathrm{T}}(\mathrm{aq}) \, \rho_{1}^{*}(\ell)-\kappa_{\mathrm{T} 1}^{*}(\ell) \, \rho(\mathrm{aq})\right] \\ +\kappa_{\mathrm{Tl}}^{*}(\ell) \, \mathrm{M}_{\mathrm{j}} / \rho_{1}^{*}(\ell) \end{array}\]Again from equation (j) \[\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\mathrm{K}_{\mathrm{T}}(\mathrm{aq})-\kappa_{\mathrm{T} 1}^{*}(\ell)\right]+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \kappa_{\mathrm{T}}(\mathrm{aq})\]Hence, \[\begin{aligned} &\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)= \\ &{\left[\frac{\rho_{1}^{*}(\ell)}{\mathrm{c}_{\mathrm{j}}}-\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \rho_{1}^{*}(\ell)\right] \,\left[\rho_{1}^{*}(\ell)\right]^{-1} \,\left[\kappa_{\mathrm{T}}(\mathrm{aq})-\kappa_{\mathrm{T} 1}^{*}(\ell)\right]} \\ &\quad+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \kappa_{\mathrm{T}}(\mathrm{aq}) \end{aligned}\]Or \[\phi\left(\mathrm{K}_{\mathrm{Tj}}\right)=\left[\mathrm{c}_{\mathrm{j}}\right]^{-1} \,\left[\mathrm{K}_{\mathrm{T}}(\mathrm{aq})-\kappa_{\mathrm{T} 1}^{*}(\ell)\right]+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \kappa_{\mathrm{T} 1}^{*}(\ell)\] F. T. Gucker, J.Am.Chem.Soc.,1933,55,2709. F. T. Gucker, Chem. Rev.,1933,13,111. From equation (h), \[\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\frac{\kappa_{\mathrm{T}}(\mathrm{aq}) \, \rho_{1}^{*}(\ell)}{\mathrm{m}_{\mathrm{j}} \, \rho(\mathrm{aq}) \, \rho_{1}^{*}(\ell)}-\frac{\kappa_{\mathrm{T} 1}^{*}(\ell) \, \rho(\mathrm{aq})}{\mathrm{m}_{\mathrm{j}} \, \rho(\mathrm{aq}) \, \rho_{1}^{*}(\ell)}+\frac{\kappa_{\mathrm{T}}(\mathrm{aq}) \, \mathrm{M}_{\mathrm{j}}}{\rho(\mathrm{aq})}\]\[\phi\left(\mathrm{K}_{\mathrm{T}}\right)=\frac{\kappa_{\mathrm{T}}(\mathrm{aq})}{\rho(\mathrm{aq})} \,\left[\frac{1}{\mathrm{~m}_{\mathrm{j}}}+\mathrm{M}_{\mathrm{j}}\right]-\frac{\kappa_{\mathrm{Tl}}^{*}(\ell)}{\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{\star}(\ell)}\]This page titled 1.7.19: Compressions- Isothermal- Solutions- Apparent Molar- Determination is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.7.20: Compressions- Isothermal- Salt Solutions

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/1.7.20%3A_Compressions-_Isothermal-_Salt_Solutions

In 1933 Gucker reviewed attempts to measure the apparent isothermal molar compressions of salts in aqueous solution, these attempts dating back to the earliest reliable measurements by Rontgen and Schneider in 1886 and 1887. Gucker showed that for several aqueous salt solutions the apparent isothermal molar compression, \(\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)\) is a linear function of the square root of the salt concentration. \[\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)^{\infty}+\mathrm{a} \, \mathrm{c}_{\mathrm{j}}^{1 / 2}\]This general equation holds for \(\mathrm{CaCl}_{2}(\mathrm{aq})\) at 60 Celsius. In general terms, \(\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)\) for salts is negative becoming less negative as the salt concentration increases. Gibson described an interesting approach which characterises salt solutions in terms of effective pressures, \(\mathrm{p}_{\mathrm{c}}\) exerted by the salt on the solvent. This effective pressure is expressed as a linear function of the product of salt and solvent concentrations. The constant of proportionality is characteristic of the salt. Leyendekkers based an analysis using the Tammann-Tait-Gibson (TTG) model, on the assertion that solutes, salts and organic solutes, exert an excess pressure on water in aqueous solution. The TTG approach described by Leyendekkers is intuitively attractive but the analysis is based on an extra - thermodynamic assumption. Calculation of an excess pressure requires an estimate of the volume of solute molecules, \(\phi_{j}\) in solution. If this property is independent of solute molality \(\mathrm{m}_{j}\), the dependence of the volume of a solution (in \(1 \mathrm{~kg}\) of water) on solute molality is described by the dependence of the ‘partial molar volume' of water. The difference between \(\left[\mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)-\mathrm{m}_{\mathrm{j}} \, \phi(\mathrm{Vj})\right]\) and \(\mathrm{M}_{1}^{-1} \, \mathrm{V}_{1}^{*}(\ell)\) is understood in terms of an effective pressure on the solvent. The assumptions underlying this calculation are not trivial. Furthermore from a thermodynamic viewpoint, the pressure is the same in every volume element of a solution.Footnotes F. T. Gucker, J. Am. Chem.Soc.,1933,55,2709. F. T. Gucker, Chem.Rev.,1933,13,111. F. T. Gucker, F. W. Lamb, G. A. Marsh and R. M. Haag, J.Am. Chem. Soc.,1950,72,310. W. C. Rontgen and J. Schneider, Wied. Ann., 1886,29,165;1887,31,36. R. E. Gibson, J.Am.Chem.Soc.,1934,56,4.;1935,57,284. J. V. Leyendekkers, J. Chem. Soc. Faraday Trans.1,1981,77,1529; 1982,78,357; 1988,84, 397,1653. J. V. Leyendekkers, Aust. J. Chem.,1981,34,1785. M. J. Blandamer, J. Burgess and A. Hakin, J. Chem. Soc. Faraday Trans.1,1986, 82,3681.This page titled 1.7.20: Compressions- Isothermal- Salt Solutions is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.7.21: Compressions- Isothermal- Binary Aqueous 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/1.7.21%3A_Compressions-_Isothermal-_Binary_Aqueous_Mixtures

A given binary aqueous mixture is prepared using \(\mathrm{n}_{1}\) moles of water (\(\ell\)) and \(\mathrm{n}_{2}\) moles of liquid 2. The volume of the mixture, \(\mathrm{V}(\mathrm{mix})\) is given by equation (a) (at fixed temperature and pressure). \[\mathrm{V}(\operatorname{mix})=\mathrm{n}_{1} \, \mathrm{V}_{1}(\operatorname{mix})+\mathrm{n}_{2} \, \mathrm{V}_{2}(\mathrm{mix})\]The mixture is perturbed by a change in pressure at fixed temperature along an equilibrium pathway where the affinity for spontaneous change remains at zero. \[\left(\frac{\partial \mathrm{V}(\mathrm{mix})}{\partial \mathrm{p}}\right)_{\mathrm{T}}=\mathrm{n}_{1} \,\left(\frac{\partial \mathrm{V}_{1}(\mathrm{mix})}{\partial \mathrm{p}}\right)_{\mathrm{T}}+\mathrm{n}_{2} \,\left(\frac{\partial \mathrm{V}_{2}(\mathrm{mix})}{\partial \mathrm{p}}\right)_{\mathrm{T}}\]The isothermal compression of the mixture \([\partial \mathrm{V}(\mathrm{mix}) / \partial \mathrm{p}]_{\mathrm{T}}\) is an extensive property. The partial differentials \(\left[\partial V_{1}(\operatorname{mix}) / \partial p\right]_{T}\) and \(\left[\partial V_{2}(\operatorname{mix}) / \partial \mathrm{p}\right]_{\mathrm{T}}\) are intensive properties. There is merit in defining an intensive molar compression using equation (c). \[\mathrm{K}_{\mathrm{Tm}}=\frac{\mathrm{K}_{\mathrm{T}}(\operatorname{mix})}{\left(\mathrm{n}_{1}+\mathrm{n}_{2}\right)}=-\left(\mathrm{n}_{1}+\mathrm{n}_{2}\right)^{-1} \,[\partial \mathrm{V}(\operatorname{mix}) / \partial \mathrm{p}]_{\mathrm{T}}\]By definition, \[\mathrm{K}_{\mathrm{T} 1}(\operatorname{mix})=-\left[\partial \mathrm{V}_{1}(\operatorname{mix}) / \partial \mathrm{p}\right]_{\mathrm{T}}\]And \[\mathrm{K}_{\mathrm{T} 2}(\operatorname{mix})=-\left[\partial \mathrm{V}_{2}(\operatorname{mix}) / \partial \mathrm{p}\right]_{\mathrm{T}}\]Hence \[\mathrm{K}_{\mathrm{Tm}}(\operatorname{mix})=\mathrm{x}_{1} \, \mathrm{K}_{\mathrm{T} 1}(\operatorname{mix})+\mathrm{x}_{2} \, \mathrm{K}_{\mathrm{T} 2}(\operatorname{mix})\]For an ideal binary mixture, \[\mathrm{V}(\operatorname{mix} ; \mathrm{id})=\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{n}_{2} \, \mathrm{V}_{2}^{*}(\ell)\]\(\mathrm{V}_{1}^{*}(\ell)\) and \(\mathrm{V}_{2}^{*}(\ell)\) are the molar volumes of the pure liquids at the same \(\mathrm{T}\) and \(\mathrm{p}\). Therefore, following the argument outlined above, \[\mathrm{K}_{\mathrm{Tm}}(\mathrm{id})=\mathrm{x}_{1} \, \mathrm{K}_{\mathrm{T} 1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{K}_{\mathrm{T} 2}^{*}(\ell)\]By definition, \[\mathrm{K}_{\mathrm{T}}^{\mathrm{E}}(\operatorname{mix})=\mathrm{K}_{\mathrm{Tm}}(\operatorname{mix})-\mathrm{K}_{\mathrm{Tm}}(\operatorname{mix} ; \mathrm{id})\]Hence the excess molar compression is given by equation (j). \[\mathrm{K}_{\mathrm{Tm}}^{\mathrm{E}}(\operatorname{mix})=\mathrm{x}_{1} \,\left[\mathrm{K}_{\mathrm{T} 1}(\operatorname{mix})-\mathrm{K}_{\mathrm{T} 1}^{*}(\ell)\right]+\mathrm{x}_{2} \,\left[\mathrm{K}_{\mathrm{T} 2}(\operatorname{mix})-\mathrm{K}_{\mathrm{T} 2}^{*}(\ell)\right]\]This page titled 1.7.21: Compressions- Isothermal- Binary Aqueous Mixtures is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.7.22: Compressions- Isothermal- Liquid Mixtures Binary- Compressibilities

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/1.7.22%3A_Compressions-_Isothermal-_Liquid_Mixtures_Binary-_Compressibilities

The isothermal compressibility of a given binary liquid mixture having ideal thermodynamic properties is related to the isothermal compressions of the liquid components using equation (a). \[\mathrm{K}_{\mathrm{T}}(\operatorname{mix} ; \mathrm{id})=\frac{\mathrm{K}_{\mathrm{T} 1}^{*}(\ell)+\mathrm{x}_{2} \,\left[\mathrm{K}_{\mathrm{T} 2}^{*}(\ell)-\mathrm{K}_{\mathrm{T} 1}^{*}(\ell)\right]}{\mathrm{V}_{1}^{*}(\ell)+\mathrm{x}_{2} \,\left[\mathrm{V}_{2}^{*}(\ell)-\mathrm{V}_{1}^{*}(\ell)\right]}\]The excess compression for a given binary liquid mixture is defined by equation (b). \[\mathrm{K}_{\mathrm{Tm}}^{\mathrm{E}}=\mathrm{K}_{\mathrm{Tm}}(\mathrm{mix})-\mathrm{K}_{\mathrm{Tn}}(\mathrm{mix} ; \mathrm{id})\]Or, \[\mathrm{K}_{\mathrm{Tm}}^{\mathrm{E}}=\mathrm{K}_{\mathrm{Tm}}(\mathrm{mix})-\left[\mathrm{x}_{1} \, \mathrm{K}_{\mathrm{T} 1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{K}_{\mathrm{T} 2}^{*}(\ell)\right]\]The isothermal compressibilities of ideal and real binary liquid mixtures are defined by equations (d) and (e) respectively. \[\kappa_{\mathrm{T}}(\operatorname{mix} ; \mathrm{id})=-\frac{1}{\mathrm{~V}(\operatorname{mix} ; \mathrm{id})} \,\left(\frac{\partial \mathrm{V}(\mathrm{mix} ; \mathrm{id})}{\partial \mathrm{p}}\right)_{\mathrm{T}}\]\[\kappa_{\mathrm{T}}(\operatorname{mix})=-\frac{1}{\mathrm{~V}(\operatorname{mix})} \,\left(\frac{\partial \mathrm{V}(\operatorname{mix})}{\partial \mathrm{p}}\right)_{\mathrm{T}}\]For a given binary liquid mixture we can define an excess compressibility using equation (f). \[\kappa_{\mathrm{T}}^{\mathrm{E}}=\kappa_{\mathrm{T}}(\operatorname{mix})-\kappa_{\mathrm{T}}(\operatorname{mix} ; \mathrm{id})\]Then \[\begin{aligned} &\kappa_{\mathrm{T}}^{\mathrm{E}}(\mathrm{mix})=-\frac{1}{\mathrm{~V}(\mathrm{mix})} \,\left(\frac{\partial \mathrm{V}(\mathrm{mix})}{\partial \mathrm{p}}\right)_{\mathrm{T}} \\ &+\frac{1}{\left[\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}(\ell)\right]} \,\left(\frac{\partial\left[\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}(\ell)\right]}{\partial \mathrm{p}}\right)_{\mathrm{T}} \end{aligned}\]A similar equation was used by Moelwyn-Hughes and Thorpe. They introduced the concept of a compressibility of the excess volume. \[\Delta \kappa_{\mathrm{T}}(\operatorname{mix})=-\frac{1}{\Delta \mathrm{V}(\operatorname{mix})} \,\left(\frac{\partial \Delta \mathrm{V}(\mathrm{mix})}{\partial \mathrm{p}}\right)_{\mathrm{T}}\]In publications by Prigogine and by Moelwyn-Hughes and Thorpe the analysis was taken a step further to facilitate analysis of experimental results. However approximations were made in both treatments. An exact formulation was given by Missen in terms of volume fractions of both components in the corresponding having ideal thermodynamic properties, \(\phi_{1}(\text { mix;id })\) and \(\phi_{2}(\text { mix;id })\). Hence, \[\kappa_{\mathrm{T}}^{\mathrm{E}}(\operatorname{mix})=-\frac{1}{\mathrm{~V}_{\mathrm{m}}(\operatorname{mix})} \,\left[\left(\frac{\partial \mathrm{V}_{\mathrm{m}}^{\mathrm{E}}}{\partial \mathrm{p}}\right)_{\mathrm{T}}+\mathrm{V}_{\mathrm{m}}^{\mathrm{E}} \, \kappa_{\mathrm{T}}(\operatorname{mix} ; \mathrm{id})\right]\]A partial compressibility was defined by Moelwyn-Hughes. For liquid 1 in a binary liquid mixture at defined \(\mathrm{T}\) and \(\mathrm{p}\), the partial compressibility is defined by equation (j). \[\kappa_{T_{1}}(\operatorname{mix})=-\frac{1}{V_{1}(\operatorname{mix})} \,\left(\frac{\partial V_{1}(\operatorname{mix})}{\partial p}\right)_{T}\]Similarly for component 2, \[\kappa_{\mathrm{T} 2}(\operatorname{mix})=-\frac{1}{\mathrm{~V}_{2}(\operatorname{mix})} \,\left(\frac{\partial \mathrm{V}_{2}(\operatorname{mix})}{\partial \mathrm{p}}\right)_{\mathrm{T}}\]The excess compressibility of a given binary liquid mixture \(\mathrm{K}_{\mathrm{T}}^{\mathrm{E}}(\operatorname{mix})\) was defined in equation (f). Hence, \[\begin{aligned} \kappa_{\mathrm{T}}^{\mathrm{E}}(\operatorname{mix})=& \phi_{1}(\operatorname{mix}) \, \kappa_{\mathrm{T} 1}^{\mathrm{E}}(\operatorname{mix})+\phi_{2}(\operatorname{mix}) \, \kappa_{\mathrm{T} 2}^{\mathrm{E}}(\operatorname{mix}) \\ &+\left[\phi_{1}(\operatorname{mix})-\phi_{1}(\operatorname{mix} ; \mathrm{id})\right] \, \kappa_{\mathrm{T} 1}^{*}(\ell)+\left[\phi_{2}(\operatorname{mix})-\phi_{2}(\operatorname{mix} ; \mathrm{id})\right] \, \kappa_{\mathrm{T} 2}^{*}(\ell) \end{aligned}\]It may be noted that ‘true’ partial properties can also be defined for the isothermal compressibility. Then the properties introduced in equations (j) and (k) would be termed specific partial isothermal compressions.It is also possible to formulate a set of equations incorporating rational activity coefficients for the two components of the binary liquid mixture. We start with the equation for the partial molar volume of component 1. \[V_{1}(\operatorname{mix})=V_{1}^{*}(\ell)+R \, T \,\left(\frac{\partial \ln \left(f_{1}\right)}{\partial p}\right)_{T}\]\[K_{T 1}(\operatorname{mix})=K_{T 1}^{*}(\ell)-R \, T \,\left(\frac{\partial^{2} \ln \left(f_{1}\right)}{\partial p^{2}}\right)_{T}\]Similarly \[\mathrm{K}_{\mathrm{T} 2}(\operatorname{mix})=\mathrm{K}_{\mathrm{T} 2}^{*}(\ell)-\mathrm{R} \, \mathrm{T} \,\left(\frac{\partial^{2} \ln \left(\mathrm{f}_{2}\right)}{\partial \mathrm{p}^{2}}\right)_{\mathrm{T}}\]Therefore \[\mathrm{K}_{\mathrm{Tm}}(\operatorname{mix})=\mathrm{K}_{\mathrm{Tm}}(\operatorname{mix} ; \mathrm{id})-\mathrm{R} \, \mathrm{T} \,\left[\mathrm{x}_{1} \,\left(\frac{\partial^{2} \ln \left(\mathrm{f}_{1}\right)}{\partial \mathrm{p}^{2}}\right)_{\mathrm{T}}+\mathrm{x}_{2} \,\left(\frac{\partial^{2} \ln \left(\mathrm{f}_{2}\right)}{\partial \mathrm{p}^{2}}\right)_{\mathrm{T}}\right]\]The two liquid components are characterised by their molar excess properties. \[\mathrm{K}_{\mathrm{T} 1}^{\mathrm{E}}(\operatorname{mix})=-\mathrm{R} \, \mathrm{T} \,\left(\frac{\partial^{2} \ln \left(\mathrm{f}_{1}\right)}{\partial \mathrm{p}^{2}}\right)_{\mathrm{T}}\]and \[\mathrm{K}_{\mathrm{T} 2}^{\mathrm{E}}(\operatorname{mix})=-\mathrm{R} \, \mathrm{T} \,\left(\frac{\partial^{2} \ln \left(\mathrm{f}_{2}\right)}{\partial \mathrm{p}^{2}}\right)_{\mathrm{T}}\]Therefore \[\mathrm{K}_{\mathrm{Tm}}^{\mathrm{E}}(\mathrm{mix})=-\mathrm{R} \, \mathrm{T} \,\left[\mathrm{x}_{1} \,\left(\frac{\partial^{2} \ln \left(\mathrm{f}_{1}\right)}{\partial \mathrm{p}^{2}}\right)_{\mathrm{T}}+\mathrm{x}_{2} \,\left(\frac{\partial^{2} \ln \left(\mathrm{f}_{2}\right)}{\partial \mathrm{p}^{2}}\right)_{\mathrm{T}}\right]\]Also \[\mathrm{K}_{\mathrm{T} 1}^{\mathrm{E}}=-\left(\frac{\partial \mathrm{V}_{1}^{\mathrm{E}}}{\partial \mathrm{p}}\right)_{\mathrm{T}} \text { and } \mathrm{K}_{\mathrm{T} 2}^{\mathrm{E}}=-\left(\frac{\partial \mathrm{V}_{2}^{\mathrm{E}}}{\partial \mathrm{p}}\right)_{\mathrm{T}}\]In other words \[\mathrm{K}_{\mathrm{Tm}}^{\mathrm{E}}=-\left(\frac{\partial \mathrm{V}_{\mathrm{m}}^{\mathrm{E}}}{\partial \mathrm{p}}\right)_{\mathrm{T}}\]Isothermal compressions of liquid mixtures can be directly measured. Hamann and Smith report measurements using binary liquid mixtures at \(303 \mathrm{~K}\) and two pressures. Hamann and Smith define excess isothermal molar compressions \(\mathrm{K}_{\mathrm{T}}^{\mathrm{E}}(\phi)\) in terms of volume fraction weighted isothermal compressions of the pure liquids. The volume fractions are defined as follows. \[\phi_{1}=\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell) /\left[\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}(\ell)\right]\]\[\phi_{2}=x_{2} \, V_{2}^{*}(\ell) /\left[x_{1} \, V_{1}^{*}(\ell)+x_{2} \, V_{2}^{*}(\ell)\right]\]Then \[\mathrm{K}_{\mathrm{T}}^{\mathrm{E}}(\phi)=\mathrm{K}_{\mathrm{Tm}}(\mathrm{mix})-\left[\phi_{1} \, \mathrm{K}_{\mathrm{T} 1}^{*}(\ell)+\phi_{2} \, \mathrm{K}_{\mathrm{T} 2}^{*}(\ell)\right]\]For most binary aqueous mixtures \(\mathrm{K}_{\mathrm{T}}^{\mathrm{E}}(\phi)\) is negative, plots of \(\mathrm{K}_{\mathrm{T}}^{\mathrm{E}}(\phi)\) against \(\phi_{2}\) being smooth curves. The minima in aqueous mixtures containing \(\mathrm{THF}\) and propanone the minima are near 0.4 and 0.6 respectively.Footnotes For a binary liquid mixture having ideal thermodynamic properties, \[\mathrm{V}_{\mathrm{m}}(\operatorname{mix} ; \mathrm{id})=\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}(\ell)\]Then \[\mathrm{K}_{\mathrm{Tm}}(\operatorname{mix} ; \mathrm{id})=\mathrm{x}_{1} \, \mathrm{K}_{\mathrm{T} 1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{K}_{\mathrm{T} 2}^{*}(\ell)\]But \[\kappa_{\mathrm{T}}(\operatorname{mix} ; \mathrm{id})=\frac{\mathrm{K}_{\mathrm{Tm}}(\operatorname{mix} ; \mathrm{id})}{\mathrm{V}_{\mathrm{m}}(\operatorname{mix} ; \mathrm{id})}\]Then, \[\kappa_{\mathrm{T}}(\operatorname{mix} ; \mathrm{id})=\frac{\mathrm{K}_{\mathrm{T} 1}^{*}(\ell)+\mathrm{x}_{2} \,\left[\mathrm{K}_{\mathrm{T} 2}^{*}(\ell)-\mathrm{K}_{\mathrm{T} 1}^{*}(\ell)\right]}{\mathrm{V}_{1}^{*}(\ell)+\mathrm{x}_{2} \,\left[\mathrm{V}_{2}^{*}(\ell)-\mathrm{V}_{1}^{*}(\ell)\right]}\] I. Prigogine, The Molecular Theory of Solutions, North Holland, Amsterdam, 1957, p.18. E. A. Moelwyn-Hughes and P. L. Thorpe, Proc. R. Soc. London, Ser. A,1964,278A, 574. R. W. Missen, Ind. Eng. Chem. Fundam., 1969,8,81. E. A. Moelwyn-Hughes, Physical Chemistry, Pergamon, London, 2nd. Edn., 1965, .817 J. C. R. Reis, J. Chem. Soc. Faraday Trans.,1998,94,2385. J. E. Stutchbury, Aust. J. Chem.,1971,24,2431. S. D. Hamann and F. Smith, Aust. J. Chem.,1971,24,2431. For a detailed report on the properties of liquid mixtures see G. M. Schneider, Pure Appl. Chem.,1983,55,479 ; and references therein.This page titled 1.7.22: Compressions- Isothermal- Liquid Mixtures Binary- Compressibilities is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.8.1: Enthalpies and Gibbs Energies

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By definition, the Gibbs energy, \[\mathrm{G}=\mathrm{U}+\mathrm{p} \, \mathrm{V}-\mathrm{T} \, \mathrm{S}\]Enthalpy, \[\mathrm{H}=\mathrm{U}+\mathrm{p} \, \mathrm{V}\]Combination of equations (a) and (b) yields an important equation relating Gibbs energy \(\mathrm{G}\) and enthalpy \(\mathrm{H}\). \[\mathrm{G}=\mathrm{H}-\mathrm{T} \, \mathrm{S}\]Just as we can never know the thermodynamic energy of a system, so we can never know the enthalpy. Consequently analysis of enthalpies is more complicated than analysis of volumetric properties, bearing in mind that the density of a solution (liquid) can be accurately measured. Differences are therefore emphasised in the context of enthalpies.A differential change in Gibbs energy at constant temperature is related to the changes in enthalpy \(\mathrm{dH}\) and entropy, \(\mathrm{dS}\). \[\mathrm{dG}=\mathrm{dH}-\mathrm{T} \, \mathrm{dS}\]For an isothermal process from state I to state II, the change in Gibbs energy \(\Delta \mathrm{G}\) is given by equation (e). \[\Delta \mathrm{G}=\Delta \mathrm{H}-\mathrm{T} \, \Delta \mathrm{S}\]Equation (e) signals how enthalpy and entropy changes determine the change in Gibbs energy.A closed system at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\) is prepared using \(\mathrm{n}_{1}\) moles of solvent (water) and \(\mathrm{n}_{j}\) moles of solute-\(j\). The system is at equilibrium such that the composition/organisation is represented by \(\xi^{\mathrm{eq}}\) and the affinity for spontaneous change is zero. Using an over-defined representation we define the system as follows. \[\mathrm{G}^{\mathrm{eq}}=\mathrm{G}^{\mathrm{eq}}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}, \xi^{\mathrm{eq}}, \mathrm{A}=0\right]\]Under such circumstances the Gibbs energy \(\mathrm{G}\) is a minimum \(\mathrm{G}^{\mathrm{eq}}\) when plotted as a function of \(\xi\). The enthalpy of this system can be defined using a similar equation. \[\mathrm{H}^{\mathrm{eq}}=\mathrm{H}^{\mathrm{eq}}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}, \xi^{\mathrm{eq}}, \mathrm{A}=0\right]\]It is unlikely that \(\mathrm{H}^{\mathrm{eq}}\) corresponds to a minimum in the plot of enthalpy \(\mathrm{H}\) against \(\xi\). Indeed the same comment applies to the entropy \(\mathrm{S}^{\mathrm{eq}}\); \[\mathrm{S}^{\mathrm{eq}}=\mathrm{S}^{\mathrm{eq}}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}, \xi^{\mathrm{eq}}, \mathrm{A}=0\right]\]The plots showing the product \(\mathrm{T} \, \mathrm{S}\) and \(\mathrm{H}\) against \(\xi\) may not show extrema though taken together they produce a minimum in \(\mathrm{G}\) at \(\xi^{\mathrm{eq}}\). \[\mathrm{G}^{e q}=\mathrm{H}^{e q}-\mathrm{T} \, \mathrm{S}^{e q}\]This page titled 1.8.1: Enthalpies and Gibbs Energies is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.8.2: Enthalpy

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There is considerable merit in identifying an extensive property of a closed system called the enthalpy, \(\mathrm{H}\). The enthalpy of a closed system is a state variable and defined by equation (a). \[\mathrm{H}=\mathrm{U}+\mathrm{p} \, \mathrm{V}\]We identify a given state by the symbol I having enthalpy \(\mathrm{H}[\mathrm{I}]\), energy \(\mathrm{U}[\mathrm{I}]\) and volume \(\mathrm{V}[\mathrm{I}]\) at pressure \(\mathrm{p}\). \[\mathrm{H}[\mathrm{I}]=\mathrm{U}[\mathrm{I}]+\mathrm{p} \, \mathrm{V}[\mathrm{I}]\]This system is displaced to a neighbouring state such that the differential change in enthalpy is \(\mathrm{dH}\). Using equation (a), \[\mathrm{dH}=\mathrm{dU}+\mathrm{p} \, \mathrm{dV}+\mathrm{V} \, \mathrm{dp}\]But according to the first law of thermodynamics, the differential change in thermodynamic energy \(\mathrm{dU}\) is given by ‘\(q-p \, d V\)’ where \(\mathrm{q}\) is the heat accompanying the change. Then, \[\mathrm{dH}=\mathrm{q}-\mathrm{p} \, \mathrm{dV}+\mathrm{p} \, \mathrm{dV}+\mathrm{V} \, \mathrm{dp}\]or, \[\mathrm{dH}=\mathrm{q}+\mathrm{V} \, \mathrm{dp}\]At constant pressure, \[\mathrm{dH}=\mathrm{q}\]For a change from state I to state II the change in enthalpy is given by equation (g). \[\Delta \mathrm{H}=\int_{\mathrm{I}}^{\mathrm{II}} \mathrm{dH}=\mathrm{H}(\mathrm{II})-\mathrm{H}(\mathrm{I})=\mathrm{q}\]In equation (g) we replace the integral of dH by the difference \(\mathrm{H}(\mathrm{II}) - \mathrm{H}(\mathrm{I})\) because enthalpy is a state variable and so \(\Delta \mathrm{H}\) is independent of the path between the two states and hence so is \(\mathrm{q}\). In liquid solutions, the recorded heat is also independent of the rate of change in chemical composition between state I and state II.This page titled 1.8.2: Enthalpy is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.8.3: Enthalpy- Thermodynamic Potential

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The enthalpy \(\mathrm{H}\) of a closed system is related by definition to the thermodynamic energy \(\mathrm{U}\); \(\mathrm{H}=\mathrm{U}+\mathrm{p} \, \mathrm{V}\). But \[\mathrm{dH}=\mathrm{q}+\mathrm{V} \, \mathrm{dp}\]From the second law of thermodynamics, \[\mathrm{T} \, \mathrm{dS}=\mathrm{q}+\mathrm{A} \, \mathrm{d} \xi ; \quad \mathrm{A} \, \mathrm{d} \xi \geq 0\]Then \[\mathrm{dH}=\mathrm{T} \, \mathrm{dS}+\mathrm{V} \, \mathrm{dp}-\mathrm{A} \, \mathrm{d} \xi ; \mathrm{A} \, \mathrm{d} \xi \geq 0\]Thus all spontaneous processes at constant entropy and pressure (i.e. isentropic and isobaric) lower the enthalpy of a closed system. This conclusion finds application in acoustics where the changes in a system perturbed by a travelling sound wave are discussed in terms of changes in enthalpy at constant entropy and pressure. Confining our attention to systems either at equilibrium (i.e. \(\mathrm{A} = 0\)) or at fixed \(\xi\), two key relationships follow from equation (c). \[\mathrm{T}=(\partial \mathrm{H} / \partial \mathrm{S})_{\mathrm{p}}\]and \[\mathrm{V}=(\partial \mathrm{H} / \partial \mathrm{p})_{\mathrm{S}}\]In these terms the extensive variable, volume, is given by theThis page titled 1.8.3: Enthalpy- Thermodynamic Potential is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.8.4: Enthalpy- Solutions- Partial Molar Enthalpies

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The enthalpy of a solution containing n1 moles of water and nj moles of solute, chemical substance j, is defined by the independent variables, \(\mathrm{T}\), \(\mathrm{p}\), \(\mathrm{n}_{1}\) and \(\mathrm{n}_{j}\). \[\mathrm{H}=\mathrm{H}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}\right]\]where, \[\mathrm{H}=\mathrm{n}_{1} \, \mathrm{H}_{1}(\mathrm{aq})+\mathrm{n}_{\mathrm{j}} \, \mathrm{H}_{\mathrm{j}}(\mathrm{aq})\]Here \(\mathrm{H}_{1}(\mathrm{aq})\) and \(\mathrm{H}_{j}(\mathrm{aq})\) are the partial molar enthalpies of water and solute \(j\) in the solution. \[\mathrm{H}_{1}(\mathrm{aq})=\left(\partial \mathrm{H} / \partial \mathrm{n}_{1}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{j})}\]\[\mathrm{H}_{\mathrm{j}}(\mathrm{aq})=\left(\partial \mathrm{H} / \partial \mathrm{n}_{\mathrm{j}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}}\]For a solution prepared using \(1 \mathrm{~kg}\) of solvent, water and \(\mathrm{m}_{j}\) moles of solute \(j\), \[\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{H}_{1}(\mathrm{aq})+\mathrm{m}_{\mathrm{j}} \, \mathrm{H}_{\mathrm{j}}(\mathrm{aq})\]The chemical potential of the solvent in an aqueous solution is related to the molality of solute \(j\), \(\mathrm{m}_{j}\) using equation (f) where \(\phi\) is the practical osmotic coefficient, a property of the solvent. \[\mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\lambda)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\]The chemical potential and partial molar enthalpy are linked using the Gibbs-Helmholtz equation such that at fixed pressure, \(\mathrm{d}\left(\mu_{1}(\mathrm{aq}) / \mathrm{T}\right) / \mathrm{dT}=-\mathrm{H}_{1}(\mathrm{aq}) / \mathrm{T}^{2}\). Hence \[\mathrm{H}_{1}(\mathrm{aq})=\mathrm{H}_{1}^{*}(\lambda)+\mathrm{R} \, \mathrm{T}^{2} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,(\mathrm{d} \phi / \mathrm{dT})_{\mathrm{p}}\]By definition the practical osmotic coefficient is unity for ideal solutions at all \(\mathrm{T}\) and \(\mathrm{p}\). Then the partial molar enthalpy of the solvent in an ideal solution, \[\mathrm{H}_{1}(\mathrm{aq}, \mathrm{id})=\mathrm{H}_{1}^{*}(\lambda)\]The definition of \(\phi\) requires that \(\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{H}_{1}(\mathrm{aq})\) equals \(\mathrm{H}_{1}^{*}(\lambda)\). We express the difference between the partial molar enthalpies of the solvent in real and ideal solutions using a relative (partial) molar enthalpy, \(\mathrm{L}_{1}(\mathrm{aq})\). \[\mathrm{L}_{1}(\mathrm{aq})=\mathrm{H}_{1}(\mathrm{aq})-\mathrm{H}_{1}^{*}(\lambda)\]In equation (i), we encounter another difference in order to take account of the fact that we cannot measure absolute enthalpies of solutions and solvents.The chemical potential of the solute \(j\) (at fixed \(\mathrm{T}\) and \(\mathrm{p}\), which is close to ambient pressure) is related to the molality \(\mathrm{m}_{j}\) using equation (j). \[\mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)\]From the Gibbs-Helmholtz Equation, \[\mathrm{H}_{\mathrm{j}}(\mathrm{aq})=\mathrm{H}_{\mathrm{j}}^{0}(\mathrm{aq})-\mathrm{R} \, \mathrm{T}^{2} \,\left(\mathrm{d} \ln \gamma_{\mathrm{j}} / \mathrm{dT}\right)_{\mathrm{p}}\]But activity coefficient \(\gamma_{j}\) is defined such that \(\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{\mathrm{j}}=1.0\) at all \(\mathrm{T}\) and \(\mathrm{p}\). Moreover for an ideal solution, \(\gamma_{j} = 1.0\). Hence, \[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{H}_{\mathrm{j}}(\mathrm{aq})=\mathrm{H}_{\mathrm{j}}^{0}(\mathrm{aq})=\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})\]In other words, with increasing dilution \(\mathrm{H}_{j}(\mathrm{aq})\) approaches a limiting partial molar enthalpy \(\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})\) which equals the partial molar enthalpy of the solute in an ideal solution. We identify a relative (partial) molar enthalpy of solute \(j\), \(\mathrm{L}_{\mathrm{j}}(\mathrm{aq})\). \[\mathrm{L}_{\mathrm{j}}(\mathrm{aq})=\mathrm{H}_{\mathrm{j}}(\mathrm{aq})-\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})\]Hence, at fixed \(\mathrm{T}\) and \(\mathrm{p}\) \[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{L}_{\mathrm{j}}(\mathrm{aq})=0\]Therefore for simple solutes in solution in the limit of infinite dilution the relative partial molar enthalpy of solute \(j\) is zero.Footnotes \([\mathrm{J}]=[\mathrm{mol}] \,\left[\mathrm{J} \mathrm{mol}^{-1}\right]+[\mathrm{mol}] \,\left[\mathrm{J} \mathrm{mol}^{-1}\right]\) \(\left[\mathrm{J} \mathrm{kg}^{-1}\right]=\left[\mathrm{kg} \mathrm{mol}^{-1}\right]^{-1} \,\left[\mathrm{J} \mathrm{mol}^{-1}\right]+\left[\mathrm{mol} \mathrm{kg}^{-1}\right] \,\left[\mathrm{J} \mathrm{mol}^{-1}\right]\) Note the advantage of expressing the composition in terms of molalities rather than in concentrations for which we would have to take account of the dependence of volume on temperature. An interesting comparison is the molar enthalpy of water(\(\lambda\)) and the limiting molar enthalpy of solute water in a solvent such as methanol. We define a transfer quantity, \(\Delta_{\mathrm{tr}} \mathrm{H}^{0}\) [ \(= \mathrm{H}^{\infty}\) (\(\mathrm{H}_{2}\mathrm{O}\) as solute in a defined solvent) \(-\mathrm{H}_{1}^{*}\left(\lambda \mathrm{H}_{2} \mathrm{O}\right)\)], characterizing the difference in molar enthalpy of liquid water and the limiting partial molar enthalpy of solute water at ambient pressure and \(298.15 \mathrm{~K}\)_. \(\Delta_{\mathrm{tr}} \mathrm{H}^{0}\) is \(0.85\), \(4.05\) and \(10.11 \mathrm{~kJ mol}^{-1}\) in \(\mathrm{CH}_{3}\mathrm{OH}(\lambda)\), \(\mathrm{C}_{7}\mathrm{H}_{15}\mathrm{OH}(\lambda)\) and \(\mathrm{C}_{2}\mathrm{H}_{4}(\mathrm{O.CO.C}_{3}\mathrm{H}_{7})_{2} (\lambda)\) respectively. S.-O. Nilsson, J. Chem. Thermodyn., 1986, 18, 1115.This page titled 1.8.4: Enthalpy- Solutions- Partial Molar Enthalpies is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.8.5: Enthalpies- Solutions- Equilibrium and Frozen Partial Molar Enthalpies

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A given system at fixed \(\mathrm{T}\) and \(\mathrm{p}\) is at thermodynamic equilibrium. The enthalpy of the system is perturbed by adding \(\delta \mathrm{n}_{j}\) moles of chemical substance \(j\). We imagine two possible limiting changes to the system. In one limit the enthalpy of the system changes to a neighbouring state where the extent of chemical reaction remains constant; i.e. at fixed \(\xi\). In another limit the enthalpy of the system changes to a neighbouring state where the affinity for spontaneous change \(\mathrm{A}\) remains constant. The two differential changes in enthalpy are related. \[\left(\frac{\partial H}{\partial n_{j}}\right)_{A}=\left(\frac{\partial H}{\partial n_{j}}\right)_{\xi}-\left(\frac{\partial \mathrm{A}}{\partial n_{j}}\right)_{\xi} \,\left(\frac{\partial \xi}{\partial A}\right)_{n_{j}} \,\left(\frac{\partial H}{\partial \xi}\right)_{n_{j}}\]We identify the state being perturbed as the equilibrium state where \(\mathrm{A} = 0\) and the composition-organisation is represented by \(\xi^{\mathrm{eq}\). We identify two quantities describing the impact of adding \(\delta \mathrm{n}_{j}\) moles of chemical substance \(j\).Equilibrium partial molar enthalpy, \[\mathrm{H}_{\mathrm{j}}(\mathrm{A}=0)=\left(\frac{\partial \mathrm{H}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{A}=0}\]Frozen partial molar enthalpy, \[\mathrm{H}_{\mathrm{j}}\left(\xi^{\mathrm{eq}}\right)=\left(\frac{\partial \mathrm{H}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \xi^{\mathrm{eq}}}\]Because the triple product term on the r.h.s. of equation (a) is not zero at equilibrium (i.e. at \(\mathrm{A} = \text { zero}\) and \(\xi = \xi^{\mathrm{eq}}\)), then \(\mathrm{H}_{\mathrm{j}}(\mathrm{A}=0)\) is not equal to. By convention, the term ‘ partial molar enthalpy is taken to mean \(\mathrm{H}_{\mathrm{j}}(\mathrm{A}=0)\).This page titled 1.8.5: Enthalpies- Solutions- Equilibrium and Frozen Partial Molar Enthalpies is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.8.6: Enthalpies- Neural Solutes

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.08%3A_Enthalpy/1.8.06%3A_Enthalpies-_Neural_Solutes

A given aqueous solution at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\) (close to ambient pressure \(\mathrm{p}^{0}\)) is prepared using \(\mathrm{n}_{1}\) moles of water and \(\mathrm{n}_{j}\) moles of a solute, chemical substance-\(j\). The enthalpy of this solution \(\mathrm{H}(\mathrm{aq})\) is given by equation (a) where \(\mathrm{H}_{1}(\mathrm{aq})\) and \(\mathrm{H}_{j}(\mathrm{aq})\) are the (equilibrium) partial molar enthalpies of solvent and solute respectively \[\mathrm{H}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{H}_{1}(\mathrm{aq})+\mathrm{n}_{\mathrm{j}} \, \mathrm{H}_{\mathrm{j}}(\mathrm{aq})\]\[\mathrm{H}_{1}(\mathrm{aq})=\left(\frac{\partial \mathrm{H}}{\partial \mathrm{n}_{1}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{j})}\]\[\mathrm{H}_{\mathrm{j}}(\mathrm{aq})=\left(\frac{\partial \mathrm{H}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}}\]Then, \[\mathrm{H}_{1}(\mathrm{aq})=\mathrm{H}_{1}^{*}(\lambda)+\mathrm{R} \, \mathrm{T}^{2} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\frac{\partial \phi}{\partial \mathrm{T}}\right)_{\mathrm{p}}\]\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{H}_{1}(\mathrm{aq})=\mathrm{H}_{1}^{*}(\lambda)\]Similarly for the solute, chemical substance \(j\) (assuming ambient pressure \(\mathrm{p}\) is close to the standard pressure), \[\mathrm{H}_{\mathrm{j}}(\mathrm{aq})=\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})-\mathrm{R} \, \mathrm{T}^{2} \,\left(\frac{\partial \ln \left(\gamma_{\mathrm{j}}\right)}{\partial \mathrm{T}}\right)_{\mathrm{p}}\]\(\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})\) is the limiting (infinite dilution) partial molar enthalpy of solute \(j\). The enthalpy of a solution prepared using \(\mathrm{n}_{1}\) moles of water and \(\mathrm{n}_{j}\) moles of solute is given by equation (g). \[\begin{aligned} \mathrm{H}(\mathrm{aq})=\mathrm{n}_{1} \,\left[\mathrm{H}_{1}^{*}(\lambda)+\mathrm{R} \, \mathrm{T}^{2} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,(\partial \phi / \partial \mathrm{T})_{\mathrm{p}}\right] \\ &\left.+\mathrm{n}_{\mathrm{j}} \, \mid \mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})-\mathrm{R} \, \mathrm{T}^{2} \,\left(\partial \ln \gamma_{\mathrm{j}} / \partial \mathrm{T}\right)_{\mathrm{p}}\right] \end{aligned}\]For a solution in \(1 \mathrm{~kg}\) of water, \[\begin{aligned} \mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=&\left(1 / \mathrm{M}_{1}\right) \,\left[\mathrm{H}_{1}^{*}(\lambda)+\mathrm{R} \, \mathrm{T}^{2} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,(\partial \phi / \partial \mathrm{T})_{\mathrm{p}}\right] \\ &+\mathrm{m}_{\mathrm{j}} \,\left[\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})-\mathrm{R} \, \mathrm{T}^{2} \,\left(\partial \ln \gamma_{\mathrm{j}} / \partial \mathrm{T}\right)_{\mathrm{p}}\right] \end{aligned}\]We re-arrange equation (h). \[\begin{aligned} \mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1}=\right.&1 \mathrm{~kg})=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{H}_{1}^{*}(\lambda) \\ &+\mathrm{m}_{\mathrm{j}} \,\left[\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})+\mathrm{R} \, \mathrm{T}^{2} \,(\partial \phi / \partial \mathrm{T})_{\mathrm{p}}-\mathrm{R} \, \mathrm{T}^{2} \,\left(\partial \ln \gamma_{\mathrm{j}} / \partial \mathrm{T}\right)_{\mathrm{p}}\right] \end{aligned}\]Equation (i) is interesting because inside the brackets [….] we have the limiting partial molar enthalpy of the solute and two terms which describe the extent to which the enthalpic properties of the solution differ from those of the corresponding ideal solution. We find it advantageous to describe the property in the brackets […] as the apparent molar enthalpy of the solution, \(\phi\left(\mathrm{H}_{\mathrm{j}}\right)\). By definition, \[\phi\left(\mathrm{H}_{\mathrm{j}}\right)=\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})-\mathrm{R} \, \mathrm{T}^{2} \,\left(\mathrm{d} \ln \gamma_{\mathrm{j}} / \mathrm{dT}\right)_{\mathrm{p}}+\mathrm{R} \, \mathrm{T}^{2} \,(\mathrm{d} \phi / \mathrm{dT})_{\mathrm{p}}\]For a solution prepared using \(1 \mathrm{~kg}\) of solvent water. \[\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{H}_{1}^{*}(\lambda)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{H}_{\mathrm{j}}\right)\]But at all \(\mathrm{T}\) and \(\mathrm{p}\), \[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{\mathrm{j}}=1.0 ; \ln \left(\gamma_{\mathrm{j}}\right)=0 ; \phi=1.0\]Hence, \[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right)\left[\partial \ln \left(\gamma_{\mathrm{j}}\right) / \partial \mathrm{T}\right]_{\mathrm{p}}=[\partial \phi / \partial \mathrm{T}]_{\mathrm{p}}=0\]\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi\left(\mathrm{H}_{\mathrm{j}}\right)=\phi\left(\mathrm{H}_{\mathrm{j}}\right)^{\infty}=\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})\]We recognize a crucial complication in the treatment of the enthalpies of solutions. Unlike volumetric properties of solutions, we cannot measure the enthalpy of a solution. In other words we need to examine differences. Based on equation (k) we form an equation for the enthalpy of the corresponding solution having thermodynamic properties which are ideal. \[\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg} ; \mathrm{id}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{H}_{1}^{*}(\lambda)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{H}_{\mathrm{j}}\right)^{\infty}\]The difference between the two enthalpies is given by equation (p) \[\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)-\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg} ; \mathrm{id}\right)=\mathrm{m}_{\mathrm{j}} \,\left[\phi\left(\mathrm{H}_{\mathrm{j}}\right)-\phi\left(\mathrm{H}_{\mathrm{j}}\right)^{\infty}\right]\]Interesting descriptions of the enthalpies of solutions containing simple solutes are based on the concept of excess thermodynamic properties and pairwise solute-solute interaction parameters. Equation (k) describes the enthalpy of a solution prepared using \(1 \mathrm{~kg}\) of water whereas equation (o) describes the enthalpy of the corresponding solution where the thermodynamic properties are ideal. The excess enthalpy \(\mathrm{H}^{\mathrm{E}}\left(\mathrm{aq} ; \mathrm{m}_{\mathrm{j}}\right)\) is given by equation (q). \[\mathrm{H}^{\mathrm{E}}\left(\mathrm{aq} ; \mathrm{m}_{\mathrm{j}}\right)=\mathrm{h}_{\mathrm{ij}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2}+\mathrm{h}_{\mathrm{iji}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{3} \ldots \ldots\]Footnotes For the solvent in solutions ( at constant pressure), \[\mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\lambda)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\]But \[-\frac{\mathrm{H}_{1}(\mathrm{aq})}{\mathrm{T}^{2}}=\frac{\partial\left[\mu_{1}(\mathrm{aq}) / \mathrm{T}\right]}{\partial T}\]Then \(-\frac{\mathrm{H}_{1}(\mathrm{aq})}{\mathrm{T}^{2}}=-\frac{\mathrm{H}_{1}^{*}(\lambda)}{\mathrm{T}^{2}}-\mathrm{R} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\frac{\partial \phi}{\partial \mathrm{T}}\right)_{\mathrm{p}}\) \(\mathrm{R} \, \mathrm{T}^{2} \,\left(\frac{\partial \phi}{\partial \mathrm{T}}\right)_{\mathrm{p}}=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}]^{2} \,[\mathrm{K}]^{-1} =\left[\mathrm{J} \mathrm{mol}^{-1}\right]\) From \(\mu_{j}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)\) Then, \(-\frac{\mathrm{H}_{\mathrm{j}}(\mathrm{aq})}{\mathrm{T}^{2}}=-\frac{\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})}{\mathrm{T}^{2}}+\mathrm{R} \,\left(\frac{\partial \ln \left(\gamma_{\mathrm{j}}\right)}{\partial \mathrm{T}}\right)_{\mathrm{p}}\) \(\mathrm{R} \, \mathrm{T}^{2} \,\left(\frac{\partial \ln \left(\gamma_{\mathrm{j}}\right)}{\partial \mathrm{T}}\right)_{\mathrm{p}}=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}]^{2} \,[\mathrm{K}]^{-1}=\left[\mathrm{J} \mathrm{mol}{ }^{-1}\right]\) See for example,This page titled 1.8.6: Enthalpies- Neural Solutes is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.8.7: Enthalpies- Solutions- Dilution- Simple Solutes

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.08%3A_Enthalpy/1.8.07%3A_Enthalpies-_Solutions-_Dilution-_Simple_Solutes

A given (old) aqueous solution is prepared using \(\mathrm{n}_{1}\)(old) moles of water(\(\lambda\)) and \(\mathrm{n}_{j}\) moles of a simple neutral solute at fixed \(\mathrm{T}\) and \(\mathrm{p}\). The enthalpy \(\mathrm{H}(\mathrm{aq} ; \mathrm{old})\) of this solution is expressed in terms of the molar enthalpy of water(\(\lambda\)), \(\mathrm{H}_{1}^{*}(\lambda)\) and the apparent molar enthalpy of the solute \(\phi\left(\mathrm{H}_{\mathrm{j}} ; \text { old }\right)\). \[\mathrm{H}(\mathrm{aq} ; \text { old })=\mathrm{n}_{1}(\text { old }) \, \mathrm{H}_{1}^{*}(\lambda)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{H}_{\mathrm{j}} ; \text { old }\right)\]We use the description ‘old’ because we envisage preparing a ‘new’ solution by adding \(\mathrm{n}_{1}\)(added) moles of water, enthalpy \(\mathrm{n}_{1}(\text { added }) \, \mathrm{H}_{1}^{*}(\lambda)\). \[\mathrm{H}(\text { added })=\mathrm{n}_{1}(\text { added }) \, \mathrm{H}_{1}^{*}(\lambda)\]The enthalpy of the resultant solution is \(\mathrm{H}(\mathrm{aq} ; \text { new })\); \[\mathrm{H}(\text { aq; new })=\left[\mathrm{n}_{1}(\mathrm{old})+\mathrm{n}_{1}(\text { added })\right] \, \mathrm{H}_{1}^{*}(\lambda)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{H}_{\mathrm{j}} ; \text { new }\right)\]In effect the ‘old’ solution has been diluted. \[\left.\Delta \mathrm{H}(\text { old } \rightarrow \text { new })=\mathrm{H}(\mathrm{aq} ; \text { new })-\mathrm{H}(\text { aq } ; \text { old })-\left[\mathrm{n}_{1} \text { (added }\right) \, \mathrm{H}_{1}^{*}(\lambda)\right]\]Hence, \[\Delta \mathrm{H}(\text { old } \rightarrow \text { new })=\mathrm{n}_{\mathrm{j}} \,\left[\phi\left(\mathrm{H}_{\mathrm{j}} \text { (new }\right)-\phi\left(\mathrm{H}_{\mathrm{j}} \text { (old }\right)\right]\]An isobaric calorimeter measures heat \(\mathrm{q}\) characterising the dilution. \[\begin{aligned} \mathrm{q} / \mathrm{n}_{\mathrm{j}} &=\Delta_{\mathrm{dil}} \mathrm{H}\left[\mathrm{m}_{\mathrm{j}}(\mathrm{old}) \rightarrow \mathrm{m}_{\mathrm{j}}(\text { new })\right] \\ &=\Delta \mathrm{H}(\text { old } \rightarrow \text { new }) / \mathrm{n}_{\mathrm{j}}=\phi\left(\mathrm{H}_{\mathrm{j}}(\mathrm{j} ; \text { new })\right)-\phi\left(\mathrm{H}_{\mathrm{j}}(\mathrm{j} ; \mathrm{old})\right) \end{aligned}\]We imagine a series of experiments in which the molality of solute at the start of the experiment is \(\mathrm{m}_{j}\)(I). Following dilution the molality is \(\mathrm{m}_{j}\)(II). \[\Delta_{\mathrm{dil}} \mathrm{H}\left[\mathrm{m}_{\mathrm{j}}(\mathrm{I}) \rightarrow \mathrm{m}_{\mathrm{j}}(\mathrm{II})\right]=\phi\left(\mathrm{H}_{\mathrm{j}} ; \mathrm{m}_{\mathrm{j}} ; \mathrm{II}\right)-\phi\left(\mathrm{H}_{\mathrm{j}} ; \mathrm{m}_{\mathrm{j}} ; \mathrm{I}\right)\]In a calorimetric experiment we record heat \(\mathrm{q}\) accompanying a second dilution. Hence, \[\Delta_{\mathrm{dll}} \mathrm{H}\left[\mathrm{m}_{\mathrm{j}}(\mathrm{II}) \rightarrow \mathrm{m}_{\mathrm{j}}(\mathrm{III})\right]=\phi\left(\mathrm{H}_{\mathrm{j}} ; \mathrm{m}_{\mathrm{j}} ; \text { III }\right)-\phi\left(\mathrm{H}_{\mathrm{j}} ; \mathrm{m}_{\mathrm{j}} ; \text { II }\right)\]In a third dilution we have that \[\Delta_{\mathrm{dil}} \mathrm{H}\left[\mathrm{m}_{\mathrm{j}}(\mathrm{III}) \rightarrow \mathrm{m}_{\mathrm{j}}(\mathrm{IV})\right]=\phi\left(\mathrm{H}_{\mathrm{j}} ; \mathrm{m}_{\mathrm{j}} ; \mathrm{IV}\right)-\phi\left(\mathrm{H}_{\mathrm{j}} ; \mathrm{m}_{\mathrm{j}} ; \text { III }\right)\]In this experiment the molality of the solution in the sample cell is gradually falling. Combination of the results described by equations (g), (h) and (i) yields the set, \(\Delta_{\mathrm{dil}} \mathrm{H}\left[\mathrm{m}_{\mathrm{j}}(\mathrm{I}) \rightarrow \mathrm{m}_{\mathrm{j}}(\mathrm{II})\right], \Delta_{\mathrm{dil}} \mathrm{H}\left[\mathrm{m}_{\mathrm{j}} \text { (II) } \rightarrow \mathrm{m}_{\mathrm{j}}(\mathrm{III})\right], \Delta_{\mathrm{dil}} \mathrm{H}\left[\mathrm{m}_{\mathrm{j}}(\mathrm{III}) \rightarrow \mathrm{m}_{\mathrm{j}}(\mathrm{IV})\right] \ldots\) This set is expanded with further dilutions until by extrapolation we obtain for solution \(\Delta_{\text {dil }} \mathrm{H}\left[\mathrm{m}_{\mathrm{j}}(\mathrm{I}) \rightarrow \text { infinite dilution }\right]\). We obtain the enthalpies of dilution for all dilutions in a given set of experiments; i.e. for dilution for solutions II, III, IV…Alternatively a given solution is diluted by increasing amounts of solvent; e.g. adding ethanol(\(\lambda\)) to a solution of urea in ethanol.In the analysis of enthalpies of solutions simplification of the algebra is achieved by defining a number of L-variables, signalling differences in enthalpies. The relative enthalpy L describes the difference between the enthalpies of real and ideal solutions. For a solution prepared using \(\mathrm{w}_{1} \mathrm{~kg}\) of solvent (e.g. water), \[\mathrm{L}\left(\mathrm{aq} ; \mathrm{w}_{1} \mathrm{~kg}\right)=\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1} \mathrm{~kg}\right)-\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1} \mathrm{~kg} ; \mathrm{id}\right)\]By definition for the solvent, \[\mathrm{L}_{1}(\mathrm{aq})=\mathrm{H}_{1}(\mathrm{aq})-\mathrm{H}_{1}^{*}(\lambda)\]For the solute \(j\), \[\mathrm{L}_{\mathrm{j}}(\mathrm{aq})=\mathrm{H}_{\mathrm{j}}(\mathrm{aq})-\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})\]\(\mathrm{L}_{1}(\mathrm{aq})\) and \(\mathrm{L}_{j}(\mathrm{aq})\) are the relative partial molar enthalpies of solvent and solute respectively. Similarly in terms of apparent properties, \[\phi\left(\mathrm{L}_{\mathrm{j}}\right)=\phi\left(\mathrm{H}_{\mathrm{j}}\right)-\phi\left(\mathrm{H}_{\mathrm{j}}\right)^{\infty}\]\[\mathrm{L}=\mathrm{n}_{1} \, \mathrm{L}_{1}+\mathrm{n}_{\mathrm{j}} \, \mathrm{L}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{L}_{\mathrm{j}}\right)\]\(\phe(\mathrm{L}_{j})\) is the apparent relative molar enthalpy of solute \(j\) in solution at molality \(\mathrm{m}_{j}\), describing the difference between the apparent molar enthalpies of solute \(j\) in real and ideal solutions. In other words we have a direct probe of the role of solute-solute interactions in solution. Both \(\mathrm{L}\) and \(\phi(\mathrm{L}_{j})\) are (by definition) zero for solutions where the thermodynamic properties are ideal.This galaxy of variables is clarified if we return to a calorimetric experiment where a solution is diluted. An (old) aqueous solution is prepared using \(\mathrm{n}_{1}\) moles of water(\(\lambda\)) and \(\mathrm{n}_{j}\) moles of solute producing a solution having enthalpy \(\mathrm{H}(\mathrm{aq} ; \mathrm{old})\). \[\mathrm{H}(\mathrm{aq} ; \text { old })=\mathrm{n}_{1}(\text { old }) \, \mathrm{H}_{1}^{*}(\lambda)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{H}_{\mathrm{j}} ; \text { old }\right)\]To this solution we add (at the same \(\mathrm{T}\) and \(\mathrm{p}\)) \(\mathrm{n}_{1}\)(added) moles of water(\(\lambda\)). \[\mathrm{H}(\text { added })=\mathrm{n}_{1}(\text { added }) \, \mathrm{H}_{1}^{*}(\lambda)\]in the limit that \(\mathrm{n}_{1}\)(added) is sufficiently large that the molality \(\mathrm{m}_{j}\) of the ‘new’ solution is negligibly small, then \[\operatorname{limit}\left(\text { new } ; \mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi\left(\mathrm{H}_{\mathrm{j}} ; \text { new }\right)=\phi\left(\mathrm{H}_{\mathrm{j}}\right)^{\infty}\]\[\Delta \mathrm{H}(\text { old } \rightarrow \text { new })=-\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{L}_{\mathrm{j}}\right)\]By definition, \[\Delta_{\text {dil }} \mathrm{H}=\Delta \mathrm{H}(\text { old } \rightarrow \text { new }) / \mathrm{n}_{j}\]\[\Delta_{\mathrm{dil}} \mathrm{H}=-\phi\left(\mathrm{L}_{\mathrm{j}}\right)\]Consistent with our definitions of heat \(\mathrm{q}\) and enthalpy change, a positive \(\Delta_{\mathrm{dil}}\mathrm{H}\) indicates that dilution is endothermic.We have not commented on the dependence of either \(\phi\left(\mathrm{L}_{\mathrm{j}}\right)\) or \(\phi\left(\mathrm{H}_{\mathrm{j}}\right)\) on molality of solute. In order to say something about these variables we need explicit equations for these dependences on composition of solution.An important approach to the description of the properties of solutions uses excess thermodynamic functions. The quantity \(\mathrm{L}(\mathrm{aq})\) defined in equation (n) refers to a solutions prepared using \(\mathrm{n}_{1}\) moles of solvent and \(\mathrm{n}_{j}\) moles of solute, contrasting the properties of real and ideal solutions. The excess enthalpy \(\mathrm{H}^{\mathrm{E}}\) refers to the corresponding solutions prepared using \(1 \mathrm{~kg}\) of water and \(\mathrm{m}_{j}\) moles of solute \(j\). \[\mathrm{H}^{\mathrm{E}}=\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg} ; \mathrm{m}_{\mathrm{j}}\right)-\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg} ; \mathrm{m}_{\mathrm{j}} ; \mathrm{id}\right)\]thus, \[\mathrm{H}^{\mathrm{E}}= \left[\left(1 / \mathrm{M}_{1}\right) \, \mathrm{H}_{1}^{*}(\lambda)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{H}_{\mathrm{j}}\right)\right]-\left[\left(1 / \mathrm{M}_{1}\right) \, \mathrm{H}_{1}^{*}(\lambda)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{H}_{\mathrm{j}}\right)^{\infty}\right]\]Or, \[\mathrm{H}^{\mathrm{E}}=\mathrm{m}_{\mathrm{j}} \,\left[\phi\left(\mathrm{H}_{\mathrm{j}}\right)-\phi\left(\mathrm{H}_{\mathrm{j}}\right)^{\infty}\right]\]Therefore \[\mathrm{H}^{\mathrm{E}}=\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{L}_{\mathrm{j}}\right)\]Again the development of equation (x) reflects our continuing interest in differences with respect to enthalpies. Nevertheless the key isobaric calorimetric equation requires that the measured ratio (\(\mathrm{q} / \mathrm{~n}_{j}\)) for the process solvent + solute forming an ideal solution (at fixed \(\mathrm{T}\) and \(\mathrm{p}\)) equals the standard enthalpy of solution for pure substance \(j\), \(\Delta_{s \ln } \mathrm{H}^{0}\). For neutral solutes, the dependence of partial molar enthalpy of solute \(\mathrm{H}_{j}(\mathrm{aq})\) on solute molality mj is small such that the recorded (\(\mathrm{q} / \mathrm{~n}_{j}\)) for real solutions can often be equated to the corresponding limiting enthalpy of solution, \(\Delta_{s \ln } \mathrm{H}^{0}\) because in an ideal solution the standard partial molar enthalpy of a solute equals the partial molar enthalpy of the solute at infinite dilution. For solute \(j\), \[\Delta_{\mathrm{s} \ln } \mathrm{H}_{\mathrm{j}}^{0}=\Delta_{\mathrm{s} \ln } \mathrm{H}_{\mathrm{j}}^{\infty}=\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})-\mathrm{H}_{\mathrm{j}}^{*}(\mathrm{~s})\]Significantly modern calorimeters are sufficiently sensitive to measure heat \(\mathrm{q}\) when a known but small amount of substance \(j\) is dissolved in a known amount of solvent. In many cases the dependence of \(\Delta_{\sin } \mathrm{H}\) ion solute molality is, for small neutral solutes, negligibly small such that \(\Delta_{\sin } \mathrm{H}\) is assumed to equal \(\Delta_{\sin } \mathrm{H}^{0}\).Heats of solution can be analysed in terms of group contributions to the enthalpy of solution for a given series of solutes. Moreover the dependence of \(\Delta_{\sin } \mathrm{H}^{\infty}\) for a given solute on temperature yields the corresponding limiting isobaric heat capacity of solution, \(\Delta_{s \ln } C_{p}^{\infty}\). In fact by measuring \(\Delta_{\sin } \mathrm{H}^{\infty}\) for solutes in two solvents, the derived property is the standard enthalpy of transfer. \[\begin{aligned} \Delta_{s \ln } \mathrm{H}_{\mathrm{j}}^{\infty} &\text { solvent } \mathrm{B} \rightarrow \text { solvent } \mathrm{A}) \\ &=\Delta_{\mathrm{s} \ln } \mathrm{H}_{\mathrm{j}}^{\infty}(\text { solvent } \mathrm{A})-\Delta_{\mathrm{s} \ln } \mathrm{H}_{\mathrm{j}}^{\infty}(\text { solvent } \mathrm{B}) \end{aligned}\]Such a study identified a quite striking extremum for limiting partial molar enthalpies of solution for \(\mathrm{NaBH}_{4}\) in water + 2-methylpropan-2-ol mixtures at low alcohol mole fractions and \(298.2 \mathrm{~K}\) and hence a quite striking reversal of sign in limiting partial molar isobaric heat capacities for \(\mathrm{NaBPh}_{4}\) in this binary aqueous mixture; see also data for dialkyl sulfonates in alcohol + water mixtures and tri-n-alkyl phosphates in water + DMF binary mixtures. Indeed an extensive literature describes the enthalpies of solution for neutral solutes and, where the results concern one solute in two or more solvents, the corresponding enthalpy of transfer; cf. equation (z).Where the results describe a series of closely related neutral solutes, it is often possible to estimate contributions from individual groups (e.g. \(\mathrm{CH}_{2}\) and \(\mathrm{OH}\)) to a given limiting enthalpy of transfer.In many reports, the results of calorimetric experiments show clear evidence of a dependence of partial molar enthalpy of a given solute on molality of the solution. One of the first reports of such a dependence for neutral solutes was published in 1940. Hence a direct signal is obtained of enthalpic solute-solute interactions in solution.An aqueous solution is prepared using water(\(\mathrm{w}_{1} = 1 \mathrm{~kg}\)) and \(\mathrm{m}_{j}\) moles of solute \(j\) at defined \(\mathrm{T}\) and \(\mathrm{p}\). \[\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{H}_{1}(\mathrm{aq})+\mathrm{m}_{\mathrm{j}} \, \mathrm{H}_{\mathrm{j}}(\mathrm{aq})\]In the event that the thermodynamic properties of this solutions are ideal the enthalpy of the solution is given by equation (zb). \[\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg} ; \mathrm{id}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{H}_{1}^{*}(\lambda)+\mathrm{m}_{\mathrm{j}} \, \mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})\]The excess enthalpy \(\mathrm{H}^{\mathrm{E}}\left(\mathrm{aq} ; \mathrm{m}_{\mathrm{j}}\right)\) is given by equation (zc). \[\mathrm{H}^{\mathrm{E}}\left(\mathrm{aq} ; \mathrm{m}_{\mathrm{j}}\right)=\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)-\left(1 / \mathrm{M}_{1}\right) \, \mathrm{H}_{1}^{*}(\lambda)-\mathrm{m}_{\mathrm{j}} \, \mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})\]\(\mathrm{H}^{\mathrm{E}}\left(\mathrm{aq} ; \mathrm{m}_{\mathrm{j}}\right)\) can be expressed as a power series in molality \(\mathrm{m}_{j}\). \[\mathrm{H}^{\mathrm{E}}\left(\mathrm{aq} ; \mathrm{m}_{\mathrm{j}}\right)=\mathrm{h}_{\mathrm{jj}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2}+\mathrm{h}_{\mathrm{ij}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{3}\]A given solution contains solute \(j\) such that an isobaric calorimeter is used to measure the heat of dilution. We obtain the enthalpy per mole of solute on going from molality \(\mathrm{m}_{j}\)(initial) to \(\mathrm{m}_{j}\)(final), \(\Delta_{\mathrm{dil}} \mathrm{H}\). \[\Delta_{\mathrm{dil}} \mathrm{H}=\mathrm{H}^{\mathrm{E}}\left(\mathrm{m}_{\mathrm{j}} \text { - final }\right) / \mathrm{m}_{\mathrm{j}}(\text { final })-\mathrm{H}^{\mathrm{E}}\left(\mathrm{m}_{\mathrm{j}}-\text { initial }\right) / \mathrm{m}_{\mathrm{j}} \text { (initial) }\]Equations (zd) and (ze) yield an equation for measured \(\Delta_{\mathrm{dil}} \mathrm{H}\) in terms of enthalpic solute-solute pairwise and triplet interaction parameters. \[\begin{aligned} \Delta_{\mathrm{dil}} \mathrm{H}=\mathrm{h}_{\mathrm{ij}} \,\left[\mathrm{m}_{\mathrm{j}}(\text { final })\right.&\left.-\mathrm{m}_{\mathrm{j}}(\text { initial })\right] / \mathrm{m}^{0} \\ &+\mathrm{h}_{\mathrm{jij}} \,\left[\left\{\mathrm{m}_{\mathrm{j}}(\text { final })\right\}^{2}-\left\{\mathrm{m}_{\mathrm{j}}(\text { initial })\right\}^{2}\right] /\left(\mathrm{m}^{0}\right)^{2} \end{aligned}\]In most cases, authors concentrate attention on pairwise interaction parameters between identical (homotactic) and different (heterotactic) solute molecules in a given solution. The concept of solute-solute pairwise (and higher order) interaction parameters allows quite detailed patterns to emerge from enthalpies of dilution of neutral solutes in salt solution.Footnotes E.g. adding ethanol(\(\lambda\)) to a solution of urea in ethanol(\(\lambda\)) ; D. Hamilton and R. H. Stokes, J. Solution Chem.,1972,1,223. E. M. Arnett and D. R. McKelvey, J. Am. Chem. Soc., 1966,88, 5031 C. V Krishnan and H. L. Friedman, J Solution Chem.,1973,2,37. F. T. Gucker and H. B. Pickard, J. Am.Chem.Soc.,1940,62,1464.This page titled 1.8.7: Enthalpies- Solutions- Dilution- Simple Solutes is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.8.8: Enthalpies- Solutions- Simple Solutes- Interaction Parameters

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.08%3A_Enthalpy/1.8.08%3A_Enthalpies-_Solutions-_Simple_Solutes-_Interaction_Parameters

The excess Gibbs energy \(\mathrm{G}^{\mathrm{E}}\) for a dilute aqueous solution containing a simple solute \(j\) prepared using \(1 \mathrm{~kg}\) of solvent, water is given by equation (a). \[\mathrm{G}^{\mathrm{E}}=\mathrm{R} \, \mathrm{T} \, \mathrm{m}_{\mathrm{j}} \,\left[1-\phi+\ln \left(\gamma_{\mathrm{j}}\right)\right]\]In terms of Gibbs energies pairwise solute-solute interaction parameters, \[\mathrm{G}^{\mathrm{E}}=\mathrm{g}_{\mathrm{jj}} \,\left[\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right]^{2}\]The excess enthalpy \[\mathrm{H}^{\mathrm{E}}=\mathrm{h}_{\mathrm{ij}} \,\left[\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right]^{2}\]where [cf. Gibbs –Helmholtz Equation], \[\mathrm{h}_{\mathrm{ij}}=-\mathrm{T}^{2} \,\left\{\partial\left[\mathrm{g}_{\mathrm{jj}} / \mathrm{T}\right] / \partial \mathrm{T}\right\}_{\mathrm{p}}\]Here \(\mathrm{h}_{\mathrm{jj}}\) is the pairwise solute-solute enthalpic interaction parameter. For the solvent, \[\mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\lambda)-\mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}-\mathrm{M}_{1} \, \mathrm{g}_{\mathrm{j}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2}\]Using the Gibbs-Helmholtz Equation, \[\mathrm{H}_{1}(\mathrm{aq})=\mathrm{H}_{1}^{*}(\lambda)-\mathrm{M}_{1} \, \mathrm{h}_{\mathrm{ij}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2}\]For the solute, \[\mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)+2 \, \mathrm{g}_{\mathrm{ij}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{m}_{\mathrm{j}}\]Then using the Gibbs-Helmholtz Equation \[\mathrm{H}_{\mathrm{j}}(\mathrm{aq})=\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})+2 \, \mathrm{h}_{\mathrm{ij}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{m}_{\mathrm{j}}\]Alternatively we may express the enthalpy of the solution in terms of the apparent molar enthalpy of the solute, \(\phi\left(\mathrm{H}_{\mathrm{j}}\right)\). \[\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{H}_{1}^{*}(\lambda)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{H}_{\mathrm{j}}\right)\]For the ideal solution, \[\mathrm{H}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{H}_{1}^{*}(\lambda)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{H}_{\mathrm{j}}\right)^{\infty}\]where \(\phi\left(\mathrm{H}_{\mathrm{j}}\right)^{\infty}=\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})\). Then \[\mathrm{H}^{\mathrm{E}}=\mathrm{m}_{\mathrm{j}} \,\left[\phi\left(\mathrm{H}_{\mathrm{j}}\right)-\phi\left(\mathrm{H}_{\mathrm{j}}\right)^{\infty}\right]\]Hence using equation (c), \[\phi\left(\mathrm{H}_{\mathrm{j}}\right)=\phi\left(\mathrm{H}_{\mathrm{j}}\right)^{\infty}+\mathrm{h}_{\mathrm{ji}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{m}_{\mathrm{j}}\]We use these equations in the analysis of a calorimetric data where a given solution is diluted. The solution is prepared using \(\mathrm{n}_{1}\) moles of solvent (water) and \(\mathrm{n}_{j}\) moles of a simple solute \(j\). Then \[\mathrm{H}(\mathrm{I} ; \mathrm{aq})=\mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\lambda)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{H}_{\mathrm{j}} ; \mathrm{I} ; \mathrm{aq}\right)\]A new solution is prepared by adding (in the calorimeter) \(\Delta \mathrm{n}_{1}\) moles of solvent, Then \[\mathrm{H}(\mathrm{II} ; \mathrm{aq})=\left(\mathrm{n}_{1}+\Delta \mathrm{n}_{1}\right) \, \mathrm{H}_{1}^{*}(\lambda)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{H}_{\mathrm{j}} ; \mathrm{II} ; \mathrm{aq}\right)\]Thus the molality of solute \(j\) changes from \(\mathrm{m}_{j}\)(I) \(\left[=\mathrm{n}_{\mathrm{j}} / \mathrm{n}_{1} \, \mathrm{M}_{1}\right]\) to \(\mathrm{m}_{j}\)(II) \(\left[=\mathrm{n}_{\mathrm{j}} /\left(\mathrm{n}_{1}+\Delta \mathrm{n}_{1}\right) \, \mathrm{M}_{1}\right]\). Therefore, \[\phi\left(\mathrm{H}_{\mathrm{j}} ; \mathrm{I} ; \mathrm{aq}\right)=\phi\left(\mathrm{H}_{\mathrm{j}}\right)^{\infty}+\left[\mathrm{h}_{\mathrm{ij}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{n}_{\mathrm{j}} / \mathrm{n}_{1} \, \mathrm{M}_{1}\right]\]And \[\phi\left(\mathrm{H}_{\mathrm{j}} ; \mathrm{II} ; \mathrm{aq}\right)=\phi\left(\mathrm{H}_{\mathrm{j}}\right)^{\infty}+\left[\mathrm{h}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{n}_{\mathrm{j}} /\left(\mathrm{n}_{1}+\Delta \mathrm{n}_{1}\right) \, \mathrm{M}_{1}\right]\]In fact we record the heat \(\mathrm{q}\) (at constant pressure) when \(\Delta \mathrm{n}_{1}\) moles of solvent are added to solution I to form solution II. Thus, \[\mathrm{q}=\mathrm{H}(\mathrm{II} ; \mathrm{aq})-\mathrm{H}(\mathrm{I} ; \mathrm{aq})-\Delta \mathrm{n} \, \mathrm{H}_{1}^{*}(\lambda)\]Footnotes From equation (a) and (b) \(\mathrm{G}^{\mathrm{E}}=\left[\mathrm{J} \mathrm{kg}^{-1}\right]\) From equation (c) \(\mathrm{H}^{\mathrm{E}}=\left[\mathrm{J} \mathrm{kg}^{-1}\right]\) A check on the equations with reference to solution prepared using \(1 \mathrm{~kg}\) of solvent. \[\begin{aligned} \mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \,\left[\mathrm{H}_{1}^{*}(\lambda)-\mathrm{M}_{1} \, \mathrm{h}_{\mathrm{ji}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2}\right] \\ &+\mathrm{m}_{\mathrm{j}} \,\left[\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})+2 \, \mathrm{h}_{\mathrm{jj}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{m}_{\mathrm{j}}\right] \end{aligned}\]Or, \[\begin{aligned} \mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{H}_{1}^{*} &(\lambda)-\mathrm{h}_{\mathrm{jj}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2} \\ &+\mathrm{m}_{\mathrm{j}} \, \mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})+2 \, \mathrm{h}_{\mathrm{ij}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2} \end{aligned}\]Then \[\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{H}_{1}^{*}(\lambda)+\mathrm{m}_{\mathrm{i}} \, \mathrm{H}_{\mathrm{i}}^{\infty}(\mathrm{aq})+\mathrm{H}^{\mathrm{E}}\]This page titled 1.8.8: Enthalpies- Solutions- Simple Solutes- Interaction Parameters is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.8.9: Enthalpies- Salt Solutions- Apparent Molar- Partial Molar and Relative Enthalpies

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.08%3A_Enthalpy/1.8.09%3A_Enthalpies-_Salt_Solutions-_Apparent_Molar-_Partial_Molar_and_Relative_Enthalpies

Description of the enthalpies of salt solutions is similar to that given for neutral solutes except that account is taken of the fact that one mole of a given salt can with complete dissociation produce \(v\) moles of ions. The chemical potential of the solvent in an aqueous salt solution (at constant temperature and ambient pressure) is given by equation (a). \[\mu_{1}(\mathrm{aq})=\mu_{1}^{\star}(\lambda)-\mathrm{v} \, \phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\]Here \(\phi\) is the practical osmotic coefficient where \(\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi=1.0\) at all \(\mathrm{T}\) and \(\mathrm{p}\). Using the Gibbs-Helmholtz Equation, \[\mathrm{H}_{1}(\mathrm{aq})=\mathrm{H}_{1}^{*}(\lambda)+\mathrm{v} \, \mathrm{R} \, \mathrm{T}^{2} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,(\partial \phi / \partial \mathrm{T})_{\mathrm{p}}\]Also \[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{H}_{1}(\mathrm{aq})=\mathrm{H}_{1}^{*}(\lambda)\]By definition, \[\mathrm{L}_{1}(\mathrm{aq})=\mathrm{H}_{1}(\mathrm{aq})-\mathrm{H}_{1}^{*}(\lambda)\]The chemical potential of a salt \(j\) in aqueous solution is given by equation (e). \[\mu_{j}(a q)=\mu_{j}^{0}(a q)+v \, R \, T \, \ln \left(Q \, m_{j} \, \gamma_{\pm} / m^{0}\right)\]where, at all \(\mathrm{T}\) and \(\mathrm{p}\), \[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{\pm}=1.0\]Using the Gibbs-Helmholtz Equation, \[\mathrm{H}_{\mathrm{j}}(\mathrm{aq})=\mathrm{H}_{\mathrm{j}}^{0}(\mathrm{aq})-\mathrm{v} \, \mathrm{R} \, \mathrm{T}^{2} \,\left[\partial \ln \left(\gamma_{\pm}\right) / \partial \mathrm{T}\right]_{\mathrm{p}}\]For a salt solution having ideal thermodynamic properties, \[\mathrm{H}_{\mathrm{j}}(\mathrm{aq} ; \mathrm{id})=\mathrm{H}_{\mathrm{j}}^{0}(\mathrm{aq})=\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})\]By definition, the relative partial molar enthalpy of the salt, \[\mathrm{L}_{\mathrm{j}}(\mathrm{aq})=\mathrm{H}_{\mathrm{j}}(\mathrm{aq})-\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})\]In the limit of infinite dilution the relative partial molar enthalpy of a salt is zero. Thus \[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{L}_{\mathrm{j}}(\mathrm{aq})=0\]For a solution prepared using \(\mathrm{w}_{1} \mathrm{~kg}\) of water(\(\lambda\)), \[\begin{aligned} \mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1} \mathrm{~kg}\right) &=\mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\lambda)+\mathrm{n}_{1} \, \mathrm{v} \, \mathrm{R} \, \mathrm{T}^{2} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,(\partial \phi / \partial \mathrm{T})_{\mathrm{p}} \\ +\mathrm{n}_{\mathrm{j}} \, \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{aq})-\mathrm{n}_{\mathrm{j}} \, \mathrm{v} \, \mathrm{R} \, \mathrm{T}^{2} \,\left[\partial \ln \left(\gamma_{\pm}\right) / \partial \mathrm{T}\right]_{\mathrm{p}} \end{aligned}\]But \[\mathrm{n}_{1} \, \mathrm{m}_{\mathrm{j}}=\mathrm{n}_{1} \, \mathrm{n}_{\mathrm{j}} / \mathrm{w}_{1}=\mathrm{n}_{\mathrm{j}} / \mathrm{M}_{1}\]\[\begin{aligned} \mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1} \mathrm{~kg}\right)=\mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\lambda)+\mathrm{n}_{\mathrm{j}} \, \mathrm{v} \, \mathrm{R} \, \mathrm{T}^{2} \,(\partial \phi / \partial \mathrm{T})_{\mathrm{p}} \\ &+\mathrm{n}_{\mathrm{j}} \, \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{aq})-\mathrm{n}_{\mathrm{j}} \, \mathrm{v} \, \mathrm{R} \, \mathrm{T}^{2} \,\left[\partial \ln \left(\gamma_{\pm}\right) / \partial \mathrm{T}\right]_{\mathrm{p}} \end{aligned}\]\[\begin{aligned} \mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1} \mathrm{~kg}\right)=& \mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\lambda)+\mathrm{n}_{\mathrm{j}} \,\left\{\mathrm{v} \, \mathrm{R} \, \mathrm{T}^{2} \,(\partial \phi / \partial \mathrm{T})_{\mathrm{p}}\right.\\ &\left.+\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})-\mathrm{v} \, \mathrm{R} \, \mathrm{T}^{2} \,\left[\partial \ln \left(\gamma_{\pm}\right) / \partial \mathrm{T}\right]_{\mathrm{p}}\right\} \end{aligned}\]The term in the brackets {….} defines the apparent molar enthalpy of salt \(j\), \(\phi\left(\mathrm{H}_{\mathrm{j}}\right)\). \[\phi\left(\mathrm{H}_{\mathrm{j}}\right)=\mathrm{v} \, \mathrm{R} \, \mathrm{T}^{2} \,(\partial \phi / \partial \mathrm{T})_{\mathrm{p}}+\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})-\mathrm{v} \, \mathrm{R} \, \mathrm{T}^{2} \,\left[\partial \ln \left(\gamma_{\pm}\right) / \partial \mathrm{T}\right]_{\mathrm{p}}\]Using equation (o), \[\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1} \mathrm{~kg}\right)=\mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\lambda)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{H}_{\mathrm{j}}\right)\]In other words we have grouped all the parameters describing the properties of the salt in a real solution under a single term, \(\phi(\mathrm{H}_{j})\). For a solution prepared using \(1 \mathrm{~kg}\) of water, \[\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{H}_{1}^{*}(\lambda)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{H}_{\mathrm{j}}\right)\]At all \(\mathrm{T}\) and \(\mathrm{p}\), \[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{\pm}=1 ; \ln \left(\gamma_{\pm}\right)=0 ; \phi=1\]Hence, \[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right)\left[\partial \ln \left(\gamma_{\pm}\right) / \partial \mathrm{T}\right]_{\mathrm{p}}=[\partial \phi / \partial \mathrm{T}]_{\mathrm{p}}=0\]\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi\left(\mathrm{H}_{\mathrm{j}}\right)=\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})\]This page titled 1.8.9: Enthalpies- Salt Solutions- Apparent Molar- Partial Molar and Relative Enthalpies is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.8.10: Enthalpy of Solutions- Salts

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The chemical substance \(\mathrm{NaCl}(\mathrm{s})\) is a hard crystalline solid with a high melting point, \(1074 \mathrm{K}\). All the more remarkable therefore is the observation when a few crystals are dropped into water(\(\ell\)) the crystals disintegrate into ions with no dramatic change in the temperature of the water. One concludes that the intensity of interactions between ions in the crystal is comparable to that between ions and water molecules in the aqueous solution. Not surprisingly therefore enthalpies of solutions have been extensively investigated.The enthalpy of an aqueous solution prepared at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\) using \(\mathrm{n}_{1}\) moles of water and \(\mathrm{n}_{j}\) moles of salt is given by equation (a) where \(\phi\left(\mathrm{H}_{j}\right)\) is the apparent molar enthalpy of salt \(j\) in solution. \[\mathrm{H}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{H}_{\mathrm{j}}\right)\]Before the solution was prepared the enthalpy of the system, \(\mathrm{H}(\mathrm{no}-\mathrm{mix})\) is given by equation (b) where \(\mathrm{H}_{\mathrm{j}}^{*}(\mathrm{~s})\) is the molar enthalpy of solid salt \(j\). \[\mathrm{H}(\mathrm{no}-\operatorname{mix})=\mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \mathrm{H}_{\mathrm{j}}^{*}(\mathrm{~s})\]Using an isobaric calorimeter, heat \(\mathrm{q}\) is recorded for the solution process. \[\mathrm{q}=\mathrm{H}(\mathrm{aq})-\mathrm{H}(\mathrm{no}-\mathrm{mix})=\mathrm{n}_{\mathrm{j}} \,\left[\phi\left(\mathrm{H}_{\mathrm{j}}\right)-\mathrm{H}_{\mathrm{j}}^{*}(\mathrm{~s})\right]\]Or \[\phi\left(H_{j}\right)-H_{j}^{*}(s)=q / n_{j}\]In many studies using sensitive calorimeters (\(\mathrm{q}/\mathrm{n}_{j}\)) can be recorded for the production of quite dilute solutions such that \(\phi\left(\mathrm{H}_{j}\right)\) is effectively equal to \(\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})\). In other cases \(\Delta_{s \ln } \mathrm{H}(\mathrm{s} \rightarrow \mathrm{aq})\) is found to depend on the molality of the resultant solution. One procedure fits the measured enthalpy of solution to a quadratic in the molality of salt. \[\Delta_{s \ln } \mathrm{H}(\mathrm{s} \rightarrow \mathrm{aq})=\Delta_{\mathrm{sin}} \mathrm{H}^{0}(\mathrm{~s} \rightarrow \mathrm{aq})+\mathrm{A} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)+\mathrm{B} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2}\]In other cases the Debye-Huckel limiting law is used as a basis for extrapolating \(\Delta_{\mathrm{s} \ln } \mathrm{H}(\mathrm{s} \rightarrow \mathrm{aq})\) to the required infinite dilution value.Footnotes C. V. Krishnan and H. L. Friedman, J. Phys.Chem.,1970,74,3900 . e.g. Om. N. Bhatnagar and C. M. Criss, J. Phys Chem.,1969,73,174.This page titled 1.8.10: Enthalpy of Solutions- Salts is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.8.11: Enthalpies- Salt Solutions- Dilution

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One mole of salt in solution can, with complete dissociation, produce \(ν\) moles of ions. Hence for a given solution prepared using \(\mathrm{n}_{1}\) moles of water(\(\ell\)) and \(\mathrm{n}_{j}\) moles of salt, the enthalpy \(\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1} \mathrm{~kg}\right)\) is given by equation (a). \[\begin{aligned} &\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1} \mathrm{~kg}\right)= \\ &\begin{aligned} \mathrm{n}_{1} \, & {\left[\mathrm{H}_{1}^{*}(\ell)+\mathrm{v} \, \mathrm{R} \, \mathrm{T}^{2} \, \mathrm{m}_{\mathrm{j}} \,(\partial \phi / \partial \mathrm{T})_{\mathrm{p}}\right] } \\ &+\mathrm{n}_{\mathrm{j}} \,\left[\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})-\mathrm{v} \, \mathrm{R} \, \mathrm{T}^{2} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\partial \ln \gamma_{\pm} / \partial \mathrm{T}\right)_{\mathrm{p}}\right] \end{aligned} \end{aligned}\]With a little re-arrangement, \[\begin{aligned} &\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1} \mathrm{~kg}\right)= \\ &\quad \mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\ell) \\ &\quad+\mathrm{n}_{\mathrm{j}} \,\left[\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})-\mathrm{v} \, \mathrm{R} \, \mathrm{T}^{2} \,\left(\partial \ln \gamma_{\pm} / \partial \mathrm{T}\right)_{\mathrm{p}}+\mathrm{v} \, \mathrm{R} \, \mathrm{T}^{2} \,(\partial \phi / \partial \mathrm{T})_{\mathrm{p}}\right] \end{aligned}\]The terms within the brackets [….] define the apparent molar enthalpy of salt \(j\) in aqueous solution, \(\phi\left(\mathrm{H}_{j}\right)\). \[\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1} \mathrm{~kg}\right)=\mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{H}_{\mathrm{j}}\right)\]\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi\left(\mathrm{H}_{\mathrm{j}}\right)=\phi\left(\mathrm{H}_{\mathrm{j}}\right)^{\infty}=\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})\]By definition \[\mathrm{L}\left(\mathrm{aq} ; \mathrm{w}_{1} \mathrm{~kg}\right)=\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1} \mathrm{~kg}\right)-\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1} \mathrm{~kg} ; \mathrm{id}\right)\]\[\mathrm{L}_{1}(\mathrm{aq})=\mathrm{H}_{1}(\mathrm{aq})-\mathrm{H}_{1}^{*}(\ell)\]\[\mathrm{L}_{\mathrm{j}}(\mathrm{aq})=\mathrm{H}_{\mathrm{j}}(\mathrm{aq})-\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})\]\[\mathrm{L}\left(\mathrm{aq} ; \mathrm{w}_{1} \mathrm{~kg}\right)=\mathrm{n}_{1} \, \mathrm{L}_{1}(\mathrm{aq})+\mathrm{n}_{\mathrm{j}} \, \mathrm{L}_{\mathrm{j}}(\mathrm{aq})=\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{L}_{\mathrm{j}}\right)\]Thus, \[\mathrm{L}\left(\mathrm{aq} ; \mathrm{w}_{1} \mathrm{~kg} ; \mathrm{id}\right)=0\]Equation (e) forms the basis of comments on changes in enthalpy when a salt solution is diluted by adding \(\Delta \mathrm{n}_{1}\) moles of water(\(\ell\)). Hence \[\begin{aligned} \Delta_{\text {dil }} \mathrm{H}=\left[\left(\mathrm{n}_{1}\right.\right.&\left.\left.+\Delta \mathrm{n}_{1}\right) \, \mathrm{H}_{1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{H}_{\mathrm{j}}-\text { final }\right)\right] \\ &-\left[\mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{H}_{\mathrm{j}}-\text { initial }\right)\right]-\Delta \mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\ell) \end{aligned}\]Or, \[\Delta_{\text {dil }} \mathrm{H}=\mathrm{n}_{\mathrm{j}} \,\left[\phi\left(\mathrm{H}_{\mathrm{j}}-\text { final }\right)-\phi\left(\mathrm{H}_{\mathrm{j}}-\text { initial }\right)\right]\]If in a given experiment where ‘\(\mathrm{n}_{j} = 1 \mathrm{~mol}\)’ and \(\Delta \mathrm{n}_{1}\) is large such that \(\phi\left(\mathrm{H}_{\mathrm{j}}-\text { final }\right)\) equals \(\phi\left(\mathrm{H}_{\mathrm{j}}-\text { final }\right)^{\infty}\), equation (k) is re-written as shown in equation (l). Then, \[\Delta_{\text {dil }} \mathrm{H}\left(\mathrm{n}_{\mathrm{j}}=1 \mathrm{~mol}\right)=-\left[\phi\left(\mathrm{H}_{\mathrm{j}}\right)-\phi\left(\mathrm{H}_{\mathrm{j}}\right)^{\infty}\right]\]Or, \[\Delta_{\mathrm{dil}} \mathrm{H}\left(\mathrm{n}_{\mathrm{j}}=1 \mathrm{~mol}\right)=-\phi\left(\mathrm{L}_{\mathrm{j}}\right)\]If for such a dilution, heat passes from the surroundings into the system , \(\Delta_{\text {dil }} \mathrm{H}\left(\mathrm{n}_{\mathrm{j}}=1 \mathrm{~mol}\right)\) is positive and \(\phi\left(L_{j}\right)\) is negative. Thus direct calorimetric measurement of \(\Delta_{\mathrm{dil}} \mathrm{H}\left(\mathrm{n}_{\mathrm{j}}=1 \mathrm{~mol}\right)\) yields the relative apparent molar enthalpy of the salt in solution at molality \(\mathrm{m}_{j}\).However we need to comment in more detail on the analysis of heats of dilution for salt solutions. We envisage a situation where a calorimeter records the heat associated with dilution of a given salt solution from an initial molality \(\mathrm{m}_{\mathrm{i}}\) to a final molality \(\mathrm{m}_{\mathrm{f}}\). A data set often includes pairs of \(\mathrm{m}_{\mathrm{i}}-\mathrm{m}_{\mathrm{f}}\) values together with the accompanying enthalpy change, \(\Delta \mathrm{H}(\text { old } \rightarrow \text { new })\) which yields the difference in apparent molar enthalpies of the two salt solutions, cf. equation (k). Thus \[\Delta \mathrm{H}(\text { old } \rightarrow \text { new })=\mathrm{n}_{\mathrm{j}} \,\left[\phi\left(\mathrm{H}_{\mathrm{j}} ; \text { new }\right)-\phi\left(\mathrm{H}_{\mathrm{j}} ; \text { old }\right)\right]\]Or, \[\left[\phi\left(\mathrm{H}_{\mathrm{j}} ; \text { new }\right)-\phi\left(\mathrm{H}_{\mathrm{j}} ; \text { old }\right)\right]=\Delta \mathrm{H}(\text { old } \rightarrow \text { new }) / \mathrm{n}_{\mathrm{j}}\]We note that the molalities of the ‘new’ and ‘old’ solutions differ and therefore the contributions of ion-ion interactions to the apparent molar enthalpies differ. In the event that sufficient solvent is added that \(\mathrm{m}_{\mathrm{f}}\) is effectively zero, then \(\phi\left(\mathrm{H}_{\mathrm{j}} ; \text { new }\right)\) is the infinitely dilute property \(\phi\left(\mathrm{H}_{\mathrm{j}}\right)^{\infty}\).The excess enthalpy \(\mathrm{H}^{\mathrm{E}}\) is given by equation (p). \[\mathrm{H}^{\mathrm{E}}=\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{L}_{\mathrm{j}}\right)=\mathrm{m}_{\mathrm{j}} \,\left[\phi\left(\mathrm{H}_{\mathrm{j}}\right)-\phi\left(\mathrm{H}_{\mathrm{j}}\right)^{\infty}\right]\]For salt solutions \(\mathrm{H}^{\mathrm{E}}\) is not negligible as a consequence of intense ion-ion interaction. However in order to calculate \(\mathrm{H}^{\mathrm{E}}\) and hence obtain an indication of the strength of these interactions we return to equation (m) and note that experiment yields the difference between \(\phi\left(\mathrm{H}_{\mathrm{j}} ; \text { new }\right)\) and \(\phi\left(\mathrm{H}_{\mathrm{j}} ; \text { old }\right)\). Since there are no ion-ion interactions at infinite dilution, the difference \(\left[\phi\left(\mathrm{H}_{\mathrm{j}}\right)-\phi\left(\mathrm{H}_{\mathrm{j}}\right)^{\infty}\right]\left\{\text { i.e. } \phi\left(\mathrm{L}_{\mathrm{j}}\right)\right\}\) is obtained as a function of \(\mathrm{m}_{j}\)(old).A key component of the difference \(\left[\phi\left(\mathrm{H}_{\mathrm{j}} ; \text { new }\right)-\phi\left(\mathrm{H}_{\mathrm{j}} ; \text { old }\right)\right]\) is charge-charge interaction in the real solutions which is calculated using, for example, the Debye-Huckel equations. These equations start out with a relation between \(\ln \left(\gamma_{\pm}\right)\) where \(\gamma_{\pm}\) is the mean ionic activity coefficient and I the ionic strength (or, in a simple solution, molality \(\mathrm{m}_{j}\)). These equations are differentiated with respect to temperature (at fixed pressure) requiring therefore the corresponding dependences of molar volume \(\mathrm{V}_{1}^{*}(\ell)\) and relative permittivity \(\varepsilon_{\mathrm{r}}^{*}(\ell)\) of the solvent. Not surprisingly a large chemical literature describes a range of procedures for analysing the calorimetric results. In most cases the starting point is the Debye-Huckel Limiting Law.For \(\mathrm{Bu}_{4}\mathrm{N}^{+}\mathrm{Br}^{-}(\mathrm{aq})\), the dependence of \(\phi\left(\mathrm{L}_{\mathrm{j}}\right)\) on \(\mathrm{m}_{j}\) was expressed using equation (q). \(\mathrm{S}_{\mathrm{H}}\) was taken from the compilation published by Helgeson and Kirkham. \[\phi\left(L_{j}\right)=S_{H} \,\left(m_{j} / m^{0}\right)^{1 / 2}+\sum B_{i} \,\left(m_{j} / m^{0}\right)^{(i+1) / 2}\]For \(\left(\mathrm{HOC}_{2}\mathrm{H}_{4}\right)_{4}\mathrm{N}^{+}\mathrm{Br}^{-}(\mathrm{aq})\), an extended Debye –Huckel equation was used having the following form. \[\begin{gathered} \phi\left(\mathrm{L}_{\mathrm{j}}\right)=\mathrm{S}_{\mathrm{H}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2} \,\left[\frac{1}{1+\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}}-\frac{\sigma \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}}{3}\right] \\ +\mathrm{B} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)+\mathrm{C} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{3 / 2} \end{gathered}\]The dependence of \(\phi\left(\mathrm{L}_{\mathrm{j}}\right)\) on \(\mathrm{m}_{j}\) for 1,1’-dimethyl-4,4’-dipyridinium dichloride(aq; \(298 \mathrm{~K}\)) was expressed using a simple polynomial in \(\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}\).The Pitzer equations describing the properties of salt solutions also provide a basis for examining the enthalpies of dilution of, for example, \(\mathrm{NaCl}(\mathrm{aq})\). An interesting group of papers compares relative apparent molar enthalpies of salts in \(\mathrm{D}_{2}\mathrm{O}\) and \(\mathrm{H}_{2}\mathrm{O}\); i.e. \(\phi\left(\mathrm{L}_{\mathrm{j}} ; \mathrm{D}_{2} \mathrm{O}\right)-\phi\left(\mathrm{L}_{\mathrm{j}} ; \mathrm{H}_{2} \mathrm{O}\right)\). The compositions of the salt solutions are expressed in aquamolalities; i.e. \(\mathrm{m}_{j}\) moles of salt in \(55.1\) moles of solvent. The difference is expressed as a quadratic in aqueous molality using Kerwin’s equation. \[\phi\left(\mathrm{L}_{\mathrm{j}} ; \mathrm{H}_{2} \mathrm{O} \rightarrow \mathrm{D}_{2} \mathrm{O}\right)=\mathrm{k}_{1} \, \mathrm{m}_{\mathrm{j}}+\mathrm{k}_{2} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2}\]Further examples are listed in reference.Footnotes J. E. Mayrath and R. H. Wood, J. Chem. Thermodyn., 1983,15,625; and references therein. H. C. Helgeson and D. H Kirkham, Am. J. Sci.,1974,274,1199. G. Perron and J. E. Desnoyers, J. Solution Chem.,1972,1,537. R. H. Busey, H. F. Holmes and R. E. Mesmer, J.Chem.Thermodyn., 1984,16, 343.This page titled 1.8.11: Enthalpies- Salt Solutions- Dilution is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.8.12: Enthalpies- Born-Bjerrum Equation- Salt Solutions

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It is generally assumed that the Born Equation yields a difference in Gibbs energies rather than Helmholtz energies and so one can use the Gibbs-Helmholtz Equation for the dependence on temperature at fixed pressure to yield the Born-Bjerrum Equation, assuming that (\(\mathrm{dr}_{\mathrm{j}} / \mathrm{dT}\)) is zero. \[\begin{aligned} &\Delta(\mathrm{pfg} \rightarrow \mathrm{s} \ln ) \mathrm{H}_{\mathrm{j}}\left(\mathrm{c}_{\mathrm{j}}=1 \mathrm{moldm} \mathrm{dm}^{-3} ; \mathrm{id} ; \mathrm{T} ; \mathrm{p}\right)= \\ &\quad-\left[\mathrm{N}_{\mathrm{A}} \,\left(\mathrm{z}_{\mathrm{j}} \, \mathrm{e}\right)^{2} / 8 \, \pi \, \mathrm{r}_{\mathrm{j}} \, \varepsilon_{0}\right] \,\left[1-\left(1 / \varepsilon_{\mathrm{r}}\right)-\left(\mathrm{T} / \varepsilon_{\mathrm{r}}\right) \,\left(\partial \ln \varepsilon_{\mathrm{r}} / \partial \mathrm{T}\right)_{\mathrm{p}}\right] \end{aligned}\]In fact an early calorimetric study showed that in terms of predicting the enthalpies of solution for salts, the Born equation is inadequate, often predicting the wrong sign.Differentiation of equation (a) with respect to temperature yields an equation for the partial molar isobaric heat capacity of ion \(j\) in a solution having ideal thermodynamic properties. \[\begin{aligned} &C_{p j}\left(\operatorname{sln} ; c_{j}=1 \mathrm{~mol} \mathrm{dm}{ }^{-3} ; \mathrm{id} ; \mathrm{T} ; \mathrm{p}\right) \\ &=-\left[\mathrm{N}_{\mathrm{A}} \,\left(\mathrm{z}_{\mathrm{j}} \, \mathrm{e}\right)^{2} / 8 \, \pi \, \mathrm{r}_{\mathrm{j}} \, \varepsilon_{0}\right] \,\left[\partial\left\{\left(1 / \varepsilon_{\mathrm{r}}\right)+\left(\mathrm{T} / \varepsilon_{\mathrm{r}}\right) \, \partial \ln \varepsilon_{\mathrm{r}} / \partial \mathrm{T}\right\} / \partial \mathrm{T}\right] \end{aligned}\]Footnotes F. A. Askew, E. Bullock, H. T. Smith, R. K. Tinkler, O. Gatty and J. H. Wolfenden, J. Chem. Soc., 1934, 1368. For estimation of single in enthalpies see M. Booij and G. Somsen, Electrochim Acta,1983,28,1883.This page titled 1.8.12: Enthalpies- Born-Bjerrum Equation- Salt Solutions is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.8.13: Enthalpies- Liquid Mixtures

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For an ideal binary liquid mixture the Gibbs energy at temperature T is given by equation (a). \[\mathrm{G}(\operatorname{mix} ; \mathrm{id})=\mathrm{n}_{1} \,\left[\mu_{1}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right)\right]+\mathrm{n}_{2} \,\left[\mu_{2}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{2}\right)\right]\]From the Gibbs-Helmholtz equation, \[\mathrm{H}(\operatorname{mix} ; \mathrm{id})=\mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\ell)+\mathrm{n}_{2} \, \mathrm{H}_{2}^{*}(\ell)\]Hence for an ideal binary liquid mixture, \[\mathrm{H}_{1}(\operatorname{mix} ; \mathrm{id})=\mathrm{H}_{1}^{*}(\ell) \text { and } \mathrm{H}_{2}(\operatorname{mix} ; \mathrm{id})=\mathrm{H}_{2}^{*}(\ell)\]The molar enthalpy of a real binary liquid mixture is given by equation (d). \[\mathrm{H}_{\mathrm{m}}=\mathrm{x}_{1} \, \mathrm{H}_{1}(\operatorname{mix})+\mathrm{x}_{2} \, \mathrm{H}_{2}(\operatorname{mix})\]Therefore the molar enthalpy of mixing for a real binary liquid mixture is given by equation (e). \[\Delta_{\text {mix }} H_{m}=x_{1} \,\left[H_{1}(\operatorname{mix})-H_{1}^{*}(\ell)\right]+x_{2} \,\left[H_{2}(\operatorname{mix})-H_{2}^{*}(\ell]\right.\]Significantly equations (b) and (e) show that the molar enthalpy of mixing of an ideal binary liquid mixture, \(\Delta_{\text {mix }} H_{m}(\mathrm{id})\) is zero. The latter condition offers an important point of reference for isobaric calorimetry. If we discover that the mixing of two liquids (at constant pressure) is not zero, the measured molar heat of mixing [\(=\Delta_{\text {mix }} \mathrm{H}_{\mathrm{m}}\)] is an immediate indicator of the extent to which the properties of a given mixture are not ideal.Nevertheless it is important to set down a link between the measured enthalpies of mixing with the activity coefficients of two liquid components. To this end we start with the equation for the chemical potentials of liquid component 1 in a liquid mixture at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\) (which is close to ambient); equation (f). \[\mu_{1}(\mathrm{mix})=\mu_{1}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right)\]where \[\operatorname{limit}\left(x_{1} \rightarrow 1\right) f_{1}=1 \text { at all } T \text { and } p \text {. }\]The Gibbs - Helmholtz equation yields an equation for the partial molar enthalpy of component 1 in the liquid mixture. Thus \[\mathrm{H}_{1}(\operatorname{mix})=\mathrm{H}_{1}^{*}(\ell)-\mathrm{R} \, \mathrm{T}^{2} \,\left[\partial \ln \left(\mathrm{f}_{1}\right) / \partial \mathrm{T}\right]_{\mathrm{p}}\]Similarly, \[\mathrm{H}_{2}(\operatorname{mix})=\mathrm{H}_{2}^{*}(\ell)-\mathrm{R} \, \mathrm{T}^{2} \,\left[\partial \ln \left(\mathrm{f}_{2}\right) / \partial \mathrm{T}\right]_{\mathrm{p}}\]Hence, \[\begin{aligned} &\mathrm{H}_{\mathrm{m}}(\operatorname{mix})= \\ &\quad \mathrm{H}_{\mathrm{m}}(\operatorname{mix} ; \mathrm{id})-\mathrm{R} \, \mathrm{T}^{2} \,\left\{\mathrm{x}_{1} \,\left[\partial \ln \left(\mathrm{f}_{1}\right) / \partial \mathrm{T}\right]_{\mathrm{p}}+\mathrm{x}_{2} \,\left[\partial \ln \left(\mathrm{f}_{2}\right) / \partial \mathrm{T}\right]_{\mathrm{p}}\right\} \end{aligned}\]We also obtain equations for the excess molar enthalpies of the two components (at defined \(\mathrm{T}\) and \(\mathrm{p}\)). \[\mathrm{H}_{1}^{\mathrm{E}}(\mathrm{mix})=-\mathrm{R} \, \mathrm{T}^{2} \,\left[\partial \ln \left(\mathrm{f}_{1}\right) / \partial \mathrm{T}\right]_{\mathrm{p}}\]and \[\mathrm{H}_{2}^{\mathrm{E}}(\mathrm{mix})=-\mathrm{R} \, \mathrm{T}^{2} \,\left[\partial \ln \left(\mathrm{f}_{2}\right) / \partial \mathrm{T}\right]_{\mathrm{p}}\]The excess molar enthalpy, \[\mathrm{H}_{\mathrm{m}}^{\mathrm{E}}(\text { mix })=-\mathrm{R} \, \mathrm{T}^{2} \,\left\{\mathrm{x}_{1} \,\left[\partial \ln \left(\mathrm{f}_{1}\right) / \partial \mathrm{T}\right]_{\mathrm{p}}+\mathrm{x}_{2} \,\left[\partial \ln \left(\mathrm{f}_{2}\right) / \partial \mathrm{T}\right]_{\mathrm{p}}\right\}\]At fixed pressure, the differential dependence of \(\mathrm{H}_{\mathrm{m}}^{\mathrm{E}}(\operatorname{mix})\) on temperature yields the corresponding excess isobaric heat capacity of mixing.Footnotes J. B. Ott and C. J. Wormald, Experimental Thermodynamics, IUPAC Chemical Data Series, No. 39, ed. K. N. Marsh and P. A. G. O’Hara, Blackwell, Oxford, 1994, chapter 8.This page titled 1.8.13: Enthalpies- Liquid Mixtures is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.9.1: Entropy - Second Law of Thermodynamics

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A closed system (in addition to the thermodynamic energy \(\mathrm{U}\)) is characterised by two functions of state.The concept of entropy is particularly valuable in commenting on the direction of spontaneous chemical reaction.The Second Law of Thermodynamics states that for spontaneous chemical reaction in a closed system. \[\mathrm{T} \, \mathrm{dS}=\mathrm{q}+\mathrm{A} \, \mathrm{d} \xi\]where \[\mathrm{A} \, \mathrm{d} \xi \geq 0\] These two equations comprise the Second Law. The product of the affinity for spontaneous chemical reaction and the extent of chemical reaction (i.e. accompanying change in composition) can never be negative. This is the thermodynamic `selection rule' for which there are absolutely no exceptions. Chemists base their analysis of chemical processes on the certainty of this rule (or axiom). The key point is the sense of direction of spontaneous change which emerges.In the event that the affinity for spontaneous change is zero, no change in chemical composition occurs in a closed system; i.e. \(\mathrm{d}\xi\) is zero and the rate of change \(\mathrm{d}\xi / \mathrm{dt}\) is zero. The system and surroundings are in equilibrium. Hence \[\mathrm{T} \, \mathrm{dS}=\mathrm{q} \quad(\text { at } \mathrm{A}=0)\]The latter equation has a particular set of applications. We imagine a closed system for which the affinity for spontaneous change is zero. We perturb the system by a change in pressure such that there is a corresponding change in composition-organisation in the system. However as we change the pressure along a certain pathway, we assert that the affinity for spontaneous change is always zero. Then between states I and II, at constant temperature \(\mathrm{T}\) \[\mathrm{T} \, \mathrm{S}(\mathrm{II})-\mathrm{T} \, \mathrm{S}(\mathrm{I})=\mathrm{T} \, \int_{\text {statel }}^{\text {state II }} \mathrm{dS}=\mathrm{q}\]The pathway between these two states is called reversible or an equilibrium transformation. In fact the change in pressure must be carried out infinitely slowly because we must allow the chemical composition/molecular organisation to hold to the condition that there is no affinity for spontaneous change.All processes in the real world (i.e. all natural processes) are irreversible; there is a defined direction for spontaneous changes. Many authors offer an explanation of the property, entropy. One view is that to attempt an explanation of the "meaning" of entropy is a complete waste of time (M. L. McGlashan, J. Chem. Educ., 1966,43, 226). A wide-ranging discussion is given by P.L. Huyskens and G.G. Siegel,Bull. Soc.Chem.Belg., 1988,97, 809, 815 and 823.E.A. Guggenheim [Thermodynamics, North-Holland, Amsterdam, 1950]. This monograph is often cited for the following bold statement (page 11): "There exists a function \(\mathrm{S}\) of the state of a system called the entropy of the system .....".H. Margenau [The Nature of Physical Reality, McGraw-Hill, New York, 1950] states 'Entropy is as definite and clear a thing as other thermodynamic quantities'.The common view in introductory chemistry textbooks for many years has been that entropy is a measurement of randomness and/or disorder. However this view is unhelpful if not meaningless [E. T. Jaynes, Am. J.Phys.,1965,33,391; F. L. Lambert, J. Chem. Educ.,1999,76,1385; 2002,79,187.] indeed a myth [W Brostow, Science 1972,178,211.] and an educational disaster [M. Sozlibir, J.K.Bennett, J. Chem. Educ.,2007,84,1204.]The generally accepted view [F. L. Lambert, J.Chem.Educ.,2002,79,1241] is that an entropy increase results from the energy of molecular motion becoming more dispersed or ’spread out’; e.g. in the two classic examples of a system being warmed by hotter surroundings or, isothermally, when a system’s molecules have greater volume for their energetic movement the energy of molecular motion becoming more dispersed or ‘spread out’; e.g. in the two classic examples of as system being warmed by hotter surroundings or, isothermally, when a system’s molecules have greater volume for their energetic movement. The concept is exceptionally valuable because entropy increase can be seen by chemists as simply involving the energy associated with mobile molecules spreading out more in three-dimensional space, whether a new total system of ‘less hot plus once-cooler’ or isothermally in a larger volume. This simple view is equivalent to the dispersal of energy in phase space. In quantum mechanical terms, ’energy dispersal’ means that a system will come to equilibrium in a final state that is optimal because it affords a maximal number of accessible energy arrangements. Even though the system can be in only one arrangement at one instant, its energy is truly dispersed because at the next instant it can be in a different arrangement: this amounts to a ‘temporal dance’ over a very small fraction of the hyper-astronomical number of microstates predicted by the Boltzmann relation. The account given here is based on a written comments in correspondence from F. L. Lambert. From equation (a), \(\mathrm{T}. \left.\mathrm{dS}=[\mathrm{K}]-\left[\mathrm{JK}^{-1}\right]=[]\right]+\left[\mathrm{J} \mathrm{} \mathrm{mol}^{-1}\right]=[\mathrm{J}]\) Equations (a) and (b) can be re-expressed in terms of the contribution to the change in entropy \(\mathrm{dS}\) by a process (e.g. chemical reaction) within the system \(\mathrm{d}_{\mathrm{i}}\mathrm{S}\). Then \[\mathrm{T} \, \mathrm{dS}=\mathrm{q}+\mathrm{T} \, \mathrm{d}_{\mathrm{s}} \mathrm{S}\]where \[\mathrm{d}_{\imath} \mathrm{S}>0\]Equation (B) is the Second Law in that \(\mathrm{d}_{\mathrm{i}}\mathrm{S}\) cannot be negative. For a reversible process \(\mathrm{d}_{\mathrm{i}}\mathrm{S}=0\). But for all processes in the real world, \(\mathrm{d}_{\mathrm{i}}\mathrm{S}\) is positive. In other words all spontaneous processes occur in the direction whereby there is a positive contribution from \(\mathrm{d}_{\mathrm{i}}\mathrm{S}\) to the change in entropy \(\mathrm{dS}\). This concept of spontaneous change, coupled with the idea that changes occur in a predefined direction is linked with the idea that time is "one-sided". (a) I. Prigogine, From Being to Becoming, Freeman, San Francisco, 1980, page 6. (b) see also G. Nicolis and I. Prigogine, Self-Organization in Non-Equilibrium Systems, Wiley, New York, 1977. Equation (a) forms the basis of an oft-quoted comment. For an isolated system, \(\mathrm{q}\) is zero. Then \(\mathrm{T} \, \mathrm{dS}=\mathrm{A} \, \mathrm{d} \xi\) where \(\mathrm{A} \, \mathrm{d} \xi \geq 0\)So for all spontaneous processes in an isolated system, \(\mathrm{dS} >0\). This is the basis of the statement that the entropy of the universe is increasing if the universe can be treated as an isolated system. But these comments stray from immediate interests of chemists.This page titled 1.9.1: Entropy - Second 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.

1.9.2: Entropy- Dependence on Temperature and Pressure

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The volume of a given closed system at equilibrium prepared using \(\mathrm{n}_{1}\) moles of solvent (water) and \(\mathrm{n}_{j}\) moles of solute-\(j\) is defined by the set of independent variables shown in equation (a). \[\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}, \mathrm{A}=0, \xi^{\mathrm{eq}}\right]\]The same set of independent variables defines the entropy \(\mathrm{S}\). \[\mathrm{S}=\mathrm{S}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}, \mathrm{A}=0, \xi^{\mathrm{eq}}\right]\]We envisage that the system is displaced by a change in pressure along a path where the system remains at equilibrium (i.e. \(\mathrm{A} = 0\)) and the volume remains the same as defined by equation (a). In a plot of entropy against \(\mathrm{p}\), the gradient of the plot at the point defined by the independent variables, \(\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}, \mathrm{A}=0, \boldsymbol{\xi}^{\mathrm{eq}}\right]\) is given by equation (c).Isochoric \[\left(\frac{\partial S}{\partial p}\right)_{V, A=0}\]The set of derivatives is completed by the following partial derivatives.Isothermal \[\left(\frac{\partial \mathrm{S}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{A}=0}\]Isobaric \[\left(\frac{\partial S}{\partial T}\right)_{p, A=0}\]This page titled 1.9.2: Entropy- Dependence on Temperature and Pressure is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.9.3: Entropy and Spontaneous Reaction

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It is often stated that the entropy of a system is a maximum at equilibrium. This is not generally true and is certainly not the case for closed systems at either (a) fixed \(\mathrm{T}\) and \(\mathrm{p}\), or (b) fixed \(\mathrm{T}\) and \(\mathrm{V}\).We rewrite the Master Equation in the following way: \[\mathrm{dS}=(1 / \mathrm{T}) \, \mathrm{dU}+(\mathrm{p} / \mathrm{T}) \, \mathrm{dV}+(\mathrm{A} / \mathrm{T}) \, \mathrm{d} \xi ; \mathrm{A} \, \mathrm{d} \xi \geq \text { zero }\]Temperature \(\mathrm{T}\) is positive and non-zero. At constant energy and constant volume (i.e. isoenergetic and isochoric), spontaneous processes are accompanied by an increase in entropy. This statement is important in statistical thermodynamics where the condition, ‘constant \(\mathrm{U}\) and constant \(\mathrm{V}\)’ is important.The following equation defines the enthalpy \(\mathrm{H}\) of a closed system. \[\mathrm{H}=\mathrm{U}+\mathrm{p} \, \mathrm{V}\]Then \[\mathrm{dU}=\mathrm{dH}-\mathrm{p} \, \mathrm{dV}-\mathrm{V} \, \mathrm{dp}\]From equation (a), \[\mathrm{dS}=(1 / \mathrm{T}) \, \mathrm{dH}-(\mathrm{p} / \mathrm{T}) \, \mathrm{dV}-(\mathrm{V} / \mathrm{T}) \, \mathrm{dp}+(\mathrm{p} / \mathrm{T}) \, \mathrm{dV}+(\mathrm{A} / \mathrm{T}) \, \mathrm{d} \xi \text { with } \mathrm{A} \, \mathrm{d} \xi \geq \text { zero }\]Hence, \[\mathrm{dS}=(1 / \mathrm{T}) \, \mathrm{dH}-(\mathrm{V} / \mathrm{T}) \, \mathrm{dp}+(\mathrm{A} / \mathrm{T}) \, \mathrm{d} \xi ; \mathrm{A} \, \mathrm{d} \xi \geq \text { zero }\]Temperature \(\mathrm{T}\) is always positive. Hence at constant enthalpy and pressure (i.e. iso-enthalpic and isobaric) all spontaneous processes produce an increase in entropy.We have identified two sets of conditions under which an increase in entropy accompanies a spontaneous process. If we follow through a similar argument with respect to the Gibbs energy, the outcome is not straightforward. By definition, \[\mathrm{G}=\mathrm{H}-\mathrm{T} \, \mathrm{S}\]Then \[\mathrm{dG}=\mathrm{dH}-\mathrm{T} \, \mathrm{dS}-\mathrm{S} \, \mathrm{dT}\]Or, \[\mathrm{S}=-\mathrm{dG} / \mathrm{dT}+\mathrm{dH} / \mathrm{dT}-\mathrm{T} \, \mathrm{dS} / \mathrm{dT}\]But from equation (e) \[\mathrm{dH} / \mathrm{dT}=\mathrm{T} \, \mathrm{dS} / \mathrm{dT}+(\mathrm{V} / \mathrm{T}) \, \mathrm{dp} / \mathrm{dT}-(\mathrm{A} / \mathrm{T}) \, \mathrm{d} \xi / \mathrm{dT}\]Hence, \[\mathrm{S}=-(\mathrm{dG} / \mathrm{dT})+\mathrm{V} \,(\mathrm{dp} / \mathrm{dT})-\mathrm{A} \,(\mathrm{d} \xi / \mathrm{dT}) \text { with } \mathrm{A} \, \mathrm{d} \xi \geq \text { zero }\]Clearly no definite conclusions can be drawn about changes in entropy S under isobaric - isothermal conditions. We stress these points because again it is often tempting to link, misguidedly, entropies to the degree of ‘muddled-up-ness’. This is the basis of many explanations of entropy. For example, neither the volume nor energy of a deck of cards change on shuffling. Whether what actually happens on shuffling a new well-ordered deck of cards clarifies the meaning of entropy seems doubtful.This page titled 1.9.3: Entropy and Spontaneous Reaction is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.9.4: Entropy- Dependence on Temperature

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Using a calculus operation, the isochoric dependence of entropy of temperature is related to the corresponding isobaric dependence. Thus \[\left(\frac{\partial S}{\partial T}\right)_{V}=\left(\frac{\partial S}{\partial T}\right)_{p}-\left(\frac{\partial S}{\partial p}\right)_{T} \,\left(\frac{\partial p}{\partial V}\right)_{T} \,\left(\frac{\partial V}{\partial T}\right)_{p}\]But \[\left(\frac{\partial S}{\partial p}\right)_{T}=-\left(\frac{\partial V}{\partial T}\right)_{p}\]Hence, \[\left(\frac{\partial \mathrm{S}}{\partial \mathrm{T}}\right)_{\mathrm{V}}=\left(\frac{\partial \mathrm{S}}{\partial \mathrm{T}}\right)_{\mathrm{p}}+\left(\frac{\partial \mathrm{p}}{\partial \mathrm{V}}\right)_{\mathrm{T}} \,\left[\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{p}}\right]^{2}\]Or, \[\left(\frac{\partial S}{\partial T}\right)_{V}=\left(\frac{\partial S}{\partial T}\right)_{p}-\frac{\left(E_{p}\right)^{2}}{K_{T}}\]The final term in equation (c) contains the variable \(\mathrm{p}-\mathrm{V}-\mathrm{T}\).This page titled 1.9.4: Entropy- 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.

1.9.5: Entropies- Solutions- Limiting Partial Molar Entropies

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A key equation relates the chemical potential and partial molar entropy of solute-\(j\). For a given solute \(j\) in an aqueous solution, \[\mathrm{S}_{\mathrm{j}}(\mathrm{aq})=-\left[\partial \mu_{\mathrm{j}}(\mathrm{aq}) / \partial T\right]_{\mathrm{p}}\]In order to appreciate the importance of equation (a) we initially confine our attention to the properties of a solution whose thermodynamic properties are ideal. A given aqueous solution contains solute \(j\) at temperature \(\mathrm{T}\) and ambient pressure (which is close to the standard pressure). \[\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{id} ; \mathrm{T} ; \mathrm{p})=\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)\]Using equation (a), \[\mathrm{S}_{\mathrm{j}}(\mathrm{aq} ; \mathrm{id} ; \mathrm{T} ; \mathrm{p})=\mathrm{S}_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})-\mathrm{R} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)\]\(\mathrm{S}_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\) is the partial molar entropy of solute \(j\) in an ideal aqueous solution having unit molality. Therefore for both real and ideal solutions, \[\operatorname{limit}\left(m_{j} \rightarrow 0\right) S_{j}(a q ; T ; p)=+\infty\]In other words the limiting partial molar entropy for solute \(j\) is infinite. Interestingly if the aqueous solution contains two solutes \(j\) and \(\mathrm{k}\), then the following condition holds for solutions at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\). \[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0 ; \mathrm{m}_{\mathrm{k}} \rightarrow 0\right)\left[\mathrm{S}_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})-\mathrm{S}_{\mathrm{k}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\right]=\mathrm{S}_{\mathrm{j}}^{0}(\mathrm{aq})-\mathrm{S}_{\mathrm{k}}^{0}(\mathrm{aq})\]Similarly \[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0 ; \mathrm{m}_{\mathrm{k}} \rightarrow 0\right)\left[\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})-\mu_{\mathrm{k}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\right]=\mu_{\mathrm{j}}^{0}(\mathrm{aq})-\mu_{\mathrm{k}}^{0}(\mathrm{aq})\]The partial molar entropy of solute \(j\) in a real solution is given by equation (g). \[\begin{aligned} &S_{j}(a q ; T ; p)= \\ &S_{j}^{0}(a q ; T ; p)-R \, \ln \left(m_{j} / m^{0}\right)-R \, \ln \left(\gamma_{j}\right)-R \, T \,\left[\frac{\partial \ln \left(\gamma_{j}\right)}{\partial T}\right]_{p} \end{aligned}\]A given aqueous solution having thermodynamic properties which are ideal contains a solute \(j\), molality \(\mathrm{m}_{j}\). The partial molar entropy of the solvent is given by equation (h). \[\mathrm{S}_{1}(\mathrm{aq} ; \mathrm{id} ; \mathrm{T} ; \mathrm{p})=\mathrm{S}_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})+\mathrm{R} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\]Hence, \[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{S}_{1}(\mathrm{aq} ; \mathrm{id} ; \mathrm{T} ; \mathrm{p})=\mathrm{S}_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})\]When an ideal solution is diluted the partial molar entropy of the solvent approaches that of the pure solvent.A given aqueous solution is prepared using \(1 \mathrm{~kg}\) of solvent and \(\mathrm{m}_{j}\) moles of solute \(j\) at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\), the latter being close to the standard pressure. The entropy of the solution is given by equation (j). \[\mathrm{S}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\mathrm{M}_{1}^{-1} \, \mathrm{S}_{1}(\mathrm{aq})+\mathrm{m}_{\mathrm{j}} \, \mathrm{S}_{\mathrm{j}}(\mathrm{aq})\]In the event that the thermodynamic properties of the solution are ideal the entropy of the solution is given by equation (k). \[\begin{aligned} &\mathrm{S}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)= \\ &\quad \mathrm{M}_{1}^{-1} \,\left[\mathrm{S}_{1}^{*}(\ell)+\mathrm{R} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\right]+\mathrm{m}_{\mathrm{j}} \,\left[\mathrm{S}_{\mathrm{j}}^{0}(\mathrm{aq})-\mathrm{R} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)\right] \end{aligned}\]Interestingly, \[\begin{aligned} &\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{S}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)= \\ &\mathrm{M}_{1}^{-1} \, \mathrm{S}_{1}^{*}(\ell)+ \,\left[\mathrm{S}_{\mathrm{j}}^{0}(\mathrm{aq})-\mathrm{R} \, \ln \left(0 / \mathrm{m}^{0}\right)\right] \end{aligned}\]But \[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{m}_{\mathrm{j}} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)=0\]In other words the entropy for an ideal solution in the limit of infinite dilution is given by the entropy of the pure solvent. For a real solution, \[\mathrm{S}_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mathrm{S}_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})+\phi \, \mathrm{R} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}+\mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\frac{\partial \phi}{\partial \mathrm{T}}\right)_{\mathrm{p}}\]Hence, \[\begin{aligned} &\mathrm{S}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg} ; \mathrm{T} ; \mathrm{p}\right) \\ &=\mathrm{M}_{1}^{-1} \,\left[\mathrm{S}_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})+\phi \, \mathrm{R} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}+\mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\frac{\partial \phi}{\partial \mathrm{T}}\right)_{\mathrm{p}}\right] \\ &\quad+\mathrm{m}_{\mathrm{j}} \,\left[\mathrm{S}_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})-\mathrm{R} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)-\mathrm{R} \, \mathrm{T} \,\left(\frac{\partial \ln \left(\gamma_{\mathrm{j}}\right)}{\partial \mathrm{T}}\right)_{\mathrm{p}}\right] \end{aligned}\]The difference \(\left[\mathrm{S}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg} ; \mathrm{T} ; \mathrm{p}\right)-\mathrm{S}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{w}_{1}=1 \mathrm{~kg} ; \mathrm{T} ; \mathrm{p}\right)\right]\) yields the excess entropy, \(\mathrm{S}^{\mathrm{E}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg} ; \mathrm{T} ; \mathrm{p}\right)\).Footnotes For a salt solution, the standard partial molar entropy of the salt is given by the sum of standard partial molar entropies of the ions. For a 1:1 salt, \(S_{j}^{0}(a q)=S_{+}^{0}(a q)+S_{-}^{0}(a q)\) Y. Marcus and A. Loewenschuss, Annu. Rep. Prog. Chem., Ser. C, Phys. Chem., 1984, 81, chapter 4. For comments on the entropy of dilution of salt solutions see (a classic paper), H. S. Frank and A. L. Robinson, J. Chem. Phys.,1940,8,933. For comments on partial molar entropies of apolar solutes in aqueous solutions see, H. S. Frank and F. Franks, J. Chem. Phys.,1968,48,4746.This page titled 1.9.5: Entropies- Solutions- Limiting Partial Molar Entropies is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.9.6: Entropies- Liquid Mixtures

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The chemical potential of liquid component 1 in a binary liquid mixture (at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\), close to the standard pressure \(\mathrm{p}^{o}\)) is related to the mole fraction \(\mathrm{x}_{1}\) using equation (a). \[\mu_{1}(\operatorname{mix} ; \mathrm{id})=\mu_{1}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right)\]But \[\mathrm{S}_{1}(\operatorname{mix})=-\left[\partial \mu_{1}(\operatorname{mix}) / \partial \mathrm{T}\right]_{\mathrm{p}}\]Then, \[\mathrm{S}_{1}(\operatorname{mix} ; \mathrm{id})=\mathrm{S}_{1}^{*}(\ell)-\mathrm{R} \, \ln \left(\mathrm{x}_{1}\right)\)Hence the molar entropy of mixing of an ideal binary liquid mixture (at defined \(\mathrm{T}\) and \(\mathrm{p}\)) is given by equation (d). \[\Delta_{\text {mix }} S_{m}(\text { id })=-R \,\left[x_{1} \, \ln \left(x_{1}\right)+x_{2} \, \ln \left(x_{2}\right)\right]\]The chemical potential of component 1 in a real binary liquid mixture (at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\), near the standard pressure) is given by equation (e). \[\mu_{1}(\operatorname{mix})=\mu_{1}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right)\]Then \[\mathrm{S}_{1}(\mathrm{mix})=\mathrm{S}_{1}^{*}(\ell)-\mathrm{R} \, \ln \left(\mathrm{x}_{1}\right)-\mathrm{R} \, \ln \left(\mathrm{f}_{1}\right)-\mathrm{R} \, \mathrm{T} \,\left[\partial \ln \left(\mathrm{f}_{1}\right) / \partial \mathrm{T}\right]_{\mathrm{p}}\]\[S_{1}(\operatorname{mix})=S_{1}(\operatorname{mix} ; \text { id })-R \, \ln \left(f_{1}\right)-R \, T \,\left[\partial \ln \left(f_{1}\right) / \partial T\right]_{p}\]Similarly, \[\mathrm{S}_{2}(\mathrm{mix})=\mathrm{S}_{2}(\mathrm{mix} ; \mathrm{id})-\mathrm{R} \, \ln \left(\mathrm{f}_{2}\right)-\mathrm{R} \, \mathrm{T} \,\left[\partial \ln \left(\mathrm{f}_{2}\right) / \partial \mathrm{T}\right]_{\mathrm{p}}\]The extent to which the partial molar entropies for each liquid component in a given liquid mixture differs from that in the corresponding ideal mixture depends on the rational activity coefficient and its dependence on temperature. Hence we define excess partial molar entropies for both liquid components. \[S_{1}^{E}=-R \, \ln \left(f_{1}\right)-R \, T \,\left[\partial \ln \left(f_{1}\right) / \partial T\right]_{p}\]and \[S_{2}^{E}=-R \, \ln \left(f_{2}\right)-R \, T \,\left[\partial \ln \left(f_{2}\right) / \partial T\right]_{p}\]For the binary mixture, \[\begin{aligned} S_{m}^{E}=-R\left\{x_{1} \, \ln \left(f_{1}\right)\right.&+x_{1} \,\left[\partial \ln \left(f_{1}\right) / \partial T\right]_{p} \\ &\left.+x_{2} \, \ln \left(f_{2}\right)+x_{2} \,\left[\partial \ln \left(f_{2}\right) / \partial T\right]_{p}\right\} \end{aligned}\]This page titled 1.9.6: Entropies- Liquid Mixtures is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.