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9.3: Fluctuations

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Time_Dependent_Quantum_Mechanics_and_Spectroscopy_(Tokmakoff)/09%3A_Irreversible_and_Random_Processes/9.03%3A_Fluctuations

“Fluctuations” refers to the random or noisy time evolution of a microscopic subsystem imbedded an actively evolving environment. Randomness is a property of all chemical systems to some degree, but we will focus on an environment that is at or near thermal equilibrium. Systems at thermal equilibrium are macroscopically time-invariant; however, they are microscopically dynamic, with molecules exploring the range of microstates that are thermally accessible. Local variations in energy result in changes in molecular position, orientation, and structure, and are responsible for the activation events that allow chemical equilibria to be established.If we wish to describe an internal variable \(A\) for a system at thermal equilibrium, we can obtain the statistics of \(A\) by performing ensemble averages described above. The resulting averages would be time-invariant. However, if we observe a member of the ensemble as a function of time, \(A_i(t)\), the behavior is generally is observed to fluctuate randomly. The fluctuations in \(A_i(t)\) vary about a mean value \(\langle A \rangle\), sampling thermally accessible values which are described by an equilibrium probability distribution function \(P(A)\). \(P(A)\) describes the potential of mean force, the free energy projected as a function of \(A\):\[F ( A ) = - k _ {B} T \ln P ( A ) \label{8.19}\]Given enough time, we expect that one molecule in a homogeneous medium will be able to sample all available configurations of the system. Moreover, a histogram of the values sampled by one molecule is expected to be equal to \(P(A)\). Such a system is referred to as ergodic. Specifically, in an ergodic system, it is possible describe the macroscopic properties either by averaging over all possible values for a given member of the ensemble, or by performing an average over the realizations of \(A\) for the entire ensemble at one point in time. That is, the statistics for \(A\) can be expressed as a time-average or an ensemble average. For an equilibrium system, the ensemble average is\[\langle A \rangle = \operatorname {Tr} \left( \rho _ {e q} A \right) = \sum _ {n} \frac {e^{- \beta E _ {n}}} {Z} \langle n | A | n \rangle \label{8.20}\]and the time average is \[\overline {A} = \lim _ {T \rightarrow \infty} \frac {1} {T} \int _ {0}^{T} d t \, A _ {i} (t) \label{8.21}\]These quantities are equal for an ergodic system:\[\langle A \rangle = \overline {A}\]Equilibrium systems are ergodic. From Equation \ref{8.21}, we see that the term ergodic also carries a dynamical connotation. A system is ergodic if one member of the ensemble has evolved long enough to sample the equilibrium probability distribution. Experimental observations on shorter time scales view a nonequilibrium system.This page titled 9.3: Fluctuations is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Andrei Tokmakoff via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

InfoPage

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/00%3A_Front_Matter/02%3A_InfoPage

This text is disseminated via the Open Education Resource (OER) LibreTexts Project and like the hundreds of other texts available within this powerful platform, it is freely available for reading, printing and "consuming." Most, but not all, pages in the library have licenses that may allow individuals to make changes, save, and print this book. Carefully consult the applicable license(s) before pursuing such effects.Instructors can adopt existing LibreTexts texts or Remix them to quickly build course-specific resources to meet the needs of their students. Unlike traditional textbooks, LibreTexts’ web based origins allow powerful integration of advanced features and new technologies to support learning. The LibreTexts mission is to unite students, faculty and scholars in a cooperative effort to develop an easy-to-use online platform for the construction, customization, and dissemination of OER content to reduce the burdens of unreasonable textbook costs to our students and society. The LibreTexts project is a multi-institutional collaborative venture to develop the next generation of open-access texts to improve postsecondary education at all levels of higher learning by developing an Open Access Resource environment. The project currently consists of 14 independently operating and interconnected libraries that are constantly being optimized by students, faculty, and outside experts to supplant conventional paper-based books. These free textbook alternatives are organized within a central environment that is both vertically (from advance to basic level) and horizontally (across different fields) integrated.The LibreTexts libraries are Powered by NICE CXOne and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. This material is based upon work supported by the National Science Foundation under Grant No. 1246120, 1525057, and 1413739.Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation nor the US Department of Education.Have questions or comments? For information about adoptions or adaptions contact More information on our activities can be found via Facebook , Twitter , or our blog .This text was compiled on 07/13/2023

Preface

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/00%3A_Front_Matter/04%3A_Preface

Over three hundred Topics in Thermodynamics (which can addressed individually) describe the thermodynamic properties of aqueous solutions and aqueous mixtures. The Gibbs energies of these systems are discussed leading through successive differential operations to enthalpies, volumes, heat capacities, compressibilities (isobaric and isentropic) and expansibilities. These properties are linked quantitatively to corresponding partial molar properties including chemical potentials and generally partial molar volumes, partial molar enthalpies, partial molar heat capacities, partial molar expansibilities and compressibilities. Key equations link experimentally determined variables (e.g. densities) and partial molar properties ( e.g. partial molar volumes) of the components in an aqueous solution/mixture. Extensive references are given to published papers describing application of the equations described in The Topics.Further Topics describe application of thermodynamic equations to descriptions of chemical equilibria for many classes of systems together with the dependences of equilibrium constants on temperature and pressure. The analysis is extended to a consideration of rate constants for chemical reactions between solutes in solution.The theoretical basis is described for the Debye-Huckel treatment of salt solutions, Bjerrum equation for ion association, Euler’s theorem, Legendre (thermodynamic) transformations, Lewisian variables, L’Hospital’s Rules. Related Topics describe electrical units, axioms , equilibrium and frozen properties.In each Topic, special attention is given to the units of parameters involved in equations, ensuring that the derived property has self consistent units as required in the SI system. Related Topics describe electrical units, axioms and both equilibrium and frozen properties. The thermodynamic analysis is extended to a consideration of several special Topics including thermodynamic stability, the Law of Mass Action, Adsorption, Isochoric and Equilibrium properties, extrathermodynamic analysis of acid strengths and solvent polarities, Hildebrand Solubility parameters, internal pressure of liquids, ion association, surfactants, Gibbs Adsorption Isotherm, Phase Rule, thermal stability and hydrogen ions in aqueous systems.In the context of quantitative description of aqueous systems, key references are given to the properties of water including molar volume, viscosity, relative permittivity and self-dissociation.Cross references are given to relevant subject matter in other Topics.

1.1: Activity

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.01%3A_Activity

1.1.1: Acid Dissociation Constants- Weak Acids- Debye-Huckel Limiting Law1.1.2: Acquisitive Convention1.1.3: Activity- Solutions and Liquid Mixtures1.1.4: Activity of Aqueous Solutions1.1.5: Activity of Solvents1.1.6: Activity of Solvents- Classic Analysis1.1.7: Activity of Water - Foods1.1.8: Activity of Water - One Solute1.1.9: Activity of Water - Two Solutes1.1.10: Activity Coefficient- Two Neutral Solutes- Solute + Trace Solute i1.1.11: Activity Coefficients1.1.12: Activity Coefficients- Salt Solutions- Ion-Ion Interactions1.1.13: Activity of Water - Salt Solutions This page titled 1.1: Activity is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.2: Affinity for Spontaneous Chemical Reaction

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

1.2.1: Affinity for Spontaneous Chemical Reaction1.2.2: Affinity for Spontaneous Reaction- Chemical Potentials1.2.3: Affinity for Spontaneous Chemical Reaction- Phase Equilibria1.2.4: Affinity for Spontaneous Reaction- General Differential1.2.5: Affinity for Spontaneous Reaction- Dependence on Temperature1.2.6: Affinity for Spontaneous Reaction - Dependence on Pressure1.2.7: Affinity for Spontaneous Reaction - Stability1.2.8: Affinity for Spontaneous Chemical Reaction - Law of Mass Action1.2.9: Affinity for Spontaneous Chemical Reaction - Isochoric Condition and Controversy This page titled 1.2: Affinity for Spontaneous Chemical Reaction is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.3: Calorimeter

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

1.3.1: Calorimeter- Isobaric1.3.2: Calorimetry- Isobaric- General Operation1.3.3: Calorimetry- Solutions- Isobaric1.3.4: Calorimetry- Solutions- Adiabatic1.3.5: Calorimetry- Solutions - Heat Flow1.3.6: Calorimetry - Titration Microcalorimetry1.3.7: Calorimeter- Titration Microcalorimetry- Enzyme-Substrate Interaction1.3.8: Calorimetry- Titration Microcalorimetry- Micelle Deaggregation1.3.9: Calorimetry- Scanning1.3.10: Calorimetry- Solutions- Flow Microcalorimetry This page titled 1.3: Calorimeter is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.4: Chemical Equilibria

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

1.4.1: Chemical Equilibria- Solutions1.4.2: Chemical Equilibria- Solutions- Derived Thermodynamic Parameters1.4.3: Chemical Equilibria- Solutions- Simple Solutes1.4.4: Chemical Equilibria- Solutions- Ion Association1.4.5: Chemical Equilibria- Solutions- Sparingly Soluble Salt1.4.6: Chemical Equilibria- Cratic and Unitary Quantities1.4.7: Chemical Equilibria- Composition- Temperature and Pressure Dependence1.4.8: Chemical Equilibrium Constants- Dependence on Temperature at Fixed Pressure1.4.9: Chemical Equilibria- Dependence on Pressure at Fixed Temperature This page titled 1.4: Chemical Equilibria is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.5: Chemical Potentials

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

1.5.1: Chemical Potentials, Composition and the Gas Constant1.5.2: Chemical Potentials- Gases1.5.3: Chemical Potentials- Solutions- General Properties1.5.4: Chemical Potentials- Solutions- Composition1.5.5: Chemical Potentials- Solutions- Partial Molar Properties1.5.6: Chemical Potentials- Liquid Mixtures- Raoult's Law1.5.7: Chemical Potentials- Solutions- Raoult's Law1.5.8: Chemical Potentials- Solutions- Osmotic Coefficient1.5.9: Chemical Potentials; Excess; Aqueous Solution1.5.10: Chemical Potentials- Solutions- Henry's Law1.5.11: Chemical Potentials- Solutes1.5.12: Chemical Potentials- Solute; Molality Scale1.5.13: Chemical Potentials- Solutes- Mole Fraction Scale1.5.14: Chemical Potentials; Solute; Concentration Scale1.5.15: Chemical Potentials- Solute- Concentration and Molality Scales1.5.16: Chemical Potentials- Solute- Molality and Mole Fraction Scales1.5.17: Chemical Potentials- Solutions- Salts1.5.18: Chemical Potentials- Solutions- 1-1 Salts1.5.19: Chemical Potentials- Solutions- Salt Hydrates in Aqueous Solution1.5.20: Chemical Potentials- Salt Solutions- Ion-Ion Interactions1.5.21: Chemical Potentials- Salt Solutions- Debye-Huckel Equation This page titled 1.5: Chemical Potentials is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.6: Composition

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

1.6.1: Composition- Mole Fraction- Molality- ConcentrationChemists are particularly expert at identifying the number and chemical formulae of chemical substances present in a given closed system. Here we explore how the chemical composition of a given system is expressed.1.6.2: Composition- Scale Conversions- Molality1.6.3: Composition- Scale Conversion- Solvent Mixtures This page titled 1.6: Composition is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.7: 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

1.7.1: Compressions and Expansions- Liquids1.7.2: Compressibilities (Isothermal) and Chemical Potentials- Liquids1.7.3: Compressions- Isentropic- Solutions- General Comments1.7.4: Compressibilities- Isentropic- Related Properties1.7.5: Compressions- Isentropic- Solutions- Partial and Apparent Molar1.7.6: Compressions- Isentropic- Neutral Solutes1.7.7: Compressions- Isentropic- Salt Solutions1.7.8: Compresssions- Isentropic- Aqueous Solution1.7.9: Compressions- Isentropic and Isothermal- Solutions- Approximate Limiting Estimates1.7.10: Compressions- Desnoyers - Philip Equation1.7.11: Compression- Isentropic- Apparent Molar Volume1.7.12: Compressions- Isentropic- Binary Liquid Mixtures1.7.13: Compressions- Isothermal- Equilibrium and Frozen1.7.14: Compressions- Ratio- Isentropic and Isothermal1.7.15: Compression- Isentropic and Isothermal- Solutions- Limiting Estimates1.7.16: Compressions- Isentropic and Isothermal- Apparent Molar Volume1.7.17: Compressions- Isothermal- Solutes- Partial Molar Compressions1.7.18: Compressions- Isothermal- Apparent Molar Compression1.7.19: Compressions- Isothermal- Solutions- Apparent Molar- Determination1.7.20: Compressions- Isothermal- Salt Solutions1.7.21: Compressions- Isothermal- Binary Aqueous Mixtures1.7.22: Compressions- Isothermal- Liquid Mixtures Binary- Compressibilities This page titled 1.7: Compressions is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.8: Enthalpy

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

1.8.1: Enthalpies and Gibbs Energies1.8.2: Enthalpy1.8.3: Enthalpy- Thermodynamic Potential1.8.4: Enthalpy- Solutions- Partial Molar Enthalpies1.8.5: Enthalpies- Solutions- Equilibrium and Frozen Partial Molar Enthalpies1.8.6: Enthalpies- Neural Solutes1.8.7: Enthalpies- Solutions- Dilution- Simple Solutes1.8.8: Enthalpies- Solutions- Simple Solutes- Interaction Parameters1.8.9: Enthalpies- Salt Solutions- Apparent Molar- Partial Molar and Relative Enthalpies1.8.10: Enthalpy of Solutions- Salts1.8.11: Enthalpies- Salt Solutions- Dilution1.8.12: Enthalpies- Born-Bjerrum Equation- Salt Solutions1.8.13: Enthalpies- Liquid Mixtures This page titled 1.8: Enthalpy is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.1.1: Acid Dissociation Constants- Weak Acids- Debye-Huckel Limiting Law

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.01%3A_Activity/1.1.01%3A_Acid_Dissociation_Constants-_Weak_Acids-_Debye-Huckel_Limiting_Law

For a weak acid HA in aqueous solution at temperature T and pressure p (which is ambient pressure and so close to the standard pressure) the following chemical equilibrium is established.\[\mathrm{HA}(\mathrm{aq})+\mathrm{H}_{2} \mathrm{O} \rightleftharpoons \mathrm{H}_{3} \mathrm{O}^{+}(\mathrm{aq})+\mathrm{A}^{-}(\mathrm{aq})\]The r.h.s. of equation (1.1.3) describes a 1:1 ‘salt’ in aqueous solution. At equilibrium (i.e. at a minimum in Gibbs energy), the thermodynamic description of the solution takes the following form.\[\mu^{e q}(\mathrm{HA} ; \mathrm{aq} ; \mathrm{T} ; \mathrm{p})+\mu^{\mathrm{eq}}\left(\mathrm{H}_{2} \mathrm{O} ; \mathrm{aq} ; \mathrm{T} ; \mathrm{p}\right)=\mu^{\mathrm{eq}}\left(\mathrm{H}_{3} \mathrm{O}^{+} \mathrm{A}^{-} ; \mathrm{aq} ; \mathrm{T} ; \mathrm{p}\right)\]We express µeq(H2O;aq;T;p) in terms of the practical osmotic coefficient φ for the solution.\[\mu^{e q}\left(\mathrm{H}_{2} \mathrm{O} ; \mathrm{aq} ; \mathrm{T} ; \mathrm{p}\right)=\mu^{*}\left(\mathrm{H}_{2} \mathrm{O} ; \lambda ; \mathrm{T} ; \mathrm{p}\right)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \,\left[\mathrm{m}(\mathrm{HA})+2 \, \mathrm{m}_{\mathrm{j}}\right]^{e q}\]Here mj is the molality of the ‘salt’ H3O+A-. The latter yields 2 moles of ions for each mole of H3O+A-. A full description of the solution takes the following form.\[\begin{aligned} \mu^{0}(\mathrm{HA} ; \mathrm{aq})+& \mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{m}(\mathrm{HA}) \, \gamma(\mathrm{HA}) / \mathrm{m}^{0}\right]^{e q} \\ &+\mu^{*}\left(\mathrm{H}_{2} \mathrm{O} ; \lambda\right)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \,\left[\mathrm{m}(\mathrm{HA})+2 \, \mathrm{m}_{\mathrm{j}}\right]^{\mathrm{eq}} \\ &=\mu^{0}\left(\mathrm{H}_{3} \mathrm{O}^{+} \mathrm{A}^{-} ; \mathrm{aq}\right)+2 \, \mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{m}_{\mathrm{j}} \, \gamma_{\pm}\right]^{\mathrm{eq}} \quad \end{aligned}\]The practical osmotic coefficient φ describes the properties of solvent, water in the aqueous solution; γ± is the mean ionic activity coefficient for the ‘salt’ H3O+A-. By definition, if ambient pressure p is close to the standard pressure p0, the standard Gibbs energy of acid dissociation,\[\begin{gathered} \Delta_{\mathrm{d}} \mathrm{G}^{0}=\mu^{0}\left(\mathrm{H}_{3} \mathrm{O}^{+} \mathrm{A}^{-} ; \mathrm{aq} ; \mathrm{T} ; \mathrm{p}\right)-\mu^{0}(\mathrm{HA} ; \mathrm{aq} ; \mathrm{T} ; \mathrm{p})-\mu^{0}\left(\mathrm{H}_{2} \mathrm{O} ; \lambda ; \mathrm{T} ; \mathrm{p}\right) \\ =-\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{K}_{\mathrm{A}}^{0}\right) \end{gathered}\]\(\mathrm{K}_{\mathrm{A}}^{0}\) is the acid dissociation constant. Combination of equations (1.1.4) and (1.1.5) yields equation (1.1.6).\[\mathrm{K}_{\mathrm{A}}^{0} = \frac{\left[\mathrm{m}\left(\mathrm{H}_{3} \mathrm{O}^{+} \, \mathrm{A}^{-}\right)^{\mathrm{cq}} \, \gamma_{\pm}\left(\mathrm{H}_{3} \mathrm{O}^{+} \mathrm{A}^{-}\right)^{\mathrm{eq}} / \mathrm{m}^{0}\right]^{2} \, \exp \left[\phi \, \mathrm{M}_{1} \,\left(\mathrm{m}(\mathrm{HA})+2 \, \mathrm{m}_{\mathrm{j}}\right)\right]^{\mathrm{cq}}}{\left[\mathrm{m}(\mathrm{HA}) \, \gamma(\mathrm{HA}) / \mathrm{m}^{0}\right]^{\mathrm{eq}}}\]For dilute aqueous solutions, several approximations are valid. The exponential term and γ(HA)eq are close to unity. There are advantages in defining a quantity \(\mathrm{K}_{\mathrm{A}}^{0}\) (app) . Further, γ±(H3O+A-) is obtained using the Debye - Huckel Limiting Law, DHLL.By definition,\[\mathrm{K}_{\mathrm{A}}(\mathrm{app})=\left[\mathrm{m}\left(\mathrm{H}^{+} \mathrm{A}^{-}\right)^{\mathrm{eq}}\right]^{2} / \mathrm{m}(\mathrm{HA})^{\mathrm{eq}} \, \mathrm{m}^{0}\]Then\[\ln \mathrm{K}_{\mathrm{A}}(\operatorname{app})=\ln \mathrm{K}_{\mathrm{A}}^{0}+2 \, \mathrm{S}_{\gamma}\left(\mathrm{I} / \mathrm{m}^{0}\right)^{1 / 2}\]In other words, with increase in ionic strength \(I\), \(\mathrm{K}_{\mathrm{A}}(\operatorname{app})\) increases as a consequence of ion - ion interactions which stabilize the dissociated form of the acid.This page titled 1.1.1: Acid Dissociation Constants- Weak Acids- Debye-Huckel Limiting Law is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.1.2: Acquisitive Convention

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.01%3A_Activity/1.1.02%3A_Acquisitive_Convention

This convention refers to the sign associated with communication between a system and its surroundings. The convention describes changes as seen from the standpoint of the system. The convention guides chemists concerning the ‘sign’ of changes in thermodynamic variables00for a given system. For example, if heat q flows from the surroundings into a system, q is positive. If thermodynamic energy U is lost by a system to the surroundings, ∆U is negative. In fact this convention is intuitively attractive to chemists. For example, when told that the volume of a system increases during a given process, then chemists conclude that the volume of the surroundings (i.e. the rest of the universe!) decreases.This page titled 1.1.2: Acquisitive Convention is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.1.3: Activity- Solutions and Liquid Mixtures

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.01%3A_Activity/1.1.03%3A_Activity-_Solutions_and_Liquid_Mixtures

The concept of activity was introduced by Lewis in descriptions of the properties of liquid mixtures and solutions. By way of illustration we consider the chemical potential of chemical substance \(j\), \(\mu_{j}(\text { system })\) present in a solution at fixed pressure \(p\) and temperature \(T\). By definition, \[\mu_{j}(\text { system })=\mu_{j}(\text { ref })+R \, T \, \ln \left(a_{j}\right)\]While we can never know either \(\mu_{j}(\text { system })\) or \(\mu_{j}(\text { ref })\), the difference is related to the activity \(a_{j}\), a dimensionless function of the composition of the system. We as observers of the system are required to define the reference state where the chemical potential of chemical substance \(j\) can be clearly defined. Nevertheless the terms \(\mu_{j}(\text { system })\), \(\mu_{j}(\text { ref })\) and \(a_{j}\) in equation (a) are based on somewhat abstract concepts. The link with practical chemistry is made through the differential of equation (a) with respect to pressure at constant temperature. \[\text { Then, } \mathrm{V}_{\mathrm{j}}(\text { system })=\mathrm{V}_{\mathrm{j}}(\text { ref })+\mathrm{R} \, \mathrm{T} \,\left[\partial \ln \left(\mathrm{a}_{\mathrm{j}}\right) / \partial \mathrm{p}\right]_{\mathrm{T}}\]\(\mathrm{V}_{\mathrm{j}}(\text { system })\) and \(\mathrm{V}_{\mathrm{j}}(\text { ref })\) are, respectively, the partial molar volumes of chemical substance \(j\) in the system and in a convenient reference state. The term \(\mathrm{R} \, \mathrm{T} \,\left[\partial \ln \left(\mathrm{a}_{\mathrm{j}}\right) / \partial \mathrm{p}\right]_{\mathrm{T}}\) contrasts the role of intermolecular interactions in the two states. Four applications of the concept of activity make the point.For the binary liquid mixture, ethanol + water at defined temperature and pressure, the activity of, for example, water (substance 1) \(a_{1}\) is given by the product, \(x_{1} \, f_{1}\) where \(f_{1}\) is the (rational) activity coefficient and \(x_{1}\) is the mole fraction of water. \[a_{1}=x_{1} \, f_{1}\]The activity of urea (chemical substance \(j\)) in an aqueous solution is related to the product of activity coefficient \(\gamma_{i}\) and molality \(m_{j}\) using the reference molality \(m^{0}\), namely \(1 \mathrm{ mol kg}^{–1}\). \[a_{j}=\left(m_{j} / m^{0}\right) \, \gamma_{j}\]If the concentration of urea in the solution equals \(c_{j} \mathrm{ mol dm}^{-3}\), then the activity \(a_{j}\) is given by equation (e) where \(c_{r}\) is the reference concentration \(c_{r}\), \(1 \mathrm{ mol dm}^{–3}\), and \(y_{j}\) is the solute activity coefficient. \[a_{j}=\left(c_{j} / c_{r}\right) \, y_{j}\]If \(x_{j}\) is the mole fraction of urea and \(\mathrm{f}_{\mathrm{j}}^{*}\) is the asymmetric activity coefficient, the activity of urea is given by equation (d). \[a_{j}=x_{j} \, f_{j}^{*}\]Equations (d) to (f) describe the same property, namely activity \(a_{j}\) of solute \(j\) in a given solution.Footnotes G. N. Lewis, Proc. Am. Acad. Arts Sci.,1907,43,259.This page titled 1.1.3: Activity- Solutions and Liquid Mixtures is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.1.4: Activity of Aqueous Solutions

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.01%3A_Activity/1.1.04%3A_Activity_of_Aqueous_Solutions

A given aqueous solution is prepared using 1 kg of water(\(\lambda\), molar mass \(\mathrm{M}_{1}\), and \(m_{j}\) moles of solute at temperature \(\mathrm{T}\) and pressure \(p\) (which is close to the standard pressure \(p_{0}\)) At fixed \(\mathrm{T}\) and \(p\), the activity of water \(a_{1}(\mathrm{aq})\) is related to the chemical potential of water in the aqueous solution using equation (a) where \(\mu_{1}^{*}(\lambda)\) is the chemical potential of water(\(\lambda\)) at the same \(\mathrm{T}\) and \(p\). \[\mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\lambda)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{a}_{1}\right)\]Further equation (b) relates \(\mu_{1}(\mathrm{aq})\) to the molality of a simple solute, \(\mathrm{m}_{j}\) (e.g. urea) where \(\mathrm{R}\) is the gas constant (\(=8.314 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\)). \[\mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\lambda)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\]Here \(\mathrm{m}_{j}\) is the molality of solute \(j\) and \(\phi\) is the practical osmotic coefficient. If the thermodynamic properties of the aqueous solution are ideal (i.e. no solute-solute interactions) the practical osmotic coefficient is unity. \[\text { At fixed } \mathrm{T} \text { and } \mathrm{p}, \operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi=1.0\]\[\text { Hence, } \mu_{1}(\mathrm{aq} ; \mathrm{id})=\mu_{1}^{*}(\lambda)-\mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\]Therefore in the case of an ideal solution, addition of a solute, molality \(\mathrm{m}_{j}\), stabilises the solvent; i.e. lowers the chemical potential of the solvent. In the event that solute \(j\) is a salt which forms with complete dissociation \(ν\) ions for each mole of salt in solution, \(\mu_{1}(\mathrm{aq})\) is given by equation (e). \[\mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\lambda)-\mathrm{v} \, \phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\]We consider an aqueous solution containing a simple neutral solute \(j\). In order to understand the properties of this solution, we need to consider water-solute interactions and solute \(j\) – solute \(j\) interactions. Solute-solute interactions determine the extent to which the properties of a given solution differ from those of the corresponding solution having thermodynamic properties which are ideal.The extent to which the thermodynamic properties of solutions are not ideal also reflects in part the role of water-solute interactions. For example the extent to which urea-urea interactions differ from ethanol-ethanol interactions in aqueous solutions reflects the different hydration characteristics of urea and ethanol.Comparison of equations (a) and (b) yields the following important equation relating activity of solvent, water, and the molality of simple neutral solute \(j\). \[\text { Thus, } \quad \ln \left(\mathrm{a}_{1}\right)=-\phi \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\]The minus signs in equations (b) and (f) are extremely significant. If the thermodynamic properties of the solutions are ideal, \(\phi\) is unity. \[\text { Then, } \ln \left(\mathrm{a}_{1}\right)^{\mathrm{id}}=-\mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}=-\mathrm{M}_{1} \, \mathrm{n}_{\mathrm{j}} / \mathrm{n}_{1}^{0} \, \mathrm{M}_{1}=-\mathrm{n}_{\mathrm{j}} / \mathrm{n}_{1}^{0}\]Here \(n_{1}{}^{0}\) is the amount of solvent, water, molar mass \(\mathrm{M}_{1}\); \(n_{j}\) is the amount of solute \(j\). Therefore a plot of \(\ln \left(\mathrm{a}_{1}\right)^{\mathrm{id}}\) against molality \(\mathrm{m}_{j}\) is linear with slope ‘\(-\mathrm{M}){1}\)’. Furthermore the plots for a range of neutral solutes will be super-imposable. In other words\(\ln \left(\mathrm{a}_{1}\right)^{\mathrm{id}}\) is related to the ratio of amounts of solute to solvent. By adding a solute to a fixed amount of (solvent) water(\(\lambda\)) we lower the activity of water(\(\lambda\)) , (i.e. the chemical potential of water, µ(aq) in an aqueous solution) and stabilise the solvent.The chemical potential of solute \(j\) in an aqueous solution \(\mu_{\mathrm{j}}(\mathrm{aq})\) is related to the molality of solute mj using equation (h) where \(\mu_{j}^{0}(\mathrm{aq})\) is the chemical potential of solute \(j\) in an aqueous solution, molality \(\mathrm{m}_{\mathrm{j}}=1 \mathrm{~mol} \mathrm{~kg}^{-1}\) and \(\gamma_{j}=1\) at all \(\mathrm{T}\) and \(p\), (taken as close to the standard pressure \(p^{0}\) ). \[\mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)\]\[\text { By definition, at all T and } p \operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{\mathrm{j}} \rightarrow 1\]The role of water activity in determining enzyme activity is an important consideration. .Footnotes R. A. Robinson and R. H. Stokes, Electrolyte Solutions, Butterworths, London 2nd edn. Revised, 1965. G. Bell, A. E. M. Janssen and P. J. Halling, Enzyme and Microbial Technology, 1997,20,471.This page titled 1.1.4: Activity of Aqueous Solutions is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.1.5: Activity of Solvents

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Classically, the colligative properties of non-ionic solutions were used to determine the molar mass of solutes. For example, the depression \(\Delta \mathrm{T}_{f}\) of the freezing point of water \(\mathrm{T}_{f}\) at a given molalilty \(\mathrm{m}_{j}\) of solute-\(j\) yields an estimate of the relative molar mass of the solute \(\mathrm{M}_{j}\). Key thermodynamic assumptions require that (a) on cooling only pure solvent separates out as the solid phase and (b) the thermodynamic properties of the solution are ideal. The key relationship emerges from the Schroder- van Laar equation. The common assumption is that the thermodynamic properties of the solution are ideal. If the properties of a given aqueous solutions are determined to a significant extent by solute-solute interactions, a measured relative molar mass will be in error. Indeed McGlashan was dismissive of the procedures based on Beckmann’s apparatus for the determination of the relative molar mass of solute using freezing point measurements.The chemical potential of water in an aqueous solution, \(\mu_{1}(\mathrm{aq})\) at temperature \(\mathrm{T}\) and pressure \(p\) (assumed to be close to the standard pressure, \(p^{0}\)) is related to the molality of solute \(j\), \(\mathrm{m}_{j}\) using equation (a) where \(\mathrm{R}\) is the gas constant, \(\phi\) is the practical osmotic coefficient and \(\mathrm{M}_{1}\) is the molar mass of water, \(0.018015 \mathrm{ kg mol}^{-1}\) where \(\mu_{1}^{*}(\lambda)\) is the chemical potential of water(\(\lambda\)) at the same \(\mathrm{T}\) and \(p\). \[\mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\lambda)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\]Chemical potential \(\mu_{1}(\mathrm{aq})\) at temperature \(\mathrm{T}\) is also related to \(\mu_{1}^{*}(\lambda)\) using equation (b) where \(a_{1}\) is the activity of water in the aqueous solution. \[\mu_{1}\left(\mathrm{aq} ; \mathrm{m}_{\mathrm{j}}\right)=\mu_{1}^{*}(\lambda)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{a}_{1}\right)\]Comparison of equations (a) and (b) shows that \(\ln \left(a_{1}\right)\) is related to the molality of solute \(\mathrm{m}_{j}\) using equation (c). \[\ln \left(a_{1}\right)=-\phi \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\]For a solution having thermodynamic properties which are ideal, the practical osmotic coefficient is unity. \[\text { Then, } \ln \left(\mathrm{a}_{1}\right)^{\mathrm{id}}=-\mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\]Hence for a solution having thermodynamic properties which are ideal, \(\ln \left(a_{1}\right)\) is a linear function of molality \(m_j\), the plot having slope, \(-(\mathrm{M}_{1})\). Equation (d) forms a reference for a consideration of the properties of real solutions. For a solution having thermodynamic properties which are ideal, the solvent, water in an aqueous solution is at a lower chemical potential than the pure liquid. This observation is at the heart of the terms ‘depression of freezing point’ and ‘elevation of boiling point’. In the event that the thermodynamic properties of a given solution are not ideal then the form of the plot showing \(\ln \left(a_{1}\right)\) as a function of molality \(\mathrm{m}_{j}\) is determined by \(\phi\) which is, in turn, a function of \(\mathrm{m}_{j}\). The dependence of \(\phi\) on \(\mathrm{m}_{j}\) for a given solute in aqueous solutions (at fixed \(\mathrm{T}\) and \(p\)) is not defined ‘a priori’.Bower and Robinson report the dependence of osmotic coefficients for urea (aq) at 298 K over the range \(0 \leq \mathrm{m}_{\mathrm{j}} / \mathrm{mol} \mathrm{kg}^{-1} \leq 20.0\); \(\phi\) decreases with increase in \(\mathrm{m}_{j}\). Similarly Stokes and Robinson report the dependence of \(\phi\) on solute molality for sucrose(aq), glucose(aq) and glycerol(aq) over the range \(0 \leq \mathrm{m}_{\mathrm{j}} / \mathrm{mol} \mathrm{kg}^{-1} \leq 7.5\).For \(\mathrm{m}(\text { urea })=8 \mathrm{~mol} \mathrm{~kg}^{-1}\), \(\ln \left(a_{1}\right) \text { equals }-12 \times 10^{-2}\) whereas \(\ln \left(a_{1}\right)^{i d}\) equals approx. \(-15 \times 10^{-2}\). At this molality for urea(aq), \(\mu_{1}(\mathrm{aq})>\mu_{1}(\mathrm{aq})^{\mathrm{id}}\) indicating that water in urea(aq) at this molality is at a higher chemical potential than would be the case for a solution where the thermodynamic properties are ideal. On the other hand for the hydrophilic solute sucrose where \(\mathrm{m}(\text { sucrose })=6 \mathrm{~mol} \mathrm{~kg}^{-1}, \ln \left(\mathrm{a}_{1}\right) \text { is }-15 \times 10^{-2}\) whereas \(\ln \left(a_{1}\right)^{i d}\) equals approx. \(-11 \times 10^{-2}\) indicating that adding sucrose at this molality to water lowers the chemical potential of water relative to that for a solution having ideal properties.For a dilute solution of simple neutral solutes the difference between ideal and real properties can be understood in terms of the dependence of pairwise Gibbs energy interaction parameters \(g_{jj}\) on molality using equation (e) where \(\mathrm{m}^{0}=1 \mathrm{~mol} \mathrm{~kg}^{-1}\); the units of \(g_{jj}\) are \(\mathrm{ J kg}^{-1}\). \[1-\phi=-(1 / R \, T) \, g_{i j} \,\left(1 / m^{0}\right)^{2} \, m_{j}\]Using equation (c),\[\ln \left(a_{1}\right)=-M_{1} \, m_{j} \,\left[1+(R \, T)^{-1} \, g_{i j} \,\left(m^{0}\right)^{-2} \, m_{j}\right]\]or\[\ln \left(a_{1}\right)+M_{1} \, m_{j}=-M_{1} \,(R \, T)^{-1} \, g_{i j} \,\left(m^{0}\right)^{-2} \,\left(m_{j}\right)^{2}\]Hence for dilute solutions \(\left[\ln \left(a_{1}\right)+M_{1} \, m_{j}\right]\) is a linear function of \(\left(\mathrm{m}_{\mathrm{j}}\right)^{2}\), the gradient of the plot yielding the pairwise Gibbs energy interaction parameter \(g_{jj}\). If, for example, \(g_{jj}\) is positive indicating solute-solute repulsion, \(\left[\ln \left(\mathrm{a}_{1}\right)+\mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\right]\) decreases with increase in \(\mathrm{m}_{j}\) such that \(\mu_{1}(\mathrm{aq})>\mu_{1}(\mathrm{aq} ; \mathrm{id})\). In the event that solute-solute interactions are attractive, \(g_{jj}\) is negative. Hence the difference between the properties of real and ideal solutions is highlighted by the contrast between equations (d) and (f).The analysis described above is readily extended to aqueous solutions containing two solutes; e.g. urea(aq) + sucrose(aq), and glucose(aq) + sucrose(aq).A given aqueous salt solution contains a single salt j; µ1(aq) and µj(aq) are the chemical potentials of water and salt respectively in the closed system. For water, \[\text { For water, } \mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\lambda)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{v} \, \mathrm{m}_{\mathrm{j}}\]\[\text { And } \mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\lambda)-\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{a}_{1}\right)\]Here \(ν\) is the stoichiometric parameter, the number of moles of ions produced by complete dissociation of one mole of salt \(j\); for a 1:1 salt, \(ν\) equals 2. According to equations (h) and (i), \[\ln \left(a_{1}\right)=-\phi \, M_{1} \, v \, m_{j}\]\[\ln \left(a_{1}\right)^{\text {id }}=-M_{1} \, v \, m_{j}\]\[\text { If we confine attention to } 1: 1 \text { salts, } \ln \left(a_{1}\right)^{\text {id }}=-2 \, M_{1} \, m_{j}\]With increase in \(\mathrm{m}_{j}\), \(\ln \left(a_{1}\right)^{i d}\) decreases linearly. With reference to equation (j), a ) \(\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi=1.0\). With dilution of a salt solution, a plot of ln(a1) against mj approaches a linear dependence.\[\text { For the salt } \mathrm{j}, \mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{v} \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{Q} \, \mathrm{m}_{\mathrm{j}} \, \gamma_{\pm} / \mathrm{m}^{0}\right)\]Here \(ν_{+}\) and \(ν_{-}\) are the number of moles of cations and anions respectively produced by one mole of salt \(j\); \(v=v_{+}+v_{-} ; \gamma_{\pm}\) is the mean ionic activity γ coefficient of salt \(j\). By definition, \(\mathrm{Q}^{\mathrm{v}}=\mathrm{v}_{+}^{\mathrm{v}(+)} \, \mathrm{V}_{-}^{\mathrm{v}(-)}\). Also \(\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{\pm}=1.0\) at all \(\mathrm{T}\) and \(p\); \(\mu_{\mathrm{j}}^{0}(\mathrm{aq})\) is the chemical potential of salt \(j\) in a solution where \(\mathrm{m}_{\mathrm{j}}=1 \mathrm{~mol} \mathrm{~kg}^{-1}\) and the thermodynamic properties of the solute are ideal; i.e. no ion-ion interactions.For a 1:1 salt (e.g. \(\mathrm{KBr}\)), \[\mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+2 \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\pm} / \mathrm{m}^{0}\right)\]For a 1:1 salt where the thermodynamic properties of the solution are ideal, \[\mu_{j}(\mathrm{aq} ; \mathrm{id})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+2 \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)\]According to the Debye-Huckel Limiting Law, for very dilute solutions, \[\ln \left(\gamma_{\pm}\right)=-S_{\gamma} \,\left(m_{j} / m^{0}\right)^{1 / 2}\]where \(S_{\gamma}=f\left(T, p, \varepsilon_{r}\right)\) and \(\varepsilon_{\mathrm{r}}\) is the relative permittivity of the solvent at the same \(\mathrm{T}\) and \(p\). \[\text { Further }, 1-\phi=\left(\mathrm{S}_{\gamma} / 3\right) \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}\]At 298.15 K and ambient pressure, \(\mathrm{S}_{\gamma}=0.5115\). In other words, \[\phi^{\mathrm{dh} l l}=1-\left(\mathrm{S}_{\gamma} / 3\right) \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}\]Hence using equation (j), for a 1:1 salt, \[\ln \left(\mathrm{a}_{1}\right)^{\mathrm{dhll}}+2 \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}=\left[2 \,\left(\mathrm{S}_{\gamma} / 3\right) \, \mathrm{M}_{1} \,\left(\mathrm{m}^{0}\right)^{-1 / 2}\right] \,\left(\mathrm{m}_{\mathrm{j}}\right)^{3 / 2}\]Then \(\ln \left(a_{1}\right)^{\text {dhll }}\) indicates that for a salt solution, molality \(\mathrm{m}_{j}\), \(\ln \left(a_{1}\right)\) exceeds that in the corresponding salt solution having ideal thermodynamic properties . In other words the activity of the solvent, water, is enhanced above that for water having ideal thermodynamic properties. For very dilute solutions \(\left[\ln \left(\mathrm{a}_{1}\right)^{\mathrm{dhll}}+2 \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\right]\) is a linear function of \(\left(\mathrm{m}_{\mathrm{j}}\right)^{3 / 2}\). However other than for very dilute solutions equation (q) is inadequate and so a more sophisticated equation is required relating \(\phi\) and \(\mathrm{m}_{j}\). Extensive compilations of \(\phi\) for salt solutions are given in references and.Footnotes I. Prigogine and R. Defay, Chemical Thermodynamics, trans. D. H. Everett, Longmans Green, London, 1954, equation 22.5. M. L. McGlashan, Chemical Thermodynamics, Academic Press, London, 1979, page 307. V. E. Bower and R. A. Robinson, J. Phys. Chem.,1963,67,1524. R. H. Stokes and R. A. Robinson, J. Phys. Chem.,1963,67,2126. J. J. Savage and R. H. Wood, J. Solution Chem., 1976,5,733. M. J. Blandamer, J. Burgess, J. B. F. N. Engberts and W. Blokzijl, Annu. Rep. Prog. Chem., Sect C., Phys.Chem.,1990,87,45. H. Ellerton and P. J. Dunlop. J. Phys.Chem.,1966,70,1831. R. A.Robinson and R. H. Stokes, Electrolyte Solutions, Butterworths, London, 2nd edn., 1965, chapter 8. K. S. Pitzer, Thermodynamics, McGraw-Hill, New York, 3rd. edition, 1995. S. Lindenbaum and G. E. Boyd, J. Phys.Chem.,1964,68,911.This page titled 1.1.5: Activity of Solvents is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.1.6: Activity of Solvents- Classic Analysis

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Thermodynamics underpins a classic topic in physical chemistry concerning the depression of freezing point, \(\Delta \mathrm{T}_{\mathrm{f}}\) of a liquid by added solute. We note the superscript ‘id’ in equation (a) relating the activity of a solvent in an ideal solution, molality \(\mathrm{m}_{j}\). \[\ln \left(\mathrm{a}_{1}\right)^{\mathrm{id}}=-\mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}=-\mathrm{M}_{1} \, \mathrm{n}_{\mathrm{j}} / \mathrm{n}_{1}^{0} \, \mathrm{M}_{1}=-\mathrm{n}_{\mathrm{j}} / \mathrm{n}_{1}^{0}\]If the properties of a given aqueous solution are determined to a significant extent by solute-solute interactions, a determined molar mass for a given solute will be in error. Otherwise an observed depression is not a function of solute-solute interactions. Glasstone comments that the ratio \(\Delta \mathrm{T}_{\mathrm{f}} / \mathrm{m}_{\mathrm{j}}\) decreases with increasing concentration of solute, emphasising that a simple analysis is only valid for dilute solutions. Nevertheless the general idea is that the depression for a given mj is not a function of the hydration of a solute. Barrow noted that the ratio \(\Delta \mathrm{T}_{\mathrm{f}} / \mathrm{m}_{\mathrm{j}}\) for mannitol(aq) in very dilute solutions is effectively constant. A similar opinion is advanced by Adam who nevertheless comments on the importance of the condition ’dilute solution’ in a determination of the molar mass of a given solute.In summary, classic physical chemistry emphasises the importance of the superscript ‘id’ in equation (a). For very dilute solutions in a given solvent \(\ln \left(a_{1}\right)\) is linear function of \(\mathrm{m}_{j}\), leading to description of such properties as ‘depression of freezing point ‘ and ‘elevation of boiling point ’ under the heading ‘colligative properties. Only the molality of solute mj is important; solute-solute interactions and hydration characteristics of solutes apparently play no part in determining these colligative properties.The key to these statements is provided by the Gibbs-Duhem equation. For a solution prepared using 1 kg of water(\(\lambda\)) and \(\mathrm{m}_{j}\) moles of a simple solute \(j\), the Gibbs energy is given by equation (b). \[\mathrm{G}(\mathrm{aq})=\left(1 / \mathrm{M}_{1}\right) \, \mu_{1}(\mathrm{aq})+\mathrm{m}_{\mathrm{j}} \, \mu_{\mathrm{j}}(\mathrm{aq})\]Then, \(\mathrm{G}(\mathrm{aq})=\left(1 / \mathrm{M}_{1}\right) \,\left[\mu_{1}^{*}(\lambda)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\right]\) \[+\mathrm{m}_{\mathrm{j}} \,\left[\mu_{\mathrm{j}}^{0}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{m}_{\mathrm{j}}=1\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)\right] \text { (c) }\]According to the Gibbs - Duhem Equation, the chemical potentials of solvent and solute are linked. At fixed \(\mathrm{T}\) and \(p\), \[\mathrm{n}_{1} \, \mathrm{d} \mu_{1}(\mathrm{aq})+\mathrm{n}_{\mathrm{j}} \, \mathrm{d} \mu_{\mathrm{j}}(\mathrm{aq})=0\]Or, \(\left(1 / \mathrm{M}_{1}\right) \, \mathrm{d}\left[\mu_{1}^{*}(\lambda)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\right]\) \[+\mathrm{m}_{\mathrm{j}} \, \mathrm{d}\left[\mu_{\mathrm{j}}^{0}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{m}_{\mathrm{j}}=1\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)\right]=0\]\[\text { Or, } \quad-\mathrm{d}\left(\phi \, \mathrm{m}_{\mathrm{j}}\right)+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)=0\]\[\text { Or, } \quad-\mathrm{d}\left(\phi \, \mathrm{m}_{\mathrm{j}}\right)+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)=0\]\[\text { Hence, } \quad(\phi-1) \, \frac{\mathrm{dm}_{\mathrm{j}}}{\mathrm{m}_{\mathrm{j}}}+\mathrm{d} \phi=\mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right)\]The latter equation forms the basis of the oft-quoted statement that if the thermodynamic properties of a solute are ideal then so are the properties of the solvent. Similarly if the thermodynamic properties of the solute are ideal then so are the properties of the solvent. \[\text { From equation }(h), \phi=1+\frac{1}{m_{j}} \, \int_{0}^{m(j)} m_{j} \, d \ln \left(\gamma_{j}\right)\]The importance of equation (i) emerges from the idea that \(\gamma_{j}\) describes the impact of solute-solute interactions on the properties of a given solution. If we can formulate an equation for \(\ln \left(\gamma_{j}\right)\) in terms of the properties of the solution, we obtain \(\phi\) from equation (i). If the properties of a real solution containing a neutral solute are not ideal, both \(\gamma_{j}\) and \(\phi\) are linked functions of the solute molality. Pitzer suggests equations (j) and (k) for \(\ln \left(\gamma_{j}\right)\) and \(\phi\) in terms of solute molalities using two parameters, \(\lambda\) and \(\mu\). \[\ln \left(\gamma_{\mathrm{j}}\right)=2 \, \lambda \, \mathrm{m}_{\mathrm{j}}+3 \, \mu \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2}\]\[\phi-1=\lambda \, \mathrm{m}_{\mathrm{j}}+2 \, \mu \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2}\]For example in the case of mannitol(aq) and butan-1-ol(aq), Pitzer suggests the following equations for \(\ln \left(\gamma_{j}\right)\). \[\text { For mannitol(aq) } \quad \ln \left(\gamma_{\mathrm{j}}\right)=-0.040 \, \mathrm{m}_{\mathrm{j}}\]\[\text { For butan-1-ol(aq) } \ln \left(\gamma_{\mathrm{j}}\right)=-0.38 \, \mathrm{m}_{\mathrm{j}}+0.51 \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2}\]Guggenheim using the mole fraction scale suggests equation (n) where \(\mathrm{A}\) and \(\mathrm{B}\) are characteristic of the solute. \[1-\phi=A \, x_{j}+B \,\left(x_{j}\right)^{2}\]Prigogine and Defay comment that the non-ideal properties of solutions can be understood in terms of the different molecular sizes of solute and solvent. A similar comment is made by Robinson and Stokes who use a parameter describing the ratio of molar volumes of solute and solvent. The extent to which the properties of a solution differ from ideal can often be traced to a variety of causes including solvation, molecular size and shape.Footnotes I. Prigogine and R. Defay, Chemical Thermodynamics, trans. D. H. Everett, Longmans Green , London 1953. S. Glasstone, Physical Chemistry, McMillan, London, 2nd. edn., 1948,page 645. G. M. Barrow, Physical Chemistry, McGraw-Hill, New York, 4th edn.,1979, page 298. L. H. Adams, J. Am. Chem. Soc.,1915,37,481. Neil Kensington Adam, Physical Chemistry, Oxford,1956, page 284. K. S. Pitzer, Thermodynamics , McGraw-Hill, New York, 3rd. edn., 1995. E. A. Guggenheim, Thermodynamics, North-Holland Publishing Co., Amsterdam, 1950, pages 252-3 R. A. Robinson and R. H. Stokes, Electrolyte Solutions, Butterworths, London 2nd edn. Revised, 1965.This page titled 1.1.6: Activity of Solvents- Classic Analysis is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.1.7: Activity of Water - Foods

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An important scientific literature comments on the activity of water in the context of biochemistry and of the very important industry concerned with foods.Scott identified the importance of water activity and microbial growth on foodstuffs; e.g. chilled beef. Hartel reviews the problem of freezing of water in, for example, ice cream; see also comments on water activity in sucrose+ water systems. The importance of water activity in sensory crispness and mechanical deformation of snack products is discussed by Katz and Labula. Water activity is an important variable in fungal spoilage of food.Berg and Bruin review the role of activity, not necessarily the water content, in the context of the deterioration of food, a matter of concern for humans from earliest times. In fact water activity, a1 is a major control variable in food preservation although the chemistry of food deterioration is complicated.Crucially important in this context are publications produced by the National Institute of Standards and Technology .Footnotes P. Walstra, Physical Chemistry of Foods, M. Dekker, New York,2003. Water Activity: Influences on Food Quality, ed. L. B. Rockland and G. F. Stewart, Academic Press, New York,1981. A. J. Fontana Jr., Cereal Foods World,2000,45,7. A. J. Fontana and C. S. Campbell, Handbook of Food Analysis, volume 1, Physical Characterization and Nutrient Analysis , 2nd edn (revised and expanded) e.d. L. M.L. Nollet, 2004, M. Dekker, 46,403. R. Beauchat, J. Food Protection, 1983.46,135. W. H. Sperber, J. Food Protection, 1983.46,142. C. van den Berg and S. Bruin, Water Activity : Influence on Food Quality , ed. L. B. Rockford and G. F. Stewart, Academic Press, New York1981, pages 1-62. U. S. Food and Drug Administration, Title 21, Code of Federal Regulations, Parts 108, 110, 113 and 114, U. S. Government Printing Office, Washington, DC,1998.This page titled 1.1.7: Activity of Water - Foods is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.1.8: Activity of Water - One Solute

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Classical accounts of the physical chemical properties of solutions concentrate attention on the properties of solutes. Many experiments set out to determine activity coefficients \(\gamma_{j}\) for solutes in solution; (see also mean activity coefficients \(\gamma_{\pm}\) for salts). Information concerning solvent activity is obtained by exploiting the Gibbs-Duhem equation which (at fixed \(\mathrm{T}\) and \(p\)) links the properties of solute and solvent. However recent technological developments allow the activity of water in an aqueous solution to be measured.The classic analysis of colligative properties of solutions by van’t Hoff and others in the 19th Century is successful for dilute solutions. The importance of solute-solute interactions was generally recognised very early on in the 20th Century. However the role of solute-solvent interactions was perhaps underplayed.We develop a model for a given solution prepared by dissolving \(n_{j}\) moles of neutral solute \(j\), molar mass \(\mathrm{M}_{j}\), in \(n_{1}{}^{0}\) moles of water(\(\lambda\)), molar mass \(\mathrm{M}_{1}\). The molality of solute as prepared is given by equation (a). \[\mathrm{m}_{\mathrm{j}}(\text { prepared })=\mathrm{n}_{\mathrm{j}} / \mathrm{n}_{1}^{0} \, \mathrm{M}_{1}\]Then \(\ln \left(a_{1} ; \text { prepared }\right)^{\text {id }}\) is given by equation (b). \[\ln \left(a_{1}\right)^{i d}=-\(\mathrm{M}_{1} \, \(\mathrm{m}_{j}=-n_{j} / n_{1}^{0}]However in another description of the solution under investigation each mole of solute \(j\) is strongly hydrated by \(h\) moles of water. The mass of solvent water, \(\mathrm{w}_{1}\) is given by equation (c). \[\mathrm{w}_{1}=\left[\mathrm{n}_{1}^{0}-\mathrm{h} \, \mathrm{n}_{\mathrm{j}}\right] \, \mathrm{M}_{1}\]Hence the molality of hydrated solute is given by equation (d). \[\mathrm{m}_{\mathrm{j}}(\text { hydrated solute })=\mathrm{n}_{\mathrm{j}} /\left[\mathrm{n}_{1}^{0}-\mathrm{h} \, \mathrm{n}_{\mathrm{j}}\right] \, \mathrm{M}_{1}]In effect the molality of the solute increases because there is less ‘solvent water’ \[\text { By analogy with equation }(b), \ln \left(a_{1} ; \text { hyd }\right)=-n_{j} /\left(n_{1}^{0}-h \, n_{j}\right)\]Therefore for a range of solutions containing different solutes but prepared using the same amount of solute, the activities are a function of the different extents of hydration of the solutes. With increase in \(h\), \(\ln \left(a_{1} ; \text { hyd }\right)\) decreases (i.e. becomes more negative) indicative of increasing stabilisation of the water in the system by virtue of solute-water hydration interaction.In a real solution, the properties of a solute are not ideal because there exist solute-solute communication by virtue of the fact that each solute molecule is ‘aware’ that some of the solvent has been ‘removed’ by solute hydration. The amount of solvent ‘available ‘ to each molecule has been depleted by hydration of all solutes in solution. In other words a Gibbs-Duhem communication operates in the solution.The model of an aqueous solution described in conjunction with equation (a) is used to obtain the ratio, \(n_{1}^{0} / n_{j}\); equation (f). \[\text { Then, } \frac{n_{1}^{0}}{n_{j}}=\frac{1 \mathrm{~kg}}{M_{1}} \, \frac{M_{j}}{w_{j}}\]Thus \(n_{1}^{0} / n_{j}\) describes the solution as prepared using \(n_{j}\) and \(n_{1}{}^{0}\) moles of solutes and solvent respectively. We assert that by virtue of solute hydration an amount of water is removed from the ‘solvent’. In solution the mole fraction of (solvent) water is \(x_{1}\), the mole fraction of hydrated solute is \(x_{j}\). \[\text { Then, } \quad x_{w}+x_{j}=1\]The mole fraction ratio \(\mathbf{X}_{w} / \mathbf{X}_{j}\) is given by equation (h). \[\frac{\mathbf{X}_{w}}{X_{j}}=\frac{X_{w}}{1-X_{w}}]Equation (h) forms the basis of a treatment described by Scatchard in 1921, nearly a century ago. The model proposed by Scatchard was based on a model for water(\(\lambda\)), described as a mixture of hydrols; monohydrols and polymerised water. Scatchard discusses hydration of solutes although not all solutes in a given solution are seen as hydrated; i.e. the solution contains various hydrates. In fact a given solute may be hydrated to varying degrees; i.e. a given solution contains various hydrates. However Scatchard envisaged that one hydrate may be dominant. We note the date when the model was proposed by Scatchard. The concept of hydrogen bonding in aqueous solutions has its origin in a paper published by Latimer and Roedbush in 1920; see also; i.e. the previous year to publications by Scatchard.Scatchard invoked an assumption called the ‘semi-ideal’ assumption in which mole fraction \(x_{\mathrm{w}\) in equation (h) is replaced by the activity of the solvent, water \(\mathrm{a}_{\mathrm{w}\). Hence, from equation (h), \[\text { Hence, from equation (h), } \frac{\mathrm{x}_{\mathrm{w}}}{\mathrm{x}_{\mathrm{j}}}=\frac{\mathrm{a}_{\mathrm{w}}}{1-\mathrm{a}_{\mathrm{w}}}\]The difference between amounts of water defined by equations (f) and (i) yields the ‘average degree of hydration’, \(h\) of solute \(j\). \[\text { Then, } \quad \mathrm{h}=\frac{(1.0 / 0.0180)}{\mathrm{m}_{\mathrm{j}}}-\frac{\mathrm{a}_{\mathrm{w}}}{1-\mathrm{a}_{\mathrm{w}}}\]Equation (j) is Scatchard’s equation. If one can measure \(\mathrm{a}_{\mathrm{w}\) for an aqueous solution molality \(\mathrm{m}_{j}\) we obtain the ‘average degree of hydration’ for solute \(j\). Scatchard using vapour pressure data obtained an estimate of \(h\) for sucrose at 30 Celsius. In the case of a solution containing \(34 \mathrm{~g}\) of sucrose in \(100 \mathrm{~g}\) water(\(\lambda\)), \(h\) equals \(4.46\), decreasing with increase in the ‘strength ‘ of the solution. The term ‘semi-ideal’ proposed by Scatchard emerges from the identification of solvent activity with mole fraction of depleted solvent. In summary Scatchard obtained a property \(h\) but there is no indication of the stability of the hydrate.Stokes and Robinson extended the Scatchard analysis using a chemical equilibrium involving solute hydrates. The hydration of a given solute is described by a set of equilibrium constants describing i-hydration steps. For solute \(\mathrm{S}\), \[\mathrm{S}_{\mathrm{i}-1}+\mathrm{H}_{2} \mathrm{O} \Leftrightarrow \Rightarrow \mathrm{S}_{\mathrm{i}} \quad(\mathrm{i}=1,2 \ldots \ldots \mathrm{n})\]Each step is described by an equilibrium constant, \(\mathrm{K}_{i}\). So for a solute hydrated by 3 water molecules there are 3 equilibrium constants. Stokes and Robinson set \(n\) equal to \(11\) for sucrose. However in this case Stokes and Robinson simplify the analysis by assuming that the equilibrium constants for all hydration steps are equal. The resulting equations are as follows. \[\frac{\left(1 / \mathrm{M}_{1}\right)}{\mathrm{m}_{\mathrm{j}}}=\frac{\mathrm{a}_{\mathrm{w}}}{1-\mathrm{a}_{\mathrm{w}}}+\frac{\sigma}{\Sigma}\]\[\text { where } \quad \sigma=\mathrm{K} \, \mathrm{a}_{\mathrm{w}}+\ldots \ldots \ldots+\mathrm{K} \,\left(\mathrm{a}_{\mathrm{w}}\right)^{\mathrm{n}}\]\[\text { and } \quad \Sigma=1+K \, a_{w}+\ldots \ldots . .+\left(K \, a_{w}\right)^{n}\]Equation (\(\lambda\)) is interesting because for a given solute, the dependence of \(\mathrm{a}_{\mathrm{w}\) on \(\mathrm{m}_{j}\) yields two interesting parameters, \(n\) and \(\mathrm{K}\), describing hydration of a given solute \(j\). The equilibrium constants defined above are dimensionless.Stokes and Robinson describe a method of data analysis but modern computer-based methods should lighten the arithmetic drudgery. For sucrose(aq) at \(298.15 \mathrm{~K}\) Stokes and Robinson estimate that \(n = 11\) and \(\mathrm{K} = 0.994\). For glucose(aq) \(n = 6\) with \(\mathrm{K} = 0.786\). Finally we note that the composition of the solutions should be expressed in molalities; i.e. each solution made up by weight. If this is not done, conversion of concentrations to molalities is required. Possibly the literature will yield the required densities of the solutions. The worst approximation sets the density of the solutions at the density of water(\(\lambda\)) at the same temperature.Footnotes Water Activity Meter, Decagon Devices Inc. WA 99163, USA. G. Scatchard, J.Am.Chem.Soc.,1921,43,2387.,2408. W. M. Latimer and W. H. Rodebush, J. Am. Chem.Soc.,1920,42,1419. L. Pauling, The Nature of the Chemical Bond, Cornell Univ. Press, Ithaca, New York, 3rd edn.,1960, chapter 12. R. H. Stokes and R.A Robinson, J. Phys.Chem.,1966, 70,2126.This page titled 1.1.8: Activity of Water - One Solute is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.1.9: Activity of Water - Two Solutes

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A given solution contains two neutral (i.e. non-ionic) solutes, solute-\(i\) and solute-\(j\). We anticipate, for example, activity coefficient \(\gamma_{i}\) for solute –\(i\) is a function of the molalities of both solutes, \(\mathrm{m}_{i}\) and \(\mathrm{m}_{j}\). The thermodynamic properties of this class of solutions are discussed by Bower and Robinson and by Ellerton and Dunlop. Because the analysis discussed by these authors concerns the properties of solvent, water in aqueous solutions, the starting point is isopiestic vapour pressure measurements. Analysis of the thermodynamic properties of these mixed aqueous solutions has four themes which we develop separately, drawing the analysis together in a final section.Solution I is prepared by dissolving ni moles of solute-\(i\) in water, mass \(\mathrm{w}_{1}(\mathrm{I})\) at temperature \(\mathrm{T}\) and pressure \(p\)( which is close to the standard pressure \(p^{0}\)); \(\mathrm{M}_{1}\) is the molar mass of water and \(\phi(I)\) is the practical osmotic coefficient of the solvent, water, in solution(I). The contribution \(\mathrm{G}_{1}(\mathrm{I})\) of the solvent to the Gibbs energy of the solution is given by equation(a). \[\mathrm{G}_{1}(\mathrm{I})=\left[\mathrm{w}_{1}(\mathrm{I}) / \mathrm{M}_{1}\right] \,\left\{\mu_{1}^{*}(\lambda)-\left[\phi(\mathrm{I}) \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{i}}(\mathrm{I})\right]\right\}\]Solution (II) is similarly prepared using \(n_{j}\) moles of solute-\(j\) in water, mass \(\mathrm{w}_{1}(\mathrm{II})\). \[\mathrm{G}_{1}(\mathrm{II})=\left[\mathrm{w}_{1}(\mathrm{II}) / \mathrm{M}_{1}\right] \,\left\{\mu_{1}^{*}(\lambda)-\left[\phi(\mathrm{II}) \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}(\mathrm{II})\right]\right\}\]We add a sample of solution (I) containing \(1 \mathrm{~kg}\) of water to a sample of solution (II) also prepared using \(1 \mathrm{~kg}\) of water. The resulting solution contains \(2 \mathrm{~kg}\) of water and the initial molalities \(\mathrm{m}_{i}(\mathrm{I})\) and \(\mathrm{m}_{j}(\mathrm{II})\) will be reduced by a half. Then we imagine that \(1 \mathrm{~kg}\) of water is withdrawn from the solution. This concentration process restores the original molalities of solutes \(i\) and \(j\). The letter ‘F’ identifies the new solution. \[\mathrm{G}_{1}\left(\text { total } ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mu_{1}^{*}(\lambda)-\mathrm{R} \, \mathrm{T} \, \phi(\mathrm{F}) \,\left[\mathrm{m}_{\mathrm{i}}(\mathrm{I})+\mathrm{m}_{\mathrm{j}}(\mathrm{II})\right]\]\(\phi(\mathrm{F}\)) is the practical osmotic coefficient of the solution prepared using solutions I and II from which \(1 \mathrm{~kg}\) of solvent has been removed. The results of the analysis given above can be summarised in three equations describing the activities of water in the three solutions. \[\ln \left[\mathrm{a}_{1}(\mathrm{I})\right]=-\phi(\mathrm{I}) \, \mathrm{m}_{\mathrm{i}}(\mathrm{I}) \, \mathrm{M}_{1}\]\[\ln \left[\mathrm{a}_{1}(\mathrm{II})\right]=-\phi(\text { II }) \, \mathrm{m}_{\mathrm{j}}(\mathrm{II}) \, \mathrm{M}_{1}\]\[\ln \left[\mathrm{a}_{1}(\mathrm{~F})\right]=-\phi(\mathrm{F}) \,\left[\mathrm{m}_{\mathrm{i}}(\mathrm{I})+\mathrm{m}_{\mathrm{j}}(\mathrm{II})\right] \, \mathrm{M}_{1}\]The molalities remain the same as in the original solutions; i.e. \(\mathrm{m}_{\mathrm{i}}(\mathrm{F})=\mathrm{m}_{\mathrm{i}}(\mathrm{I})\) and \(\mathrm{m}_{\mathrm{j}}(\mathrm{F})=\mathrm{m}_{\mathrm{j}}(\mathrm{II})\). \[\text { By definition, } \Delta \equiv \phi(\mathrm{F}) \,\left[\mathrm{m}_{\mathrm{i}}(\mathrm{I})+\mathrm{m}_{\mathrm{j}}(\mathrm{II})\right]-\left[\phi(\mathrm{I}) \, \mathrm{m}_{\mathrm{i}}(\mathrm{I})+\phi(\mathrm{II}) \, \mathrm{m}_{\mathrm{j}}(\mathrm{II})\right]\]Experiments based on isopiestic measurements using the equilibrium between reference and mixed solutions and independently determined \(\phi(\mathrm{I})\) and \(\phi(\mathrm{II})\) yield the quantity \(\Delta\).The starting points are general equations for the activity coefficients \(\gamma_{i}\) and \(\gamma_{j}\) for solutes \(i\) and \(j\) respectively as a function of the molalities \(\mathrm{m}_{i}\) and \(\mathrm{m}_{j}\) in the mixed solutions. Two equations based on Taylor series are used. \[\ln \left(\gamma_{i}\right)=\sum_{k=0}^{k=\infty} \sum_{\lambda=0}^{\lambda=\infty} A_{k \lambda} \,\left(m_{i} / m^{0}\right)^{k} \,\left(m_{j} / m^{0}\right)^{\lambda}\]\[\ln \left(\gamma_{k}\right)=\sum_{k=0}^{k=\infty} \sum_{\lambda=0}^{\lambda=\infty} B_{k \lambda} \,\left(m_{i} / m^{0}\right)^{k} \,\left(m_{j} / m^{0}\right)^{\lambda}\]With reference to equations (h) and (i), both \(\mathrm{A}_{00}\) and \(\mathrm{B}_{00}\) are zero. The dimensionless coefficients \(\mathrm{A}_{k} \lambda\) and \(\mathrm{B}_{k} \lambda\) are interlinked by the Gibbs-Duhem equation. It also turns out that the series up to and including ‘\(k = 4\)’ and ‘\(\lambda = 4\)’ are sufficient in the analysis of experimental results.According to equation (h), a description of the properties of solute-\(i\) is given by equation (j). \[\begin{aligned} \ln \left(\gamma_{\mathrm{i}}\right)=& \mathrm{A}_{10} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)+\mathrm{A}_{01} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)+\mathrm{A}_{20} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{2} \\ &+\mathrm{A}_{11} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right) \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right) \\ &+\mathrm{A}_{02} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2}+\mathrm{A}_{30} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{3}+\mathrm{A}_{21} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{2} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right) \\ &+\mathrm{A}_{12} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right) \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2}+\mathrm{A}_{03} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{3}+\mathrm{A}_{40} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{4} \\ &+\mathrm{A}_{31} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{3} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)+\mathrm{A}_{22} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{2} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2} \\ &+\mathrm{A}_{13} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right) \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{3}+\mathrm{A}_{04} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{4} \end{aligned}\]In the event that \(\mathrm{m}_{j}\) is zero, \[\begin{aligned} \ln \left[\gamma_{\mathrm{i}}\left(\mathrm{m}_{\mathrm{j}}=0\right)\right]=\mathrm{A}_{10} \, &\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)+\mathrm{A}_{20} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{2} \\ &+\mathrm{A}_{30} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{3}+\mathrm{A}_{40} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{4} \end{aligned}\]Moreover \(\ln \left[\gamma_{i}\left(m_{j}=0\right)\right]\) can be calculated from the measured properties of aqueous solutions containing only solute-\(i\). Therefore the dependence of \(\ln \left[\gamma_{i}\left(m_{j}=0\right)\right]\) on \(\mathrm{m}_{i}\) can be analysed using a linear least squares procedure to yield the coefficients \(\mathrm{A}_{k 0}\) for \(k=1- 4\). Hence \(\ln \left(\gamma_{i}\right)\) for the mixed solute system is given by a combination of equations (j) and (k) to yield equation (\(\lambda\)). \[\begin{aligned} &\ln \left(\gamma_{\mathrm{i}}\right)=\ln \left[\gamma_{\mathrm{i}}\left(\mathrm{m}_{\mathrm{j}}=0\right)\right] \\ &+\mathrm{A}_{01} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)+\mathrm{A}_{11} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right) \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)+\mathrm{A}_{02} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2} \\ &+\mathrm{A}_{21} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{2} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)+\mathrm{A}_{12} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right) \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2} \\ &\quad+\mathrm{A}_{03} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{3}+\mathrm{A}_{31} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{3} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right) \\ &\quad+\mathrm{A}_{22} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{2} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2} \\ &\quad+\mathrm{A}_{13} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right) \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{3}+\mathrm{A}_{04} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{4} \end{aligned}\]According to equation (\(\lambda\)), the dependence of \(\gamma_{i}\) on \(\mathrm{m}_{j}\) at fixed \(\mathrm{m}_{i}\) is given by equation (m). \[\begin{aligned} {\left[\frac{\partial \ln \left(\gamma_{\mathrm{i}}\right)}{\partial \mathrm{m}_{\mathrm{j}}}\right]_{\mathrm{m}(\mathrm{i})} } &=\mathrm{A}_{01} \,\left(\mathrm{m}^{0}\right)^{-1}+\mathrm{A}_{11} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right) \,\left(\mathrm{m}^{0}\right)^{-1}+\mathrm{A}_{02} \, 2 \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-2} \\ &+\mathrm{A}_{21} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-1}+\mathrm{A}_{12} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right) \, 2 \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-2} \\ &+\mathrm{A}_{03} \, 3 \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-3}+\mathrm{A}_{31} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{3} \,\left(\mathrm{m}^{0}\right)^{-1} \\ &+\mathrm{A}_{22} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{2} \, 2 \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{2}+\mathrm{A}_{13} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right) \, 3 \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-3} \\ &+\mathrm{A}_{04} \, 4 \,\left(\mathrm{m}_{\mathrm{j}}\right)^{3} \,\left(\mathrm{m}^{0}\right)^{-4} \end{aligned}\]A cross-differential link yields the following interesting equation. \[\left[\frac{\partial \ln \left(\gamma_{\mathrm{i}}\right)}{\partial \mathrm{m}_{\mathrm{j}}}\right]_{\mathrm{m}(\mathrm{i})}=\left[\frac{\partial \ln \left(\gamma_{\mathrm{j}}\right)}{\partial \mathrm{m}_{\mathrm{i}}}\right]_{\mathrm{m}(\mathrm{j})}\]We combine equations (m) and (n). \[\begin{aligned} {\left[\frac{\partial \ln \left(\gamma_{\mathrm{j}}\right)}{\partial \mathrm{m}_{\mathrm{i}}}\right]_{\mathrm{m}(\mathrm{j})} } &=\mathrm{A}_{01} \,\left(\mathrm{m}^{0}\right)^{-1}+\mathrm{A}_{11} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right) \,\left(\mathrm{m}^{0}\right)^{-1}+\mathrm{A}_{02} \, 2 \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-2} \\ &+\mathrm{A}_{21} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-1}+\mathrm{A}_{12} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right) \, 2 \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-2} \\ &+\mathrm{A}_{03} \, 3 \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-3}+\mathrm{A}_{31} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{3} \,\left(\mathrm{m}^{0}\right)^{-1} \\ &+\mathrm{A}_{22} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{2} \, 2 \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-2}+\mathrm{A}_{13} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right) \, 3 \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-3} \\ &+\mathrm{A}_{04} \, 4 \,\left(\mathrm{m}_{\mathrm{j}}\right)^{3} \,\left(\mathrm{m}^{0}\right)^{-4} \end{aligned}\]We integrate the latter equation to yield an equation for \(\gamma_{j}\left(m_{i}=0\right)\) where at ‘\(\mathrm{m}_{i} = 0\)’ , \(\gamma_{j}\) is represented as \(\gamma_{j}\left(\mathrm{~m}_{\mathrm{i}}=0\right)\). The outcome is an equation for \(\ln \left(\gamma_{j}\right)\) in terms of the \(\mathrm{A}_{i}\)-variables, making the \(\mathrm{B}_{i}\) variables somewhat redundant. \[\begin{aligned} \ln \left(\gamma_{\mathrm{j}}\right)=& \ln \left[\gamma_{\mathrm{j}}\left(\mathrm{m}_{\mathrm{i}}=0\right)\right] \\ +& \mathrm{A}_{01} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)+\left(\mathrm{A}_{11} / 2\right) \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{2}+2 \, \mathrm{A}_{02} \,\left(\mathrm{m}_{\mathrm{i}} \, \mathrm{m}_{\mathrm{j}}\right) \,\left(\mathrm{m}^{0}\right)^{-2} \\ &+\left(\mathrm{A}_{21} / 3\right) \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{3}+\mathrm{A}_{12} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{2} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-1} \\ &+3 \, \mathrm{A}_{03} \, \mathrm{m}_{\mathrm{i}} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-3} \\ &+(1 / 4) \, \mathrm{A}_{31} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{4} \\ &+\left(2 \, \mathrm{A}_{22} / 3\right) \,\left(\mathrm{m}_{\mathrm{i}}\right)^{3} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-4}+\left(3 \, \mathrm{A}_{13} / 2\right) \,\left(\mathrm{m}_{\mathrm{i}}\right)^{2} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-4} \\ &+4 \, \mathrm{A}_{04} \, \mathrm{m}_{\mathrm{i}} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{3} \,\left(\mathrm{m}^{0}\right)^{-4} \end{aligned}\]For an aqueous solution at fixed \(\mathrm{T}\) and \(\mathrm{p}\) containing the single solute-\(i\), the Gibbs-Duhem equation yields the following relationship. \[\mathrm{n}_{1} \, \mathrm{d} \mu_{1}(\mathrm{aq})+\mathrm{n}_{\mathrm{i}} \, \mathrm{d} \mu_{\mathrm{i}}(\mathrm{aq})=0\]\[\text { Then, } \begin{aligned} \frac{1}{\mathrm{M}_{1}} \, \mathrm{d}\left[\mu_{1}^{*}(\lambda)-\phi_{\mathrm{i}} \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{i}}\right] \\ &+\mathrm{m}_{\mathrm{i}} \, \mathrm{d}\left[\mu_{\mathrm{i}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{i}} \, \gamma_{\mathrm{i}} / \mathrm{m}^{0}\right)\right]=0 \end{aligned}\]The symbol \(\phi_{i}\) identifies the practical osmotic coefficient in a solution containing solute-\(i\). \[\text { Hence }, \quad \mathrm{d}\left[\phi_{i} \, \mathrm{m}_{\mathrm{i}}\right]=\mathrm{dm} \mathrm{m}_{\mathrm{i}}+\mathrm{m}_{\mathrm{i}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{i}}\right)\]Similarly for an aqueous solutions containing solute-\(j\), \[\mathrm{d}\left[\phi_{\mathrm{j}} \, \mathrm{m}_{\mathrm{j}}\right]=\mathrm{dm} \mathrm{m}_{\mathrm{j}}+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right)\]From the Gibbs-Duhem equation (at fixed \(\mathrm{T}\) and \(p\)) \[n_{1} \, d \mu_{1}(a q)+n_{i} \, d \mu_{i}(a q)+n_{j} \, d \mu_{j}(a q)=0\]Then, \[\begin{aligned} &\frac{1}{\mathrm{M}_{1}} \, \mathrm{d}\left[\mu_{1}^{*}(\lambda)-\phi_{\mathrm{ij}} \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \,\left(\mathrm{m}_{\mathrm{i}}+\mathrm{m}_{\mathrm{j}}\right)\right] \\ &\quad+\mathrm{m}_{\mathrm{i}} \, \mathrm{d}\left[\mu_{\mathrm{i}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{i}} \, \gamma_{\mathrm{i}} / \mathrm{m}^{0}\right)\right] \\ &\quad+\mathrm{m}_{\mathrm{j}} \, \mathrm{d}\left[\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)\right]=0 \end{aligned}\]The practical osmotic coefficient \(\phi_{ij}\) identifies a solution containing two solutes, \(i\) and \(j\). \[\text { Hence, } \mathrm{d}\left[\phi_{\mathrm{ij}} \,\left(\mathrm{m}_{\mathrm{i}}+\mathrm{m}_{\mathrm{j}}\right)\right]=\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)+\mathrm{m}_{\mathrm{i}} \, \mathrm{d} \ln \left(\mathrm{m}_{\mathrm{i}} \, \gamma_{\mathrm{i}} / \mathrm{m}^{0}\right)\]\[\text { Therefore } \mathrm{d}\left[\phi_{\mathrm{ij}} \,\left(\mathrm{m}_{\mathrm{i}}+\mathrm{m}_{\mathrm{j}}\right)\right]=\mathrm{d}\left(\mathrm{m}_{\mathrm{i}}+\mathrm{m}_{\mathrm{j}}\right)+\mathrm{m}_{\mathrm{i}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right)+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right)\]According to equation (g) \[\Delta \equiv \phi(\mathrm{F}) \,\left[\mathrm{m}_{\mathrm{i}}(\mathrm{I})+\mathrm{m}_{\mathrm{j}}(\mathrm{II})\right]-\left[\phi(\mathrm{I}) \, \mathrm{m}_{\mathrm{i}}(\mathrm{I})+\phi(\mathrm{II}) \, \mathrm{m}_{\mathrm{j}}(\mathrm{II})\right]\]\[\text { Then } \mathrm{d} \Delta=\mathrm{d}\left\{\phi(\mathrm{F}) \,\left[\mathrm{m}_{\mathrm{i}}(\mathrm{I})+\mathrm{m}_{\mathrm{j}}(\mathrm{II})\right]-\mathrm{d}\left[\phi(\mathrm{I}) \, \mathrm{m}_{\mathrm{i}}(\mathrm{I})\right]-\mathrm{d}\left[\phi(\mathrm{II}) \, \mathrm{m}_{\mathrm{j}}(\mathrm{II})\right]\right.\]Labels (I) and (II) can be dropped when applied to solute molalities. Then using equations (s), (t) and (x), \[\begin{array}{r} \mathrm{d} \Delta=\mathrm{d}\left(\mathrm{m}_{\mathrm{i}}+\mathrm{m}_{\mathrm{j}}\right)+\mathrm{m}_{\mathrm{i}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{i}}(\mathrm{F})\right]+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left[\gamma_{\mathrm{j}}(\mathrm{F})\right] \\ \quad-\mathrm{dm}_{\mathrm{i}}-\mathrm{m}_{\mathrm{i}} \, \mathrm{d} \ln \left[\gamma_{\mathrm{i}}(\mathrm{I})\right]-\mathrm{dm}_{\mathrm{j}}-\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left[\gamma_{\mathrm{j}}[\mathrm{II}]\right. \end{array}\]\[\text { Or, } \mathrm{d} \Delta=\mathrm{m}_{\mathrm{i}} \, \mathrm{d}\left\{\ln \left(\gamma_{\mathrm{i}}(\mathrm{F})-\ln \left[\gamma_{\mathrm{j}}(\mathrm{I})\right]\right\}+\mathrm{m}_{\mathrm{j}} \, \mathrm{d}\left\{\ln \left[\gamma_{\mathrm{j}}(\mathrm{F})\right]-\mathrm{d} \ln \left[\gamma_{\mathrm{j}}[\mathrm{II}]\right\}\right.\right.\]In equation (\(\lambda\)) we identify \(\ln \left[\gamma_{i}\left(m_{j}=0\right)\right]\) with \(\ln \left[\gamma_{i}(\mathrm{I})\right]\). Similarly \(\ln \left[\gamma_{\mathrm{j}}\left(\mathrm{m}_{\mathrm{i}}=0\right)\right]\) in equation (p) equals \(\ln \left[\gamma_{\mathrm{j}}(\mathrm{II})\right]\) in equation (za). Therefore \[\begin{aligned} \Delta / & \mathrm{m}_{\mathrm{i}} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-1} \\ =& \mathrm{A}_{01}+\mathrm{A}_{11} \, \mathrm{m}_{\mathrm{i}} \,\left(\mathrm{m}^{0}\right)^{-1}+2 \, \mathrm{A}_{02} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-1}+\mathrm{A}_{21} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-2} \\ &+(3 / 2) \, \mathrm{A}_{12} \, \mathrm{m}_{\mathrm{i}} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-2}+3 \, \mathrm{A}_{03} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-2}+\mathrm{A}_{31} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{3} \\ &+(4 / 3) \, \mathrm{A}_{22} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{2} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-3}+2 \, \mathrm{A}_{13} \, \mathrm{m}_{\mathrm{i}} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-3} \\ &+4 \, \mathrm{A}_{04} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{3} \,\left(\mathrm{m}^{0}\right)^{-3} \end{aligned}\]Footnotes V. E. Bower and R. A. Robinson , J.Phys.Chem.,1963,67,1524. H. D. Ellerton and P. J. Dunlop, J. Phys Chem.1966,70,1831. H. D. Ellerton, G. Reinfelds, D. E. Mulcahy and P. J. Dunlop, J. Phys. Chem., 1964, 68,398. Hence, \(-\mathrm{d}\left[\phi_{\mathrm{i}} \, \mathrm{m}_{\mathrm{i}}\right]+\mathrm{m}_{\mathrm{i}} \, \mathrm{d} \ln \left(\mathrm{m}_{\mathrm{i}} \, \gamma_{\mathrm{i}} / \mathrm{m}^{0}\right)=0\)Then, \(-\mathrm{d}\left[\phi_{\mathrm{i}} \, \mathrm{m}_{\mathrm{i}}\right]+\mathrm{m}_{\mathrm{i}} \, \mathrm{d}\left[\ln \left(\mathrm{m}_{\mathrm{i}}\right)+\ln \left(\gamma_{\mathrm{i}}\right)-\ln \left(\mathrm{m}^{0}\right)\right]=0\)Or, \(\mathrm{d}\left[\phi_{\mathrm{i}} \, \mathrm{m}_{\mathrm{i}}\right]=\mathrm{m}_{\mathrm{i}} \, \frac{1}{\mathrm{~m}_{\mathrm{i}}} \mathrm{dm} \mathrm{m}_{\mathrm{i}}+\mathrm{m}_{\mathrm{i}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{i}}\right)\)Then, \(\mathrm{d}\left[\phi_{\mathrm{i}} \, \mathrm{m}_{\mathrm{i}}\right]=\mathrm{dm}_{\mathrm{i}}+\mathrm{m}_{\mathrm{i}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{i}}\right)\) Or, \[\begin{aligned} &\mathrm{d}\left[\phi_{\mathrm{ij}} \,\left(\mathrm{m}_{\mathrm{i}}+\mathrm{m}_{\mathrm{j}}\right)\right] \\ &\quad=\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right)+\mathrm{m}_{\mathrm{i}} \, \mathrm{d} \ln \left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right) \end{aligned}\]Or \[\begin{aligned} &\mathrm{d}\left[\phi_{\mathrm{ij}} \,\left(\mathrm{m}_{\mathrm{i}}+\mathrm{m}_{\mathrm{j}}\right)\right] \\ &\quad=\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}_{\mathrm{j}}\right) \, \mathrm{d}\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right)+\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}_{\mathrm{i}}\right) \, \mathrm{d}\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right) \end{aligned}\]Or, \[\begin{aligned} &\mathrm{d}\left[\phi_{\mathrm{ij}} \,\left(\mathrm{m}_{\mathrm{i}}+\mathrm{m}_{\mathrm{j}}\right)\right] \\ &\quad=\mathrm{d}\left(\mathrm{m}_{\mathrm{j}}+\mathrm{m}_{\mathrm{i}}\right)+\mathrm{m}_{\mathrm{i}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{i}}\right)+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right) \end{aligned}\] Differentiation of equation (\(\lambda\)) yields \[\begin{aligned} &\mathrm{d}\left\{\ln \left[\gamma_{\mathrm{i}}(\mathrm{F})-\ln \left[\gamma_{\mathrm{i}}(\mathrm{I})\right]\right\}=\right. \\ &\mathrm{A}_{01} \,\left(\mathrm{m}^{0}\right)^{-1} \, \mathrm{dm}_{\mathrm{j}}+\mathrm{A}_{11} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{dm}_{\mathrm{i}}+\mathrm{A}_{11} \, \mathrm{m}_{\mathrm{i}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{dm}_{\mathrm{j}} \\ &+2 \, \mathrm{A}_{02} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{dm}_{\mathrm{j}}+2 \, \mathrm{A}_{21} \, \mathrm{m}_{\mathrm{i}} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-3} \, \mathrm{dm}_{\mathrm{i}} \\ &+\mathrm{A}_{21} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-3} \, \mathrm{dm} \mathrm{m}_{\mathrm{j}}+\mathrm{A}_{12} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-3} \, \mathrm{dm}_{\mathrm{i}} \\ &+2 \, \mathrm{A}_{12} \, \mathrm{m}_{\mathrm{i}} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-3} \, \mathrm{dm} \mathrm{j}_{\mathrm{j}}+3 \, \mathrm{A}_{03} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-3} \, \mathrm{dm} \mathrm{m}_{\mathrm{j}} \\ &+3 \, \mathrm{A}_{31} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{2} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{i}}+\mathrm{A}_{31} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{3} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{j}} \\ &+2 \, \mathrm{A}_{22} \, \mathrm{m}_{\mathrm{i}} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{i}}+2 \, \mathrm{A}_{22} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{2} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{j}} \\ &+\mathrm{A}_{13} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{3} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{i}}+3 \, \mathrm{A}_{13} \, \mathrm{m}_{\mathrm{i}} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{j}} \\ &+4 \, \mathrm{A}_{04} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{3} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{j}} \end{aligned}\]Similarly from equation (p), \[\begin{aligned} &\mathrm{d}\left\{\ln \left[\gamma_{\mathrm{j}}(\mathrm{F})-\ln \left[\gamma_{\mathrm{j}}(\mathrm{I})\right]\right\}=\right. \\ &\mathrm{A}_{01} \,\left(\mathrm{m}^{0}\right)^{-1} \, \mathrm{dm}_{\mathrm{i}}+\mathrm{A}_{11} \, \mathrm{m}_{\mathrm{i}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{dm}_{\mathrm{i}} \\ &+2 \, \mathrm{A}_{02} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{dm}_{\mathrm{i}}+2 \, \mathrm{A}_{02} \, \mathrm{m}_{\mathrm{i}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{dm}_{\mathrm{j}} \\ &+\mathrm{A}_{21} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-3} \, \mathrm{dm}_{\mathrm{i}}+2 \, \mathrm{A}_{12} \, \mathrm{m}_{\mathrm{i}} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-3} \, \mathrm{dm}_{\mathrm{i}} \\ &+\mathrm{A}_{12} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-3} \, \mathrm{dm} \mathrm{m}_{\mathrm{j}}+3 \, \mathrm{A}_{03} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-3} \, \mathrm{dm}_{\mathrm{i}} \\ &+6 \, \mathrm{A}_{03} \, \mathrm{m}_{\mathrm{i}} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-3} \, \mathrm{dm}_{\mathrm{j}}+\mathrm{A}_{31} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{3} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{i}} \\ &+\mathrm{A}_{22} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{2} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm} \mathrm{m}_{\mathrm{i}}+(2 / 3) \, \mathrm{A}_{22} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{3} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm} \\ &\mathrm{j} \\ &+3 \, \mathrm{A}_{13} \, \mathrm{m}_{\mathrm{i}} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{i}} \\ &+3 \, \mathrm{A}_{13} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{2} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{j}}+4 \, \mathrm{A}_{04} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{3} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{i}} \\ &+12 \, \mathrm{A}_{04} \, \mathrm{m}_{\mathrm{i}} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{j}} \end{aligned}\]But according to equation (za), \[\mathrm{d} \Delta=\mathrm{m}_{\mathrm{i}} \, \mathrm{d}\left\{\ln \left(\gamma_{\mathrm{i}}(\mathrm{F})-\ln \left[\gamma_{\mathrm{j}}(\mathrm{I})\right]\right\}+\mathrm{m}_{\mathrm{j}} \, \mathrm{d}\left\{\ln \left[\gamma_{\mathrm{j}}(\mathrm{F})\right]-\mathrm{d} \ln \left[\gamma_{\mathrm{j}}[\mathrm{II}]\right\}\right.\right.\]After rearranging one obtains the following equation. \[\begin{aligned} &\mathrm{d} \Delta= \\ &\mathrm{A}_{01} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-1} \, \mathrm{dm}_{\mathrm{i}}+\mathrm{A}_{01} \, \mathrm{m}_{\mathrm{i}} \,\left(\mathrm{m}^{0}\right)^{-1} \, \mathrm{dm}_{\mathrm{j}} \\ &+2 \, \mathrm{A}_{11} \, \mathrm{m}_{\mathrm{i}} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{dm}_{\mathrm{i}}+\mathrm{A}_{11} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{dm}_{\mathrm{j}} \\ &+2 \, \mathrm{A}_{02} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{dm}_{\mathrm{i}}+4 \, \mathrm{A}_{02} \, \mathrm{m}_{\mathrm{i}} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{dm}_{\mathrm{j}} \\ &+3 \, \mathrm{A}_{21} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{2} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-3} \, \mathrm{dm}_{\mathrm{i}}+\mathrm{A}_{21} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{3} \,\left(\mathrm{m}^{0}\right)^{-3} \, \mathrm{dm}_{\mathrm{j}} \\ &+3 \, \mathrm{A}_{12} \, \mathrm{m}_{\mathrm{i}} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-3} \, \mathrm{dm}_{\mathrm{i}}+3 \, \mathrm{A}_{12} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{2} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-3} \, \mathrm{dm}_{\mathrm{j}} \\ &+3 \, \mathrm{A}_{03} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{3} \,\left(\mathrm{m}^{0}\right)^{-3} \, \mathrm{dm}_{\mathrm{i}}+9 \, \mathrm{A}_{03} \, \mathrm{m}_{\mathrm{i}} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-3} \, \mathrm{dm}_{\mathrm{j}} \\ &+4 \, \mathrm{A}_{31} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{3} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{i}}+\mathrm{A}_{31} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{4} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{j}} \\ &+4 \, \mathrm{A}_{22} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{2} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{i}}+(8 / 3) \, \mathrm{A}_{22} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{3} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{j}} \\ &+4 \, \mathrm{A}_{13} \, \mathrm{m}_{\mathrm{i}} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{3} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{i}}+6 \, \mathrm{A}_{13} \,\left(\mathrm{m}_{\mathrm{i}}\right)^{2} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{j}} \\ &+4 \, \mathrm{A}_{04} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{4} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{i}}+16 \, \mathrm{A}_{04} \, \mathrm{m}_{\mathrm{i}} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{3} \,\left(\mathrm{m}^{0}\right)^{-4} \, \mathrm{dm}_{\mathrm{j}} \end{aligned}\]Term by term integration of the latter equation yields \[\Delta=\int \mathrm{d} \Delta\]As an example we cite the terms containing the coefficient \(\mathrm{A}_{11}\). \[\begin{aligned} &\int 2 \, A_{11} \, m_{i} \, m_{j} \,\left(m^{0}\right)^{-2} \, d m_{i}+\int A_{11} \,\left(m_{i}\right)^{2} \,\left(m^{0}\right)^{-2} \, d m_{j} \\ &=A_{11} \,\left(m^{0}\right)^{-2} \, d m_{i} \, \int\left[2 \, m_{i} \, m_{j} \, d m_{i}+\left(m_{i}\right)^{2} \, d m_{j}\right] \\ &=A_{11} \,\left(m^{0}\right)^{-2} \, \int d\left[\left(m_{i}\right)^{2} \, m_{j}\right] \\ &=A_{11} \,\left(m^{0}\right)^{-2} \,\left(m_{i}\right)^{2} \, m_{j} \end{aligned}\]This page titled 1.1.9: Activity of Water - Two Solutes is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.1.10: Activity Coefficient- Two Neutral Solutes- Solute + Trace Solute i

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.01%3A_Activity/1.1.10%3A_Activity_Coefficient-_Two_Neutral_Solutes-_Solute__Trace_Solute_i

A given solution is prepared using n1 moles of water(\(\lambda\)) together with \(n_{i}\) and \(n_{j}\) moles of neutral solutes \(i\) and \(j\) respectively at temperature \(\mathrm{T}\) and pressure \(p\) (which is close to the standard pressure \(p^{0}\)). The mass of water is \(\mathrm{n}_{1} \, \mathrm{M}_{1}\) where \(\mathrm{M}_{1}\) is the molar mass of water. Hence the molalities of solutes \(i\) and \(j\) are \(\mathrm{m}_{\mathrm{i}} \left(=\mathrm{n}_{\mathrm{i}} / \mathrm{n}_{1} \, \mathrm{M}_{1}\right)\) and \( \mathrm{m}_{\mathrm{j}}\left(=\mathrm{n}_{\mathrm{j}} / \mathrm{n}_{1} \, \mathrm{M}_{1}\right)\) respectively. The chemical potential of water in the aqueous solution \(\mu_{1}(\mathrm{aq})\) is related to the molality of solutes using equation (a) where \(\phi\) is the practical osmotic coefficient and \(\mu_{1}^{*}(\lambda)\) is the chemical potential water(\(\lambda\)) at the same \(\mathrm{T}\) and \(p\). \[\mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\lambda)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \,\left(\mathrm{m}_{\mathrm{i}}+\mathrm{m}_{\mathrm{j}}\right)\]The chemical potentials of the two solutes, \(\mu_{\mathrm{i}}(\mathrm{aq})\) and \(\mu_{\mathrm{j}}(\mathrm{aq})\), are related to \(\mathrm{m}_{i}\) and \(\mathrm{m}_{j}\) together with corresponding activity coefficients, \(\gamma_{i}\) and \(\gamma_{j}\) using equations (b) and (c). \[\mu_{\mathrm{i}}(\mathrm{aq})=\mu_{\mathrm{i}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{i}} \, \gamma_{\mathrm{i}} / \mathrm{m}^{0}\right)\]\[\mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)\]Here \(\mu_{i}^{0}(\mathrm{aq})\) is the reference chemical potential for solute \(i\) in a solution where \(\mathrm{m}_{\mathrm{j}}=0 \mathrm{~mol} \mathrm{~kg}{ }^{-1}\), \(\mathrm{m}_{\mathrm{i}}=1 \mathrm{~mol} \mathrm{~kg}\) and \(\gamma_{i} = 1\). A similar definition operates for solute \(j\). For the mixed solution at all \(\mathrm{T}\) and \(p\), \[\operatorname{limit}\left(\mathrm{m}_{\mathrm{i}} \rightarrow 0 ; \mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{\mathrm{i}}=1\]\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{i}} \rightarrow 0 ; \mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{\mathrm{j}}=1\]In these terms, the thermodynamic properties of solute \(i\) are not ideal as a consequence of \(i-i\), \(j-j\) and \(i-j\) solute-solute interactions. A similar comment concerns the thermodynamic properties of solute \(j\). With increase in molalities \(\mathrm{m}_{i}\) and \(\mathrm{m}_{j}\), so the extent to which the thermodynamic properties deviate from ideal increases; i.e. for solute \(j\), \(\gamma_{j} \neq 1\) and for solute \(i\), \(\gamma{i} \neq 1\). Such deviations can be understood in terms of \(i-i\), \(j-j\), and \(i-j\) solute-solute interactions.In some applications of this analysis, solute \(i\) is present in trace amounts and so \(\gamma_{i}\) in the absence of solute \(j\) would be close to unity. However as the molality of solute \(j\) is increased, the thermodynamic properties of solute \(i\) deviate increasingly from ideal as a result of solute \(i\) \(\rightleftarrows\) solute \(j\) interactions. This feature can be described quantitatively using equation \((\mathrm{f})\) where \(\beta_{1}, \beta_{2} \ldots\) describe the role of pairwise i-j , triplet \(i-j-j \ldots \ldots\) solute-solute interactions. \[\ln \left(\gamma_{i}\right)=\beta_{1} \,\left(m_{j} / m^{0}\right)+\beta_{2} \,\left(m_{j} / m^{0}\right)^{2}+\ldots \ldots\]Depending on the signs of the \(\beta\) - coefficients , added solute \(j\) can either stabilise or destabilise solute-\(i\); i.e. either lower or raise \(\mu_{i}(a q)\) relative to that in a solution having ideal thermodynamic properties.Footnotes Based on an analysis suggested by C. Wagner, Thermodynamics of Alloys, Addison-Wesley, Reading, Mass., 1952, pages 19-22. As quoted by G. N. Lewis and M. Randall, Thermodynamics, 2nd edn., revised by K. S. Pitzer and L Brewer, McGraw-Hill, New York, 1961, page 562.This page titled 1.1.10: Activity Coefficient- Two Neutral Solutes- Solute + Trace Solute i is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.1.11: Activity Coefficients

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In descriptions of the properties of the components of liquid mixtures, rational activity coefficients are given the symbols \(\mathrm{f}_{1}, \mathrm{f}_{2}, \mathrm{f}_{3} \ldots\). With reference to solvents their thermodynamic properties are described by rational activity coefficients \(\mathrm{f}_{1}, \mathrm{f}_{2}, \mathrm{f}_{3} \ldots\) and (practical) osmotic coefficient, \(\phi\). The properties of solutes in solutions are described using activity coefficients which are linked to the descriptions of the composition of solutions: molality scale, \(\gamma_{j}\); concentration scale \(\gamma_{j}\); mole fraction scale \(\mathrm{f}_{\mathrm{j}}^{*}\).These coefficients are intimately related to the concept of activity. Their significance is clarified by equations relating chemical potentials to the composition of a given system; e.g.In all cases they summarise the extent to which the thermodynamic properties of liquid mixtures and solutions are not ideal. The challenge for chemists is to understand in terms of molecular properties why the thermodynamic properties of real systems are not ideal. It has to be admitted that activity coefficients have a ‘bad press’ as far as most chemists are concerned. All too often their importance is ignored. But ‘learn to love activity coefficients! Perhaps the importance of activity coefficients can be understood in the following terms.The chemical potential of urea as a solute in aqueous solutions, \(\mu_{j}(\mathrm{aq})\) at temperature \(\mathrm{T}\) and pressure \(p\) ( \(\approx\) the standard pressure \(p^{0}\)) is related to the molality of urea \(\mathrm{m}_{j}\) using equation (a). \[\begin{aligned} &\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T}, \mathrm{p})=\\ &\mu_{\mathrm{j}}^{0}\left(\mathrm{aq} ; \mathrm{T}, \mathrm{p}, \mathrm{m}_{\mathrm{j}}=1 \mathrm{~mol} \mathrm{~kg} \mathrm{~kg}^{-1}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\gamma_{\mathrm{j}}\right) \end{aligned}\]The term \(\mu_{\mathrm{j}}^{0}\left(\mathrm{aq} ; \mathrm{T}, \mathrm{p}, \mathrm{m}_{\mathrm{j}}=1 \mathrm{~mol} \mathrm{\textrm {kg } ^ { - 1 } )}\right.\) describes the chemical potential of solute, urea in an aqueous solution having unit molality where there are no urea - urea (i.e. solute – solute) interactions. Each urea molecule in these terms is unaware of the presence of other urea molecules in the aqueous solution. In a solution having ideal thermodynamic properties \(\mu_{j}(\mathrm{aq} ; \mathrm{T}, \mathrm{p} ; \mathrm{id})\) depends on the molality of solute mj. Thus the osmotic pressure of this solution is a function of the molality of urea. For such an ideal solution there are no urea-urea interactions although there are important urea-water interactions, the hydration of urea. But in a real solution there are also solute-solute interactions. Each solute molecule ‘knows’ there are other solute molecules in the solution. Indeed the extent to which \(\mu_{j}(\mathrm{aq} ; \mathrm{T}, \mathrm{p})\) differs from \(\mu_{j}(\mathrm{aq} ; \mathrm{T}, \mathrm{p} ; \mathrm{id})\) is a function of the molality of the solute, \(\mathrm{m}_{j}\).This page titled 1.1.11: Activity Coefficients is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.1.12: Activity Coefficients- Salt Solutions- Ion-Ion Interactions

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For most dilute aqueous salt solutions (at ambient temperature and pressure), mean ionic activity coefficients γ± are less than unity. Thus ion-ion interactions within a real solution lower chemical potentials below those of salts in the corresponding ideal solutions. Clearly a quantitative treatment of this stabilisation is enormously important. In fact for almost the whole of the 20th Century, scientists offered theoretical bases for expressing \(\ln \left(\gamma_{\pm}\right)\) as a function of the composition of a salt solution, temperature, pressure and electric permittivity of the solvent.In effect we offer as much information as demanded by the theory (e.g. molality of salt, nature of salt, permittivity of solvent, ion sizes, temperature, pressure, .....). We set the apparently simple task - please calculate the mean activity coefficient of the salt in this solution.Many models and treatments have been proposed. Most models start by considering a reference j-ion in an aqueous salt solution. In order to calculate the electric potential at the j-ion arising from all other ions in solution, we need to know the distribution of these ions about the j-ion. Unfortunately this distribution is unknown and so we need a model for this distribution. Further activity coefficients reflect the impact of ions on water-water interactions in aqueous solutions .Footnotes (a) R. A. Robinson and R. H. Stokes, Electrolyte Solutions, Butterworths, London, 2nd. edition revised,1965. (b) H. S. Harned and B. B. Owen., The Physical Chemistry of Electrolytic Solutions ,Reinhold, New York, 2nd. edn.,1950, chapter The analysis presented by Harned and Owen anticipates application to irreversible processes; e.g. electric conductance of salt solutions. Here we confine attention to equilibrium properties. H. S. Frank, Z. Phys. Chem., 1965,228,364.This page titled 1.1.12: Activity Coefficients- Salt Solutions- Ion-Ion Interactions is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.10.1: Gibbs Energy

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The Gibbs energy is an extensive state variable defined by the following equation. \[\mathrm{G}=\mathrm{U}+\mathrm{p} \, \mathrm{V}-\mathrm{T} \, \mathrm{S}\]Instead of Gibbs energy the terms Gibbs free energy and Gibbs function are often used. Physicists prefer the term Gibbs function. The term ‘free energy’ is not encouraged. Everyday experience tells us that no energy is ‘free’.Footnote Nevertheless the French term ‘enthalpie libre’ ( i.e. free enthalpy) for \(\mathrm{G}\) has merit. Enthalpy is defined by \(\mathrm{H} = \mathrm{ U} + \mathrm{p V}\). Then \(\mathrm{G} = \mathrm{ H} – \mathrm{ TS}\). The product \(\mathrm{TS}\) is the linked energy in a system from which no work can be produced. Hence the available or ‘free’ part of the enthalpy is the Gibbs energy.This page titled 1.10.1: Gibbs Energy is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.10.2: Gibbs Energy- Thermodynamic Potential

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The Gibbs energy of a system, \[\mathrm{G}=\mathrm{U}+\mathrm{p} \, \mathrm{V}-\mathrm{T} \, \mathrm{S}\]For a closed single phase system, changes in thermodynamic energy \(\mathrm{dU}\) and Gibbs energy \(\mathrm{dG}\) are related by the following equation. \[\mathrm{dG}=\mathrm{dU}+\mathrm{p} \, \mathrm{dV}+\mathrm{V} \, \mathrm{dp}-\mathrm{T} \, \mathrm{dS}-\mathrm{S} \, \mathrm{dT}\]The change in thermodynamic energy \(\mathrm{dU}\) is related to the affinity for spontaneous change using the Master Equation. \[\mathrm{dU}=\mathrm{T} \, \mathrm{dS}-\mathrm{p} \, \mathrm{dV}-\mathrm{A} \, \mathrm{d} \xi ; \quad \mathrm{A} \, \mathrm{d} \xi \geq 0\]We use equation (b) by substituting for \(\mathrm{dU}\) in equation (a). Hence, \[\mathrm{dG}=-\mathrm{S} \, \mathrm{dT}+\mathrm{V} \, \mathrm{dp}-\mathrm{A} \, \mathrm{d} \xi ; \quad \mathrm{A} \, \mathrm{d} \xi \geq 0\]Chemists carry out most of their experiments under the twin conditions, constant pressure (usually ambient) and constant temperature (often near room temperature). Hence we can see why the latter equation is so important. At fixed \(\mathrm{T}\) and \(\mathrm{p}\), \[\mathrm{dG}=-\mathrm{A} \, \mathrm{d} \xi ; \quad \mathrm{A} \, \mathrm{d} \xi \geq 0\]Hence under common laboratory conditions the direction of spontaneous change (e.g. chemical reaction) is in the direction for which \(\mathrm{G}\) decreases. The spontaneous ‘flow’ of a chemical reaction (at fixed \(\mathrm{T}\) and \(\mathrm{p}\)) is down the plot of \(\mathrm{G}\) against extent of reaction, \(\xi\); high to low \(\mathrm{G}\). This statement opens the door to the quantitative study of chemical reactions. Thus from equation (e), \[\mathrm{A}=-\left(\frac{\partial \mathrm{G}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}\]The Gibbs energy decreases until the affinity for spontaneous change is zero; i.e. equilibrium. Then, \[\left(\frac{\partial \mathrm{G}}{\partial \xi}\right)_{\mathrm{t}, \mathrm{p}}^{\mathrm{eq}}=0\]At equilibrium the Gibbs energy is a minimum. In general terms, a thermodynamic potential is an extensive property of a closed system which reaches an extremum at equilibrium under specified conditions. For processes in closed systems at fixed \(\mathrm{T}\) and \(\mathrm{p}\), the thermodynamic potential is \(\mathrm{G}\). Thus \(\mathrm{T}\) and \(\mathrm{p}\) are the natural variables for \(\mathrm{G}\).Experience shows that for a given system there is one unique composition which corresponds to the minimum in Gibbs energy (at fixed \(\mathrm{T}\) and \(\mathrm{p}\)). In fact chemistry would be a very difficult subject (and it is difficult as it is) if there were many minima such that it was just a matter of chance which minimum a system ended up in following spontaneous chemical reaction.The conclusions advanced above refer to the Gibbs energy of a closed system; i.e. a macroscopic property. We cannot at this stage draw conclusions about the properties of the chemical substances making up the system. At the molecular level a whole range of processes may be taking place; chemical reaction, diffusion, molecular collisions. We cannot comment on these using equation (g). It may be that one or more of these processes contributes towards an increase in Gibbs energy. However these processes operate in such a way that the fluctuations in Gibbs energies in small domains are opposed, holding the overall system at a minimum in \(\mathrm{G}\).The Gibbs energy is a contrived property. It is not the ‘energy’ of the system. Nevertheless we can begin to ‘understand’ this property by returning to equation (d). Consider a system at equilibrium and at constant temperature ; i.e. \(\mathrm{A}\) = 0\) and \(\mathrm{dT} =0\). Then \[V=\left(\frac{\partial G}{\partial p}\right)_{T, A=0}\]The familiar property, volume, is the differential dependence of Gibbs energy on pressure at constant temperature and at equilibrium. If we can assume that the coffee mug on this desk is at equilibrium, although I do not know (and can never know) its Gibbs energy, I know that the volume offers a direct measure of the dependence of its Gibbs energy on pressure. Indeed the link between a property which can be readily measured (e.g. volume or density) offers chemists a pathway into the Gibbs energy and a detailed thermodynamic analysis.Footnote G. Willis and D. Ball, J.Chem.Educ.,1984,61,173.This page titled 1.10.2: Gibbs Energy- Thermodynamic Potential is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.10.3: Gibbs Energies- Solutions- Solvent and Solute

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A given solution (at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\), where the latter is close to the standard pressure) is prepared using \(1 \mathrm{~kg}\) of water and \(\mathrm{m}_{\mathrm{j}}\) moles of a simple solute. We consider the differential dependence of the excess Gibbs energy for the solution \(\mathrm{G}^{\mathrm{E}}\) on molality \(\mathrm{m}_{\mathrm{j}}\). \[\mathrm{G}_{\mathrm{m}}^{\mathrm{E}}=\mathrm{R} \, \mathrm{T} \, \mathrm{m}_{\mathrm{j}} \,\left[1-\phi+\ln \left(\gamma_{\mathrm{j}}\right)\right]\]Hence, at fixed \(\mathrm{T}\) and \(\mathrm{p}\), \[\begin{aligned} (1 / \mathrm{R} \, \mathrm{T}) \,\left[\mathrm{dG}^{\mathrm{E}} / \mathrm{dm}_{\mathrm{j}}\right]=\left[1-\phi+\ln \left(\gamma_{\mathrm{j}}\right)\right]-\mathrm{m}_{\mathrm{j}} \,\left[\mathrm{d} \phi / \mathrm{dm}_{\mathrm{j}}\right] \\ &+\mathrm{m}_{\mathrm{j}} \,\left[\mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right) / \mathrm{dm}_{\mathrm{j}}\right] \end{aligned}\]But according to the Gibbs-Duhem equation, \[-\phi-\mathrm{m}_{\mathrm{j}} \,\left[\mathrm{d} \phi / \mathrm{dm}_{\mathrm{j}}\right]+1+\mathrm{m}_{\mathrm{j}} \,\left[\mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right) / \mathrm{dm}_{\mathrm{j}}\right]=0\]Hence, we obtain an equation for \(\ln \left(\gamma_{j}\right)\) as a function of the differential dependence of \(\mathrm{G}^{\mathrm{E}}\) on \(\mathrm{m}_{\mathrm{j}}\). \[\ln \left(\gamma_{\mathrm{j}}\right)=(1 / \mathrm{R} \, \mathrm{T}) \,\left[\mathrm{dG}^{\mathrm{E}} / \mathrm{dm}_{\mathrm{j}}\right]\]If we substitute for \(\ln \left(\gamma_{j}\right)\) in the equation for \(\mathrm{G}^{\mathrm{E}}\), an equation for \(\phi\) in terms of \(\mathrm{G}^{\mathrm{E}}\) is obtained. \[1-\phi=(1 / \mathrm{R} \, \mathrm{T}) \,\left[\mathrm{G}^{\mathrm{E}} / \mathrm{m}_{\mathrm{j}}-\mathrm{dG}^{\mathrm{E}} / \mathrm{dm}_{\mathrm{j}}\right]\]A more elegant derivation of equation (e) starts out with the equation (a) for the excess Gibbs energy written in the following form. \[\left[\mathrm{G}^{\mathrm{E}} / \mathrm{m}_{\mathrm{j}}\right] / \mathrm{R} \, \mathrm{T}=1-\phi+\ln \left(\gamma_{\mathrm{j}}\right)\]Then at fixed \(\mathrm{T}\) and \(\mathrm{p}\), \[(1 / \mathrm{R} \, \mathrm{T}) \,\left\{\mathrm{d}\left[\mathrm{G}^{\mathrm{E}} / \mathrm{m}_{\mathrm{j}}\right] / \mathrm{dm}_{\mathrm{j}}\right\}=-\left(\mathrm{d} \phi / \mathrm{dm}_{\mathrm{j}}\right)+\mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right) / \mathrm{dm}_{\mathrm{j}}\]But according to the Gibbs-Duhem equation, \[-\left(\mathrm{d} \phi / d m_{\mathrm{j}}\right)+\left(\mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right) / \mathrm{dm} \mathrm{m}_{\mathrm{j}}\right)=(\phi-1) / \mathrm{m}_{\mathrm{j}}\]Then, \[1-\phi=-(1 / \mathrm{R} \, \mathrm{T}) \,\left\{\mathrm{d}\left[\mathrm{G}^{\mathrm{E}} / \mathrm{m}_{\mathrm{j}}\right] / \mathrm{dm}_{\mathrm{j}}\right\} \, \mathrm{m}_{\mathrm{j}}\]Or, \[1-\phi=-(1 / \mathrm{R} \, \mathrm{T}) \,\left[\mathrm{dG}^{\mathrm{E}} / \mathrm{dm}_{\mathrm{j}}\right] \, \mathrm{m}_{\mathrm{j}}\]The latter equation does not however require that \((1-\phi)\) is a linear function of \(\mathrm{m}_{\mathrm{j}}\). The actual form of this dependence has to be obtained by experiment.Footnotes \(\ln \left(\gamma_{\mathrm{j}}\right)=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right]^{-1} \,\left[\mathrm{K}^{-1} \,\left[\mathrm{J} \mathrm{kg}^{-1}\right] \,\left[\mathrm{mol} \mathrm{kg}^{-1}\right]^{-1}=\right.\) \((1-\phi)=\left[\frac{1}{\left[\mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}]}\right] \,\left[\frac{\left[\mathrm{J} \mathrm{kg}^{-1}\right]}{\left[\mathrm{mol} \mathrm{kg}^{-1}\right.}+\frac{\left[\mathrm{J} \mathrm{kg}^{-1}\right]}{\left[\mathrm{mol} \mathrm{kg}^{-1}\right]}\right]=\)This page titled 1.10.3: Gibbs Energies- Solutions- Solvent and Solute is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.10.4: Gibbs Energies- Equilibrium and Spontaneous Change

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The Gibbs energy of a closed system at temperature \(\mathrm{T}\) is related to the enthalpy \(\mathrm{H}\) using equation (a). \[\mathrm{G}=\mathrm{H}-\mathrm{T} \, \mathrm{S}\]The differential change in Gibbs energy at constant temperature is related to the change in enthalpy \(\mathrm{dH}\) using equation (b). \[\mathrm{dG}=\mathrm{dH}-\mathrm{T} \, \mathrm{dS}\]For a process taking place in a closed system involving a change from state I to state II, the change in Gibbs energy is given by equation (c). \[\Delta \mathrm{G}=\Delta \mathrm{H}-\mathrm{T} \, \Delta \mathrm{S}\]The latter equation signals how changes in enthalpy and entropy determine the change in Gibbs energy. A given closed system at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\) is prepared using \(\mathrm{n}_{1}\) moles of solvent (water) and \(\mathrm{n}_{\mathrm{j}}\) 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. We summarise this state of affairs as follows. \[\mathrm{G}^{e q}=\mathrm{G}^{\mathrm{eq}}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}, \xi^{\mathrm{eq}}, \mathrm{A}=0\right]\]In a plot of \(\mathrm{G}\) against \(\xi\) (at fixed \(\mathrm{T}\) and \(\mathrm{p}\)) the Gibbs energy is a minimum at \(\xi^{\mathrm{eq}}\). The enthalpy \(\mathrm{H}^{\mathrm{eq}}\) of the equilibrium state can be represented in a similar fashion. \[\mathrm{H}^{e q}=\mathrm{H}^{e q}\left[\mathrm{~T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}, \xi^{\mathrm{eq}}, \mathrm{A}=0\right]\]However it is unlikely that \(\mathrm{H}^{\mathrm{eq}}\) at \(\xi^{\mathrm{eq}}\) corresponds to a minimum in enthalpy \(\mathrm{H}\) when \(\mathrm{H}\) is plotted as a function of \(\xi\). A similar comment applies to the entropy \(\mathrm{S}^{\mathrm{eq}}\); \[\mathrm{S}^{\mathrm{eq}}=\mathrm{S}^{\mathrm{eq}}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}, \xi^{\mathrm{eq}}, \mathrm{A}=0\right]\]However taken together \(\mathrm{H}^{\mathrm{eq}}\) and \(\mathrm{S}^{\mathrm{eq}}\) produce the minimum in \(\mathrm{G}\) at \(\mathrm{G}^{\mathrm{eq}}\). \[\mathrm{G}^{\mathrm{eq}}=\mathrm{H}^{\mathrm{eq}}-\mathrm{T} \, \mathrm{S}^{\mathrm{eq}}\]In summary; at thermodynamic equilibriumThe latter condition emerges from the conclusion that this rate is zero if there is no affinity for change. For a given system at defined \(\mathrm{T}\) and \(\mathrm{p}\), the state for which \(\mathrm{G}\) is a minimum is unique. Indeed if this was not the case, chemistry would be a very difficult subject. In a given spontaneous chemical reaction proceeds until the composition/organisation reaches \(\xi^{\mathrm{eq}}\). In other words the Gibbs energy is the important thermodynamic potential, certainly forming the basis of treatments of chemical reactions in closed systems at fixed \(\mathrm{T}\) and \(\mathrm{p}\). However a word of caution is in order. The Gibbs energy of a system differs from the thermodynamic energy \(\mathrm{U}\). In fact the Gibbs energy is a somewhat contrived property but aimed at a description of closed systems at fixed \(\mathrm{T}\) and \(\mathrm{p}\). Nevertheless the Gibbs energy can be given practical significance. We consider a system at equilibrium (i.e. \(\mathrm{A} = 0\)) at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\) where in this state (state I) the Gibbs energy is \(\mathrm{G}[\mathrm{I}]\). The system is displaced by a change in pressure to a neighbouring equilibrium state (at constant \(\mathrm{T}\)). The equilibrium isothermal dependence of Gibbs energy \(\mathrm{G}[\mathrm{I}]\) on pressure equals the volume of the system, \(\mathrm{V}[\mathrm{I}]\). \[V=\left[\frac{\partial G}{\partial p}\right]_{T, A=0}\]In other words we may not know the Gibbs energy of a system (in fact never know) at least we know that the pressure dependence is the volume which we can readily measure. The isobaric dependence of \(\mathrm{G}[\mathrm{I}]\) on temperature for an equilibrium displacement yields the entropy. \[\mathrm{S}=-\left[\frac{\partial \mathrm{G}}{\partial \mathrm{T}}\right]_{\mathrm{p}, \mathrm{A}=0}\]Four key points can be made.Footnotes \[\begin{aligned} &\mathrm{G}=[\mathrm{J}] ; \mathrm{T} \, \mathrm{S}=[\mathrm{K}] \,\left[\mathrm{J} \mathrm{K}^{-1}\right]=[\mathrm{J}] ; \mathrm{p} \, \mathrm{V}=\left[\mathrm{N} \mathrm{m}^{-2}\right] \,\left[\mathrm{m}^{3}\right]=[\mathrm{N} \mathrm{m}]=[\mathrm{J}] \\ &\mathrm{A} \, \xi=\left[\mathrm{J} \mathrm{mol}^{-1}\right] \,[\mathrm{mol}]=[\mathrm{J}] \end{aligned}\] See for example This point is discussed in the monograph, F. Van Zeggeren and S. H. Story, The Computation of Chemical Equilibria, Cambridge University Press, 1970. The dependence of rate of reaction on composition is described using the law of mass action and rate constants. The law of mass action is in these terms, extrathermodynamic , meaning that the law does not follow from the first and second laws. For completeness we consider the case where in equilibrium state [I] at \(\xi^{\mathrm{eq}}[\mathrm{I}]\), the system is displaced by a change to a neighbouring state having the same composition/organisation, \(\xi[\mathrm{I}]\); i.e. the system is ‘frozen’. The isothermal dependence of \(\mathrm{G}[\mathrm{I}]\) on pressure at constant composition equals the volume. Thus, \(\mathrm{V}[\mathrm{I}]=\left[\frac{\partial \mathrm{G}}{\partial \mathrm{p}}\right]_{\mathrm{T}, \xi[\mathrm{I}]}\) Similarly, \(\mathrm{S}[\mathrm{I}]=-\left[\frac{\partial \mathrm{G}}{\partial \mathrm{T}}\right]_{\mathrm{p}, \xi[\mathrm{I}]}\)The identities of \(\mathrm{S}\) and \(\mathrm{V}\) at constant ‘\(\mathrm{A}=0\)’ and at \(\xi^{\mathrm{eq}}[\mathrm{I}]\) arise from the fact that \(\mathrm{V}\) and \(\mathrm{S}\) are strong state variables. Consider a simple cup in which we have placed (delicately) a steel ball near the top edge.This page titled 1.10.4: Gibbs Energies- Equilibrium and Spontaneous Change is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.10.5: Gibbs Energies- Raoult's Law

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.10%3A_Gibbs_Energies/1.10.05%3A_Gibbs_Energies-_Raoult's_Law

We consider a closed system containing a (homogeneous) mixture of two volatile liquids. The closed system is connected to a pressure measuring device which records that at temperature \(\mathrm{T}\) the pressure inside the closed system is \(\mathrm{p}(\text{tot})\). The composition of the liquid mixture is assayed; the mole fractions of the two components of the liquid are \(\mathrm{x}_{1}\) and \(\mathrm{x}_{2}\) (where \(\mathrm{x}_{2} = 1 – \mathrm{x}_{1}\)). Thus the system contains two components so that in terms of the Phase Rule, \(\mathrm{C} = 2\). There are two phases, vapour and liquid, so \(\mathrm{P}\) equals \(2\). Thus in terms of the Rule, \(\mathrm{P} + \mathrm{~F} = \mathrm{~C} + 2\), we have fixed the composition and the temperature using up the two degrees of freedom. Hence the pressure \(\mathrm{p}(\text{tot})\) is fixed.The foundation of thermodynamics is experiment. So, in considering the properties of water in dilute aqueous solutions, we take account of the observation that the equilibrium vapour pressure \(\mathrm{p}_{1}^{\mathrm{eq}}(\mathrm{aq})\) of water in equilibrium with water in an aqueous solution (at fixed temperature) is approximately a linear function of the mole fraction of water in the solution; equation (a). \[\mathrm{p}_{1}^{\mathrm{eq}}(\mathrm{aq}) \cong \mathrm{p}_{1}^{*}(\ell) \, \mathrm{x}_{1}\]Thus as mole fraction \(\mathrm{x}_{1}\) approaches unity (the composition of the solution approaches pure water where \(\mathrm{x}_{1}\) is unity), the equilibrium vapour pressure \(\mathrm{p}_{1}^{\mathrm{eq}}(\mathrm{aq})\) approaches the vapour pressure of pure liquid water \(\mathrm{p}_{1}^{*}(\ell)\) at the same temperature. At this stage we introduce the concept of an ideal solution. We assert that for an ideal solution the approximation (a) is an equation. Thus \[\mathrm{p}_{1}^{\mathrm{eq}}(\mathrm{aq} ; \mathrm{id})=\mathrm{p}_{1}^{*}(\ell) \, \mathrm{x}_{1}\]In other words, \(\mathrm{p}_{1}^{\mathrm{eq}}(\mathrm{aq} ; \mathrm{id})\) is a linear function of mole fraction composition of the solution. We have linked the (equilibrium) vapour pressure of the solvent to the composition of the solution. Returning to the results of experiments, we invariably find that as real solution becomes more dilute (i.e. as \(\mathrm{x}_{1}\) approaches unity) \(\mathrm{p}_{1}^{\mathrm{eq}}(\mathrm{aq})\) for real solutions approaches \(\mathrm{p}_{1}^{\mathrm{eq}}(\mathrm{aq} ; \mathrm{id})\) for the corresponding ideal solution at the same temperature. Therefore, we rewrite equation (b) as an equation for real solutions by introducing a new quantity called the (rational) activity coefficient, \(\mathrm{f}_{1}\). Then, \[\mathrm{p}_{1}^{\mathrm{eq}}(\mathrm{aq}) \cong \mathrm{p}_{1}^{*}(\ell) \, \mathrm{x}_{1} \, \mathrm{f}_{1}\]where, by definition, \[\operatorname{limit}\left(\mathrm{x}_{1} \rightarrow 0\right) \mathrm{f}_{1}=1.0\]Although equations (c) and (d) have simple forms, rational activity coefficients carry a heavy load in terms of information. Thus for a given solution \(\mathrm{f}_{1}\) describes the extent to which interactions involving solvent water in the real solution differ from those in the corresponding ideal solution. The challenge of expressing this information in molecular terms is formidable.Footnotes Note that, \(\mathrm{p}_{1}^{\mathrm{eq}}(\mathrm{aq} ; \mathrm{id})-\mathrm{p}_{1}^{*}(\ell)<0\). Thus adding a solute lowers the vapour pressure of the solvent. However the total vapour pressure of a binary liquid mixture can be either increased or decreased by adding a small amount of solute, the change being characteristic of the solute; G. Bertrand and C. Treiner, J. Solution Chem.,1984,13, 43.This page titled 1.10.5: Gibbs Energies- Raoult's Law is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.10.6: Gibbs Energies- Solutions- Pairwise Solute Interaction Parameters

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.10%3A_Gibbs_Energies/1.10.06%3A_Gibbs_Energies-_Solutions-_Pairwise_Solute_Interaction_Parameters

A given solution is prepared using \(1 \mathrm{~kg}\) of water(\(\ell\)) and \(\mathrm{m}_{\mathrm{j}}\) moles of solute \(j\). The chemical potential of the solvent water is related to \(\mathrm{m}_{\mathrm{j}}\) using equation (a) where pressure \(\mathrm{p}\) is close to the standard pressure, \(\mathrm{p}^{0}\) \[\mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\ell)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{m}_{\mathrm{j}} \, \mathrm{M}_{1}\]Here \(\mu_{1}^{*}(\ell)\) is the chemical potential of solvent water at the same \(\mathrm{T}\) and \(\mathrm{p}\); \(\phi\) is the practical osmotic coefficient which is unity for a solution having thermodynamic properties which are ideal; \(\mathrm{M}_{1}\) is the molar mass of water. The chemical potential of the solute \(\mathrm{j}\), \(\mu_{j}(\mathrm{aq})\) is related to the molality \(\mathrm{m}_{\mathrm{j}}\) using equation (b). \[\mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)\]The chemical potentials of solute and solvent are linked by the Gibbs-Duhem equation which for aqueous solutions (at fixed \(\mathrm{T}\) and \(\mathrm{p}\)) containing \(1 \mathrm{~kg}\) of water takes the following form. \[\left(1 / \mathrm{M}_{1}\right) \, \mathrm{d} \mu_{1}(\mathrm{aq})+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \mu_{\mathrm{j}}(\mathrm{aq})=0\]We draw equations (a) and (b) together in an equation for the Gibbs energy of a solution prepared using \(1 \mathrm{~kg}\) of solvent water. Then, \[\mathrm{G}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mu_{1}(\mathrm{aq})+\mathrm{m}_{\mathrm{j}} \, \mu_{\mathrm{j}}(\mathrm{aq})\]Or, \[\begin{aligned} \mathrm{G}\left(\mathrm{aq} ; \mathrm{w}_{1}=\right.&1 \mathrm{~kg})=\left(1 / \mathrm{M}_{1}\right) \,\left[\mu_{1}^{*}(\ell)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{m}_{\mathrm{j}} \, \mathrm{M}_{1}\right] \\ &+\mathrm{m}_{\mathrm{j}} \,\left[\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)\right] \end{aligned}\]If the thermodynamic properties of the solution are ideal, both \(\phi\) and \(\gamma_{\mathrm{j}}\) are unity. \[\begin{aligned} \mathrm{G}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg} ; \mathrm{id}\right)=\left(1 / \mathrm{M}_{1}\right) \,\left[\mu_{1}^{*}(\ell)-\mathrm{R} \, \mathrm{T} \, \mathrm{m}_{\mathrm{j}} \, \mathrm{M}_{1}\right] \\ &+\mathrm{m}_{\mathrm{j}} \,\left[\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)\right] \end{aligned}\]The difference between \(\mathrm{G}(\mathrm{aq})\) and \(\mathrm{G}(\mathrm{aq} ; \mathrm{id})\) is the excess Gibbs energy \(\mathrm{G}^{\mathrm{E}}(\mathrm{aq})\); \[\mathrm{G}^{\mathrm{E}}(\mathrm{aq})=\mathrm{R} \, \mathrm{T} \, \mathrm{m}_{\mathrm{j}} \,\left[1-\phi+\ln \left(\gamma_{\mathrm{j}}\right)\right]\]A related quantity is the excess molar Gibbs energy \(\mathrm{G}_{\mathrm{m}}^{\mathrm{E}}(\mathrm{aq}) \quad\left\{=\mathrm{G}^{\mathrm{E}} / \mathrm{m}_{\mathrm{j}}\right\}\]. Then \[\mathrm{G}_{\mathrm{m}}^{\mathrm{E}}(\mathrm{aq})=\mathrm{R} \, \mathrm{T} \,\left[1-\phi+\ln \left(\gamma_{\mathrm{j}}\right)\right]\]The dependence of \(\mathrm{G}^{\mathrm{E}}(\mathrm{aq})\) on \(\mathrm{m}_{\mathrm{j}}\) emerges from equation (g). \[(1 / \mathrm{R} \, \mathrm{T}) \, \mathrm{d}\left[\mathrm{G}^{\mathrm{E}}(\mathrm{aq}) / \mathrm{m}_{\mathrm{j}}\right]=\mathrm{d}(1-\phi)+\mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right)\]But, from the Gibbs-Duhem equation, \[\mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right)=-\left(1 / \mathrm{m}_{\mathrm{j}}\right) \, \mathrm{d}\left[\mathrm{m}_{\mathrm{j}} \,(1-\phi)\right]\]Then \[(1 / \mathrm{R} \, \mathrm{T}) \, \mathrm{d}\left[\mathrm{G}^{\mathrm{E}}(\mathrm{aq}) / \mathrm{m}_{\mathrm{j}}\right]=-\left[(1-\phi) / \mathrm{m}_{\mathrm{j}}\right] \, \mathrm{dm}_{\mathrm{j}}\]Equation (k) relates \((1-\phi)\) to the dependence of \(\mathrm{G}^{\mathrm{E}}(\mathrm{aq})\) on molality \(\mathrm{m}_{\mathrm{j}}\). The relationship between \(\mathrm{G}^{\mathrm{E}}(\mathrm{aq})\) and \(\gamma_{\mathrm{j}}\) is given by equation (l). \[\ln \left(\gamma_{\mathrm{j}}\right)=(1 / \mathrm{R} \, \mathrm{T}) \,\left[\mathrm{dG}^{\mathrm{E}}(\mathrm{aq}) / \mathrm{dm}_{\mathrm{j}}\right]\]At this point we make a key extrathermodynamic assumption. We assert that (at fixed temperature and pressure) the excess Gibbs energy \(\mathrm{G}^{\mathrm{E}}\) is related to molality \(\mathrm{m}_{\mathrm{j}}\) of neutral solute \(\mathrm{j}\) using equation (m). Thus \[\mathrm{G}^{\mathrm{E}}=\mathrm{g}_{\mathrm{ij}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2}+\mathrm{g}_{\mathrm{ij}}+\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{3} \ldots \ldots\]Here \(g_{j j}, g_{j j j} \ldots\) are the coefficients in a virial type of equation. Thus \(g_{j j}\) measures the contribution of pairwise solute-solute interactions to \(\mathrm{G}^{\mathrm{E}} (\mathrm{aq})\); \(g_{j j j}\) is a triplet interaction term. For quite dilute solutions the dependence of \(\mathrm{G}^{\mathrm{E}} (\mathrm{aq})\) on \(\mathrm{m}_{\mathrm{j}}\) is effectively described by the pairwise term, \(g_{j j}\). \[\mathrm{G}^{\mathrm{E}}=\mathrm{g}_{\mathrm{j}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2}\]Here \(g_{j j}\) is expressed in \(\left[\mathrm{J kg}^{-1}\right]\), being the (Gibbs) energy of interaction in a solution containing \(1 \mathrm{~kg}\) of water. Pairwise solute-solute Gibbs energy interaction parameters are characteristic of solute \(j\), temperature and pressure.At this stage we have not defined either the sign or magnitude of \(g_{j j}\). Clearly if pairwise solute-solute interactions are attractive/cohesive, both \(g_{j j}\) and \(\mathrm{G}^{\mathrm{E}}\) are negative. In the next stage of the analysis we use equation (l) to obtain an equation for \(\ln \left(\gamma_{j}\right)\) in terms of \(g_{j j}\) and molality \(\mathrm{m}_{\mathrm{j}}\). Thus \[\ln \left(\gamma_{\mathrm{j}}\right)=[2 / \mathrm{R} \, \mathrm{T}] \, \mathrm{g}_{\mathrm{jj}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{m}_{\mathrm{j}}\]Hence equation (o) requires that (at fixed \(\mathrm{T}\) and \(\mathrm{p}\)) if \(g_{j j}\) is negative \(\ln \left(\gamma_{\mathrm{j}}\right)\) decreases with increase in \(\mathrm{m}_{\mathrm{j}}\) whereby \(\mu_{j}(\mathrm{aq})<\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{id})\). We anticipate that the sign and magnitude of \(g_{j j}\) reflect the hydration characteristics of the two solute molecules because these characteristics determine the impact of cosphere overlap on the properties of the solution.We turn to the properties of the solvent. Thus \[(1-\phi)=-(1 / \mathrm{R} \, \mathrm{T}) \, \mathrm{g}_{\mathrm{ij}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{m}_{\mathrm{j}}\]Or, \[\phi=1+(1 / R \, T) \, g_{j} \,\left(m^{0}\right)^{-2} \, m_{j}\]From the equation for \(\left[\mu_{1}(\mathrm{aq})-\mu_{1}(\mathrm{aq} ; \mathrm{id})\right]\), the difference in chemical potentials of solvent water in real and ideal solutions, it follows that negative \(g_{j j}\) requires that \(\mu_{1}(\mathrm{aq})>\mu_{1}(\mathrm{aq} ; \mathrm{id})\), the solvent in the real solution being at a higher chemical potential than in the corresponding ideal solution. In other words the non-ideal properties of the solvent are also related to the pairwise interaction parameter \(g_{j j}\) and \(\mathrm{m}_{\mathrm{j}}\).As a check on the procedures described above we draw the equations together to recover the original equation for \(\mathrm{G}^{\mathrm{E}} (\mathrm{aq})\). \[\begin{aligned} &\mathrm{G}^{\mathrm{E}}(\mathrm{aq})=\mathrm{m}_{\mathrm{j}} \, \mathrm{R} \, \mathrm{T} \,\left[1-\phi+\ln \left(\gamma_{\mathrm{j}}\right)\right] \\ &=\mathrm{m}_{\mathrm{j}} \, \mathrm{R} \, \mathrm{T} \,\left[-(1 / \mathrm{R} \, \mathrm{T}) \, \mathrm{g}_{\mathrm{jj}} \,\left(1 / \mathrm{m}^{0}\right)^{2} \, \mathrm{m}_{\mathrm{j}}+(2 / \mathrm{R} \, \mathrm{T}) \, \mathrm{g}_{\mathrm{j}} \, \mathrm{m}_{\mathrm{j}} \,\left(1 / \mathrm{m}^{0}\right)^{2}\right] \\ &=\mathrm{g}_{\mathrm{jj}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2} \end{aligned}\]Thus for dilute solutions both \(\phi\) and \(\operatorname{ln}\left(\gamma_{\mathrm{j}}\right)\) are linear functions of \(\mathrm{m}_{\mathrm{j}}\). Equation (n) forms the basis for understanding the properties of dilute aqueous solutions where the solutes are non-ionic. The underlying theme is the idea that solute -solute interactions in these solutions can be understood in terms of cosphere- cosphere interactions. Description of the properties of real solutions based on equation (a) is closely related to descriptions of dilute solutions developed for metallurgical systems. Similarly procedures are discussed using site-site pair correlation functions for molecular interaction energies and using quasi-chemical models.Footnotes \(\mathrm{G}\left(\mathrm{aq} ; \mathrm{W}_{1}=1 \mathrm{~kg}\right]=\left[\mathrm{kg} \mathrm{mol}^{-1}\right]^{-1} \,\left[\mathrm{J} \mathrm{mol}^{-1}\right]+\left[\mathrm{mol} \mathrm{kg}^{-1}\right] \,\left[\mathrm{J} \mathrm{mol}^{-1}\right]=\left[\mathrm{J} \mathrm{kg}^{-1}\right]\) \(\mathrm{G}^{\mathrm{E}}(\mathrm{aq})=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}] \,\left[\mathrm{mol} \mathrm{kg}{ }^{-1}\right] \,[\mathrm{I}]=\left[\mathrm{J} \mathrm{kg}^{-1}\right]\) \(\mathrm{G}_{\mathrm{m}}{ }^{\mathrm{E}}(\mathrm{aq})=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}] \,[\mathrm{I}]=\left[\mathrm{J} \mathrm{mol}^{-1}\right]\) \[\begin{gathered} (1 / \mathrm{R} \, \mathrm{T}) \, \mathrm{d}\left[\mathrm{G}^{\mathrm{E}}(\mathrm{aq}) / \mathrm{m}_{\mathrm{j}}\right]=\mathrm{d}(1-\phi)-\left(\mathrm{l} / \mathrm{m}_{\mathrm{j}}\right) \, \mathrm{d}\left[\mathrm{m}_{\mathrm{j}} \,(1-\phi)\right] \\ (1 / \mathrm{R} \, \mathrm{T}) \, \mathrm{d}\left[\mathrm{G}^{\mathrm{E}}(\mathrm{aq}) / \mathrm{m}_{\mathrm{j}}\right]=\mathrm{d}(1-\phi)-\mathrm{d}(1-\phi)-\left[(1-\phi) / \mathrm{m}_{\mathrm{j}}\right] \, \mathrm{dm}_{\mathrm{j}} \\ (1 / \mathrm{R} \, \mathrm{T}) \, \mathrm{d}\left[\mathrm{G}^{\mathrm{E}}(\mathrm{aq}) / \mathrm{m}_{j}\right]=-\left[(1-\phi) / \mathrm{m}_{\mathrm{j}}\right] \, \mathrm{dm}_{\mathrm{j}} \end{gathered}\] From equation (g), \(\mathrm{G}^{\mathrm{E}}(\mathrm{aq})=\mathrm{R} \, \mathrm{T} \, \mathrm{m}_{\mathrm{j}} \,\left[1-\phi+\ell \mathrm{n}\left(\gamma_{\mathrm{j}}\right)\right]\) \[(1 / \mathrm{R} \, \mathrm{T}) \, \mathrm{dG}^{\mathrm{E}}(\mathrm{aq}) / \mathrm{dm}_{\mathrm{j}}=(1-\phi)-\mathrm{m}_{\mathrm{j}} \,\left(\mathrm{d \phi} / \mathrm{dm}_{\mathrm{j}}\right)+\ell \mathrm{n}\left(\gamma_{\mathrm{j}}\right)+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ell \mathrm{n}\left(\gamma_{\mathrm{j}}\right) / \mathrm{dm}_{\mathrm{j}}\]But from the Gibbs-Duhem equation, \(\mathrm{d}\left[\mathrm{m}_{\mathrm{j}} .(1-\phi)\right]+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ell \mathrm{n}\left(\gamma_{\mathrm{j}}\right)=0\)Or, \[-\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \phi+(1-\phi) \, \mathrm{dm}_{\mathrm{j}}+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ell \mathrm{n}\left(\gamma_{j}\right)=0\]Or, \[-\mathrm{m}_{\mathrm{j} \,} \,\left(\mathrm{d} \phi / \mathrm{dm} \mathrm{m}_{\mathrm{j}}\right)+(1-\phi)+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ell \mathrm{n}\left(\gamma_{\mathrm{j}}\right) / \mathrm{dm}_{\mathrm{j}}=0\] \[\begin{aligned} &\ell \mathrm{n}\left(\gamma_{j}\right)=(1 / \mathrm{R} \, \mathrm{T}) \, \mathrm{d}\left[\mathrm{g}_{\mathrm{jj}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2}\right] / \mathrm{dm}_{\mathrm{j}} \\ &\ell \mathrm{n}\left(\gamma_{\mathrm{j}}\right)=(2 / \mathrm{R} \, \mathrm{T}) \, \mathrm{g}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{m}_{\mathrm{j}} \end{aligned}\]Or, \(= \,\left[\mathrm { J } \mathrm { mol } ^ { - 1 } \mathrm { K } ^ { - 1 } \, \left[\mathrm{K}^{-1} \,\left[\mathrm{J} \mathrm{kg}^{-1}\right] \,\left[\mathrm{mol} \mathrm{kg}^{-1}\right]^{-2} \,\left[\mathrm{mol} \mathrm{kg}^{-1}\right]\right.\right.\) From, \(\mathrm{G}^{\mathrm{E}}(\mathrm{aq})=\mathrm{R} \, \mathrm{T} \, \mathrm{m}_{\mathrm{j}} \,\left[1-\phi+\ln \left(\gamma_{\mathrm{j}}\right)\right]\) But \(\mathrm{G}^{\mathrm{E}}(\mathrm{aq})=\mathrm{g}_{i \mathrm{j}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2}\)Then, \(\mathrm{g}_{\mathrm{j}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2} / \mathrm{R} \, \mathrm{T}=\mathrm{m}_{\mathrm{j} \,}(1-\phi)+(2 / \mathrm{R} \, \mathrm{T}) \, \mathrm{g}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-2} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{2}\)Then \((1-\phi)=-(1 / R \, T) \, g_{i j} \,\left(m^{0}\right)^{-2} \, m_{j}\) M. J. Blandamer, J. Burgess, J. B. F. N. Engberts and W. Blokzijl, Ann. Rep. Progr. Chem., Sect. C, Phys. Chem., C, 1990,87,45. R. W. Gurney, Ionic Processes in Solution, McGraw-Hill, New York, 1953. See for example, W. G. McMillan Jr. and J. E. Mayer, J. Chem. Phys., 1945, 13, 276. L. S. Darken, Trans.Metallurg. Soc.AIME, 1967, 239, 80. R. P. Currier and J. P. O'Connell, Fluid Phase Equilib., 1987,33, 245.[141 J. Abusleme and J. H. Vera, Can. J. Chem. Eng., 1985, 63,845.This page titled 1.10.6: Gibbs Energies- Solutions- Pairwise Solute Interaction Parameters is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.10.7: Gibbs Energies- Solutions- Parameters Phi and ln(gamma)

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The practical osmotic coefficient can be calculated knowing the dependence of \(\gamma_{\mathrm{j}}\) on molality of solute \(j\). Of course at this stage we do not know the form of the dependence of \(\gamma_{\mathrm{j}}\) on \(\mathrm{m}_{\mathrm{j}}\). In fact \(\gamma_{\mathrm{j}}\) also depends on the solute, temperature and pressure. But for a given system (at fixed \(\mathrm{T}\) and \(\mathrm{p}\)) we might express \(\phi\) as a series expansion of the molality \(\mathrm{m}_{\mathrm{j}}\). Thus, \[\phi=1+a_{1} \, m_{j}+a_{2} \, m_{j}^{2}+a_{3} \, m_{j}^{3}+\ldots \ldots\]Interestingly this assumed dependence is equivalent to a series expansion in mole fraction of solute \(\mathrm{x}_{\mathrm{j}}\) for \(1 \mathrm{nf}_{1}\), where \(\mathrm{f}_{1}\) is the (rational) activity coefficient for the solvent. \[\operatorname{lnf}_{1}=\mathrm{b}_{1} \, \mathrm{x}_{\mathrm{j}}^{2}+\mathrm{b}_{1} \, \mathrm{x}_{\mathrm{j}}^{3}+\mathrm{b}_{3} \, \mathrm{x}_{\mathrm{j}}^{4}+\ldots \ldots\]Here \(\mathrm{b}_{1}, \mathrm{~b}_{2}, \mathrm{~b}_{3} \ldots\) depend on the solute (for given \(\mathrm{T}\) and \(\mathrm{p}\)). The link between the two equations can be expressed as follows. \[\mathrm{b}_{1}=-\left[(1 / 2)+\mathrm{M}_{1}^{-1} \, \mathrm{a}_{1}\right]\]\[\mathrm{b}_{2}=-\left[(2 / 3)+2 \, \mathrm{M}_{1}^{-1} \, \mathrm{a}_{1}+\mathrm{M}_{1}^{-2} \, \mathrm{a}_{2}\right]\]\[\mathrm{b}_{3}=-\left[(3 / 4)+3 \, \mathrm{M}_{1}^{-1} \, \mathrm{a}_{1}+3 \, \mathrm{M}_{1}^{-2} \, \mathrm{a}_{2}+\mathrm{a}_{3} \, \mathrm{M}_{1}^{-3}\right]\]Footnotes J. J. Kozak, W. S. Knight and W. Kauzmann, J. Chem. Phys., 1968,48, 675. By definition, for a solution j in solvent, chemical substance 1, \[\mathrm{x}_{\mathrm{j}}=\mathrm{m}_{\mathrm{j}} /\left(\mathrm{M}_{\mathrm{l}}^{-1}+\mathrm{m}_{\mathrm{j}}\right)\]where \(\mathrm{M}_{1}\) is the molar mass of solvent expressed in \(\mathrm{kg mol}^{-1}\). Hence molality of solute \(j\), \[\mathrm{m}_{\mathrm{j}}=\mathrm{x}_{\mathrm{j}} \, \mathrm{M}_{\mathrm{l}}^{-1} \,\left(1-\mathrm{x}_{\mathrm{j}}\right)^{-1}\]We expand \(\left(1-x_{j}\right)^{-1}\) based on the premise that \(0<\mathrm{x}_{\mathrm{j}}<<1.0\) for dilute solutions. Then, \[\mathrm{m}_{\mathrm{j}}=\mathrm{x}_{\mathrm{j}} \mathrm{M}_{1}^{-1} \,\left[1+\mathrm{x}_{\mathrm{j}}+\mathrm{x}_{\mathrm{j}}^{2}+\mathrm{x}_{\mathrm{j}}^{3}+\ldots \ldots\right]\]or, m x j jj jj M x x x = ⋅+ + ++ − 1 1 234 [ .....] (d)Here we carry all terms up to and including the fourth power of xj. But from the two methods for relating µ1(aq) to the composition of a solution, 1nx f M m 11 1 j ( ) ⋅ =−⋅ ⋅ φ (e)Then, 1 1 11 1 1 2 2 3 3 nx f M m m m m j jjj ( ) [ .....] ⋅ =− ⋅ ⋅ + ⋅ + ⋅ + ⋅ + a a a (f)or, 1 11 1 11 2 2 3 3 4 nf n M m m m m j jj j j x a a a =− − − ⋅ + ⋅ + ⋅ + ⋅ + ( ) [ .....] (g)But for dilute solutions, −− =+ + + 11 2 3 4 2 34 nx x x x j jj j j ( ) ( /) ( /) ( /) x (h)Using equation (c) for mj as a function of xj in the context of equation (f), we obtain an equation for ln(f1). 1 234 1 2 34 nf x x x jj j j ( ) ( /) ( /) ( /) x =+ + + −− − − xxxx jj jj 234 1 4 j 1 1 1 3 j 1 1 1 2 j 1 1 − M ⋅ x ⋅ a − 2 ⋅ M ⋅ x ⋅ a − 3⋅ M ⋅ x ⋅ a − − − − ⋅ ⋅ −⋅ ⋅ ⋅ − − M xa M xa 1 j j 2 3 2 1 2 4 2 3 − ⋅⋅ − M xa 1 j 3 4 3 (i)Hence, 1n(f1) = { [( / ) ( )] } − +⋅ ⋅ − 1 2 1 1 1 2 a M x j } { [(2 / 3) (2 a M ) (a M )] x3 j 2 2 1 1 1 1 + − + ⋅ ⋅ + ⋅ ⋅ − − +− + ⋅ ⋅ + ⋅ ⋅ + ⋅ ⋅ − −− { [ / ) ( ) ( ) ( )] } 34 3 3 1 1 1 2 1 2 3 1 3 4 aM aM aM x j (j)This page titled 1.10.7: Gibbs Energies- Solutions- Parameters Phi and ln(gamma) is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.10.8: Gibbs Energies- Solutions- Solute-Solute Interactions

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.10%3A_Gibbs_Energies/1.10.08%3A_Gibbs_Energies-_Solutions-_Solute-Solute_Interactions

In real solutions, solute molecules are not infinitely far apart. With increase in solute concentration, the mean separation of solute molecules decreases. Deviations in the properties of real solutions of neutral solutes from ideal can be understood in term of contact, overlap and interaction between cospheres of solvent surrounding solute molecules. Two limiting cases can be identified. In one case overlap occurs between cospheres for which the organisation of solvent molecules are compatible, leading to attractive interaction between two solute molecules; a stabilising effect. In the opposite case the organisation of solvent in the cospheres is incompatible leading to repulsion between the solute molecules; i.e a destabilising effect. These ideas can be formulated quantitatively leading to an understanding of the factors controlling the properties of solutes in aqueous solution.A given solution is prepared using \(1 \mathrm{~kg}\) of water(\(\ell\)) and \(\mathrm{m}_{j}\) moles of solute \(j\). The chemical potential of the solvent water is related to \(\mathrm{m}_{j}\) using equation (a). \[\mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\ell)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{m}_{\mathrm{j}} \, \mathrm{M}_{1}\]Here \(\mu_{1}^{*}(\ell)\) is the chemical potential of solvent water at the same \(\mathrm{T}\) and \(\mathrm{p}\); \(\phi\) is the practical osmotic coefficient which is unity for a solution having thermodynamic properties which are ideal; \(\mathrm{M}_{1}\) is the molar mass of water. The difference \(\left[\mu_{1}(\mathrm{aq})-\mu_{1}^{*}(\ell)\right]\) equals \(\left[-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{m}_{\mathrm{j}} \, \mathrm{M}_{1}\right]\). Hence for an ideal solution where \(\phi\) is unity addition of a solute lowers the chemical potential of the solvent; i.e. stabilises the solvent.The chemical potential of the solute \(\mu_{\mathrm{j}}(\mathrm{aq})\) is related to the molality \(\mathrm{m}_{j}\) using equation (b). \[\mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)\]We rewrite equation (b) in an extended form. \[\mu_{j}(a q)=\mu_{j}^{0}(a q)+R \, T \, \ln \left(m_{j} / m^{0}\right)+R \, T \, \ln \left(\gamma_{j}\right)\]Or, \[\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{id})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)\]Hence \(\gamma_{j}\) measures the extent to which the chemical potential of solute \(j\) in the real solution differs from that in an ideal solution. If \(\gamma_{j} > 1\) [and hence \(\ln \left(\gamma_{j}\right)>0\)] \(\mu_{j}(\mathrm{aq})<\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{id})\), solute-solute interactions destabilise the solute. If \(\gamma_{\mathrm{j}}<1\) [and hence \(\ln \left(\gamma_{\mathrm{j}}\right)<0\)] \(\mu_{\mathrm{j}}(\mathrm{aq})>\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{id})\), and so these interaction stabilise the solute. [\(\mathrm{NB} \gamma_{j}\) cannot be negative.]The chemical potentials of solute and solvent are linked by the Gibbs – Duhem equation which for aqueous solutions (at fixed \(\mathrm{T}\) and \(\mathrm{p}\)) containing \(1 \mathrm{~kg}\) of water takes the following form. \[\left(1 / M_{1}\right) \, d \mu_{1}(a q)+m_{j} \, d \mu_{j}(a q)=0\]Then, \[\begin{aligned} \left(1 / \mathrm{M}_{1}\right) \, \mathrm{d}\left[\mu_{1}^{*}(\ell)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{m}_{\mathrm{j}} \, \mathrm{M}_{1}\right] \\ &+\mathrm{m}_{\mathrm{j}} \, \mathrm{d}\left[\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)\right]=0 \end{aligned}\]Or, \[-\mathrm{d}\left[\phi \, \mathrm{m}_{\mathrm{j}}\right]+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)=0\]The latter equation links changes in the osmotic coefficient \(\phi\) and \(\gamma_{j}\). In other words, a perturbation which affects the solvent feeds back on to the properties the solute. This is Gibbs-Duhem communication. In the present context equation (g) explains why interaction between cospheres feeds back to the properties of the solute. Consequently the Gibbs-Duhem equation is used to switch between equations describing \(\phi\) and \(\gamma_{j}\). Thus from equation (g), \[\mathrm{d}\left[\mathrm{m}_{\mathrm{j}} \,(1-\phi)\right]+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right)=0\]Equation (h) can be written in two forms depending which direction we wish to proceed. Thus from equation (h), \[(1-\phi)=-\left(1 / m_{j}\right) \, \int_{m(j)=0}^{m(j)} m_{j} \, d \ln \left(\gamma_{j}\right)\]In other words we have an equation for \(\phi\) and in terms of \(\gamma_{j}\). Alternatively we can express \(\gamma_{j}\) in terms of \(\phi\) and its dependence on \(\mathrm{m}_{j}\). \[\left.\int_{m(j)=0}^{m(j)} d \ln \left(\gamma_{j}\right)=\int_{m(j)=0}^{m(j)} d(1-\phi)+\int\left[(1-\phi) / m_{j}\right)\right] \, d m_{j}\]Then \[\ln \left(\gamma_{\mathrm{j}}\right)=(\phi-1)+\int_{\mathrm{m}(\mathrm{j})=0}^{\mathrm{m}(\mathrm{j})}(\phi-1) \, \mathrm{d} \ln \left(\mathrm{m}_{\mathrm{j}}\right)\]Hence \(\gamma_{j}\) can be calculated from knowing \(\phi\) and its dependence on dependence on \(\mathrm{m}_{j}\).Footnotes R. A. Robinson and R. H. Stokes, Electrolyte Solutions, Butterworths, London, 2nd. edn.,1959, chapter 1. J. J. Kozak, W. S. Knight and W. Kauzmann, J. Chem. Physics,1968,48,675. \(\begin{aligned} &-\phi \, d m_{j}-m_{j} \, d \phi+m_{j} \, d m_{j} / m_{j}+m_{j} \, \ln \left(\gamma_{j}\right)=0\\ &\text { Or, } \mathrm{dm}_{\mathrm{j}}-\phi \, d \mathrm{~m}_{\mathrm{j}}-\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \phi+\mathrm{m}_{\mathrm{j}} \, \ln \left(\gamma_{\mathrm{j}}\right)=0 \end{aligned}\) \(\begin{array}{r} \int_{\mathrm{m}(\mathrm{j} j=0}^{\mathrm{m}(\mathrm{j})} \mathrm{d}\left[\mathrm{m}_{\mathrm{j}} \,(1-\phi)\right]=-\int_{\mathrm{m}(\mathrm{j})=0}^{\mathrm{m}(\mathrm{j})} \mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right) \\ \mathrm{m}_{\mathrm{j}} \,(1-\phi)=-\int_{\mathrm{m}(\mathrm{j})=0}^{\mathrm{m}(\mathrm{j})} \mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right) \end{array}\) From equation (h), \(\mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right)=\left(1 / \mathrm{m}_{\mathrm{j}}\right) \, \mathrm{d}\left[\mathrm{m}_{\mathrm{j}} \,(\phi-1)\right]\)Or, \[\left.\int_{m(j)=0}^{m(j)} d \ln \left(\gamma_{j}\right)=\int_{m(j)=0}^{m(j)} d(\phi-1)+\int\left[(\phi-1) / m_{j}\right)\right] \, d m_{j}\]This page titled 1.10.8: Gibbs Energies- Solutions- Solute-Solute Interactions is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.10.9: Gibbs Energies- Solutes- Cospheres

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The chemical potential of solute \(j\) in aqueous solution, molality \(\mathrm{m}_{j}\), at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\) (which is close to ambient) is given by equation (a). \[\mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)\]In developing an understanding the of factors which contribute to \(\mu_{\mathrm{j}}(\mathrm{aq})\), a model for solutions developed by Gurney is often helpful.A co-sphere is identified around each solute molecule \(j\) where the organization of solvent molecules differs from that in the bulk solvent at the same \(\mathrm{T}\) and \(\mathrm{p}\). In a solution where the thermodynamic properties of the solute \(j\) are ideal, there are no solute-solute interactions such that the activity coefficient \(\gamma_{j}\) is unity. In real solutions the fact that \(\gamma_{j} \neq 1\) can be understood in terms of co-sphere---co-sphere interactions together for salt solutions strong charge-charge interactions.The model identifies two zones. Zone A describes solvent molecules close to the solute molecule, the number of such solvent molecules being the primary hydration number. Zone B describes the solvent molecules outside Zone A. Their organization differs from that in the bulk solvent as a consequence of the presence of solute molecule (or, ion) \(j\). Zone C lies beyond zone B where the organization of solvent is effectively the same as that in pure solvent at the same \(\mathrm{T}\) and \(\mathrm{p}\). There is merit in not being too pedantic concerning the definitions of zones A, B and C.For real solutions co-sphere----co-sphere interactions are accounted for using for example the term \(\left[\partial \ln \left(\gamma_{\mathrm{j}}\right) / \partial \mathrm{p}\right]_{\mathrm{T}}\) in the equation describing the partial molar volume \(\mathrm{V}_{\mathrm{j}}(\mathrm{aq})\) for solute \(j\) in a real solution.Footnotes R. W. Gurney, Ionic Processes in Solution, McGraw-Hill, New York, 1953. H. S. Frank and W.-Y. Wen, Discuss. Faraday Trans.,1957,24,756.This page titled 1.10.9: Gibbs Energies- Solutes- Cospheres is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.10.10: Gibbs Energies- Solutions- Cosphere-Cosphere Interactions

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For neutral solutes in aqueous solutions, the solvent plays a key role in determining the form and magnitude of solute-solute interactions. One description of these interactions uses the Gurney model for solute cospheres. Cosphere-cosphere interaction, involving the solvent, feeds back to the properties of the solute; i.e. Gibbs - Duhem communication. Two limiting cases are identified.Footnotes See for example, R. P. Currier and J. P. O’Connell, Fluid Phase Equilibria,1987,33,245.This page titled 1.10.10: Gibbs Energies- Solutions- Cosphere-Cosphere Interactions is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.10.11: Gibbs Energies- Solutions- Solute-Solute Interactions- Pairwise

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Analysis of the thermodynamic properties of aqueous solutions was taken a step further by Savage and Wood who envisage two solute molecules A and B in aqueous solution. The total pairwise interaction between these molecules is described in terms of pairwise group-group interaction parameters. Then, for example, the pairwise enthalpic solute-solute interaction parameter \(\mathrm{H}_{\mathrm{AB}\) is written as the sum of products, \(\mathrm{n}_{\mathrm{i}}^{\mathrm{A}} \, \mathrm{n}_{\mathrm{j}}^{\mathrm{B}} \, \mathrm{h}_{\mathrm{ij}}\) where \(\mathrm{n}_{\mathrm{i}}^{\mathrm{A}}\) is the number of A-groups in solute molecule \(\mathrm{i}\) and \(\mathrm{n}_{\mathrm{j}}^{\mathrm{B}}\) is the number of B-groups in solute molecule \(\mathrm{j}\) where \(\mathrm{h}_{\mathrm{ij}}\) is a pairwise enthalpic group interaction parameter. A similar analysis is carried out for interaction Gibbs energies leading to pairwise Gibbs energy parameters \(\mathrm{g}_{\mathrm{ij}}\). So, for example, \(\mathrm{g}(\mathrm{OH}-\mathrm{OH})\) is negative characteristic of a hydrophilic-hydrophilic interaction. Whereas \(\mathrm{g}\left(\mathrm{OH}-\mathrm{CH}_{2}\right)\) is positive indicating 'repulsion' within hydrophobic-hydrophilic pairs. Interestingly \(\mathrm{g}\left(\mathrm{CH}_{2}-\mathrm{CH}_{2}\right)\) is negative" which is indicative of a hydrophobic-hydrophobic attraction (cf. hydrophobic bonding); the corresponding enthalpic pairwise parameter is positive. Thus it is tempting to speculate that hydrophobic attraction is entropy driven; for further comments see references.The general approach is readily extended to a consideration of pairwise interactions between added solutes and both initial and transitions states for given chemical reactions in aqueous solution.Footnotes J.J. Savage and R. H. Wood, J. Solution Chem.,1976,5,733. J.J. Spitzer, S. K. Suri and R. H. Wood, J. Solution Chem.,1985,14,5; and references therein. S. K. Suri and R. H. Wood, J. Solution Chem.,1986,15,705. S. K. Suri, J.J.Spitzer, R. H. Wood, E.G.Abel and P.T. Thompson, J. Solution Chem.,1986,14,781. A. L. Harris, P. T. Thompson and R. H. Wood, J. Solution Chem.,1980, 9,305. For the role of solute stereochemistry see F. Franks and M. D. Pedley, J. Chem. Soc. Faraday Trans. 1, 1983,79,2249. Amides in N,N-dimethvl formamide: M. Bloemendal and G. Somsen, J. Solution Chem.,1983,12,83. Solutions in DMF with a modification of the role of the solvent; M. Bloemendal and G. Somsen, J. Solution Chem.,1987,16,367 Interaction between amides and urea in aqueous solution; P. J. Cheek and T. H. Lilley, J. Chem. Soc. Faraday Trans.1, 1988,84,1927. Urea and polyols(aq); G. Barone, V. Elia and E. Rizzo, J. Solution Chem.,1982,11,687. Volumes and heat capacities of aromatic solutes(aq); S. Cabani, P. Gianni,V. Mollica and L. Lepori, J. Solution Chem., 1981,10,563. Small peptides(aq); enthalpies; O. V. Kulikov, A. Zielenkiewicz, W. Zielenkiewicz. V. G. Badelin and A.Krestov , J. Solution Chem., 1993,22,59. M. J. Blandamer, J. Burgess, I . M. Horn, J. B. F. N. Engberts and P. Warrick Jr., Colloids and Surfaces, 1990,48,139. M. J. Blandamer, J. Burgess, J. B. F. N. Engberts and W. Blokzijl, Annu. Rep. Prog. Chem., Sect C, Phys. Chem.,1990,87,45. W. Blokzijl, J. B. F. N. Engberts, J. Jager and M. J. Blandamer, J. Phys. Chem.,1987, 91,6022. M. J. Blandamer, J. Burgess and J. B. F. N. Engberts, Chem. Soc Rev.,1985,14,237. M. J. Blandamer and J. Burgess, Pure Appl. Chem.,1982, 54,2285.This page titled 1.10.11: Gibbs Energies- Solutions- Solute-Solute Interactions- Pairwise is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.10.12: Gibbs Energies- Solutions- Two Neutral Solutes

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A given aqueous solution containing two neutral solutes, \(\mathrm{i}\) and \(\mathrm{j}\), (e.g. urea and sucrose) was prepared using \(1 \mathrm{~kg}\) of water at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\) ( which was close to ambient). The molalities of the two solutes were \(\mathrm{m}_{\mathrm{i}}\) and \(\mathrm{m}_{\mathrm{j}}\) The chemical potential of the solvent in the mixed aqueous solution is given by equation (a) where for an ideal solution the practical osmotic coefficient \(\phi\) is unity. Then, \[\mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\ell)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \,\left(\mathrm{m}_{\mathrm{i}}+\mathrm{m}_{\mathrm{j}}\right)\]The chemical potentials of the two solutes are related to their molalities using equations (b) and (d). \[\mu_{\mathrm{i}}(\mathrm{aq})=\mu_{\mathrm{i}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{i}} \, \gamma_{\mathrm{i}} / \mathrm{m}^{0}\right)\]where \[\left.\operatorname{limit}\left(\mathrm{m}_{\mathrm{i}} \rightarrow 0 ; \mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{\mathrm{i}}=1.0 \quad \text { (at all } \mathrm{T} \text { and } \mathrm{p}\right)\]Similarly, \[\mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)\]where \[\left.\operatorname{limit}\left(\mathrm{m}_{\mathrm{i}} \rightarrow 0 ; \mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{\mathrm{j}}=1.0 \quad \text { (at all T and } \mathrm{p}\right)\]Therefore the Gibbs energy of a solution prepared using \(1 \mathrm{~kg}\) of water is given by equation (f). \[\begin{aligned} \mathrm{G}\left(\mathrm{aq} ; \mathrm{w}_{1} / \mathrm{kg}=1.0\right)=\left(1 / \mathrm{M}_{1}\right) \, & {\left[\mu_{1}^{*}(\ell)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \,\left(\mathrm{m}_{\mathrm{i}}+\mathrm{m}_{\mathrm{j}}\right)\right] } \\ +\mathrm{m}_{\mathrm{i}} \,\left[\mu_{\mathrm{i}}^{0}(\mathrm{aq})+\right.&\left.\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{i}} \, \gamma_{\mathrm{i}} / \mathrm{m}^{0}\right)\right] \\ &+\mathrm{m}_{\mathrm{j}} \,\left[\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)\right] \end{aligned}\]For the corresponding solution having ideal thermodynamic properties \[\begin{aligned} \mathrm{G}\left(\mathrm{aq} ; \mathrm{w}_{1} / \mathrm{kg}=1.0\right)=\left(1 / \mathrm{M}_{1}\right) \,\left[\mu_{1}^{*}(\ell)-\mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \,\left(\mathrm{m}_{\mathrm{i}}+\mathrm{m}_{\mathrm{j}}\right)\right] \\ +\mathrm{m}_{\mathrm{i}} \,\left[\mu_{\mathrm{i}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)\right] \\ &+\mathrm{m}_{\mathrm{j}} \,\left[\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)\right] \end{aligned}\]But the excess Gibbs energy is defined by equation (h). \[\mathrm{G}^{\mathrm{E}}(\mathrm{aq})=\mathrm{G}(\mathrm{aq})-\mathrm{G}(\mathrm{aq} ; \mathrm{id})\]Hence \[\mathrm{G}^{\mathrm{E}}(\mathrm{aq}) / \mathrm{R} \, \mathrm{T}=\left(\mathrm{m}_{\mathrm{i}}+\mathrm{m}_{\mathrm{j}}\right) \,(1-\phi)+\mathrm{m}_{\mathrm{j}} \, \ln \left(\gamma_{\mathrm{j}}\right)+\mathrm{m}_{\mathrm{i}} \, \ln \left(\gamma_{\mathrm{i}}\right)\]Actually \(\phi\), \(\gamma_{\mathrm{i}}\) and \(\gamma_{\mathrm{j}}\) are linked. They cannot change independently. According to the Gibbs-Duhem equation (at fixed \(\mathrm{T}\) and \(\mathrm{p}\)), \[\left(1 / M_{1}\right) \, d \mu_{1}(a q)+m_{j} \, d \mu_{j}+m_{i} \, d \mu_{i}=0\]We consider the case where the solution is perturbed by a change in molality, \(\mathrm{dm}_{\mathrm{j}}\), recognising that the chemical potentials of solvent and solutes change. Thus, \[\left(1 / M_{1}\right) \,\left(d \mu_{1}(a q) / d m_{j}\right)+m_{j} \,\left(d \mu_{j} / d m_{j}\right)+m_{i} \,\left(d \mu_{i} / d m_{j}\right)=0\]Therefore from equation (j), \[\begin{aligned} &\left(1 / \mathrm{M}_{1}\right) \, \frac{\mathrm{d}}{\mathrm{dm}_{\mathrm{j}}}\left[\mu_{1}^{*}(\ell)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \,\left(\mathrm{m}_{\mathrm{i}}+\mathrm{m}_{\mathrm{j}}\right)\right] \\ &+\mathrm{m}_{\mathrm{i}} \, \frac{\mathrm{d}}{\mathrm{dm}_{\mathrm{j}}}\left[\mu_{\mathrm{i}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{i}} \, \gamma_{\mathrm{i}} / \mathrm{m}^{0}\right)\right] \\ &\quad+\mathrm{m}_{\mathrm{j}} \, \frac{\mathrm{d}}{\mathrm{dm}_{\mathrm{j}}}\left[\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)\right]=0 \end{aligned}\]In the case considered here molality \(\mathrm{m}_{\mathrm{i}}\) does not change. Then (at constant, \(\mathrm{m}_{\mathrm{i}}\), \(\mathrm{T}\), \(\mathrm{p}\) and mass of solvent), \[-\left(\mathrm{m}_{\mathrm{i}}+\mathrm{m}_{\mathrm{j}}\right) \,\left[\frac{\partial \phi}{\partial \mathrm{m}_{\mathrm{j}}}\right]_{\mathrm{m}(\mathrm{i})}+(1-\phi)+\mathrm{m}_{\mathrm{j}} \,\left[\frac{\partial \ln \left(\gamma_{\mathrm{j}}\right)}{\partial \mathrm{m}_{\mathrm{j}}}\right]_{\mathrm{m}(\mathrm{i})}+\mathrm{m}_{\mathrm{i}} \,\left[\frac{\partial \ln \left(\gamma_{\mathrm{i}}\right)}{\partial \mathrm{m}_{\mathrm{j}}}\right]_{\mathrm{m}(\mathrm{i})}=0\]With the help of the latter equation we explore how the Gibbs energy depends on the molality of solute \(j\) at constant \(\mathrm{m}_{\mathrm{i}}\). Then from equation (i), \[\begin{aligned} \frac{1}{R \, T} \,\left[\frac{\partial G^{\mathrm{E}}(\mathrm{aq})}{\partial \mathrm{m}_{\mathrm{j}}}\right] &=-\left(\mathrm{m}_{\mathrm{i}}+\mathrm{m}_{\mathrm{j}}\right) \,\left[\frac{\partial \phi}{\partial \mathrm{m}_{\mathrm{j}}}\right]_{\mathrm{m}(\mathrm{i})}+(1-\phi) \\ &+\mathrm{m}_{\mathrm{j}} \,\left[\frac{\partial \ln \left(\gamma_{\mathrm{j}}\right)}{\partial \mathrm{m}_{\mathrm{j}}}\right]_{\mathrm{m}(\mathrm{i})}+\ln \left(\gamma_{\mathrm{j}}\right)+\mathrm{m}_{\mathrm{i}} \,\left[\frac{\partial \ln \left(\gamma_{\mathrm{i}}\right)}{\partial \mathrm{m}_{\mathrm{j}}}\right]_{\mathrm{m}(\mathrm{i})} \end{aligned}\]Comparison of equations (m) and (n) shows that the differential dependence of \(\mathrm{G}^{\mathrm{E}}(\mathrm{aq})\) on \(\mathrm{m}_{\mathrm{j}}\) at constant \(\mathrm{m}_{\mathrm{i}}\) is related to \(\ln \left(\gamma_{\mathrm{s}}\right)\); \[\ln \left(\gamma_{\mathrm{j}}\right)=\frac{1}{\mathrm{R} \, \mathrm{T}} \,\left[\frac{\partial \mathrm{G}^{\mathrm{E}}(\mathrm{aq})}{\partial \mathrm{m}_{\mathrm{j}}}\right]\]If the solution is dilute then \(\mathrm{G}^{\mathrm{E}}(\mathrm{aq})\) can be described using pairwise Gibbs energy interaction parameters. Thus \[\mathrm{G}^{\mathrm{E}}(\mathrm{aq})=\mathrm{g}_{\mathrm{ji}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2}+2 \, \mathrm{g}_{\mathrm{ij}} \, \mathrm{m}_{\mathrm{i}} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-2}+\mathrm{g}_{\mathrm{ii}} \,\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{2}\]Here \(\mathrm{g}_{i i}\) and \(\mathrm{g}_{j j}\) are homotactic pairwise Gibbs energy interaction parameters whereas \(\mathrm{g}_{i j}\) is the corresponding heterotactic parameter. According to equation (p) the differential dependence of \(\mathrm{G}^{\mathrm{E}}(\mathrm{aq})\) on molality \(\mathrm{m}_{\mathrm{j}}\) at constant \(\mathrm{m}_{\mathrm{i}}\) is given by equation (q). Thus, \[\left[\frac{\partial \mathrm{G}^{\mathrm{E}}(\mathrm{aq})}{\partial \mathrm{m}_{\mathrm{j}}}\right]_{\mathrm{m}(\mathrm{i})}=2 \, \mathrm{g}_{\mathrm{j}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{m}_{\mathrm{j}}+2 \, \mathrm{g}_{\mathrm{ij}} \,\left(\mathrm{m}^{0}\right)^{-2} \, \mathrm{m}_{\mathrm{i}}\]Combination of equations (o) and (q) yields an equation for \(\ln \left(\gamma_{j}\right)\) as a function of two pairwise interaction parameters. Then, \[\ln \left(\gamma_{\mathrm{j}}\right)=\left[\frac{2}{R \, T}\right] \,\left[\frac{1}{m^{0}}\right]^{2} \,\left[g_{i j} \, m_{j}+g_{i j} \, m_{i}\right]\]In other words \(\ln \left(\gamma_{\mathrm{j}}\right)\) is simply related to the molality of the two solutes. In many applications we are concerned with a solution in which \(\mathrm{m}_{\mathrm{i}} >> \mathrm{m}_{\mathrm{j}}\) such that the solution contains only a trace of solute \(j\). The activity coefficient for solute \(j\) is written \(\gamma_{\mathrm{j}}^{\mathrm{T}}\). The latter describes the effect of solute-solute interactions on solute \(j\). If we set \(\mathrm{m}_{\mathrm{j}} \cong 0\), then equation (r) yields an equation for \(\gamma_{\mathrm{j}}^{\mathrm{T}}\) in terms of \(\mathrm{m}_{\mathrm{i}}\). \[\ln \left(\gamma_{\mathrm{j}}^{\mathrm{T}}\right)=\left[\frac{2}{\mathrm{R} \, \mathrm{T}}\right] \,\left[\frac{1}{\mathrm{~m}^{0}}\right]^{2} \, \mathrm{g}_{\mathrm{ij}} \, \mathrm{m}_{\mathrm{i}}\]Footnote \(\ln \left(\gamma_{\mathrm{j}}\right)=\left[\frac{2}{\left[\mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}]}\right] \,\left[\frac{1}{\left[\mathrm{~mol} \mathrm{~kg}{ }^{-1}\right]}\right]^{2} \,\left[\mathrm{J} \mathrm{kg}^{-1}\right] \,\left[\mathrm{mol} \mathrm{kg}{ }^{-1}\right]\)This page titled 1.10.12: Gibbs Energies- Solutions- Two Neutral Solutes is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.10.13: Gibbs Energies- Solutions- Hydrates in Aqueous Solution

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An explanation of the properties of a given solute j in aqueous solutions is in terms of the formation of a hydrate; j.hJ\(\mathrm{H}_{2}\mathrm{O}\) where h is the hydration number independent of temperature and pressure. In summary there are two descriptions of the solutions prepared using \(\mathrm{n}_{1}\) moles of water and \(\mathrm{n}_{j}\) moles of solute \(j\). In description A there are \(\mathrm{n}_{j}\) moles of solute, chemical substance j, and n1 moles of solvent. In description B there are nj moles of solute j.h\(\mathrm{H}_{2}\mathrm{O}\) and \(\left(n_{1}-h \, n_{j}\right)\) moles of water. At fixed \(\mathrm{T}\) and \(\mathrm{p}\) the system is at equilibrium, being therefore at a minimum in Gibbs energy. The Gibbs energy is not dependent on our description of the system; it does not know which description we favour!We imagine two open dishes in a partially evacuated chamber at constant \(\mathrm{T}\). Each dish contains the same amount of a given solution but we label one dish A and the other dish B. Further the Gibbs energies are equal; \(\mathrm{G}(\mathrm{A}) = \mathrm{G}(\mathrm{B})\). The vapour pressures are the same so that \(\mu_{1}(\mathrm{aq} ; \mathrm{A})=\mu_{1}(\mathrm{aq} ; \mathrm{B})\). For dish A, \[\mathrm{G}(\mathrm{A})=\mathrm{n}_{1} \, \mu_{1}(\mathrm{aq})+\mathrm{n}_{\mathrm{j}} \, \mu_{\mathrm{j}}(\mathrm{aq})\]For dish B. \[\mathrm{G}(\mathrm{B})=\left(\mathrm{n}_{1}-\mathrm{n}_{\mathrm{j}} \, \mathrm{h}\right) \, \mu_{1}(\mathrm{aq})+\mathrm{n}_{\mathrm{jh}} \, \mu_{\mathrm{jh}}(\mathrm{aq})\]Here \(\mu_{j \mathrm{~h}}(\mathrm{aq})\) is the chemical potential of hydrate j.h\(\mathrm{H}_{2}\mathrm{O}\) in solution. We notes that \(\mathrm{n}_{\mathrm{j}}=\mathrm{n}_{\mathrm{jh}}\). Because \(\mathrm{G}(\mathrm{A})=\mathrm{G}(\mathrm{B})\), and the chemical potentials of the solvent are the same, \(\mu_{j h}(a q)=\mu_{j}(a q)+h \, \mu_{1}(a q)\). The molality of hydrate j.h\(\mathrm{H}_{2}\mathrm{O}\), \(\mathrm{m}_{\mathrm{jh}}=\mathrm{n}_{\mathrm{jh}} /\left[\left(\mathrm{n}_{1}-\mathrm{h} \, \mathrm{n}_{\mathrm{jh}}\right) \, \mathrm{M}_{1}\right]\) whereas the molality of solute \(j\) \(\mathrm{m}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} /\left[\mathrm{n}_{1} \, \mathrm{M}_{1}\right]\). Then at fixed \(\mathrm{T}\) and \(\mathrm{p}\), \[\begin{aligned} &\mu_{\mathrm{jh}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{jh}} \, \gamma_{\mathrm{jh}} / \mathrm{m}^{0}\right)= \\ &\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)+\mathrm{h} \,\left\{\mu_{1}^{*}(\ell)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\right\} \end{aligned}\]In the \(\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{\mathrm{j}}=1.0\) at all \(\mathrm{T}\) and \(\mathrm{p}\); in the \(\operatorname{limit}\left(\mathrm{m}_{\mathrm{jh}} \rightarrow 0\right) \gamma_{\mathrm{jh}}=1.0\) at all \(\mathrm{T}\) and \(\mathrm{p}\). In the same limit, \(\phi=1\). Hence assuming \(\mathrm{h}\) is independent of \(\mathrm{m}_{j}\). \[\mu_{\mathrm{jh}}^{0}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{h} \, \mu_{1}^{*}(\ell)\]We use equation (d) and reorganise equation (c) as an equation for \(\gamma_{j}\). \[\ln \left(\gamma_{\mathrm{j}}\right)=\ln \left(\mathrm{m}_{\mathrm{jh}} / \mathrm{m}_{\mathrm{j}}\right)+\mathrm{h} \, \phi \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}+\ln \left(\gamma_{\mathrm{jh}}\right)\]We assert that the formation of hydrate by solute \(j\) accounts for the fact that the properties of solute \(j\) are not ideal. We also assert that the properties of the hydrate are ideal; \(\gamma_{\mathrm{jh}}=1\). Moreover, \(\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}_{\mathrm{jh}}\right)=1-\left(\mathrm{h} \, \mathrm{m}_{\mathrm{j}} \, \mathrm{M}_{1}\right)\) Then, \[\ln \left(\gamma_{\mathrm{j}}\right)=-\ln \left[1-\left(\mathrm{h} \, \mathrm{m}_{\mathrm{j}} \, \mathrm{M}_{\mathrm{j}}\right)\right]+\mathrm{h} \, \phi \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\]If the solution is dilute , \(\phi \cong 1\). Then, \[\ln \left(\gamma_{\mathrm{j}}\right)=2 \, h \, \mathrm{m}_{\mathrm{j}} \, \mathrm{M}_{1}\]The hydrate model for activity coefficients can be understood in the following fashion. When \(\delta n_{j}\) moles of solute are added to a solution molality \(\mathrm{m}_{j}\), \(\mathrm{h} \, \delta \mathrm{n}_{\mathrm{j}}\) moles of water are removed from ‘solvent’ and transferred to the solute. In these terms each solute molecule responds to this increased competition for solvent by other solute molecules and therefore ‘knows’ that there are other solute molecules in the solution. Any communication between solute molecules in solution is reflected in the extent to which \(\gamma_{j}\) differs from unity.Footnotes L. P. Hammett, Physical Organic Chemistry, McGraw-Hill, New York, 2nd. Edn., 1970,Section 2.13. \(\mathrm{h} \, \mathrm{n}_{\mathrm{j}}\) must be \(<\mathrm{n}_{1}\) E. Grunwald, Thermodynamics of Molecular Species, Wiley, New York, 1977, chapter 2. M. J. Blandamer, J. B. N. Engberts, P. T. Gleeson and J. C. R. Reis, Chem. Soc. Rev., submitted,.This page titled 1.10.13: Gibbs Energies- Solutions- Hydrates in Aqueous Solution is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.10.14: Gibbs Energies- Salt Hydrates

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.10%3A_Gibbs_Energies/1.10.14%3A_Gibbs_Energies-_Salt_Hydrates

An aqueous solution is prepared using \(mathrm{n}_{\mathrm{j}}\) moles of salt \(\mathrm{MX}\) and \(\mathrm{n}_{1}\) moles of water. The properties of the system are accounted for using one of two possible Descriptions.The solute \(j\) comprises a 1:1 salt MX molality \(\mathrm{m}(\mathrm{MX})\left[=\mathrm{n}(\mathrm{MX}) / \mathrm{w}_{1}\right.\) where \(\mathrm{w}_{1}\) is the mass of water]. The single ion chemical potentials, are defined in the following manner \[\begin{aligned} &\mu\left(\mathrm{M}^{+}\right)=\left[\partial \mathrm{G} / \partial \mathrm{n}\left(\mathrm{M}^{+}\right)\right]_{\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}\left(\mathrm{x}^{-}\right)} \\ &\mu\left(\mathrm{X}^{-}\right)=\left[\partial \mathrm{G} / \partial \mathrm{n}\left(\mathrm{X}^{-}\right)\right]_{\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}\left(\mathrm{M}^{+}\right)} \end{aligned}\]Then the total Gibbs energy (at fixed \(\mathrm{T}\) and \(\mathrm{p}\)) is given by equation (b). \(\mathrm{G}(\mathrm{aq} ; \mathrm{I})=\mathrm{n}_{1} \, \mu_{1}(\mathrm{aq})\) \[\begin{aligned} &+\mathrm{n}_{\mathrm{j}} \,\left\{\mu^{\#}\left(\mathrm{M}^{+} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{m}\left(\mathrm{M}^{+}\right) \, \gamma_{+}(\mathrm{I}) / \mathrm{m}^{0}\right]\right\} \\ &+\mathrm{n}_{\mathrm{j}}\left\{\mu^{\#}\left(\mathrm{X}^{-} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{m}\left(\mathrm{X}^{-}\right) \, \gamma_{-}(\mathrm{I}) / \mathrm{m}^{0}\right]\right\} \end{aligned}\]According to this Description each mole of cation is hydrated by \(\mathrm{h}_{\mathrm{m}}\left(\mathrm{H}_{2}\mathrm{O}\right)\) moles of water and each mole of anion is hydrated by \(\mathrm{h}_{\mathrm{x}}\left(\mathrm{H}_{2}\mathrm{O}\right)\) moles of water. Hence the single ion chemical potentials are defined as follows. \[\mu\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O}\right)=\left\lfloor\partial \mathrm{G} / \partial \mathrm{n}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O}\right)\right\rfloor\]at constant \(\mathrm{T}\), \(\mathrm{p}\), \(\mathrm{n}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O}\right),\left[\mathrm{n}_{1}-\mathrm{n}_{\mathrm{j}} \,\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right)\right]\left(\mathrm{H}_{2} \mathrm{O}\right)\) and, \[\mathrm{m}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O}\right)=\mathrm{n}_{\mathrm{j}} / \mathrm{M}_{1} \,\left[\mathrm{n}_{1}-\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right) \, \mathrm{n}_{\mathrm{j}}\right]\]at constant \(\mathrm{T}\), \(\mathrm{p}\), \(\mathrm{n}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O}\right),\left[\mathrm{n}_{1}-\mathrm{n}_{\mathrm{j}} \,\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right)\right]\left(\mathrm{H}_{2} \mathrm{O}\right)\) Then, \[\mathrm{m}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O}\right)=\mathrm{n}_{\mathrm{j}} / \mathrm{M}_{1} \,\left[\mathrm{n}_{1}-\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right) \, \mathrm{n}_{\mathrm{j}}\right]\]\[\mathrm{m}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O}\right)=\mathrm{n}_{\mathrm{j}} / \mathrm{M}_{1} \,\left[\mathrm{n}_{1}-\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right) \, \mathrm{n}_{\mathrm{j}}\right]\]Hence the (equilibrium) Gibbs energy (at defined \(\mathrm{T}\) and \(\mathrm{p}\)) is given by the following equation. \[\begin{aligned} &\mathrm{G}(\mathrm{aq})=\left[\mathrm{n}_{1}-\mathrm{n}_{\mathrm{j}} \,\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right)\right] \, \mu_{1}(\mathrm{aq}) \\ &\quad+\mathrm{n}_{\mathrm{j}} \,\left[\mu^{\prime \prime}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O}\right) \, \gamma_{+}(\mathrm{II}) / \mathrm{m}^{0}\right\}\right] \\ &+\mathrm{n}_{\mathrm{j}} \,\left[\mu^{\mathrm{y}}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O}\right)\right. \\ &\left.\quad+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O}\right) \, \gamma_{-} \text {(II) } / \mathrm{m}^{0}\right\}\right] \end{aligned}\]But the Gibbs energies defined by equations (b) and (g) are identical (at equilibrium at defined \(\mathrm{T}\) and \(\mathrm{p}\)). Hence, \[\begin{aligned} &\mu^{\prime \prime}\left(\mathrm{M}^{+} ; \mathrm{aq}\right)+\mu^{\prime \prime}\left(\mathrm{X}^{-} ; \mathrm{aq}\right)+2 \, \mathrm{R} \, \mathrm{T} \, \ln \left[1-\mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{X}}\right)\right]\\ &+\mathrm{R} \, \mathrm{T} \, \ln \left\{\gamma_{+}(\mathrm{I}) \, \gamma_{-}(\mathrm{I})\right\}\\ &=-\left(h_{m}+h_{X}\right) \,\left\{\mu_{1}^{*}(\ell)-2 \, \phi \, R \, T \, M_{1} \, m_{j}\right\}\\ &+\mu^{\prime \prime}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{aq}\right)+\mu^{\prime \prime}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{X}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{aq}\right)\\ &+\mathrm{R} \, \mathrm{T} \, \ln \left\{\gamma_{+} \text {(II) } \, \gamma_{-} \text {(II) }\right\} \end{aligned}\]We use the latter equation to explore what happens in the limit that \(\mathrm{n}_{j}\) approaches zero. Thus, \(\operatorname{limit}\left(\mathrm{n}_{\mathrm{j}} \rightarrow 0\right) \gamma_{+}(\mathrm{I})=1 ; \gamma_{-}(\mathrm{I})=1 ; \gamma_{+}(\mathrm{II})=1 ; \gamma_{-}(\mathrm{II})=1 ; \mathrm{m}_{\mathrm{j}}=0\) Hence, \[\begin{aligned} &\mu^{\#}\left(\mathrm{M}^{+} ; \mathrm{aq}\right)+\mu^{\#}\left(\mathrm{X}^{-} ; \mathrm{aq}\right)= \\ &\mu^{\#}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{aq}\right)+\mu^{\#}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{aq}\right) \\ &-\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right) \, \mu_{1}{ }^{*}(\ell) \end{aligned}\]We obtain an equation linking the ionic chemical potentials. Thus, \[\begin{array}{r} \ln \gamma_{+}(\mathrm{I})+\ln \gamma_{-}(\mathrm{I})=2 \, \phi \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right) \\ -2 \, \ln \left[1-\mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{X}}\right)\right] \\ +\ln _{+} \gamma_{+}(\mathrm{II})+\ln \gamma_{-}(\mathrm{II}) \end{array}\]Then in dilute solutions, \[\begin{array}{r} \ln \gamma_{+}(\mathrm{I})+\ln \gamma_{-}(\mathrm{I})=2 \,(\phi+1) \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right) \\ +\ln \gamma_{+}(\mathrm{II})+\ln \gamma_{-}(\mathrm{II}) \end{array}\]But \(\ln \gamma_{+}(\mathrm{I})+\ln \gamma_{-}(\mathrm{I})=2 \, \ln \gamma_{\pm}(\mathrm{I})\) Then, \(2 \, \ln \gamma_{\pm}(\mathrm{I})=2 \,(\phi+1) \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{x}}\right)+2 \, \ln \gamma_{\pm}(\mathrm{II})\)We identify relationships between single ion activity coefficients in an extra-thermodynamic analysis. Thus from equation (k), \[\ln \gamma_{+}(\mathrm{II})=\ln \gamma_{+}(\mathrm{I})-2 \,(\phi+1) \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \, \mathrm{h}_{\mathrm{m}}\]\[\ln \gamma_{-}(\mathrm{II})=\ln \gamma_{-}(\mathrm{I})-2 \,(\phi+1) \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \, \mathrm{h}_{\mathrm{x}}\]It is noteworthy that in these terms the solution can be ideal using description I where \(\gamma_{\pm} = 1.0\) but non-ideal using description II. Nevertheless, these equations show how the activity coefficient of the hydrated ion (description II) is related to the activity coefficient of the simple ion (description I).Footnote From equations (b) and (g), (dividing by \(\mathrm{n}_{j}\)) \[\begin{aligned} &\left[\mu^{n}\left(\mathrm{M}^{+} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{M}^{+} ; \mathrm{I}\right) \, \gamma_{+}(\mathrm{I}) / \mathrm{m}^{0}\right\}\right]\\ &+\left[\mu^{\prime \prime}\left(\mathrm{X}^{-} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{X}^{-} ; \mathrm{I}\right) \, \gamma_{-}(\mathrm{I}) / \mathrm{m}^{0}\right\}\right]=\\ &-\left(h_{m}+h_{x}\right) \, \mu_{1}(a q)+\\ &+\left[\mu^{\prime \prime}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O}\right) \, \gamma_{+} \text {(II) } / \mathrm{m}^{0}\right\}\right]\\ &+\left[\mu^{\prime \prime}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O}\right) \, \gamma_{-}(\mathrm{II}) / \mathrm{m}^{0}\right\}\right] \end{aligned}\]Then \[\begin{aligned} &\text { en }\left[\mu^{*}\left(\mathrm{M}^{+} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{M}^{+} ; \mathrm{I}\right) \, \gamma_{+}(\mathrm{I}) / \mathrm{m}^{0}\right\}\right]\\ &+\left[\mu^{*}\left(\mathrm{X}^{-} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{X}^{-} ; \mathrm{I}\right) \, \gamma_{-}(\mathrm{I}) / \mathrm{m}^{0}\right\}\right]=\\ &-\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{X}}\right) \,\left\{\mu_{1}^{*}(\ell)-2 \, \phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\right\}\\ &+\left[\mu^{*}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O}\right) \, \gamma_{+}(\mathrm{II}) / \mathrm{m}^{0}\right\}\right]\\ &+\left[\mu^{\prime \prime}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm{m}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{x}} \mathrm{H}_{2} \mathrm{O}\right) \, \gamma_{-}(\mathrm{II}) / \mathrm{m}^{0}\right\}\right] \end{aligned}\]Or, \[\begin{aligned} &{\left[\mu^{\prime \prime}\left(\mathrm{M}^{+} ; \mathrm{aq}\right)+\mu^{\#}\left(\mathrm{X}^{-} ; \mathrm{aq}\right)\right.} \\ &+\mathrm{R} \, \mathrm{T} \, \ln \left\{\mathrm { m } ( \mathrm { M } ^ { + } ; \mathrm { I } ) \, \mathrm { m } \left(\mathrm{X}^{-} ;(\mathrm{I}) /\left(\mathrm{M}^{+} ; \mathrm{II}\right) \, \mathrm{m}\left(\mathrm{X}^{-} ; \mathrm{II}\right\}\right.\right. \\ &\mathrm{R} \, \mathrm{T} \, \ln \left\{\gamma_{+}(\mathrm{I}) \, \gamma_{-}(\mathrm{I})\right\} \\ &=-\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{X}}\right) \,\left\{\mu_{1}^{*}(\ell)-2 \, \phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\right\} \\ &+\mu^{*}\left(\mathrm{M}^{+} \, \mathrm{h}_{\mathrm{m}}\left(\mathrm{H}_{2} \mathrm{O}\right) ; \mathrm{aq}\right)+\mu^{\#}\left(\mathrm{X}^{-} \, \mathrm{h}_{\mathrm{X}}\left(\mathrm{H}_{2} \mathrm{O}\right) ; \mathrm{aq}\right) \\ &+\mathrm{R} \, \mathrm{T} \, \ln \left\{\gamma_{+}(\mathrm{II}) \, \gamma_{-}(\mathrm{II})\right\} \end{aligned}\]Using the definition of \(\mu^{\prime \prime}\left(\mathrm{M}^{+} ; \mathrm{I}\right)\) and \(\mu^{\prime \prime}\left(\mathrm{X}^{-} ; \mathrm{I}\right)\) and equations (e) and (f) for description (II), \[\begin{aligned} &\frac{\mathrm{m}\left(\mathrm{M}^{+} ; \mathrm{I}\right) \, \mathrm{m}\left(\mathrm{X}^{-} ; \mathrm{I}\right)}{\mathrm{m}\left(\mathrm{M}^{+} ; \mathrm{II}\right) \, \mathrm{m}\left(\mathrm{X}^{-} ; \mathrm{II}\right)}= \\ &\frac{\mathrm{n}_{\mathrm{j}}}{\mathrm{M}_{1} \, \mathrm{n}_{1}} \, \frac{\mathrm{M}_{1} \,\left[\mathrm{n}_{1}-\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{X}}\right) \, \mathrm{n}_{\mathrm{j}}\right]}{\mathrm{n}_{\mathrm{j}}} \, \frac{\mathrm{n}_{\mathrm{j}}}{\mathrm{M}_{1} \, \mathrm{n}_{1}} \, \frac{\mathrm{M}_{1} \,\left[\mathrm{n}_{1}-\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{X}}\right) \, \mathrm{n}_{\mathrm{j}}\right]}{\mathrm{n}_{\mathrm{j}}} \end{aligned}\]Thus, \[\frac{\mathrm{m}\left(\mathrm{M}^{+} ; \mathrm{I}\right) \, \mathrm{m}\left(\mathrm{X}^{-} ; \mathrm{I}\right)}{\mathrm{m}\left(\mathrm{M}^{+} ; \mathrm{II}\right) \, \mathrm{m}\left(\mathrm{X}^{-} ; \mathrm{II}\right)}=\left[1-\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{X}}\right) \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\right]^{2}\]Therefore, \[\begin{aligned} &\mu^{\# \prime}\left(\mathrm{M}^{+} ; \mathrm{aq}\right)+\mu^{\# \prime}\left(\mathrm{X}^{-} ; \mathrm{aq}\right)+2 \, \phi \, \mathrm{R} \, \mathrm{T} \, \ln \left[1-\mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{X}}\right)\right] \\ &\quad+\mathrm{R} \, \mathrm{T} \, \ln \left\{\gamma_{+}(\mathrm{I}) \, \gamma_{-}(\mathrm{I})\right\} \\ &=-\left(\mathrm{h}_{\mathrm{m}}+\mathrm{h}_{\mathrm{X}}\right) \,\left\{\mu_{1}^{*}(\ell)-2 \, \phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\right\} \\ &\quad+\mu^{\# *}\left(\mathrm{M}^{+} ; \mathrm{h}_{\mathrm{m}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{aq}\right)+\mu^{\# \#}\left(\mathrm{X}^{-} ; \mathrm{h}_{\mathrm{X}} \mathrm{H}_{2} \mathrm{O} ; \mathrm{aq}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left\{\gamma_{+}(\mathrm{II}) \, \gamma_{-}(\mathrm{II})\right\} \end{aligned}\]This page titled 1.10.14: Gibbs Energies- Salt Hydrates is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.10.15: Gibbs Energies- Salt Solutions- Electric Neutrality

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.10%3A_Gibbs_Energies/1.10.15%3A_Gibbs_Energies-_Salt_Solutions-_Electric_Neutrality

The solid crystalline salt \(\mathrm{NaCl}\) comprises a lattice of sodium \(\mathrm{Na}^{+}\) and chloride \(\mathrm{Cl}^{-}\) ions. The charge number on each sodium ion, \(\mathrm{z}_{+}\), is \(+ 1\); the charge number on each chloride ion, \(\mathrm{z}_{-}\), is \(- 1\). The amount of sodium ions \(\mathrm{ν}_{+}\) produced by one mole of sodium chloride is 1 mol. The amount of chloride ions \(\mathrm{ν}_{-}\) produced by one mole of sodium chloride is 1 mol. The electric charge on the sodium ions in one mole of sodium chloride is \(\left(\mathrm{v}_{+} \, \mathrm{z}_{+} \, \mathrm{N}_{\mathrm{A}} \, \mathrm{e} \right) \mathrm{~C}\) where \(\mathrm{e}\) is the unit charge and \(\mathrm{N}_{\mathrm{A}}\) is the Avogadro constant. The product \(\left(\mathrm{N}_{\mathrm{A}} \, \mathrm{e}\right) \quad\left\{=[\mathrm{C}] .[\mathrm{mol}]^{-1}\right\}\) is the Faraday constant \(\left{=\left[\mathrm{C} \mathrm{mol}^{-1}\right]\right\}\). Similarly the electric charge on the chloride ions in 1 mol of sodium chloride equals, \(\left(\mathrm{v}_{-} \, \mathrm{z}_{-} \, \mathrm{N}_{\mathrm{A}} \, \mathrm{e} \right) \mathrm{~C}\). The total electric charge on one mole of solid sodium chloride equals \(\left[\left(\mathrm{v}_{+} \, \mathrm{z}_{+} \, \mathrm{N}_{\mathrm{A}} \, \mathrm{e}\right)+\left(\mathrm{v}_{-} \, \mathrm{z}_{-} \, \mathrm{N}_{\mathrm{A}} \, \mathrm{e}\right)\right] C\) which equals zero.We make these points in order to highlight the fact that the total electric charge on 1 mol of sodium cations (in for example 53 g of common salt) is enormous, being 96 500 C. Very few laboratories can handle such enormous electric charges. Chemists cope because the electric neutrality condition always operates. When we set down equations describing the properties of salt solutions we ensure that the electric neutrality condition is not violated. However when we turn to the task of developing molecular models for these systems we recognize the magnitude of the forces involved.Footnotes One model of Utopia is a society when there are equal number of men and women. It is interesting to note that from the perspective of each male, the Utopian society has a majority of women. Similarly each woman lives in a male dominated society. Life is the same for ions.This page titled 1.10.15: Gibbs Energies- Salt Solutions- Electric Neutrality is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.10.16: Gibbs Energies- Salt Solutions- Born Equation

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.10%3A_Gibbs_Energies/1.10.16%3A_Gibbs_Energies-_Salt_Solutions-_Born_Equation

The Born Equation is based on a BBB model, “brass balls in a bathtub”. The solvent is treated as a dielectric continuum characterised at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\) by its relative permittivity \(\varepsilon_{r}\). The ions are treated as hard non-polarizable spheres, having radius \(r_{j}\). The Born Equation describes the difference in thermodynamic properties of a mole of i-ions forming a perfect gas and a mole of j-ions in an ideal solution at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\). The calculation is not straightforward. What emerges is the difference in Helmholtz energies (at constant \(\mathrm{T}\) and \(\mathrm{V}\)) for a mole of \(j\) ions in incompressible liquid phases. The calculated quantities refer to the energies associated with the electric fields over the limits \(r_{j} \leq r \leq \infty\). In these terms the Born Equation describes the electrical part of the change in chemical potential on transferring an ion from the gas phase, permittivity \(\varepsilon_{0}\), to a solvent, relative (electric) permittivity \(\varepsilon_{r}\). In effect the Born Equation yields parameters characterizing the difference between the properties of one mole of \(j\) ions in ideal systems having equal concentrations at fixed \(\mathrm{T}\) and \(\mathrm{p}\). \[=-N_{A} \,\left(z_{j} \, e\right)^{2} \,\left[1-\left(1 / \varepsilon_{r}\right)\right] / 8 \, \pi \, r_{j} \, \varepsilon_{0}\]Similarly for transfer of one mole of \(j\) ions from an ideal solution in solvent \(\mathrm{s}_{1}\) to an ideal solution in solvent \(\mathrm{s}_{2}\), the transfer chemical potential is given by the Born Equation assuming \(r_{j}\) is independent of solvent. \[\mathrm{N}_{\mathrm{A}} \,\left(\mathrm{z}_{\mathrm{j}} \, \mathrm{e}\right)^{2} \,\left[\left(1 / \varepsilon_{\mathrm{r}}(\mathrm{s} 2)\right)-\left(1 / \varepsilon_{\mathrm{r}}(\mathrm{s} 1)\right)\right] / 8 \, \pi \, \mathrm{r}_{\mathrm{j}} \, \varepsilon_{0}\]Many attempts have been made to modify the Born Equation in order to attain agreement between theory and measured thermodynamic ionic properties, particularly in the case of aqueous salt solutions. A common concern is the extent to which near-neighbor water molecules form an electrostricted layer around ions in solution, namely a layer of solvent molecules having dielectric properties which differ from those of the pure solvent at the same temperature and pressure [8 - 11]. A common concern in this subject is the definition of the ionic radius for a given ion [12 - 16]. There is no agreement concerning a set of ‘absolute’ ionic radii. As Conway pointed out in 1966, ‘ …theories .. based on the Born equation seem to have reached an asymptotic level of usefulness..’.Nevertheless correlations involving thermodynamic properties of salt solutions play an important role.Standard partial molar entropies for alkali metal halides in various solvents are linear functions of the corresponding entropies in aqueous solution. A similar correlation is reported for ionic entropies in mixed aqueous solvents and the corresponding entropies in aqueous solutions. With reference to enthalpies, the analysis also suffers from the fact that the ionic radius is sensitive to temperature.Footnotes M. Born, Z. Phys., 1920, 1, 45. H. S. Frank quoted by H. L. Friedman, J. Electrochem. Soc., 1977, 124, 421c. H. S. Frank, J. Chem. Phys., 1955, 23, 2023. J. E. Desnoyers and C. Jolicoeur, Modern Aspects of Electrochem, ed. B. Conway and J.O’M. Bockris.1969, 5,1. J. E. Desnoyers, R. E. Verrall and B. E. Conway, J. Chem. Phys., 1965,43, 243. \[\begin{aligned} & Delta(\mathrm{pfg} \rightarrow \mathrm{s} \ln ) \mu_{\mathrm{j}}=\left[\mathrm{mol}^{-1}\right] \,[\mathrm{C}]^{2} \,\{-\} / \, \,[\mathrm{m}] \,\left[\mathrm{Fm}^{-1}\right]= \\ &{[\mathrm{mol}]^{-1} \,\left[\mathrm{A}^{2} \mathrm{~s}^{2}\right] /[\mathrm{m}] \,\left[\mathrm{A}^{2} \mathrm{~s}^{4} \mathrm{~kg}^{-1} \mathrm{~m}^{-3}\right]=\left[\mathrm{mol}^{-1}\right] \,\left[\mathrm{kg} \mathrm{m}^{2} \mathrm{~s}^{-2}\right]=\left[\mathrm{J} \mathrm{mol}^{-1}\right]} \end{aligned}\] M. H. Abraham, E. Matteoli and J. Liszi, J. Chem. Soc. Faraday Trans.I, 1983, 79,2781; and references therein. D. R. Rosseinsky, Chem. Rev.,1965,65,467; and references therein. A. A. Rashin and B. Hornig, J. Phys. Chem., 1985, 89,5588. T. Abe, Bull. Chem. Soc. Jpn.,1991, 64,3035. S. Goldman and R.G.Bates, J.Am.Chem.Soc.,1972,94,1476. L. Pauling, Nature of the Chemical Bond, Cornell University Press, Ithaca, 3rd edn., 1960, chapter 8. M. Bucher and T. L. Porter, J. Phys. Chem.,1986,90,3406; and references therein. M. Salomon, J.Phys.Chem.,1970,74,2519. K. H. Stern and E. S. Amis, Chem. Rev.,1959,59,1. Y. Marcus, Chem. Rev.,1988,88,1475. B. E. Conway, Annu. Rev. Phys. Chem.,1966,17,481. C. M. Criss, R. P. Held and E. Luksha, J.Phys.Chem.,1968,72,2970. F. Franks and D. S .Reid, J. Phys.Chem.,1969,73,3152. B. Roux, H.-A. Yu and M. Karplus, J. Phys. Chem.,1990, 94,4683.This page titled 1.10.16: Gibbs Energies- Salt Solutions- Born Equation is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.10.17: Gibbs Energies- Salt Solutions- Lattice Models

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Lattice models for salt solutions have attracted and continue to attract interest. Ions in a salt solution are regarded as occupying lattice sites, the lattice parameter increasing as a solution is diluted; solvent molecules occupy the interstices of the lattice. This model for salt solutions generates interest because the distribution of ions about a central reference j-ion is therefore known. This theory requires that \(\ln \gamma_{\pm}\) is a linear function of \(\left(\mathrm{m}_{\mathrm{i}} / \mathrm{m}^{0}\right)^{1 / 3}\) for salt-i; the cube-root law. This dependence is observed for reasonably concentrated salt solutions. Unfortunately convincing evidence for lattice structures is not forthcoming. For example, the electrical conductivities of salt solutions cannot be understood in terms of lattice structures.Footnotes R. A. Robinson and R. H. Stokes, Electrolyte Solutions, Butterworths, London, 2nd. edition revised,1965, pp. 226This page titled 1.10.17: Gibbs Energies- Salt Solutions- Lattice Models is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.10.18: Gibbs Energies- Salt Solutions- Debye-Huckel Equation

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The electric potential \(\phi_{j}\) (at ion-\(j\); ion-ion interaction) describes the electric potential at a given reference \(j\)-ion arising from all other \(i\)-ions in solution. The contribution to the chemical potential of one mole of \(j\)-ions is obtained using the Guntleberg charging process. Thus, \[\Delta \mu_{\mathrm{j}}(\text { ion }-\mathrm{j} \text {; ion }-\text { ion int eractions })=\int_{0}^{\mathrm{z}_{\mathrm{j}} \,{ }_{\mathrm{e}}} \varphi_{\mathrm{j}}(\text { at ion }-\mathrm{j} \text {; ion }-\text { ion }) \, \mathrm{d}\left(\mathrm{z}_{\mathrm{j}} \, \mathrm{e}\right)\]Hence, \[\Delta \mu_{\mathrm{j}}(\text { ion }-\mathrm{j} ; \text { ion }-\text { ion int eractions })=-\frac{\left(\mathrm{z}_{\mathrm{j}} \, \mathrm{e}\right)^{2} \, \mathrm{N}_{\mathrm{A}}}{8 \, \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}}} \, \frac{\kappa}{1+\kappa \, \mathrm{a}}\]The charge-charge interactions are the only source of deviations in the properties of a given solution from ideal. Then for ion-\(j\), \[\ln \left(\gamma_{\mathrm{j}}\right)=\Delta \mu_{j}(\text { ion }-\mathrm{j}<-\longrightarrow>\text { ion atmos. }) / \mathrm{R} \, \mathrm{T}\]Hence for the ionic activity coefficient \(\gamma_{j}\), \[\ln \left(\gamma_{\mathrm{j}}\right)=-\frac{\left(\mathrm{z}_{\mathrm{j}} \, \mathrm{e}\right)^{2} \, \mathrm{N}_{\mathrm{A}}}{8 \, \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{R} \, \mathrm{T}} \, \frac{\kappa}{1+\kappa \, \mathrm{a}}\]Ionic activity coefficients have no practical significance. We require an equation for the mean ionic activity coefficient for a salt in solution. For one of mole of salt in solution, \[v \, \ln \left(\gamma_{\pm}\right)=v_{+} \, \ln \left(\gamma_{+}\right)+v_{-} \, \ln \left(\gamma_{-}\right)\]or, \[\ln \left(\gamma_{\pm}\right)=\frac{1}{\left(v_{+}+v_{-}\right)}\left[v_{+} \, \ln \left(\gamma_{+}\right)+v_{-} \, \ln \left(\gamma_{-}\right)\right]\]Hence \[\ln \left(\gamma_{\pm}\right)=-\frac{1}{\left(v_{+}+v_{-}\right)} \,\left[v_{+} \, z_{+}^{2}+v_{-} \, z_{-}^{2}\right] \,\left[\frac{e^{2} \, N_{A}}{8 \, \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, R \, T} \, \frac{\kappa}{(1+\kappa \, a)}\right]\]Or, \[\ln \left(\gamma_{\pm}\right)=-\left[\frac{\left|z_{+} \, z_{-}\right| \, e^{2} \, N_{A}}{8 \, \pi \, \varepsilon_{0} \, \varepsilon_{t} \, R \, T} \, \frac{K}{(1+K \, a)}\right]\]In this connection the distance ‘a’ characterises both cations and anions in the salt. In the Debye-Huckel Limiting Law (DHLL) the term \((1+\kappa \, a)\) is approximated to unity thereby assuming that \((1>>K \, a)\). Then using the term \(\mathrm{S}_{\gamma}\), for a 1:1 salt equation (h) is rewritten as follows. \[\ln \left(\gamma_{\pm}\right)=-\left|\mathrm{z}_{+} \, \mathrm{z}_{-}\right| \, \mathrm{S}_{\gamma} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2} /\left[1+\mathrm{b} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}\right]\]where \[\mathrm{b}=\beta \, \mathrm{a}\]and \[\beta=\left[\frac{2 \, \mathrm{e}^{2} \, \mathrm{N}_{\mathrm{A}}^{2} \, \rho_{1}^{*}(\ell) \, \mathrm{m}^{0}}{\varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{R} \, \mathrm{T}}\right]^{1 / 2}\]For aqueous solutions at ambient pressure and \(298.15 \mathrm{~K}\), \(\beta=3.285 \times 10^{9} \mathrm{~m}^{-1}\).The quantity \(\mathrm{b}\) depends on a distance parameter ‘a’ which characterises salt \(j\) and reflects the role of repulsion between ions in determining the chemical potentials of a salt in solution. Hence with increase in distance ‘a’ so the denominator increases and \(\ln \left(\gamma_{j}\right)\) is not so strongly negative as predicted by the DHLL. For large ‘a’ and high ionic strengths, the salts are not stabilised to the extent required by the DHLL. The integrated form of the Gibbs-Duhem equation yields an equation for \((\phi-1)\) in terms of molality \(\mathrm{m}_{j}\). Thus, \[\mathrm{m}_{\mathrm{j}} \,(1-\phi)=-\int_{0}^{\mathrm{m}(\mathrm{j})} \mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\pm}\right)\]Hence, \[(1-\phi)=\left[\left|\mathrm{z}_{+} \, \mathrm{z}_{-}\right| \, \mathrm{S}_{\gamma} / 3\right] \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{-1} \, \sigma(\mathrm{x})\]and \[\sigma(x)=\left(3 / x^{3}\right) \,\left[(1+x)-(1+x)^{-1}-2 \, \ln (1+x)\right]\]Then the excess molar Gibbs energy for a solution containing a 1:1 salt is given by equation (o). \[\mathrm{G}_{\mathrm{m}}^{\mathrm{E}}=2 \, \mathrm{R} \, \mathrm{T} \, \mathrm{S}_{\gamma} \, \mathrm{m}_{\mathrm{j}}^{3 / 2} \,\left(\mathrm{m}^{0}\right)^{-1} \,\left[\sigma(\mathrm{x}) / 3-(1+\mathrm{x})^{-1}\right]\]Footnotes \[\begin{aligned} \Delta \mu_{j}(---) &=\frac{[\mathrm{A} \mathrm{s}]^{2} \,\left[\mathrm{mol}^{-1}\right]}{ \,\left[\mathrm{F} \mathrm{m}^{-1}\right] \,} \, \frac{\left[\mathrm{m}^{-1}\right]}{\left\{+\left[\mathrm{m}^{-1}\right] \,[\mathrm{m}]\right\}} \\ &=\frac{\left[\mathrm{A}^{2} \mathrm{~s}^{2}\right] \,\left[\mathrm{mol}^{-1}\right]}{\left[\mathrm{As} \mathrm{} \mathrm{V}^{-1}\right]}=\frac{\left[\mathrm{A}^{2} \mathrm{~s}^{2}\right] \,\left[\mathrm{mol}^{-1}\right]}{\left[\mathrm{As} \mathrm{As} \mathrm{} \mathrm{J}^{-1}\right]}=\left[\mathrm{J} \mathrm{mol}^{-1}\right] \end{aligned}\] Condition of electric neutrality; \(v_{+} \, z_{+}=-v_{-} \, z_{-}\) or, \(v_{-}=-v_{+} \, z_{+} / z_{-}\)Then, \[\begin{aligned} & \frac{1}{\left(v_{+}+v_{-}\right)} \,\left[v_{+} \, z_{+}^{2}+v_{-} \, z_{-}^{2}\right] \\ =& {\left[\frac{1}{v_{+}-v_{-} \, z_{+} / z_{-}}\right] \,\left[v_{+} \, z_{+}^{2}-v_{+} \, z_{+} \, z_{-}\right]=\frac{z_{-}}{\left(z_{-}-z_{+}\right)} \,\left[z_{+}^{2}-z_{+} \, z_{-}\right] } \\ =&-z_{+} \, z_{-}=\left|z_{+} \, z_{-}\right| \end{aligned}\] \(\ln \left(\gamma_{\pm}\right)=\frac{ \,[\mathrm{A} \mathrm{s}]^{2} \,[\mathrm{mol}]}{ \, \,\left[\mathrm{As} \mathrm{} \mathrm{J}^{-1} \mathrm{As}\right] \,\left[\mathrm{J} \mathrm{mol}^{-1} \mathrm{~K}^{-1}\right] \,[\mathrm{K}]} \, \frac{\left[\mathrm{m}^{-1}\right]}{\left\{1+\left[\mathrm{m}^{-1}\right] \,[\mathrm{m}]\right\}}=\) \[\begin{aligned} \beta=\left\{ \,[\mathrm{C}]^{2} \,\left[\mathrm{mol}^{-1}\right]^{2} \,\left[\mathrm{kg} \mathrm{m}^{-3}\right] \,\left[\mathrm{mol} \mathrm{kg}^{-1}\right]\right\}^{1 / 2} \\ /\left\{\left[\mathrm{J}^{-1} \mathrm{C}^{2} \mathrm{~m}^{-1}\right] \, \,\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}]\right\}^{1 / 2}=\left[\mathrm{m}^{-1}\right] \end{aligned}\] For DH parameters for aqueous solutions to high \(\mathrm{T}\) and \(\mathrm{p}\), see D. J. Bradley and K. S. Pitzer,J. Phys.Chem.,1979,83,1599;1983;87,3798. Parameter ‘a’ is sometimes called ‘ion size’ . But as S. Glasstone [Introduction to Electrochemistry, D.van Nostrand, New Jersey, 1943, page 145, footnote] points out ‘the exact physical significance cannot be expressed precisely’. Nevertheless an important consideration is the relative sizes of ions and solvent molecules; B. E. Conway and R. E. Verrall, J.Phys.Chem.,1966, 70,1473. From equation (i) when by definition \(\mathrm{x}=\mathrm{b} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}\) and \(\mathrm{k}=\left|\mathrm{Z}_{+} \, \mathrm{Z}_{-}\right| \, \mathrm{S}_{\gamma} / \mathrm{b}\)Hence, \[\begin{aligned} &\ln \left(\gamma_{\pm}\right)=-(b \, k) \,(x / b) /(1+x)=-k \, x /(1+x) \\ &\mathrm{d} \ln \left(\gamma_{\pm}\right)=-\mathrm{k} \,\left\{[1 /(1+\mathrm{x})]-\left[\mathrm{x} /(1+\mathrm{x})^{2}\right]\right\} \, \mathrm{dx}=-\mathrm{k} \,\left\{[1+\mathrm{x}-\mathrm{x}] /[1+\mathrm{x}]^{2}\right\} \, \mathrm{dx} \\ &\mathrm{d} \ln \left(\gamma_{\pm}\right)=-\mathrm{k} \, \mathrm{dx} /[1+\mathrm{x}]^{2} \end{aligned}\]Therefore, \[(1-\phi) \, m_{j}=-\int_{0}^{x} m_{j} \,\left\{-k /(1+x)^{2}\right\} \, d x\]Or, \((1-\phi)=\left(k / x^{2}\right) \, \int_{0}^{x}\left\{x^{2} /(1+x)^{2}\right\} \, d x\)Standard integral: \[\begin{aligned} &\int_{0}^{x}\left\{x^{2} /(a \, x+b)^{2}\right\} \, d x= \\ &\left\{(a \, x+b) / a^{3}\right\}-\left\{b^{2} / a^{3} \,(a \, x+b)\right\}-\left(2 \, b / a^{3}\right) \, \ln (a \, x+b) \end{aligned}\]With \(a=b=1\), \[\int_{0}^{x}\left\{x^{2} /(1+x)^{2}\right\} \, d x=(1+x)-[1 /(1+x)]-2 \, \ln (1+x)\]Thus, \[(1-\phi)=(k \, x / 3) \,\left\{\left(3 / x^{3}\right) \,\left[(1+x)-(1+x)^{-1}-2 \, \ln (1+x)\right]\right.\]This page titled 1.10.18: Gibbs Energies- Salt Solutions- Debye-Huckel Equation is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.10.19: Gibbs Energies- Salt Solutions- Debye-Huckel Limiting Law

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According to the Debye-Huckel analysis the mean ionic activity coefficient is given by equation (a). \[\ln \left(\gamma_{\pm}\right)=-\left[\frac{\left|z_{+} \, z_{-}\right| \, e^{2} \, N_{A}}{8 \, \pi \, \varepsilon_{0} \, \varepsilon_{t} \, R \, T} \, \frac{K}{(1+K \, a)}\right]\]In the DHLL the term \((1+\kappa \, a)\) is approximated to unity thereby assuming that \((1>>\kappa \, a)\). By definition \[\mathrm{S}_{\mathrm{\gamma}}=\left[\frac{2 \, \pi \, \mathrm{N}_{\mathrm{A}} \, \mathrm{M}_{1} \, \mathrm{m}^{0}}{\mathrm{~V}_{1}^{*}(\ell)}\right]^{1 / 2} \,\left[\frac{\mathrm{e}^{2} \, \mathrm{N}_{\mathrm{A}}}{4 \, \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{R} \, \mathrm{T}}\right]^{3 / 2}\]Then \[\ln \left(\gamma_{\pm}\right)=-\left|z_{+} \, z_{-}\right| \, S_{\gamma} \,\left(m_{j} / m^{0}\right)^{1 / 2}\]The practical osmotic coefficient for quite dilute salt solutions is also a linear function of \(\left(\mathrm{m}_{j}\right)^{1 / 2}. Indeed \(\phi\) and \(\ln \left(\gamma_{\pm}\right)\) are simply related. \[1-\phi=-(1 / 3) \, \ln \left(\gamma_{\pm}\right)\]Footnotes From, \[\mathrm{S}_{\gamma}=\frac{\mathrm{e}^{3} \,\left[2 \, \mathrm{N}_{\mathrm{A}} \, \rho_{1}^{*}(\ell)\right]^{1 / 2}}{8 \, \pi \,\left[\varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{k} \, \mathrm{T}\right]^{3 / 2}}\]Then, \(\frac{1}{\pi}=\frac{\pi^{1 / 2}}{\pi^{3 / 2}}\) and \(\frac{1}{8}=\frac{1}{4^{3 / 2}}\)This page titled 1.10.19: Gibbs Energies- Salt Solutions- Debye-Huckel Limiting Law is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.10.20: Gibbs Energies- Salt Solutions- DHLL- Derived Parameters

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Granted that the DHLL forms a starting point for understanding the role of ion-ion interactions in aqueous solutions, we use the DHLL to explain the impact of these interactions on related properties such as volumes and enthalpies. We write the Debye-Huckel coefficient as a function of three variables:Here we take account of the fact that \(\mathrm{V}_{1}^{*}(\ell)\) and \(\varepsilon_{r}\) depend on both temperature and pressure. \[S_{\gamma}=\left[\frac{2 \, \pi \, \mathrm{N}_{\mathrm{A}} \, \mathrm{M}_{1} \, \mathrm{m}^{0}}{\mathrm{~V}_{1}^{*}(\ell)}\right]^{1 / 2} \,\left[\frac{\mathrm{e}^{2} \, \mathrm{N}_{\mathrm{A}}}{4 \, \pi \, \varepsilon^{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{R} \, \mathrm{T}}\right]^{3 / 2}\]\(\mathrm{S}_{\gamma}\) is written in the following form. \[\mathrm{S}_{\gamma}=\mathrm{E} \,\left[\mathrm{V}_{1}^{*}(\ell)\right]^{-1 / 2} \,\left(\varepsilon_{\mathrm{r}}\right)^{-3 / 2} \,(\mathrm{T})^{-3 / 2}\]where \[\mathrm{E}=\left[2 \, \pi \, \mathrm{N}_{\mathrm{A}} \, \mathrm{M}_{1} \, \mathrm{m}^{0}\right]^{1 / 2} \,\left[\frac{\mathrm{e}^{2} \, \mathrm{N}_{\mathrm{A}}}{4 \, \pi \, \varepsilon^{0} \, \mathrm{R}}\right]^{3 / 2}\]Hence \[\mathrm{S}_{\gamma}=\mathrm{E} \, \mathrm{F}\]where \[\mathrm{F}=\left[\mathrm{V}_{1}^{*}(\ell)\right]^{-1 / 2} \,\left(\varepsilon_{\mathrm{r}}\right)^{-3 / 2} \,(\mathrm{T})^{-3 / 2}\]In terms of our interest in the dependence of \(\mu_{\mathrm{j}}(\mathrm{aq})\) for salt \(j\) on temperature leading to partial molar enthalpies we require \(\left[\partial \mathrm{S}_{\gamma} / \partial \mathrm{T}\right]_{\mathrm{p}}\) which is calculated using the dependences of both \(\mathrm{V}_{1}^{*}(\ell)\) and \(\varepsilon_{r}\) on temperature yielding \((\partial \mathrm{F} / \partial \mathrm{T})_{\mathrm{p}}\). For partial molar isobaric heat capacities we require the second differential \(\left(\partial^{2} \mathrm{~F} / \partial \mathrm{T}^{2}\right)_{\mathrm{P}}\). The predicted dependence by DHLL of the partial molar volume \(\mathrm{V}_{\mathrm{j}}(\mathrm{aq})\) on salt molality involves the derivative \((\partial \mathrm{F} / \partial \mathrm{p})_{\mathrm{T}}\).Calculations are considerably helped using a PC in conjunction with equations describing the \(\mathrm{T} - \mathrm{p}\) dependences of \(\mathrm{V}_{1}^{*}(\ell)\) and \(\varepsilon_{r}\).Footnotes \(\mathrm{E}=\left[ \, \,\left[\mathrm{mol}^{-1}\right] \,\left[\mathrm{kg} \mathrm{mol}^{-1}\right] \,\left[\mathrm{mol} \mathrm{kg}^{-1}\right]\right]^{1 / 2} \,\left[\frac{[\mathrm{C}]^{2} \,\left[\mathrm{mol}^{-1}\right]}{ \, \,\left[\mathrm{C}^{2} \mathrm{~J}^{-1} \mathrm{~m}^{-1}\right] \,\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right.}\right]^{3 / 2}\)or, \(\mathrm{E}=[\mathrm{mol}]^{-1 / 2} \,[\mathrm{m}]^{3 / 2} \,[\mathrm{K}]^{3 / 2}=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]^{1 / 2} \,[\mathrm{K}]^{3 / 2}\)and \(\mathrm{F}=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]^{-1 / 2} \,^{-3 / 2} \,[\mathrm{K}]^{-3 / 2}\)Hence, \(\mathrm{S}_{\gamma}=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]^{1 / 2} \,[\mathrm{K}]^{3 / 2} \,\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]^{-1 / 2} \,^{-3 / 2} \,[\mathrm{K}]^{-3 / 2}=\)This page titled 1.10.20: Gibbs Energies- Salt Solutions- DHLL- Derived Parameters is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.10.21: Gibbs Energies- Salt Solutions- DHLL- Empirical Modifications

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The success of equations based on the Debye-Huckel equations is often modest and so attempts are made to describe quantitatively the dependences of \(\phi\), \(\ln \left(\gamma_{\pm}\right)\) and \(\mathrm{G}_{\mathrm{m}}^{\mathrm{E}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\) on \(\mathrm{m}_{j}\) to higher molalities. In most cases attempts are made to moderate the stabilization of the salt with increasing ionic strength. The obvious procedure centres on incorporating a denominator into the DHLL as illustrated by the Guntleberg equation. \[\ln \left(\gamma_{\pm}\right)=-\left|\mathrm{z}_{+} \, \mathrm{z}_{-}\right| \, \mathrm{S}_{\gamma} \,\left(\mathrm{I} / \mathrm{m}^{0}\right)^{1 / 2} /\left\{1+\left(\mathrm{I} / \mathrm{m}^{0}\right)^{1 / 2}\right\}\]The Guggenheim Equation starts with equation (a) and adds a further term, linear in ionic strength. \[\ln \left(\gamma_{\pm}\right)=-\left[\left|\mathrm{z}_{+} \, \mathrm{z}_{-}\right| \, \mathrm{S}_{\gamma} \,\left(\mathrm{I} / \mathrm{m}^{0}\right)^{1 / 2} /\left\{1+\left(\mathrm{I} / \mathrm{m}^{0}\right)^{1 / 2}\right\}\right]+\mathrm{b} \,\left(\mathrm{I} / \mathrm{m}^{0}\right)\]The quantity ‘b’ is characteristic of the salt. Another obvious development uses the same approach in the context of the DHLL. An interesting equation takes the following form for the solution containing a salt \(j\). \[\ln \left(\gamma_{\pm}\right)=-\left[\left|\mathrm{z}_{+} \, \mathrm{z}_{-}\right| \, \mathrm{S}_{\gamma} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}\right]+\mathrm{B} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)\]Here \(\mathrm{B}\) describes the role of ion size and the impact of cosphere-cosphere interactions specific to a particular salt.In most approaches, the starting point in an equation for \(\ln \left(\gamma_{\pm}\right)\) as a function of ionic strength, the equation for the dependence of \(\phi\) on ionic strength being obtained using the integral of equation (c). An interesting approach suggested by Bronsted starts out with a virial equation for \(1 - \phi\) in terms of molality \(\mathrm{m}_{j}\). \[1-\phi=\alpha \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}+\beta \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)\]Hence \[\ln \left(\gamma_{\pm}\right)=-3 \, \alpha \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}-2 \, \beta \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)\]Footnote From \[\begin{aligned} &\ln \left(\gamma_{\pm}\right)=(\phi-1)+\int_{0}^{m_{j}}(\phi-1) \, \mathrm{d} \ln m_{j} \\ &\ln \left(\gamma_{\pm}\right)=(\phi-1)-\int_{0}^{\mathrm{m}_{\mathrm{j}}}\left[\left\{\alpha \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}+\beta \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)\right\} / \mathrm{m}_{\mathrm{j}}\right] \, \mathrm{dm}_{\mathrm{j}} \\ &\left.\ln \left(\gamma_{\pm}\right)=(\phi-1)-\int_{0}^{m_{j}}\left[\left\{\alpha / m_{j} \, m^{0}\right)^{1 / 2}\right\}+\left\{\beta / m^{0}\right\}\right] \, d_{j} \\ &\ln \left(\gamma_{\pm}\right)=(\phi-1)-\left[2 \, \alpha \,\left(m_{j} / m^{0}\right)^{1 / 2}+\beta \,\left(m_{j} / m^{0}\right)\right]_{0}^{m_{j}} \\ &\ln \left(\gamma_{\pm}\right)=-\alpha \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}-\beta \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)-2 \, \alpha \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}-\beta \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right) \\ & \ln \left(\gamma_{\pm}\right)=-3 \, \alpha \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}-2 \, \beta \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right) \end{aligned}\]This page titled 1.10.21: Gibbs Energies- Salt Solutions- DHLL- Empirical Modifications is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.10.22: Gibbs Energies- Salt Solutions- Solvent

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A given salt solution contains a 1:1 salt \(j\) (e.g. \(\mathrm{NaCl}\)) in which the salt, chemical substance \(j\), completely dissociates into ions. In other words, the total molality of solutes equals \(2 \, m_{j}\). By definition the chemical potential of water in this aqueous solution, \(\mu_{1}(\mathrm{aq})\) (at fixed temperature and pressure, the latter being ambient and hence close to the standard pressure \(\mathrm{p}^{0}\)) is given by equation (a). \[\mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\ell)-2 \, \phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\]For the corresponding ideal solution, \(\phi = 1.0\) at all \(\mathrm{T}\) and \(\mathrm{p}\), Hence, \[\mu_{1}(\mathrm{aq} ; \mathrm{id})=\mu_{1}^{*}(\ell)-2 \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\]Just as for solutions containing neutral solutes, the minus sign in equation (b) means that added salt stabilizes the solvent in an ideal solution; \(\mu_{1}(\mathrm{aq} ; \mathrm{id})<\mu_{1}^{*}(\ell)\).For water in an aqueous salt solution containing salt \(j\), molality \(\mathrm{m}_{j}\), where each mole of salt forms \(v\) moles of ions with complete dissociation, the chemical potential of the solvents is given by equation (c). \[\mu_{1}(\mathrm{aq})=\mu_{1}^{*}\left(\ell ; \mathrm{p}^{0}\right)-\mathrm{v} \, \phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}+\int_{\mathrm{p}}^{\mathrm{p}} \mathrm{V}_{1}^{*}(\ell) \, \mathrm{dp}\]For the ideal dilute solution, \(\phi =1.0\). Here \(\mu_{1}^{*}\left(\ell, \mathrm{p}^{0}\right)\) is the standard chemical potential of water at temperature \(\mathrm{T}\). Alternatively we may switch the reference chemical potential for the solvent to the pure liquid at the same pressure. \[\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})-\mathrm{v} \, \phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\]Footnotes For relevant Tables see;’ R. A. Robinson and R. H. Stokes, Electrolyte Solutions, 2nd edn.(revised), Butterworths, London, 1965, Appendix 8. The impact of salts on osmotic coefficients is illustrated by the properties of aqueous solutions containing alkylammonium salts.This page titled 1.10.22: Gibbs Energies- Salt Solutions- Solvent is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.10.23: Gibbs Energies- Salt Solutions- Excess Gibbs Energies

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A given salt solution contains a single salt \(j\) which completely dissociates to form \(ν\) moles of ions from one mole of salt. Then the chemical potential of the salt \(j\) in aqueous solution at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\) is given by equation (a). \[\mu_{j}(a q)=\mu_{j}^{0}(a q)+v \, R \, T \, \ln \left(Q \, m_{j} \, \gamma_{\pm} / m^{0}\right)+\int_{p^{0}}^{p} V_{j}^{\infty}(a q) \, d p\]Here \(\mu_{\mathrm{j}}^{0}(\mathrm{aq})\) is the chemical potential of the salt in solution at the same temperature and the standard pressure where molality \(\mathrm{m}_{\mathrm{j}}=1 \mathrm{~mol} \mathrm{~kg}^{-1}\) and mean ionic activity coefficient \(\gamma_{\pm} = 1\). The chemical potential of water in aqueous solution at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\) is given by equation (b) where \(\phi\) is the practical osmotic coefficient, \[\mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\ell)-\mathrm{v} \, \phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}+\int_{\mathrm{p}}^{\mathrm{p}} \mathrm{V}_{1}^{*}(\ell) \, \mathrm{dp}\]If we confine our attention to the properties of solutions at ambient pressure (which is very close to the standard pressure) then we can ignore the integrals in equations (a) and (b). Hence the Gibbs energy of the solution at the same \(\mathrm{T}\) and \(\mathrm{p}\) prepared using \(1 \mathrm{~kg}\) of water is given by equation (c). \[\begin{aligned} \mathrm{G}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \,\left[\mu_{1}^{*}(\ell)-\mathrm{v} \, \phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\right] \\ &+\mathrm{m}_{\mathrm{j}} \,\left[\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{v} \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{Q} \, \mathrm{m}_{\mathrm{j}} \, \gamma_{\pm} / \mathrm{m}^{0}\right)\right] \end{aligned}\]As for solutions containing neutral solutes we cannot put a number value to \(\mathrm{G}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)\). If the properties of this salt solution are in fact ideal (in a thermodynamic sense) then \(\mathrm{G}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}: \mathrm{id}\right)\) is given by equation (d). \[\begin{aligned} \mathrm{G}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=&\left(1 / \mathrm{M}_{1}\right) \,\left[\mu_{1}^{*}(\ell)-\mathrm{v} \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{\mathrm{l}} \, \mathrm{m}_{\mathrm{j}}\right] \\ &+\mathrm{m}_{\mathrm{j}} \,\left[\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{v} \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{Q} \, \mathrm{m}_{\mathrm{j}} \, \gamma_{\pm} / \mathrm{m}^{0}\right)\right] \end{aligned}\]Hence in the case where \(j = \mathrm{~NaCl}\), \(ν = 2\) and \(\mathrm{Q} = 1\). In the next stage we use differences between \(\mathrm{G}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)\) and \(\mathrm{G}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)\) to define excess Gibbs energies for a solution prepared using \(1 \mathrm{~kg}\) of water. Then \[\mathrm{G}^{\mathrm{E}}=\mathrm{G}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)-\mathrm{G}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)\]For salt \(j\), \[\mathrm{G}^{\mathrm{E}}=\mathrm{V} \, \mathrm{m}_{\mathrm{j}} \, \mathrm{R} \, \mathrm{T} \,\left[1-\phi+\ln \left(\gamma_{\pm}\right)\right]\]According to the Gibbs-Duhem for a solution at constant temperature and constant pressure, \[\mathrm{n}_{1} \, \mathrm{d} \mu_{1}(\mathrm{aq})+\mathrm{n}_{\mathrm{j}} \, \mathrm{d} \mu_{\mathrm{j}}(\mathrm{aq})=0\]Hence for salt \(j\), \[\begin{aligned} &\left(1 / \mathrm{M}_{1}\right) \, \mathrm{d}\left[\mu_{1}^{*}(\ell)-\mathrm{v} \, \phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\right] \\ &\quad+\mathrm{m}_{\mathrm{j}} \, \mathrm{d}\left[\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{v} \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{Q} \, \mathrm{m}_{\mathrm{j}} \, \gamma_{\pm} / \mathrm{m}^{0}\right)\right]=0 \end{aligned}\]We are concerned with the dependence of chemical potential on the molality of salt. Thus the amount of solvent and, for the salt, both \(\mathrm{Q}\) and \(v\) are fixed. Hence \[\mathrm{d}\left[-\phi \, \mathrm{m}_{\mathrm{j}}\right]+\mathrm{m}_{\mathrm{j}} \, \mathrm{d}\left[\ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\pm} / \mathrm{m}^{0}\right)\right]=0\]Hence, \[-\phi \, d m_{j}-m_{j} \, d \phi+m_{j} \, d m_{j} / m_{j}+m_{j} \, d \ln \left(\gamma_{t}\right)=0\]Then, \[(\phi-1) \, d m_{j}+m_{j} \, d \phi=m_{j} \, d \ln \left(\gamma_{\pm}\right)\]Hence we obtain an equation for \(\ln \left(\gamma_{\pm}\right)\) in terms of the dependence of \((\phi - 1)\) on molality bearing in mind that \(\ln \left(\gamma_{\pm}\right)\) equals zero and \(\phi\) equals 1 at ‘\(\mathrm{m}_{j} = 0\)’. \[\ln \left(\gamma_{\pm}\right)=(\phi-1)+\int_{0}^{m_{j}}(\phi-1) \, d \ln \left(m_{j}\right)\]From equation (f), the dependence of \(\mathrm{G}^{\mathrm{E}}\) on \(\mathrm{m}_{j}\) is given by equation (m). \[\begin{aligned} &(1 / \mathrm{V} \, \mathrm{R} \, \mathrm{T}) \, \mathrm{dG}^{\mathrm{E}} / \mathrm{dm}_{\mathrm{j}} \\ &=\left[1-\phi+\ln \left(\gamma_{\pm}\right)\right]-\mathrm{m}_{\mathrm{j}} \,\left(\mathrm{d} \phi / \mathrm{dm}_{\mathrm{j}}\right)+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\pm}\right) / \mathrm{dm}_{\mathrm{j}}=0 \end{aligned}\]But according to equation (i), \[-\phi-\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \phi / \mathrm{dm}_{\mathrm{j}}+1+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\pm}\right) / \mathrm{dm}_{\mathrm{j}}=0\]Hence, \[\ln \left(\gamma_{\pm}\right)=(1 / \mathrm{V} \, \mathrm{R} \, \mathrm{T}) \, \mathrm{dG}^{\mathrm{E}} / \mathrm{dm}_{\mathrm{j}}\]Footnotes Compilation of Data for polyvalent electrolytes; R. N. Goldberg, B. R. Staples, R. L. Nuttall and R. Arbuckle, NBS Special Publication 485, 1977. Thermal Properties of Aqueous Univalent –Univalent Electrolytes, V.B.Parker, NBS, 2, 1965. J.-L. Fortier and J. E. Desnoyers, J. Solution Chem.,1976,5,297. \[\ln \left(\gamma_{\pm}\right)=\left[\frac{1}{ \,\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}]}\right] \,\left[\frac{\mathrm{J} \mathrm{kg}^{-1}}{\mathrm{~mol} \mathrm{~kg}^{-1}}\right]=\]This page titled 1.10.23: Gibbs Energies- Salt Solutions- Excess Gibbs Energies is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.10.24: Gibbs Energies- Salt Solutions- Pitzer's Equations

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The Debye-Huckel treatment of the properties of salt solutions is based on a linearization of the Botzmann Equation leading to an equation for the radial distribution function, \(\mathrm{g}_{i j} (\mathrm{r})\). If a further term is taken into the expansion, the equation for \(\mathrm{g}_{i j} (\mathrm{r})\) takes the following form. \[g_{i j}(r)=1-q_{i j}+\left(q_{i j}^{2} / 2\right)\]When equation (a) was tested against the results of a careful Monte Carlo calculation the conclusion was drawn that the three-term equation is good approximation. The result is a set of equations for both the practical osmotic coefficient \(\phi\) and mean ionic activity coefficient for the salt in a solution having ionic strength \(\mathrm{I}\). The theory has been extended to consider the properties of salt solutions at high \(\mathrm{T}\) and \(\mathrm{p}\). In fact key parameters in Pitzer equations covering extensive ranges of \(\mathrm{T}\) and \(\mathrm{p}\) have been extensively documented. The Pitzer treatment has been extended to a consideration of the properties of mixed salt solutions.Footnotes K. S. Pitzer, Acc. Chem.Res.,1977,10,371. D. N. Card and J. P. Valleau, J. Chem. Phys.,1970,52,6232. K. S. Pitzer, J.Phys.Chem.,1973,77,268. Activity and Osmotic Coefficients for R. C. Phutela and K. S. Pitzer, J. Phys Chem.,1986, 90,895.This page titled 1.10.24: Gibbs Energies- Salt Solutions- Pitzer's Equations is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.10.25: Gibbs Energies- Salt Solutions- Cosphere - Cosphere Interactions

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The Debye-Huckel equations form the starting point for detailed analyses of the properties of salt solutions. An interesting method examines excess thermodynamic properties of salt solutions and their dependence on salt molality. For an aqueous solution containing a 1:1 salt in 1 kg of solvent, the excess Gibbs energy \(\mathrm{G}^{\mathrm{E}}\) is given by equation (a). \[\mathrm{G}^{\mathrm{E}}=2 \, \mathrm{R} \, \mathrm{T} \, \mathrm{m}_{\mathrm{j}} \,\left[1-\phi+\ln \gamma_{\pm}\right]\]The corresponding excess molar Gibbs energy is given by equation (b). \[\mathrm{G}_{\mathrm{m}}^{\mathrm{E}}=2 \, \mathrm{R} \, \mathrm{T} \,\left[1-\phi+\ln \gamma_{\pm}\right]\]The corresponding excess molar enthalpy \({\mathrm{H}_{\mathrm{m}}}^{\mathrm{E}}\) is obtained from calorimetric data. Hence the molar excess entropy is calculated using equation (c). \[\mathrm{G}_{\mathrm{m}}^{\mathrm{E}}=\mathrm{H}_{\mathrm{m}}^{\mathrm{E}}-\mathrm{T} \, \mathrm{S}_{\mathrm{m}}^{\mathrm{E}}\]A common observation is that \({\mathrm{H}_{\mathrm{m}}}^{\mathrm{E}}\) and \(\mathrm{T}.{\mathrm{S}_{\mathrm{m}}}^{\mathrm{E}}\) are more sensitive to both the molality \(\mathrm{m}_{j}\) and the salt than is \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}}\). For dilute salt solutions \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}} < 0\) as a consequence of charge-charge interactions between the ions leading to a stabilisation. Further \({\mathrm{H}_{\mathrm{m}}}^{\mathrm{E}}\) and \(\mathrm{T}.{\mathrm{S}_{\mathrm{m}}}^{\mathrm{E}}\) are more sensitive to switching the solvent from \(\mathrm{H}_{2}\mathrm{O}\) to \(\mathrm{D}_{2}\mathrm{O}\) than is \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}}\).The dependence of \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}}\) on salt at fixed \(\mathrm{m}_{j}\) shows that in addition to charge-charge interactions there are further interactions which are characteristic of the ions in a given salt. So the suggestion is that even in the absence of charge-charge interactions the properties of the solution would not be ideal by virtue of cosphere - cosphere interactions along the lines suggested by Gurney. But there are two types of solutes in a simple 1:1 salt solution so we must consider in the analysis of these properties at least \(\mathrm{g}_{++}\), \(\mathrm{g}_{+-}\) and \(\mathrm{g}_{--}\) pairwise ion-ion Gibbs energy interaction parameters. Consequently the analysis is not straightforward.An interesting approach examines patterns in \(\ln \left(\gamma_{\pm}\right)\) for a series of 1:1 salts at fixed molality \(\mathrm{m}_{j}\), temperature and pressure. A most dramatic change is observed for \(\mathrm{Pr}_{4}\mathrm{N}^{+}\) salts (aq; \(0.2 \mathrm{~mol kg}^{-1}\)). Thus \(\ln \left(\gamma_{\pm}\right)>\left(\ln \gamma_{\pm}(\mathrm{DHLL}-\text { calc })\right)\) for the fluoride salt ; i.e. a higher chemical potential than calculated simply on the basis of the DHLL-- a destabilisation. But \(\ln \left(\gamma_{\pm}\right)>\left(\ln \gamma_{\pm}(\mathrm{DHLL}-\text { calc })\right)\) for the corresponding iodide salt; i.e. lower chemical potential than calculated simply on the grounds of the DHLL--- a stabilisation. The dependence of \(\ln \gamma_{\pm}\) for \(\mathrm{K}^{+}\), \(\mathrm{Rb}^{+}\) and \(\mathrm{Cs}^{+}\) on the anion \(\mathrm{F}^{-}\), \(\mathrm{Cl}^{-}\), \(\mathrm{Br}^{-}\) and \(\mathrm{I}^{-}\) is much more modest. The pattern signals the important role of hydrophobic-hydrophobic, hydrophilic-hydrophobic and hydrophilic-hydrophilic ion-ion interactions. Indeed there is considerable merit in the approach. The pattern emerges in a comparison of salt effects on rates of hydrolysis in aqueous salt solutions. This conclusion is supported by the observation that the dependence of \(\ln \gamma_{\pm}\) on \(\left(\mathrm{m}_{\mathrm{j}}\right)^{1 / 2}\) for \(\left(\mathrm{HOCH}_{2} \mathrm{CH}_{2}\right)_{4} \mathrm{~N}^{+} \mathrm{Br}^{-}\) deviates from the DHLL pattern in a direction indicating a more hydrophilic character for the cation than in the cases of \(\mathrm{Pr}_{4}\mathrm{N}^{+}\) and \(\mathrm{Et}_{4}\mathrm{N}^{+}\).The patterns identified in \(\ln \gamma_{\pm}\) signal that the ion-ion pair potential for ions in solution comprises several components. This recognition forms the basis of the treatment developed by Friedman and coworkers. The pair potential \(\mathrm{u}_{i j}\) for two ions charge \(\mathrm{z}_{\mathrm{i}}.\mathrm{e}\) and \(\mathrm{z}_{\mathrm{j}}.\mathrm{e}\) in a solvent having relative permittivity \(\varepsilon_{r}\) is expressed in the form shown in equation (d). \[u_{i j}(r)=\left[\frac{\left(z_{i} \, e\right) \,\left(z_{j} \, e\right)}{4 \, \pi \, \varepsilon_{0} \, \varepsilon_{r} \, r}\right]+C O R_{i j}+C A V_{i j}+\mathrm{GUR}_{i j}\]The first term takes account of charge-charge interactions; i.e. the Coulombic term, COUL. The term \(\mathrm{COR}_{i j}\) is a repulsive core potential , being a function of the sizes of the ions \(i\) and \(j\). The \(\mathrm{CAV}_{i j}\) term takes account of a special effect arising from the interactions between ion + solvent cavities. The impact of cosphere overlap is taken into account by the Gurney potential, \(\mathrm{GUR}_{i j}\). The pair potential is used in conjunction with McMillan-Mayer theory. The \(\mathrm{GUR}__{i j}\) term includes an adjustable parameter \(\mathrm{A}_{i j}\), the change in Helmholtz energy, \(\mathrm{F}\) when one mole of solvent in the overlap region returns to the bulk solvent. There are therefore three such terms, \(\mathrm{A}_{++}\), \(\mathrm{A}_{+-}\), and \(\mathrm{A}_{--}\) for the three types of overlap. These terms are related to the corresponding volumetric \(\mathrm{V}_{i j}\), entropy \(\mathrm{S}_{i j}\) and energy \(\mathrm{U}_{i j}\) terms. The analysis is slightly complicated by the fact that the derived thermodynamic functions refer to a solution in osmotic equilibrium with the solvent at the standard pressure; the MM state. Conversion is required to thermodynamic parameters for a solution of the same salt at the same molality at the standard pressure.Footnotes Y.-C. Wu and H. L. Friedman, J.Phys.Chem.,1966,70,166. H. S. Frank, Z. Phys. Chem., 1965,228,364. H. S. Frank and A. L. Robinson, J.Chem.Phys.,1940,8,933; this paper concerns the dependence of partial molar entropies on composition of salt solutions; the analysis set the stage for subsequent developments in this subject. J. E. Desnoyers, M. Arel, G. Perron and C. Jolicoeur, J. Phys. Chem., 1969, 73, 3346. M. J. Blandamer, J. B. F. N. Engberts, J. Burgess, B. Clark and A. W. Hakin, J. Chem. Soc. Chem.Commun.,1985,414; see also apparent molar Cp; J. L. Fortier, P.-A. Leduc and J. E. Desnoyers, J. Solution Chem.,1974,3,323. P. S. Ramanthan and H. L. Friedman, J.Chem.Phys.,1971,54,1086. W. G. McMillan and J. E. Mayer, J.Chem.Phys.,1945,13,276. H. L. Friedman and C. V. Krishnan, Ann. N. Y. Acad. Sci.,1973,204,79.This page titled 1.10.25: Gibbs Energies- Salt Solutions- Cosphere - Cosphere Interactions is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.10.26: Gibbs Energies- Binary Liquid Mixtures

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Interest in the thermodynamic properties of binary liquid mixtures extended over most of the 19th Century. Interest was concerned with the vapour pressures of the components of a given binary mixture. Experimental work led Raoult to propose a law of partial vapour pressures which characterises a liquid mixture whose thermodynamic properties are ideal.If liquid 1 is a component of a binary liquid mixture, components liquid 1 and liquid 2, the mixture is defined as having ideal thermodynamic properties if the partial vapour pressure at equilibrium \(\mathrm{p}_{1}(\text{mix})\) is related to the mole fraction composition using equation (a). \[) p (mix) x p ( * 1 1 1 = ⋅ l (a)Here \(\mathrm{p}_{1}^{*}(\ell)\) is the vapour pressure of pure liquid 1 at the same temperature. A similar equation describes the vapour pressure of liquid 2, \(\mathrm{p}_{2}(\text{mix})\) in the mixture. Raoult’s Law as given in equation (a) forms the basis for examining the properties of real binary liquid mixtures.Binary liquid mixtures are interesting in their own right. Nevertheless chemists also use binary liquid mixtures as solvents for chemical equilibria and for media in which to carry out chemical reactions between solutes. In fact an enormous amount of our understanding of the mechanisms of inorganic and organic reactions is based on kinetic data describing the rates of chemical reactions in 80/20 ethanol and water mixtures. Because so much experimental information in the literature concerns the properties of solutes in binary aqueous mixtures we concentrate our attention on these systems. Indeed the properties of such solvent systems cannot be ignored when considering the properties of solutes in these mixtures. In examining the properties of binary aqueous mixtures we adopt a convention in which liquid water is chemical substance 1 and the non-aqueous liquid component is chemical substance 2. For the most part we describe the composition of a given liquid mixture (at defined temperature and pressure) using the mole fraction scale. Then if \(\mathrm{n}_{1}\) and \(\mathrm{n}_{2}\) are the amounts of chemical substances 1 and 2, the mole fractions are defined by equation (b). \[\mathrm{x}_{1}=\mathrm{n}_{1} /\left(\mathrm{n}_{1}+\mathrm{n}_{2}\right) \quad \mathrm{x}_{2}=\mathrm{n}_{2} /\left(\mathrm{n}_{1}+\mathrm{n}_{2}\right)\]The chemical potentials of each component in a given liquid mixture, mole fraction composition \(\mathrm{x}_{2}\left(=1-\mathrm{x}_{1}\right)\) are compared with the chemical potential of the pure liquid chemical potential at the same temperature and pressure, \(\mu_{1}^{*}(\ell)\) and \(\mu_{2}^{*}(\ell)\).The starting point is a description of the equilibrium at temperature \(\mathrm{T}\) between pure liquid and its saturated vapour in a closed system; i.e. a two phase system. [In terms of the Gibbs phase rule the number of components = 1; the number of phases = 2. Hence the number of degrees of freedom = 1. Then at a specified temperature the vapour pressure is defined.] The equilibrium is described in terms of the equality of chemical potentials of substance 1 in the two phases; equation (c). \[\mu_{1}\left(\mathrm{~g} ; \mathrm{p}_{1}^{*} ; \mathrm{T}\right)=\mu_{1}\left(\ell ; \mathrm{p}_{1}^{*} ; \mathrm{T}\right)\]Thus \(\mu_{1}\left(\ell ; \mathrm{p}_{1}^{*} ; \mathrm{T}\right)\) is the molar Gibbs energy of the pure liquid 1, otherwise the chemical potential.However our interest concerns the properties of binary liquid mixtures. Using the Gibbs phase rule, number of phases = 2; number of components = 2; hence number of degrees of freedom = 2. Then for a defined temperature and mole fraction composition the vapour pressure \(\mathrm{p}(\text{mix})\) is fixed. Equation (d) describes the equilibrium between liquid and vapour phases with reference to liquid substance 1. \[\mu_{1}\left(\mathrm{~g} ; \mathrm{p}_{1} ; \mathrm{T}\right)=\mu_{1}\left(\operatorname{mix} ; \mathrm{x}_{1} ; \mathrm{p} ; \mathrm{T}\right)\]Equations (c) and (d) offer a basis for comparing the chemical potentials of chemical substance 1 in the two phases, liquid and vapour. \[\mu_{1}\left(\mathrm{~g} ; \mathrm{p}_{1}^{*} ; \mathrm{T}\right)-\mu_{1}\left(\mathrm{~g} ; \mathrm{p}_{1} ; \mathrm{T}\right)=\mu_{1}\left(\ell ; \mathrm{p}_{1}^{*} ; \mathrm{T}\right)-\mu_{1}\left(\mathrm{mix} ; \mathrm{x}_{1} ; \mathrm{p} ; \mathrm{T}\right)\]Analysis of vapour pressure data is not straightforward because account has to be taken of the fact that the properties of real gases are not those of an ideal gas. However here we assume that the properties of chemical substance in the gas phase are ideal. Consequently we use the following equation to provide an equation for the r.h.s. of equation (e) in terms of two vapour pressures. \[\mu_{1}\left(\operatorname{mix} ; \mathrm{x}_{1} ; \mathrm{p} ; \mathrm{T}\right)=\mu_{1}\left(\ell ; \mathrm{p}_{1}^{*} ; \mathrm{T}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{p}_{1} / \mathrm{p}_{1}^{*}\right)\]Equation (a) is combined with equation (f) to yield the following equation for chemical substance 1 as component of the binary liquid mixture. \[\mu_{1}\left(\operatorname{mix} ; \mathrm{id} ; \mathrm{x}_{1}\right)=\mu_{1}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right)\]Similarly for chemical substance 2 in the liquid mixture, \[\mu_{2}\left(\operatorname{mix} ; \mathrm{id} ; \mathrm{x}_{2}\right)=\mu_{2}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{2}\right)\]Hence for an ideal binary liquid mixture, formed by mixing \(\mathrm{n}_{1}\) and \(\mathrm{n}_{2}\) moles respectively of chemical substance 1 and 2, the Gibbs energy of the mixture is given by equation (i). \[\begin{aligned} &\mathrm{G}(\operatorname{mix} ; \mathrm{id})= \\ &\quad \mathrm{n}_{1} \,\left[\mu_{1}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right)\right]+\mathrm{n}_{2} \,\left[\mu_{2}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{2}\right)\right] \end{aligned}\]Bearing in mind that Gibbs energies cannot be measured for either a pure liquid or a liquid mixture, it is useful to rephrase equation (i) in terms of the change in Gibbs energy that accompanies mixing to form an ideal binary liquid mixture. We envisage a situation where before mixing the molar Gibbs energy of the system defined as \(\mathrm{G}(\text{no}-\text{mix})\) is given by equation (j). \[\mathrm{G}(\mathrm{no}-\operatorname{mix})=\mathrm{n}_{1} \, \mu_{1}^{*}(\ell)+\mathrm{n}_{2} \, \mu_{2}^{*}(\ell)\]The change in Gibbs energy on forming the ideal binary liquid mixture \(\Delta_{\text{mix}} \mathrm{G}\) is given by the equation (k). \[\Delta_{\text {mix }} \mathrm{G}(\mathrm{id})=\mathrm{R} \, \mathrm{T} \,\left[\mathrm{n}_{1} \, \ln \left(\mathrm{x}_{1}\right)+\mathrm{n}_{2} \, \ln \left(\mathrm{x}_{2}\right)\right]\]We re-express \(\Delta_{\text{mix}} \mathrm{G}\) in terms of the Gibbs energy of mixing forming one mole of the ideal binary liquid mixture. \[\Delta_{\text {mix }} \mathrm{G}_{\mathrm{m}}=\Delta_{\text {mix }} \mathrm{G} /\left(\mathrm{n}_{1}+\mathrm{n}_{2}\right)\]Hence, \[\Delta_{\text {mix }} \mathrm{G}_{\mathrm{m}}(\mathrm{id})=\mathrm{R} \, \mathrm{T} \,\left[\mathrm{x}_{1} \, \ln \left(\mathrm{x}_{1}\right)+\mathrm{x}_{2} \, \ln \left(\mathrm{x}_{2}\right)\right]\]As required, \[\operatorname{limit}\left(\mathrm{x}_{1} \rightarrow 0\right) \Delta_{\operatorname{mix}} \mathrm{G}_{\mathrm{m}}(\mathrm{id})=0\]\[\operatorname{limit}\left(x_{2} \rightarrow 0\right) \Delta_{\text {mix }} G_{m}(\text { id })=0\]The dependence of \(\Delta_{\operatorname{mix}} \mathrm{G}_{\mathrm{m}}(\mathrm{id}) / \mathrm{R} \, \mathrm{T}\) on mole fraction composition is defined by equation (m). For a mixture where \(x_{1}= x_{2}=0.5\), \[\Delta_{\text {mix }} \mathrm{G}_{\mathrm{m}}(\mathrm{id}) / \mathrm{R} \, \mathrm{T}=2.0 \times 0.5 \times \ln (0.5)=-0.693\]In fact the molar Gibbs energy of mixing for an ideal binary liquid mixture is negative across the whole composition range.Equation (m) is the starting point of most equations used in the analysis of the properties of binary liquid mixtures. In fact most of the chemical literature concerned with liquid mixtures describes the properties of aqueous mixtures, at ambient pressure and \(298.15 \mathrm{~K}\). Nevertheless an extremely important subject concerns the properties of liquid mixtures at high pressures.Footnotes K. N. Marsh, Pure Appl. Chem., 1983, 55, 467. K. N. Marsh, Annu. Rep. Prog. Chem., Sect. C, Phys. Chem., 1994,91,209. J. S. Rowlinson and F. L. Swinton, Liquids and Liquid Mixtures, Butterworths, 3rd. edn., London, 1982. G. Scatchard, Chem.Rev., 1931,8,321;1949, 44,7. E. A. Guggenheim, Liquid Mixtures, Clarendon Press, Oxford, 1952. J. H. Hildebrand and R. L. Scott, Solubility of Non-Electrolytes, Reinhold, New York,3rd. edn.,1950. J. H. Hildebrand, J. M. Prausnitz and R. L. Scott, Regular and Related Solutions, van Nostrand Reinhold, New York,1970. A. G. Williamson, An Introduction to Non-Electrolyte Solutions, Oliver and Boyd, Edinburgh, 1967. Y. Koga, J.Phys.Chem.,1996,100,5172. L. S. Darken, Trans. Metallurg. Soc., A.M.E., 1967,239,80. A. D. Pelton and C. W Bale, Metallurg. Trans.,!986,17A,211. According to Phase Rule, P = 2 (for liquid and vapour), C = 2 then F = 2 + 2 - 2 = 2. Having defined temperature and pressure there remain no degrees of freedom - the system is completely specified. Other methods of defining the composition include the following.Mass % [or w%] Mass of component \(1=\mathrm{n}_{1} \, \mathrm{M}_{1}\) Mass of component \(2=\mathrm{n}_{2} \, \mathrm{M}_{2}\)Then \[\mathrm{w}_{1} \%=\frac{\mathrm{n}_{1} \, \mathrm{M}_{1} \, 100}{\left[\mathrm{n}_{1} \, \mathrm{M}_{1}+\mathrm{n}_{2} \, \mathrm{M}_{2}\right]} \quad \mathrm{w}_{2} \%=100-\mathrm{w}_{1} \%\]Volume % This definition often starts out by defining the volumes of the two liquid components used to prepare a given mixture at defined temperature and pressure. The definition does not normally refer to the volume of the actual mixture. The volume after mixing is often less than the sum of the component volumes before mixing. \[\begin{aligned} &\mathrm{V}_{1}^{*}(\ell)=\mathrm{n}_{1} \, \mathrm{M}_{1} / \rho_{1}^{*}(\ell) \quad \mathrm{V}_{2}^{*}(\ell)=\mathrm{n}_{2} \, \mathrm{M}_{2} / \rho_{2}^{*}(\ell) \\ &\mathrm{V}_{2} \%=\frac{\mathrm{V}_{2}^{*}(\ell)}{\mathrm{V}_{1}^{*}(\ell)+\mathrm{V}_{2}^{*}(\ell)} \end{aligned}\] G. Schneider, Pure Appl. Chem.,1983,55,479; 1976,47,277. G. Schneider, Ber. Bunsenges, Phys.Chem.,1972,76,325.This page titled 1.10.26: Gibbs Energies- Binary Liquid Mixtures is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.10.27: Gibbs Energies- Binary Liquid Mixtures- General Properties

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.10%3A_Gibbs_Energies/1.10.27%3A_Gibbs_Energies-_Binary_Liquid_Mixtures-_General_Properties

A given binary liquid mixture ( at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\), which is close to the standard pressure) mole fraction \(x_{1}\left(=1-x_{2}\right)\) can be characterised by the molar Gibbs energy of mixing, \(\Delta_{\text{mix}} \mathrm{~G}_{\mathrm{m}}\) and related molar enthalpic, volumetric and entropic properties. A corresponding set of properties exists for this mixture granted that the thermodynamic properties are ideal; e.g. \(\Delta_{\text {mix }} G_{m}(\mathrm{id})\). Hence we can define the corresponding excess molar property, \(\mathrm{G}_{\mathrm{m}}^{\mathrm{E}}=\Delta_{\operatorname{mix}} \mathrm{G}_{\mathrm{m}}-\Delta_{\operatorname{mix}} \mathrm{G}_{\mathrm{m}} \text { (id) }\). Interesting patterns emerge relating these properties and the corresponding partial molar properties; e.g. chemical potentials.At defined \(\mathrm{T}\) and \(\mathrm{p}\), the molar Gibbs energy for an ideal binary liquid mixture is given by equation (a). \[\mathrm{G}_{\mathrm{m}}(\mathrm{id})=\mathrm{x}_{1} \,\left[\mu_{1}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right)\right]+\mathrm{x}_{2} \,\left[\mu_{2}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{2}\right)\right]\]Here \(\mu_{1}^{*}(\ell)\) and \(\mu_{2}^{*}(\ell)\) are the chemical potentials of the two pure liquids 1 and 2 at the same \(\mathrm{T}\) and \(\mathrm{p}\). If the same amounts of the two liquid had not been allowed to mix, \[\mathrm{G}_{\mathrm{m}}(\text { no }-\operatorname{mix})=\mathrm{x}_{1} \, \mu_{1}^{*}(\ell)+\mathrm{x}_{2} \, \mu_{2}^{*}(\ell)\]By definition, \[\Delta_{\text {mix }} G_{m}(\mathrm{id})=x_{1} \, R \, T \, \ln \left(x_{1}\right)+x_{2} \, R \, T \, \ln \left(x_{2}\right)\]Recalling that \(x_{1}+x_{2}=1\) and \(dx_{1}=-d x_{2}\), \[\mathrm{d} \Delta_{\text {mix }} \mathrm{G}_{\mathrm{m}}(\mathrm{id}) / \mathrm{dx} \mathrm{x}_{2}=-\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right)-\mathrm{R} \, \mathrm{T}+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{2}\right)+\mathrm{R} \, \mathrm{T}\]We use equation (c) for \(\ln \left(x_{2}\right)\) and substitute in equation (d). \[\begin{aligned} &\mathrm{d} \Delta_{\text {mix }} \mathrm{G}_{\mathrm{m}}(\mathrm{id}) / \mathrm{dx} \mathrm{x}_{2}= \\ &-\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right)+\left(1 / \mathrm{x}_{2}\right) \,\left[\Delta_{\text {mix }} \mathrm{G}_{\mathrm{m}}(\mathrm{id})-\mathrm{x}_{1} \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right)\right] \end{aligned}\]\[\begin{aligned} &\mathrm{x}_{2} \, \mathrm{d} \Delta_{\operatorname{mix}} \mathrm{G}_{\mathrm{m}}(\mathrm{id}) / \mathrm{dx} \mathrm{x}_{2}= \\ &\left.-\mathrm{x}_{2} \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right)+\Delta_{\text {mix }} \mathrm{G}_{\mathrm{m}}(\mathrm{id})-\mathrm{x}_{1} \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right)\right] \end{aligned}\]Hence, \[\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right)=\Delta_{\mathrm{mix}} \mathrm{G}_{\mathrm{m}}(\mathrm{id})-\mathrm{x}_{2} \, \mathrm{d} \Delta_{\mathrm{mix}} \mathrm{G}_{\mathrm{m}}(\mathrm{id}) / \mathrm{dx} \mathrm{x}_{2}\]At all mole fractions, \(\Delta_{\operatorname{mix}} \mathrm{G}_{\mathrm{m}}(\mathrm{id})\) is negative, the plot of \(\Delta_{\operatorname{mix}} \mathrm{G}_{\mathrm{m}}(\mathrm{id})\) against \(x_{1}\) being symmetric about ‘\(x_{1}=0.5\)’. At the extreme, where \(\mathrm{d} \Delta_{\operatorname{mix}} \mathrm{G}_{\mathrm{m}}(\mathrm{id}) / \mathrm{dx} \mathrm{x}_{2}\) is zero; equation (c) shows that \(\Delta_{\operatorname{mix}} \mathrm{G}_{\mathrm{m}}(\mathrm{id}) \text { equals } [\mathrm{R} \, \mathrm{T} \, \ln (0.5)]\). At \(298 \mathrm{~K}\), the latter quantity equals \(– 1.72 \mathrm{~kJ mol}^{-1}\).Using equation (c), the ratio \(\Delta_{\operatorname{mix}} \mathrm{G}_{\mathrm{m}}(\mathrm{id}) / \mathrm{T}\) is given by equation (h) \[\Delta_{\operatorname{mix}} \mathrm{G}_{\mathrm{m}}(\mathrm{id}) / \mathrm{T}=\mathrm{R} \,\left[\mathrm{x}_{1} \, \ln \left(\mathrm{x}_{1}\right)+\mathrm{x}_{2} \, \ln \left(\mathrm{x}_{2}\right)\right]\]Hence using the Gibbs-Helmholtz it follows that the molar enthalpy of mixing is zero at all mole fractions. Similarly the molar isobaric heat capacity of mixing is zero at all mole fractions. In terms of entropies, \[\Delta_{\operatorname{mix}} \mathrm{S}_{\mathrm{m}}(\mathrm{id})=-\mathrm{R} \,\left[\mathrm{x}_{1} \, \ln \left(\mathrm{x}_{1}\right)+\mathrm{x}_{2} \, \ln \left(\mathrm{x}_{2}\right)\right]\]But at all (real) mole fractions (other than \(x_{1} =1 \text { and } x_{2} = 1\)) \(\left[x_{1} \, \ln \left(x_{1}\right)+x_{2} \, \ln \left(x_{2}\right)\right]<0\). Hence \(\Delta_{\operatorname{mix}} \mathrm{S}_{\mathrm{m}}(\mathrm{id})>0\). In the event that mixing of two liquids at temperature \(\mathrm{T}\) to produce a binary liquid mixture having ideal properties, then \(\Delta_{\text {mix }} \mathrm{G}_{\mathrm{m}}(\mathrm{id})\) is negative because \(\Delta_{\operatorname{mix}} \mathrm{S}_{\mathrm{m}}(\mathrm{id})\) is positive, \(\Delta_{\text {mix }} \mathrm{H}_{\mathrm{m}}(\mathrm{id})\) being zero. In other words we have a reference against which to examine the properties of real liquid mixtures.The molar Gibbs energy of a real binary liquid mixture is related to the mole fraction composition using equation (j). \[\begin{aligned} &\mathrm{G}_{\mathrm{m}}= \\ &\mathrm{x}_{1} \,\left[\mu_{1}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right)\right]+\mathrm{x}_{2} \,\left[\mu_{2}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{2} \, \mathrm{f}_{2}\right)\right] \end{aligned}\]For the corresponding mixture having ideal thermodynamic properties, \[\begin{aligned} &\mathrm{G}_{\mathrm{m}}(\mathrm{id})= \\ &\mathrm{x}_{1} \,\left[\mu_{1}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right)\right]+\mathrm{x}_{2} \,\left[\mu_{2}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{2}\right)\right] \end{aligned}\]The excess molar Gibbs energy \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}}\) is therefore given by equation (l). \[\mathrm{G}_{\mathrm{m}}^{\mathrm{E}}=\mathrm{R} \, \mathrm{T} \,\left[\mathrm{x}_{1} \, \ln \left(\mathrm{f}_{1}\right)+\mathrm{x}_{2} \, \ln \left(\mathrm{f}_{2}\right)\right]\]We differentiate equation (l) with respect to mole fraction, \(x_{1}\). \[\frac{1}{R \, T} \, \frac{\mathrm{dG}_{\mathrm{m}}^{\mathrm{E}}}{\mathrm{dx}_{1}}=\ln \left(\mathrm{f}_{1}\right)+\mathrm{x}_{1} \, \frac{\mathrm{d} \ln \left(\mathrm{f}_{1}\right)}{d \mathrm{x}_{1}}-\ln \left(\mathrm{f}_{2}\right)+\mathrm{x}_{2} \, \frac{\mathrm{d} \ln \left(\mathrm{f}_{2}\right)}{\mathrm{dx}_{1}}\]But from the Gibbs-Duhem equation at fixed \(\mathrm{T}\) and \(\mathrm{p}\), \[x_{1} \, \frac{d \ln \left(\mu_{1}\right)}{d x_{1}}+x_{2} \, \frac{d \ln \left(\mu_{2}\right)}{d x_{1}}=0\]Hence, \[x_{1} \, \frac{d \ln \left(f_{1}\right)}{d x_{1}}+x_{2} \, \frac{d \ln \left(f_{2}\right)}{d x_{1}}=0\]From equation (k), \[\ln \left(f_{2}\right)=\ln \left(f_{1}\right)-\frac{1}{R \, T} \, \frac{d G_{m}^{E}}{d x_{1}}\]Hence using equation (j), \[\frac{\mathrm{G}_{\mathrm{m}}^{\mathrm{E}}}{\mathrm{R} \, \mathrm{T}}=\ln \left(\mathrm{f}_{1}\right)-\frac{\mathrm{X}_{2}}{\mathrm{R} \, \mathrm{T}} \, \frac{\mathrm{dG}_{\mathrm{m}}^{\mathrm{E}}}{\mathrm{dx}}\]Or \[\ln \left(\mathrm{f}_{1}\right)=\frac{\mathrm{G}_{\mathrm{m}}^{\mathrm{E}}}{\mathrm{R} \, \mathrm{T}}=+\frac{\mathrm{x}_{2}}{\mathrm{R} \, \mathrm{T}} \, \frac{\mathrm{dG}_{\mathrm{m}}^{\mathrm{E}}}{\mathrm{dx}_{1}}\]Equation (o) has an interesting feature. At the mole fraction composition where \(\frac{\mathrm{dG}_{\mathrm{m}}^{\mathrm{E}}}{\mathrm{dx}_{1}}\) is zero, \(\frac{\mathrm{G}_{\mathrm{m}}^{\mathrm{E}}}{\mathrm{R} \, \mathrm{T}}\) offers a direct measure of \(\ln \left(f_{1}\right)\) at that mole fraction.Perhaps the most direct measure of the extent to which the thermodynamic properties of a given binary liquid mixture differs from that defined as ideal is afforded by the molar enthalpy of mixing which is therefore the excess molar enthalpy of mixing \({\mathrm{H}_{\mathrm{m}}}^{\mathrm{E}\). Thus, \[\mathrm{G}_{\mathrm{m}}^{\mathrm{E}}=\mathrm{H}_{\mathrm{m}}^{\mathrm{E}}-\mathrm{T} \, \mathrm{S}_{\mathrm{m}}^{\mathrm{E}}\]where \[\mathrm{C}_{\mathrm{pmn}}^{\mathrm{E}}=\left(\partial \mathrm{H}_{\mathrm{m}}^{\mathrm{E}} / \partial \mathrm{T}\right)_{\mathrm{p}}\]In many reports the properties of a given binary liquid mixture at \(298.15 \mathrm{~K}\) and ambient pressure are summarised in a plot showing the three properties \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}\), \({\mathrm{H}_{\mathrm{m}}}^{\mathrm{E}\) and \(\mathrm{T} \, \mathrm{S}_{\mathrm{m}}^{\mathrm{E}}\) as a function of the mole fraction composition. An enormous amount of information can be summarised in such plots. In order to understand the various patterns which emerge two liquid mixtures are often taken as models against which to compare the properties of other liquid mixtures.These two liquid mixtures provide a basis for the examination of the properties of binary aqueous mixtures for which there is an immense published information. In most cases \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}\) is either positive or negative across the mole fraction range for a given liquid mixture although plots of \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}\) and \(\mathrm{T} \, \mathrm{S}_{\mathrm{m}}^{\mathrm{E}}\) against mole fraction composition are often S-shaped, nevertheless operating to produce a smooth change in \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}\). Plots of \({\mathrm{C}_{\mathrm{pm}}}^{\mathrm{E}}\) and excess molar volume of mixing \({\mathrm{V}_{\mathrm{m}}}^{\mathrm{E}}\) against mole fraction are often quite complicated.In a few cases the plot of \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}\) against composition is S-shaped. One such system is the mixture, water + 1,1,1,3,3,3-hexafluoropropanol. Explanations of such complex patterns are not straightforward. However we might for an alcohol + water mixture envisage a switch fromDefinition of excess thermodynamic properties is not straightforward in all instances; e.g. isentropic compressibilities.Footnotes M. J. Blandamer, J. Burgess, A. Cooney, H. J. Cowles, I. M. Horn, K. J. Martin, K. W. Morcom and P. Warwick, J. Chem. Soc. Faraday Trans.,1990,86,2209` G. Douheret, C . Moreau and A. Viallard, Fluid Phase Equilibrium, 1985,22 ,277; 289.This page titled 1.10.27: Gibbs Energies- Binary Liquid Mixtures- General Properties is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.10.28: Gibbs Energies- Liquid Mixtures- Thermodynamic Patterns

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.10%3A_Gibbs_Energies/1.10.28%3A_Gibbs_Energies-_Liquid_Mixtures-_Thermodynamic_Patterns

A given binary liquid mixture (at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\), which is close to the standard pressure) mole fraction \(x_{1} \left(= 1 - x_{2}\right)\) is characterised by the molar Gibbs energy of mixing, \(\Delta_{\text{mix}}\mathrm{G}_{\mathrm{m}}\) and related molar enthalpic, volumetric and entropic properties. A corresponding set of properties for this mixture exist granted that the thermodynamic properties are ideal; e.g. \(\Delta_{\text{mix}}\mathrm{G}_{\mathrm{m}}(\text{id})\). As a consequence we define the corresponding excess molar property, \(\mathrm{G}_{\mathrm{m}}^{\mathrm{E}}=\Delta_{\operatorname{mix}} \mathrm{G}_{\mathrm{m}}-\Delta_{\operatorname{mix}} \mathrm{G}_{\mathrm{m}} \text { (id) }\). Interesting patterns emerge relating these properties and the corresponding partial molar properties; e.g. chemical potentials.For chemical substance 1, (which we will conventionally take as water) the chemical potential in the liquid mixture is related to the chemical potential of the pure liquid at the same \(\mathrm{T}\) and using equation (a) where \(x_{1}\) is the mole fraction and \(\mathrm{f}_{1}\) is the rational activity coefficient. \[\mu_{1}(\operatorname{mix})=\mu_{1}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right)\]where \[\operatorname{limit}\left(\mathrm{x}_{1} \rightarrow 1\right) \mathrm{f}_{1}=1.0 \text { at all T and } p\]Similarly for component 2, \[\mu_{2}(\operatorname{mix})=\mu_{2}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{2} \, \mathrm{f}_{2}\right)\]where \[\operatorname{limit}\left(\mathrm{x}_{2} \rightarrow 1\right) \mathrm{f}_{2}=1.0 \text { at all } \mathrm{T} \text { and } \mathrm{p}\]Here \(\mu_{1}^{*}(\ell)\) and \(\mu_{2}^{*}(\ell)\) are the chemical potentials of the two pure liquids at the same \(\mathrm{T}\) and \(\mathrm{p}\); \(\mathrm{f}_{1}\) and \(\mathrm{f}_{2}\) are rational activity coefficients. These (rational) activity coefficients approach unity at opposite ends of the mixture composition range. For the aqueous component, as \(x_{1}\) approaches 1, so \(\mathrm{f}_{1}\) approaches unity (at the same \(\mathrm{T}\) and \(\mathrm{p}\)). At the other end of the scale, as \(x_{1}\) approaches zero so the chemical potential of water in the binary system approaches ‘minus infinity’. If across the whole composition range (at all \(\mathrm{T}\) and \(\mathrm{p}\)), both \(\mathrm{f}_{1}\) and \(\mathrm{f}_{2}\) are unity, the thermodynamic properties of the liquid mixture are ideal.A given liquid mixture (at fixed \(\mathrm{T}\) and \(\mathrm{p}\)) is formed by mixing \(\mathrm{n}_{1}\) moles of liquid 1 and \(\mathrm{n}_{2}\) moles of liquid 2. Before mixing the total Gibbs energy of the system, defined as G(no-mix) is given by the following equation where \(\mu_{1}^{*}(\ell)\) and \(\mu_{2}^{*}(\ell)\) are the chemical potentials of the two pure liquids at the same \(\mathrm{T}\) and \(\mathrm{p}\). Then, \[\mathrm{G}(\mathrm{no}-\operatorname{mix})=\mathrm{n}_{1} \, \mu_{1}^{*}(\ell)+\mathrm{n}_{2} \, \mu_{2}^{*}(\ell)\]After mixing, the Gibbs energy of the mixture is given by equation (f). \[\begin{aligned} &\mathrm{G}(\operatorname{mix})= \\ &\quad \mathrm{n}_{1} \,\left[\mu_{1}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right)\right]+\mathrm{n}_{2} \,\left[\mu_{2}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{2} \, \mathrm{f}_{2}\right)\right] \end{aligned}\]By definition, \[\Delta_{\operatorname{mix}} \mathrm{G}=\mathrm{G}(\operatorname{mix})-\mathrm{G}(\mathrm{no}-\mathrm{mix})\]Hence the Gibbs energy of mixing , \[\Delta_{\text {mix }} \mathrm{G}=\mathrm{R} \, \mathrm{T} \,\left[\mathrm{n}_{1} \, \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right)\right]+\left[\mathrm{n}_{2} \, \ln \left(\mathrm{x}_{2} \, \mathrm{f}_{2}\right)\right]\]We re-express \(\Delta_{\text{mix}}\mathrm{G}\) in terms of the Gibbs energy of mixing for one mole of liquid mixture. Thus \[\Delta_{\text {mix }} \mathrm{G}_{\mathrm{m}}=\Delta_{\text {mix }} \mathrm{G} /\left(\mathrm{n}_{1}+\mathrm{n}_{2}\right)\]Hence, \[\Delta_{\operatorname{mix}} \mathrm{G}_{\mathrm{m}}=\mathrm{R} \, \mathrm{T} \,\left[\mathrm{x}_{1} \, \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right)+\mathrm{x}_{2} \, \ln \left(\mathrm{x}_{2} \, \mathrm{f}_{2}\right)\right]\]Or, \[\Delta_{\text {mix }} G_{m}=R \, T \,\left[x_{1} \, \ln \left(x_{1}\right)+x_{1} \, \ln \left(f_{1}\right)+x_{2} \, \ln \left(x_{2}\right)+x_{2} \, \ln \left(f_{2}\right)\right]]By definition, \[\Delta_{\text {mix }} \mathrm{G}_{\mathrm{m}}(\mathrm{id})=\mathrm{x}_{1} \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right)+\mathrm{x}_{2} \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{2}\right)\]Recalling that \(x_{1} + x_{2} = 1\) and \(dx_{1} = -dx_{2}\), \[\mathrm{d} \Delta_{\operatorname{mix}} \mathrm{G}_{\mathrm{m}}(\mathrm{id}) / \mathrm{dx} \mathrm{x}_{2}=-\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right)-\mathrm{R} \, \mathrm{T}+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{2}\right)+\mathrm{R} \, \mathrm{T}\]We use equation (l) for \(\ln \left(\mathrm{x}_{2}\right)\) and substitute in equation (m). \[\begin{aligned} &\mathrm{d} \Delta_{\text {mix }} \mathrm{G}_{\mathrm{m}}(\mathrm{id}) / \mathrm{dx} \mathrm{x}_{2}= \\ &-\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right)+\left(1 / \mathrm{x}_{2}\right) \,\left[\Delta_{\text {mix }} \mathrm{G}_{\mathrm{m}}(\mathrm{id})-\mathrm{x}_{1} \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right)\right] \end{aligned}\]or, \[\begin{aligned} &x_{2} \, d \Delta_{\text {mix }} G_{m}(\mathrm{id}) / d x_{2}= \\ &\left.-x_{2} \, R \, T \, \ln \left(x_{1}\right)+\Delta_{\text {mix }} G_{m} \text { (id) }-x_{1} \, R \, T \, \ln \left(x_{1}\right)\right] \end{aligned}\]Hence, \[\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right)=\Delta_{\operatorname{mix}} \mathrm{G}_{\mathrm{m}}(\mathrm{id})-\mathrm{x}_{2} \, \mathrm{d} \Delta_{\operatorname{mix}} \mathrm{G}_{\mathrm{m}}(\mathrm{id}) / \mathrm{dx} \mathrm{x}_{2}\]At all mole fractions, \(\Delta_{\operatorname{mix}} G_{m}(\text { id })\) is negative; the plot of \(\Delta_{\text {mix }} G_{m} \text { (id) }\) against \(x_{1}\) is symmetric about ‘\(x_{1} = 0.5\)’. At the extremum, where \(\mathrm{d} \Delta_{\operatorname{mix}} \mathrm{G}_{\mathrm{m}}(\mathrm{id}) / \mathrm{dx} \mathrm{x}_{2}\) is zero, equation (l) shows that \(\Delta_{\operatorname{mix}} \mathrm{G}_{\mathrm{m}}(\mathrm{id})\) equals \(\mathrm{R} \, \mathrm{T} \, \ln (0.5)\).We define an excess chemical potential for each of the two components of a binary liquid mixture. For liquid component 1, \[\mu_{1}^{\mathrm{E}}(\mathrm{mix})=\mu_{1}(\mathrm{mix})-\mu_{1}^{*}(\ell)-\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right)=\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{f}_{1}\right)\]Similarly for liquid component 2, \[\mu_{2}^{\mathrm{E}}(\mathrm{mix})=\mu_{2}(\mathrm{mix})-\mu_{2}^{*}(\ell)-\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{2}\right)=\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{f}_{2}\right)\]Rational activity coefficients \(\mathrm{f}_{1}\) and \(\mathrm{f}_{2}\) depend on mixture composition, \(\mathrm{T}\) and \(\mathrm{p}\).In summary ‘excess’ means excess over ideal. Excess properties provide a mutually consistent set of perspectives of a given liquid mixture. We define a reference state for binary liquid mixtures so that the thermodynamic properties of a given liquid mixture can be correlated with a common model.A convenient approach defines an excess molar Gibbs energy of mixing. \[\mathrm{G}_{\mathrm{m}}^{\mathrm{E}}=\Delta_{\operatorname{mix}} \mathrm{G}_{\mathrm{m}}-\Delta_{\operatorname{mix}} \mathrm{G}_{\mathrm{m}} \text { (id) }\]Then \[\mathrm{G}_{\mathrm{m}}^{\mathrm{E}}=\mathrm{R} \, \mathrm{T} \,\left[\mathrm{x}_{1} \, \ln \left(\mathrm{f}_{1}\right)+\mathrm{x}_{2} \, \ln \left(\mathrm{f}_{2}\right)\right]\]Where \[\operatorname{limit}\left(x_{1} \rightarrow 1\right) \mathrm{G}_{\mathrm{m}}^{\mathrm{E}}=0\]And \[\operatorname{limit}\left(\mathrm{x}_{2} \rightarrow 1\right) \mathrm{G}_{\mathrm{m}}^{\mathrm{E}}=0\]Other than the latter two conditions we cannot predict the dependence of \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}}\) on a mixture composition for m a given real mixture.In general terms excess molar properties of binary aqueous mixtures are expressed in terms of the following general equation with respect to the thermodynamic variable \(\mathrm{Q} (= \mathrm{~G}, \mathrm{~V}, \mathrm{~H} \text { and } \mathrm{S})\). \[\mathrm{Q}_{\mathrm{m}}^{\mathrm{E}}=\mathrm{Q}_{\mathrm{m}}(\text { mix })-\mathrm{Q}_{\mathrm{m}}(\text { mix } ; \text { ideal })\]Returning to the Gibbs energies, we differentiate equation (t) with respect to mole fraction \(x_{1}\) at fixed \(\mathrm{T}\) and \(\mathrm{p}\). \[\frac{1}{R \, T} \, \frac{d G_{m}^{E}}{d x_{1}}=\ln \left(f_{1}\right)+x_{1} \, \frac{d \ln \left(f_{1}\right)}{d x_{1}}-\ln \left(f_{2}\right)+x_{2} \, \frac{d \ln \left(f_{2}\right)}{d x_{1}}\]According to the Gibbs-Duhem equation, at fixed \(\mathrm{T}\) and \(\mathrm{p}\), \[\mathrm{x}_{1} \, \frac{\mathrm{d} \ln \left(\mu_{1}\right)}{\mathrm{dx}_{1}}+\mathrm{x}_{2} \, \frac{\mathrm{d} \ln \left(\mu_{2}\right)}{\mathrm{dx}}=0\]Hence, \[x_{1} \, \frac{d \ln \left(f_{1}\right)}{d x_{1}}+x_{2} \, \frac{d \ln \left(f_{2}\right)}{d x_{1}}=0\]From equation (x), \[\ln \left(f_{2}\right)=\ln \left(f_{1}\right)-\frac{1}{R \, T} \, \frac{d G_{m}^{E}}{d x_{1}}\]Hence using equation (t), \[\frac{\mathrm{G}_{\mathrm{m}}^{\mathrm{E}}}{\mathrm{R} \, \mathrm{T}}=\ln \left(\mathrm{f}_{1}\right)-\frac{\mathrm{x}_{2}}{\mathrm{R} \, \mathrm{T}} \, \frac{\mathrm{dG}_{\mathrm{m}}^{\mathrm{E}}}{\mathrm{dx}_{1}}\]Or \[\ln \left(f_{1}\right)=\frac{G_{m}^{E}}{R \, T}+\frac{x_{2}}{R \, T} \, \frac{d G_{m}^{E}}{d x_{1}}\]Equation (zc) has an interesting feature. At the mole fraction composition where \(\frac{\mathrm{dG}_{\mathrm{m}}^{\mathrm{E}}}{\mathrm{dx}_{1}}\) is zero, \(\frac{\mathrm{G}_{\mathrm{m}}^{\mathrm{E}}}{\mathrm{R} \, \mathrm{T}}\) offers a direct measure of \(\ln \left(\mathrm{f}_{1}\right)\) at that mole fraction. In some systems the plot of \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}}\) against composition is S-shaped so we have this information at two mole fractions.Turning to volumetric properties, the molar volume of an ideal binary liquid mixture is given by equation (zd). \[\mathrm{V}_{\mathrm{m}}=\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}(\ell)\]Hence, \[\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}=\mathrm{x}_{1} \,\left[\mathrm{V}_{1}(\mathrm{mix})-\mathrm{V}_{1}^{*}(\ell)\right]+\mathrm{x}_{2} \,\left[\mathrm{V}_{2}(\mathrm{mix})-\mathrm{V}_{2}^{*}(\ell)\right]\]\[\begin{aligned} &\frac{\mathrm{dV}_{\mathrm{m}}^{\mathrm{E}}}{\mathrm{dx} \mathrm{x}_{1}}= \\ &{\left[\mathrm{V}_{1}(\mathrm{mix})-\mathrm{V}_{1}^{*}(\ell)\right]+\mathrm{x}_{1} \, \frac{\mathrm{dV}_{1}}{\mathrm{dx}_{1}}-\left[\mathrm{V}_{2}(\mathrm{mix})-\mathrm{V}_{2}^{*}(\ell)\right]+\mathrm{x}_{2} \, \frac{\mathrm{dV}_{2}}{\mathrm{dx}_{1}}} \end{aligned}\]Using the Gibbs-Duhem equation, \[\frac{\mathrm{dV}_{\mathrm{m}}^{\mathrm{E}}}{\mathrm{dx}_{1}}=\left[\mathrm{V}_{1}(\operatorname{mix})-\mathrm{V}_{1}^{*}(\ell)\right]-\left[\mathrm{V}_{2}(\operatorname{mix})-\mathrm{V}_{2}^{*}(\ell)\right]\]or, \[\left[\mathrm{V}_{1}(\mathrm{mix})-\mathrm{V}_{1}^{*}(\ell)\right]=\frac{\mathrm{dV}_{\mathrm{m}}^{\mathrm{E}}}{\mathrm{dx} \mathrm{x}_{1}}+\left[\mathrm{V}_{2}(\mathrm{mix})-\mathrm{V}_{2}^{*}(\ell)\right]\]\[\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}=\mathrm{x}_{1} \, \frac{\mathrm{dV}_{\mathrm{m}}^{\mathrm{E}}}{\mathrm{dx} \mathrm{x}_{1}}+\mathrm{x}_{1} \,\left[\mathrm{V}_{2}(\mathrm{mix})-\mathrm{V}_{2}^{*}(\ell)\right]+\mathrm{x}_{2} \,\left[\mathrm{V}_{2}(\mathrm{mix})-\mathrm{V}_{2}^{*}(\ell)\right]\]Hence, \[\left[\mathrm{V}_{2}(\mathrm{mix})-\mathrm{V}_{2}^{*}(\ell)\right]=\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}-\mathrm{x}_{1} \, \frac{\mathrm{dV}_{\mathrm{m}}^{\mathrm{E}}}{\mathrm{dx} \mathrm{x}_{1}}\]At the composition where \(\mathrm{dV}_{\mathrm{m}}^{\mathrm{E}} / \mathrm{dx}_{1}\) is zero, \(\left[\mathrm{V}_{2}(\operatorname{mix})-\mathrm{V}_{2}^{*}(\ell)\right]=\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}\).The analysis set out above is repeated for excess molar enthalpies and excess molar isobaric heat capacities for binary liquid mixtures.Interesting proposals have been made in which the dependence of excess thermodynamic properties on mixture composition are examined in different composition domains; e.g. the four segment model.Footnotes R. Schumann, Metallurg. Trans.,B,1985,16B,807. M. I. Davis, M. C. Molina and G. Douheret, Thermochim. Acta, 1988,131,153. M. I. Davis, Thermochim. Acta 1984,77,421; 1985,90, 313; 1987, 120,299; and references therein. G. Douheret, A. H. Roux, M. I .Davis, M. E. Hernandez, H. Hoiland and E. Hogseth, J. Solution Chem.,1993,22,1041. G. Douheret, C. Moreau and A. Viallard, Fluid Phase Equilib.,1986,26,221. G. Douheret, A. Pal and M. I. Davis, J.Chem.Thermodyn., 1990,22,99. H. Hoiland, O. Anowi and M. I. Davis, J. Chem. Thermodyn., 1991,23,569. G. Douheret, A. Pal, H. Hoiland, O. Anowi and M. I. Davis, J. Chem. Thermodyn., 1991,23,569. G. Douheret, J.C.R. Reis, M. I. Davis, I. J. Fjellanger and H. Hoiland, Phys.Chem. Chem.Phys.,2004,6,784.This page titled 1.10.28: Gibbs Energies- Liquid Mixtures- Thermodynamic Patterns is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.10.29: Gibbs Energies- Binary Liquid Mixtures- Excess Thermodynamic Variables

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.10%3A_Gibbs_Energies/1.10.29%3A_Gibbs_Energies-_Binary_Liquid_Mixtures-_Excess_Thermodynamic_Variables

The properties of binary mixtures are complicated. As a point of reference the excess molar properties of two non-aqueous binary liquid mixtures are often discussed. The mixtures are (A) trichloromethane + propanone, and (B) tetrachloromethane + methanol.A snapshot of the thermodynamic properties of a given binary mixture (at fixed \(\mathrm{T}\) and \(\mathrm{p}\)) is provided by combined plots of \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}}\), \({\mathrm{H}_{\mathrm{m}}}^{\mathrm{E}}\) and \(\mathrm{T} \, \mathrm{S}_{\mathrm{m}}^{\mathrm{E}}\) as a function of mixture composition. In effect the starting point is the Gibbs energy leading to first, second, third and fourth derivatives. At this stage we make some sweeping (and dangerous) generalizations. For most binary aqueous mixtures, \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}}\) is a smooth function of water m mole fraction \(x_{1}\), with an extremum near \(x_{1} = 0.5\). Rarely for a given mixture does the sign of \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}}\) change across the mole faction range m although this feature is not unknown; e.g. water + 1,1,1,3,3,3- hexafluropropan-2-ol mixtures at \(298.15 \mathrm{~K}\) but contrast water + 2,2,2- trifluorethanol mixtures where at \(298.2 \mathrm{~K} {\mathrm{~G}_{\mathrm{m}}}^{\mathrm{E}}\) is positive across the m whole mole fraction range. However a change in sign of \({\mathrm{H}_{\mathrm{m}}}^{\mathrm{E}}\) and \(\mathrm{T} \, {\mathrm{S}_{\mathrm{m}}}^{\mathrm{E}}\) and \({\mathrm{V}_{\mathrm{m}}}^{\mathrm{E}}\) with change in mole fraction composition is quite common.For mixture A, \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}}\) is negative indicating that \(\Delta_{\text{mix}}\mathrm{G}_{\mathrm{m}}\) is more negative than in the case of an ideal binary liquid mixture. In the case of Mixture A, the negative \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}}\) is linked with a marked exothermic mixing; \({\mathrm{H}_{\mathrm{m}}}^{\mathrm{E}} < 0\). The latter is attributed to strong inter-component hydrogen bonding.For both mixtures A and B the signs of \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}}\) and \({\mathrm{H}_{\mathrm{m}}}^{\mathrm{E}}\) are the same. This feature is characteristic of binary non-aqueous liquid mixtures where in most instances, \(\left|\mathrm{H}_{\mathrm{m}}^{\mathrm{E}}\right|>\left|\mathrm{T} \, \mathrm{S}_{\mathrm{mm}}^{\mathrm{E}}\right|\). \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}}\) and \({\mathrm{H}_{\mathrm{m}}}^{\mathrm{E}}\) are both positive for mixture B. Here the pattern is understood in terms of disruption of methanol-methanol hydrogen bonding ( i.e. intracomponent interaction) by the second component. Again we note that a positive \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}}\) means that the tendency for \(\Delta_{\text{mix}}\mathrm{G}_{\mathrm{m}}\) to be negative (cf. ideal mixtures) is opposed.Through a series of mixtures with increasing \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}}\), a stage is reached where the magnitude of \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}}\) is such that phase separation occurs. For m many binary non-aqueous binary liquid mixtures the phase diagram for liquid miscibility has an upper critical solution temperature UCST. In other words only at high temperatures is the liquid mixture miscible in all proportions.Often binary aqueous mixtures are used as solvents for the following reason. The solubilities of salts in water(l) are high because ‘water is a polar solvent’ but the solubilities of apolar solutes are low. However the solubilities of apolar substances in organic solvents (e.g. ethanol) are high. If the chemical reaction being studied involves both polar and apolar solutes, judicious choice of the composition of a binary aqueous mixture leads to a solvent where the solubilities of both polar and apolar solutes are high. Nevertheless the task of accounting for the properties of binary aqueous mixtures is awesome. For this reason the classification introduced by Franks has considerable merit. A distinction is drawn between Typically Aqueous and Typically Non-Aqueous Binary Aqueous Mixtures, based on the the thermodynamic excess functions, \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}}\), \({\mathrm{H}_{\mathrm{m}}}^{\mathrm{E}}\) and \(\mathrm{T} \, {\mathrm{S}_{\mathrm{m}}}^{\mathrm{E}}\).Davis has explored how the properties of many binary aqueous mixtures can be subdivided on the basis of the ranges of mole fraction compositions.Footnotes J. S. Rowlinson and F. L. Swinton, Liquids and Liquid Mixtures, Butterworths, London, 3rd. edn., 1982. K. N. Marsh, Annu. Rep. Prog. Chem., Sect. C, Phys. Chem.,1984, 81 , 209-245; Pure Appl. Chem.,1983,55,467. G. Scatchard. Chem. Rev.,1931,8,321. G. Scatchard, Chem. Rev.,1940, 44,7. With respect to compressibilities; G. Douheret, C. Moreau and A. Viallard, Fluid Phase Equilib., 1985, 22,289. Y. Koga, J.Phys.Chem.,1996,100,5172; Y. Koga, K. Nishikawa and P. Westh, J. Phys. Chem.A,2004,108,3873. R. Jadot and M. Fraiha, J. Chem. Eng. Data, 1988, 33,237. F. Franks, in Hydrogen –Bonded Solvent Systems, ed. A. K. Covington and P. Jones, Taylor and Francis, London, 1968, pp.31-47.This page titled 1.10.29: Gibbs Energies- Binary Liquid Mixtures- Excess Thermodynamic Variables is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.10.30: Gibbs Energies- General Equations

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.10%3A_Gibbs_Energies/1.10.30%3A_Gibbs_Energies-_General_Equations

For binary liquid mixtures at fixed \(\mathrm{T}\) and \(\mathrm{p}\), an important task is to fit the dependence of \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}}\) on \(x_{2}\) to an equation in order to calculate the derivative \(\mathrm{dG}_{\mathrm{m}}^{\mathrm{E}} / \mathrm{dx}_{2}\) at required mole fractions. The Guggenheim - Scatchard (commonly called the Redlich - Kister ) equation is one such equation. This equation has the following general form. \[\mathrm{X}_{\mathrm{m}}^{\mathrm{E}}=\mathrm{x}_{2} \,\left(1-\mathrm{x}_{2}\right) \, \sum_{\mathrm{i}=1}^{\mathrm{i}=\mathrm{k}} \mathrm{A}_{\mathrm{i}} \,\left(1-2 \, \mathrm{x}_{2}\right)^{\mathrm{i}-1}\]\(\mathrm{A}_{\mathrm{i}}\) are coefficients obtained from a least squares analysis of the dependence of \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}}\) on \(x_{2}\).The equation clearly satisfies the condition that \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}}\) is zero at \(x_{2} = 0\) and at \(x_{2} = 1\). In fact the first term in the \(\mathrm{G} - \mathrm{S}\) equation has the following form. \[X_{m}^{E}=X_{2} \,\left(1-X_{2}\right) \, A_{1}\]According to equation (b) \({\mathrm{X}_{\mathrm{m}}}^{\mathrm{E}}\) is an extremum at \(x_{2} = 0.5\), the plot being symmetric about the line from \({\mathrm{X}_{\mathrm{m}}}^{\mathrm{E}}\) to ‘\(x_{2} = 0.5\)’. In fact for most systems the \(\mathrm{A}_{1}\) term is dominant. For the derivative \(\mathrm{dG}_{\mathrm{m}}^{\mathrm{E}} / \mathrm{dx} \mathrm{x}_{2}\), we write equation (a) in the following general form. \[\mathrm{X}_{\mathrm{m}}^{\mathrm{E}}=\left(\mathrm{x}_{2}-\mathrm{x}_{2}^{2}\right) \, \mathrm{Q}\]Then \[\mathrm{dX}_{\mathrm{m}}^{\mathrm{E}} / \mathrm{dx} \mathrm{x}_{2}=\mathrm{x}_{2} \,\left(1-\mathrm{x}_{2}\right) \, \mathrm{dQ} / \mathrm{dx} \mathrm{x}_{2}+\left(1-2 \, \mathrm{x}_{2}\right) \, \mathrm{Q}\]where \[\mathrm{dQ} / \mathrm{dx}_{2}=-2 \, \sum_{\mathrm{i}=2}^{\mathrm{i}=\mathrm{k}}(\mathrm{i}-1) \, \mathrm{A}_{\mathrm{i}} \,\left(1-2 \, \mathrm{x}_{2}\right)^{\mathrm{i}-2}\]Equation (a) fits the dependence with a set of contributing curves which all pass through points, \({\mathrm{X}_{\mathrm{m}}}^{\mathrm{E}}=0\) at \(x_{1} = 0\) and \(x_{1} =1\). The usual procedure involves fitting the recorded dependence using increasing number of terms in the series, testing the statistical significance of including a further term. Although equation (a) has been applied to many systems and although the equation is easy to incorporate into computer programs using packaged least square and graphical routines, the equation suffers from the following disadvantage. As one incorporates a further term in the series, (e.g. \(\mathrm{A}_{j}\)) estimates of all the previously calculated parameters (i.e. \(\mathrm{A}_{2}, \mathrm{~A}_{3 \ldots} \ldots \mathrm{A}_{\mathrm{j}-1}\)) change. For this reason orthogonal polynomials have been increasingly favoured especially where the appropriate computer software is available. The only slight reservation is that derivation of explicit equations for the required derivative \(\mathrm{d}\mathrm{X}_{\mathrm{m}}{ }^{\mathrm{E}}\) is not straightforward. The problem becomes rather more formidable when the second and higher derivatives are required. The derivative \(\mathrm{d}^{2} \mathrm{X}_{\mathrm{m}}{ }^{\mathrm{E}}\) is sometimes required by calculations concerning the properties of binary liquid mixtures.The derivative \(\mathrm{dG}_{\mathrm{m}}^{\mathrm{E}} / \mathrm{dx} \mathrm{x}_{1}\) and \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}}\) are combined to yield an equation for \(\ln\left(\mathrm{f}_{1}\right)\). \[\ln \left(f_{1}\right)=\frac{G_{m}^{E}}{R \, T}+\frac{\left(1-x_{1}\right)}{R \, T} \, \frac{d G_{m}^{E}}{d x_{1}}\]A similar equation leads to estimates of \(\ln\left(\mathrm{f}_{2}\right)\). Hence the dependences are obtained of both \(\ln\left(\mathrm{f}_{1}\right)\) and \(\ln\left(\mathrm{f}_{2}\right)\) on mixture composition. It is of interest to explore the case where the coefficients \(\mathrm{A}_{2}, \mathrm{~A}_{3} \ldots\) in equation (a) are zero. Then \[\mathrm{X}_{\mathrm{m}}^{\mathrm{E}}=\mathrm{x}_{2} \,\left(1-\mathrm{x}_{2}\right) \, \mathrm{A}_{1}\]and \[\mathrm{dX}_{\mathrm{m}}^{\mathrm{E}} / \mathrm{dx_{2 }}=\left(1-2 \, \mathrm{X}_{2}\right) \, \mathrm{A}_{1}\]With reference to the Gibbs energies, \[\ln \left(\mathrm{f}_{2}\right)=(1 / \mathrm{R} \, \mathrm{T}) \,\left[\mathrm{x}_{2} \,\left(1-\mathrm{x}_{2}\right)+\left(1-\mathrm{x}_{2}\right) \,\left(1-2 \, \mathrm{x}_{2}\right)\right] \, \mathrm{A}_{1}^{\mathrm{G}}\]\[\ln \left(\mathrm{f}_{2}\right)=\left(\mathrm{A}_{1}^{\mathrm{G}} / \mathrm{R} \, \mathrm{T}\right) \,\left[1-2 \, \mathrm{x}_{2}+\mathrm{x}_{2}^{2}\right]\]or, \[\ln \left(\mathrm{f}_{2}\right)=\left(\mathrm{A}_{1}^{\mathrm{G}} / \mathrm{R} \, \mathrm{T}\right) \,\left[1-\mathrm{x}_{2}\right]^{2}\]In fact the equation reported by Jost et al. has this form.Rather than using the Redlich-Kister equation, recently attention has been directed to the Wilson equation written in equation (l) for a two-component liquid. \[\mathrm{G}_{\mathrm{m}}^{\mathrm{E}} / \mathrm{R} \, \mathrm{T}=-\mathrm{x}_{1} \, \ln \left(\mathrm{x}_{1}+\Lambda_{12} \, \mathrm{x}_{2}\right)-\mathrm{x}_{2} \, \ln \left(\mathrm{x}_{2}+\Lambda_{21} \, \mathrm{x}_{1}\right)\]Then , for example, \[\ln \left(\mathrm{f}_{1}\right)=-\ln \left(\mathrm{x}_{1}+\Lambda_{12} \, \mathrm{x}_{2}\right)+\mathrm{x}_{2} \,\left(\frac{\Lambda_{12}}{\mathrm{x}_{1}+\Lambda_{12} \, \mathrm{x}_{2}}-\frac{\Lambda_{21}}{\Lambda_{21} \, \mathrm{x}_{1}+\mathrm{x}_{2}}\right)\]The Wilson equation forms the basis for two further developments, described as the NRTL (non-random, two-liquid) equation and the UNIQUAC equation. Nevertheless Douheret et al. show how an excess property must be carefully defined. Davis et al. have explored how excess molar properties for liquid mixtures can be analysed in terms of different mole fraction domains.Footnotes E. A. Guggenheim, Trans. Faraday Soc.,1937,33,151; equation 4.1. G. Scatchard, Chem. Rev.,1949,44,7;see page 9. O. Redlich and A. Kister, Ind. Eng. Chem.,1948,40,345; equation 8. F. Jost, H. Leiter and M. J. Schwuger, Colloid Polymer Sci., 1988, 266, 554. G. M. Wilson, J. Am. Chem. Soc.,1964,86,127. See also From equation (l), \[\begin{aligned} \frac{1}{\mathrm{R} \, \mathrm{T}} \, \frac{\mathrm{dG}_{\mathrm{m}}^{\mathrm{E}}}{\mathrm{dx}_{1}}=&-\ln \left(\mathrm{x}_{1}+\Lambda_{12} \, \mathrm{x}_{2}\right)-\frac{\mathrm{x}_{1} \,\left(1-\Lambda_{12}\right)}{\mathrm{x}_{1}+\Lambda_{12} \, \mathrm{x}_{2}} \\ &+\ln \left(\Lambda_{21} \, \mathrm{x}_{1}+\mathrm{x}_{2}\right)-\frac{\mathrm{x}_{2} \,\left(\Lambda_{21}-1\right)}{\Lambda_{21} \, \mathrm{x}_{1}+\mathrm{x}_{2}} \end{aligned}\]Then using equation (f) with \(1 − x_{1} = x_{2}\), \[\begin{aligned} \ln \left(\mathrm{f}_{1}\right)=&-\mathrm{x}_{1} \, \ln \left(\mathrm{x}_{1}+\Lambda_{12} \, \mathrm{x}_{2}\right)-\mathrm{x}_{2} \, \ln \left(\Lambda_{21} \, \mathrm{x}_{1}+\mathrm{x}_{2}\right) \\ &-\mathrm{x}_{2} \, \ln \left(\mathrm{x}_{1}+\Lambda_{12} \, \mathrm{x}_{2}\right)-\frac{\mathrm{x}_{1} \, \mathrm{x}_{2} \,\left(1-\Lambda_{12}\right)}{\mathrm{x}_{1}+\Lambda_{12} \, \mathrm{x}_{2}} \\ &+\mathrm{x}_{2} \, \ln \left(\Lambda_{21} \, \mathrm{x}_{1}+\mathrm{x}_{2}\right)+\frac{\left(\mathrm{x}_{2}\right)^{2} \,\left(1-\Lambda_{21}\right)}{\Lambda_{21} \, \mathrm{x}_{1}+\mathrm{x}_{2}} \end{aligned}\]Or, \[\begin{aligned} \ln \left(f_{1}\right) &=-\left(x_{1}+x_{2}\right) \, \ln \left(x_{1}+\Lambda_{12} \, x_{2}\right) \\ +x_{2} \,\left[\frac{\Lambda_{12} \, x_{1}-x_{1}}{x_{1}+\Lambda_{12} \, x_{2}}-\frac{\Lambda_{21} \, x_{2}-x_{2}}{\Lambda_{21} \, x_{1}+x_{2}}\right] \end{aligned}\]But \[\Lambda_{12} \, \mathrm{x}_{1}-\mathrm{x}_{1}=\Lambda_{12} \,\left(1-\mathrm{x}_{2}\right)-\mathrm{x}_{1}=\Lambda_{12}-\left(\mathrm{x}_{1}+\Lambda_{12} \, \mathrm{x}_{2}\right)\]Hence, \[\begin{aligned} \ln \left(\mathrm{f}_{1}\right) &=-\ln \left(\mathrm{x}_{1}+\Lambda_{12} \, \mathrm{x}_{2}\right) \\ &+\mathrm{x}_{2} \,\left[\frac{\Lambda_{12}-\left(\mathrm{x}_{1}+\Lambda_{12} \, \mathrm{x}_{2}\right)}{\mathrm{x}_{1}+\Lambda_{12} \, \mathrm{x}_{2}}-\frac{\Lambda_{21}-\left(\Lambda_{21} \, \mathrm{x}_{1}+\mathrm{x}_{2}\right)}{\Lambda_{21} \, \mathrm{x}_{1}+\mathrm{x}_{2}}\right] \end{aligned}\]Or, \[\ln \left(f_{1}\right)=-\ln \left(x_{1}+\Lambda_{12} \, x_{2}\right)+x_{2} \,\left[\frac{\Lambda_{12}}{x_{1}+\Lambda_{12} \, x_{2}}-\frac{\Lambda_{21}}{\Lambda_{21} \, x_{1}+x_{2}}\right]\] D. Abrams and J. M. Prausnitz, AIChE J.,1975,21,116. R. C. Reid, J. M. Prausnitz and E. B. Poling, The Properties of Gases and Liquids, McGraw-Hill, New York, 4th edn.,1987, chapter 8. J. M. Prausnitz, R. N. Lichtenthaler and E. G. de Azevedo, Molecular Thermodynamics of Fluid Phase Equilibria, Prentice –Hall, Upper Saddle River, N.J., 3rd edn.,1999,chapter 6. G. Douheret, C. Moreau and A. Viallard, Fluid Phase Equilib.,1985,22,277,287. Finally we note that the Redlich-Kister equation can be expressed in the following form. \[\mathrm{X}_{\mathrm{m}}^{\mathrm{E}}=\mathrm{x}_{1} \,\left(1-\mathrm{x}_{1}\right) \, \sum_{\mathrm{i}=1}^{\mathrm{i}=\mathrm{k}} \mathrm{B}_{\mathrm{i}} \,\left(1-2 \, \mathrm{x}_{1}\right)^{i-1}\]Then \(\mathrm{A}_{\mathrm{i}}=\mathrm{B}_{\mathrm{i}} \,(-1)^{i}\)J. D. G. de Oliveira and J. C. R.Reis, Thermochim. Acta 2008,468, 119This page titled 1.10.30: Gibbs Energies- General Equations is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.10.31: Gibbs Energies- Liquid Mixtures- Ideal

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The molar Gibbs energy of mixing for an ideal binary liquid mixture is given by equation (a); \[\Delta_{\text {mix }} \mathrm{G}_{\mathrm{m}}(\mathrm{id})=\mathrm{R} \, \mathrm{T} \,\left[\mathrm{x}_{1} \, \ln \mathrm{x}_{1}+\mathrm{x}_{2} \, \ln \mathrm{x}_{2}\right]\]\[\Delta_{\text {mix }} \mathrm{G}_{\mathrm{m}}(\mathrm{id}) / \mathrm{R} \, \mathrm{T}=\mathrm{x}_{1} \, \ln \mathrm{x}_{1}+\mathrm{x}_{2} \, \ln \mathrm{x}_{2}\]In fact the molar Gibbs energy of mixing for an ideal binary mixture is negative across the complete composition range. According to Gibbs - Helmholtz equation, the molar enthalpy of mixing for an ideal binary mixture is given by equation (c). \[\Delta_{\operatorname{mix}} \mathrm{H}_{\mathrm{m}}(\mathrm{id})=\frac{\mathrm{d}}{\mathrm{d}\left(\mathrm{T}^{-1}\right)}\left[\frac{\Delta_{\operatorname{mix}} \mathrm{G}(\mathrm{id})}{\mathrm{T}}\right]_{\mathrm{p}}\]But mole fractions are not dependent on temperature. Hence, \[\Delta_{\text {mix }} \mathrm{H}_{\mathrm{m}}(\mathrm{id})=0\]This important result offers a point of reference. At fixed pressure, the mixing of two liquids to form an ideal binary liquid mixture is athermal. Hence a recorded heat of mixing is a direct measure of the extent to which the properties of a given mixture differ from those defined as ideal. But, \[\Delta_{\text {mix }} \mathrm{G}_{\mathrm{m}}(\mathrm{id})=\Delta_{\mathrm{mix}} \mathrm{H}_{\mathrm{m}}(\mathrm{id})-\mathrm{T} \, \Delta_{\text {mix }} \mathrm{S}_{\mathrm{m}}(\mathrm{id})\]For an ideal binary liquid mixture the partial molar entropies of the two liquid components are given by the following equations. \[\mathrm{S}_{1}(\operatorname{mix} ; \mathrm{id})=\mathrm{S}_{1}^{*}(\ell)-\mathrm{R} \, \ln \left(\mathrm{x}_{1}\right)\]\[S_{2}(\operatorname{mix} ; \mathrm{id})=\mathrm{S}_{2}^{*}(\ell)-\mathrm{R} \, \ln \left(\mathrm{x}_{2}\right)\]\[\mathrm{S}_{\mathrm{m}}(\operatorname{mix} ; \mathrm{id})=\mathrm{x}_{1} \,\left[\mathrm{S}_{1}^{*}(\ell)-\mathrm{R} \, \ln \left(\mathrm{x}_{1}\right)\right]+\mathrm{x}_{2} \,\left[\mathrm{S}_{2}^{*}(\ell)-\mathrm{R} \, \ln \left(\mathrm{x}_{2}\right)\right]\]From equation (h), \[\Delta_{\operatorname{mix}} \mathrm{S}_{\mathrm{m}}(\mathrm{id})=-\mathrm{R} \,\left[\mathrm{x}_{1} \, \ln \mathrm{x}_{1}+\mathrm{x}_{1} \, \ln \mathrm{x}_{2}\right]\]or, \[\mathrm{T} \, \Delta_{\text {mix }} S_{m}(\text { id })=-R \, T\left[x_{1} \, \ln x_{1}+x_{2} \, \ln x_{2}\right]\]But across the complete mole fraction range \(\left[x_{1} \, \ln x_{1}+x_{2} \, \ln x_{2}\right] \leq 0\). Over the same range, \[\mathrm{T} \, \Delta_{\operatorname{mix}} \mathrm{S}_{\mathrm{m}}(\mathrm{id})>0\]Thus the sign and magnitude of \(\Delta_{\text{mix}}\mathrm{G}_{\mathrm{m}}(\text{id})\) and (with opposite sign) \(\mathrm{T} \, \Delta_{\mathrm{mix}} \mathrm{S}_{\mathrm{m}}(\mathrm{id})\) are defined. A further consequence of equation (d) is that the corresponding isobaric heat capacity variable, \(\Delta_{\mathrm{mix}} \mathrm{C}_{\mathrm{pm}}(\mathrm{id})\) is zero across the whole mole fraction range. Using equation (a), the molar volume of mixing is given by equation (l). Thus, \[\Delta_{\text {mix }} V_{m}(\mathrm{id})=\frac{\partial}{\partial p}\left[\Delta_{\text {mix }} G_{m}(\mathrm{id})\right]_{p}\]Hence for a binary liquid mixture having ideal properties, across the complete mole fraction range, \(\Delta_{\operatorname{mix}} \mathrm{V}_{\mathrm{m}}(\mathrm{id})=0\). The latter condition requires that the volume of a liquid mixture equals the sum of the volumes of the two liquid components used to prepare the mixture at fixed temperature and pressure. For such a mixture, \[\mathrm{V}_{\text {mix }}(\mathrm{id})=\mathrm{V}_{2}^{*}(\ell)+\mathrm{x}_{1} \,\left[\mathrm{V}_{1}^{*}(\ell)-\mathrm{V}_{2}^{*}(\ell)\right]\]This simple pattern is not observed. In fact the molar volume of a real binary mixture is usually less than \(\mathrm{V}_{\text{mix} (\text{id})\).With the benefit of hindsight, we distinguish between Gibbsian and non-Gibbsian on the one hand and between first and second law (thermodynamic) variables on the other hand. Variables \(\mathrm{H}\), \(\mathrm{V}\) and \(\mathrm{C}_{\mathrm{p}}\) are Gibbsian first law variables such that the molar property of an ideal binary liquid mixture is given by the mole fraction weighted sum of the properties of the pure liquids. However Gibbsian second law properties (e.g. entropies and Gibbs energies) require combinatorial terms arising from the irreversible entropy of mixing.These simple rules do not apply in the case of molar non-Gibbsian properties (e.g. isentropic compressions and isochoric heat capacities) of ideal mixtures.This page titled 1.10.31: Gibbs Energies- Liquid Mixtures- Ideal is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.10.32: Gibbs Energies- Liquid Mixtures- Typically Aqueous (TA)

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For many binary aqueous liquid mixtures, the pattern shown by the molar excess thermodynamic parameters is \(\mathrm{G}_{\mathrm{m}}^{\mathrm{E}}>0\); \(\left|\mathrm{T} \, \mathrm{S}_{\mathrm{m}}^{\mathrm{E}}\right|>\left|\mathrm{H}_{\mathrm{m}}^{\mathrm{E}}\right|\). This pattern of excess molar properties defines TA mixtures. \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}}\) is positive because the excess molar entropy of mixing is large in magnitude and negative in sign. In these terms mixing is dominated by the entropy change. The excess molar enthalpy of mixing is smaller in magnitude than either \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}}\) or \(\mathrm{T} \, {\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}}\) but exothermic in water-rich mixtures.The word ‘Typically’ in the description stems from observation that this pattern in thermodynamic variables is rarely shown by non-aqueous systems. At the time the classification was proposed, most binary aqueous liquid mixtures seemed to follow this pattern. Among the many examples of this class of system are aqueous mixtures formed by ethanol, 2-methyl propan-2-ol and cyclic ethers including tetrahydrofuran. In water-rich mixtures, a large in magnitude but negative in sign \(\mathrm{T} \, {\mathrm{S}_{\mathrm{m}}}^{\mathrm{E}}\) produces a large (positive) \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}}\). For mixtures rich in the apolar component m endothermic mixing produces a positive \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}}\). The key point is that in m water-rich mixtures a positive \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}}\) emerges from a negative \(\mathrm{T} \, {\mathrm{S}_{\mathrm{m}}}^{\mathrm{E}}\).With reference to volumetric properties of these system, the partial molar volume \(\mathrm{V}(\mathrm{ROH})\) for monohydric alcohols can be extrapolated to infinite dilution; i.e. \(\operatorname{limit}\left(\mathrm{x}_{2} \rightarrow 0\right) \mathrm{V}(\mathrm{ROH})=\mathrm{V}(\mathrm{ROH} ; \mathrm{aq})^{\infty}\) where \(x_{2}\) is the mole fraction of alcohol. The difference, \(\left[\mathrm{V}(\mathrm{ROH} ; \mathrm{aq})^{\infty}-\mathrm{V}^{*}(\mathrm{ROH} ; \ell)\right]\) is negative. In fact this pattern is observed for both TA and Typical Non-aqueous binary aqueous mixtures. Examples where this pattern is observed included aqueous mixtures formed by DMSO, \(\mathrm{H}_{2}\mathrm{O}_{2}\) and \(\mathrm{CH}_{3}\mathrm{CN}\). Significantly \(\left[\mathrm{V}(\mathrm{ROH} ; \mathrm{aq})-\mathrm{V}^{*}(\mathrm{ROH} ; \ell)\right]\) is negative for TA mixtures, decreasing from \(\left[\mathrm{V}(\mathrm{ROH} ; \mathrm{aq})^{\infty}-\mathrm{V}^{*}(\mathrm{ROH} ; \ell)\right]\) with increase in mole fraction of \(\mathrm{ROH}\), accompanying by a tendency to immiscibility. The initial decrease in \(\left[\mathrm{V}(\mathrm{ROH} ; \mathrm{aq})-\mathrm{V}^{*}(\mathrm{ROH} ; \ell)\right]\) with increase in \(x_{2}\) is more dramatic the more hydrophobic the non-aqueous component; for 2-methyl propan-2-ol aqueous mixtures at \(298.2 \mathrm{~K}\) and ambient pressure, the minimum occurs at an alcohol mole fraction 0.04 (at \(298.2 \mathrm{~K}\)).Many explanations have been offered for the complicated patters shown by the dependence of \(\left[\mathrm{V}(\mathrm{ROH} ; \mathrm{aq})-\mathrm{V}^{*}(\mathrm{ROH} ; \ell)\right]\) on mole fraction composition.In one model, the negative \(\left[\mathrm{V}(\mathrm{ROH} ; \mathrm{aq})-\mathrm{V}^{*}(\mathrm{ROH} ; \ell)\right]\) at low mole fractions of \(\mathrm{ROH}\) is accounted for in terms of a liquid clathrate in which part of the hydrophobic R-group ‘occupies’ a guest site in the water lattice. The decrease in \(\left[\mathrm{V}(\mathrm{ROH} ; \mathrm{aq})-\mathrm{V}^{*}(\mathrm{ROH} ; \ell)\right]\) is accounted for in terms of an increasing tendency towards a clathrate structure. But with increase in \(x_{2}\) there comes a point where there is insufficient water to construct a liquid clathrate water host. Hence \(\left[\mathrm{V}(\mathrm{ROH} ; \mathrm{aq})-\mathrm{V}^{*}(\mathrm{ROH} ; \ell)\right]\) increases.An important characteristic of TA mixtures is a tendency towards and in some cases actual decrease in liquid miscibility with increase in temperature. At ambient \(\mathrm{T}\) and \(\mathrm{p}\), the mixture 2-methyl propan-2-ol + water is miscible (but only just!) in all molar proportions. The corresponding mixtures prepared using butan-1-ol and butan-2-ol are partially miscible. TA systems are therefore often characterised by a Lower Critical Solution Temperature LCST. In fact nearly all examples quoted in the literature of systems having an LCST involve water as one component; e.g. \(\mathrm{LCST} = 322 \mathrm{~K}\) for 2-butoxyethanol + water. This tendency to partial miscibility is often signalled by the properties of the completely miscible systems.Returning to the patterns shown by relative partial molar volumes, \(\left[\mathrm{V}(\mathrm{ROH} ; \mathrm{aq})-\mathrm{V}^{*}(\mathrm{ROH} ; \ell)\right]\), a stage is reached whereby with increase in mole fraction of the non-aqueous component, this property increases after a minimum. Other properties of the mixtures also change dramatically including a marked increase in \(\left(\alpha_{a} / v^{2}\right)\) where \(\alpha_{\mathrm{a}}\) is the amplitude attenuation constant and \(ν\) is the frequency of the sound wave in the MHz range; e.g. \(70 \mathrm{~MHz}\). Actually the pattern is complicated. Over the range of mixture mole fractions \(x_{2}\) where \(\left[\mathrm{V}(\mathrm{ROH} ; \mathrm{aq})-\mathrm{V}^{*}(\mathrm{ROH} ; \ell)\right]\) decreases with increase in \(x_{2}\), the ratio \(\left(\alpha_{a} / v^{2}\right)\) hardly changes although the speed of sound increases. At a mole fraction \(x_{2}\) characteristic of the temperature and the non-aqueous component, \(\left(\alpha_{a} / v^{2}\right)\) increases sharply, reaching a maximum where the mixture has a strong tendency to immiscibility. This interplay between in-phase and out-of-phase components of the complex isentropic compressibility when the mole fraction composition of the mixture is changed highlights the molecular complexity of these systems. By way of contrast the ratio \(\left(\alpha_{a} / v^{2}\right)\) for DMSO + water mixtures (a TNAN system) changes gradually when the mole fraction of DMSO is changed.For TA mixtures where \(\left(\alpha_{a} / v^{2}\right)\) is a maximum, other evidence points to the fact these mixtures are micro-heterogeneous; cf. excess molar isobaric heat capacities. Phase separation of the mixture 2-methyl propan-2-ol is observed when butane gas is dissolved in the liquid mixture. The miscibility curve shows an LCST near \(282 \mathrm{~K}\).Footnotes F. Franks in Hydrogen-Bonded Solvent Systems, ed. A. K. Covington and P. Jones, Taylor and Francis, London,1968, pp.31-47. The following references refer to properties of TA binary liquid mixtures. F. Elizalde, J Gracia and M. Costas, J Phys.Chem.,1988,93,3565. M. J Blandamer and D. Waddington, Adv. Mol. Relax.Processes,1970,2,1. R. W. Cargill and D. E. MacPhee, J. Chem. Soc. Faraday Trans.1, 1989, 85, 2665; an excellent observation! \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}}\); M. J. Blandamer, J. Burgess, A. Cooney, H. J. Cowles, I. M. Horn, K. M. Martin, K.W. Morcom and P. Warrick, J. Chem. Soc. Faraday Trans., 1990, 86, 2209.This page titled 1.10.32: Gibbs Energies- Liquid Mixtures- Typically Aqueous (TA) is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.10.33: Gibbs Energies- Liquid Mixtures- Typically Non-Aqueous Positive; TNAP

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For this sub-group of binary aqueous liquid mixtures, \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}}\) is positive. An example of such a mixture is ‘water + ethanenitrile’. The positive \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}}\) reflects endothermic mixing across nearly all the mole fraction range. These mixtures have a tendency to be partially miscible with an Upper Critical Solution Temperature, UCST. For aqueous mixtures the composition at the UCST is often ‘water-rich’. For ethanenitrile + water, the UCST is \(272 \mathrm{~K}\). The positive \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}}\) and endothermic mixing are attributed to disruption of water-water hydrogen bonding by added mathrm{MeCN}; cf. \(\mathrm{CCl}_{4} + \mathrm{~MeOH}\).Footnotes MeCN + water. Propylene Carbonate Miscibility; N. F. Catherall and A. G. Williamson, J. Chem. Eng. Data, 1971, 16, 335.This page titled 1.10.33: Gibbs Energies- Liquid Mixtures- Typically Non-Aqueous Positive; TNAP is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.10.34: Gibbs Energies- Liquid Mixtures- Typically Non-Aqueous Negative; TNAN

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For this sub-group, \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}}\) is negative because there is strong inter- m component interaction which also produces exothermic mixing. Examples of aqueous mixtures which fall into this class areFootnotes DMSO + water Ethane-1,2- diol+ water; 2-methoxyethanol + water; M. Page, J.-Y. Huot and C. Jolicoeur, J.Chem. Thermodyn., 1993,25,139. Hydrogen peroxide +water; G. Scatchard, G. M. Kavanagh and L. B. Ticknor, J. Am. Chem.Soc.,1952,74,3715.This page titled 1.10.34: Gibbs Energies- Liquid Mixtures- Typically Non-Aqueous Negative; TNAN is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.10.35: Gibbs Energies- Liquid Mixtures- Immiscibility

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.10%3A_Gibbs_Energies/1.10.35%3A_Gibbs_Energies-_Liquid_Mixtures-_Immiscibility

For a given binary liquid mixture (at defined \(\mathrm{T}\) and \(\mathrm{p}\)) characterised by a plot of excess Gibbs energy \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}}\) against mole fraction, \({\mathrm{G}_{\mathrm{m}}}^{\mathrm{E}}\) can be positive. Indeed if \(\left[\mathrm{G}_{\mathrm{m}}^{\mathrm{E}} / \mathrm{R} \, \mathrm{T}\right]\) strongly exceeds 0.5, the mixture is partially miscible. That is to say the liquid comprises two liquid phases having different mole fraction compositions.A fascinating variety of patterns emerge in the context of partial miscibilities.Partial miscibility plots also show deuterium isotope effects. In the case of \(\mathrm{CH}_{3}\mathrm{CN}+\mathrm{H}_{2}\mathrm{O} \left(\text{component } 2 = \mathrm{~CH}_{3}\mathrm{CN}\right)\) the UCST is \(272.10 \mathrm{~K}\) at \(x_{2}=0.38\).Footnotes A. Imre and W.A. Van Hook, J. Polym Sci.; Part B; Polymer Physics,1994,32,2283. M. J. Blandamer, M. J. Foster and D. Waddington, Trans. Faraday Soc.,1970,66,1369.This page titled 1.10.35: Gibbs Energies- Liquid Mixtures- Immiscibility is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.10.36: Gibbs Energies- Salt Solutions- Aqueous Mixtures

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The solubilities of chemical substance \(j\) in two liquids \(\ell_{1}\) and \(\ell_{2}\) (at the same \(\mathrm{T}\) and \(\mathrm{p}\)) offers a method for comparing the reference chemical potentials, using the transfer parameter \(\Delta\left(\ell_{1} \rightarrow \ell_{2}\right) \mu_{\mathrm{j}}^{0}\). A similar argument is advanced in the context of salt solutions in which comparison of the solubility of salt j in two liquids leads to the transfer parameter for the salt. However the argument does not stop there. In the case of, for example a 1:1 salt \(\mathrm{M}^{+} \mathrm{X}^{-}\), the derived transfer for the salt is re-expressed as the sum of transfer parameters for the separate ions \(\mathrm{M}^{+}\) and \(\mathrm{X}^{-}\). Thus \[\Delta\left(\ell_{1} \rightarrow \ell_{2}\right) \mu^{0}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right)=\Delta\left(\ell_{1} \rightarrow \ell_{2}\right) \mu^{0}\left(\mathrm{M}^{+}\right)+\Delta\left(\ell_{1} \rightarrow \ell_{2}\right) \mu^{0}\left(\mathrm{X}^{-}\right)\]However granted that we can obtain an estimate of the transfer parameter for the salt, \(\Delta\left(\ell_{1} \rightarrow \ell_{2}\right) \mu^{0}\left(\mathrm{M}^{+} \mathrm{X}^{-}\right)\), thermodynamics does not offer a method for calculating the corresponding ionic transfer parameters. Several extra-thermodynamic procedures yield estimated single ion thermodynamic transfer parameters. The simplest approach adopts a reference ion (e.g. \(\mathrm{H}^{+}\)) and reports relative transfer ionic chemical potentials. \[\Delta\left(\ell_{1} \rightarrow \ell_{2}\right) \mu^{0}\left(\mathrm{H}^{+}\right)=0\]For example; \[\Delta\left(\ell_{1} \rightarrow \ell_{2}\right) \mu^{0}\left(\mathrm{C} \ell^{-}\right)=\Delta\left(\ell_{1} \rightarrow \ell_{2}\right) \mu^{0}(\mathrm{HC} \ell)\]A closed system (at fixed \(\mathrm{T}\) and ambient pressure) contains a solid salt \(j\) in equilibrium with salt \(j\) in aqueous solution. At equilibrium, \[\mu_{\mathrm{j}}^{*}(\mathrm{~s})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{v} \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{Q} \, \mathrm{m}_{\mathrm{j}}^{\mathrm{eq}}(\mathrm{aq}) \, \gamma_{\pm}^{\mathrm{cq}}(\mathrm{aq}) / \mathrm{m}^{0}\right)\]Similarly for an equilibrium system where the solvent is a binary aqueous mixture, mole fraction \(x_{2}\), \[\mu_{\mathrm{j}}^{*}(\mathrm{~s})=\mu_{\mathrm{j}}^{0}\left(\mathrm{~s} \ln ; \mathrm{x}_{2}\right)+\mathrm{v} \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{Q} \, \mathrm{m}_{\mathrm{j}}^{\mathrm{eq}}\left(\mathrm{s} \ln ; \mathrm{x}_{2}\right) \, \gamma_{\pm}^{\mathrm{eq}}\left(\mathrm{s} \ln ; \mathrm{x}_{2}\right) / \mathrm{m}^{0}\right)\]Then, \[\begin{aligned} \Delta\left(\mathrm{aq} \rightarrow \mathrm{x}_{2}\right) \mu_{\mathrm{j}}^{0}(\mathrm{~s} \ln ) &=\mu_{\mathrm{j}}^{0}\left(\mathrm{~s} \ln ; \mathrm{x}_{2}\right)-\mu_{\mathrm{j}}^{0}(\mathrm{aq}) \\ &=-\mathrm{v} \, \mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{m}_{\mathrm{j}}^{\mathrm{eq}}\left(\mathrm{s} \ln ; \mathrm{x}_{2}\right) \, \gamma_{\pm}^{\mathrm{eq}}\left(\mathrm{s} \ln ; \mathrm{x}_{2}\right) / \mathrm{m}_{\mathrm{j}}^{\mathrm{eq}}(\mathrm{aq}) \, \gamma_{\pm}^{\mathrm{eq}}(\mathrm{aq})\right] \end{aligned}\]A key assumption sets the ratio of mean ionic activity coefficients to unity. In effect we assume that the solubilities do not change dramatically as \(x_{2}\) is changed. Therefore, \[\Delta\left(\mathrm{aq} \rightarrow \mathrm{x}_{2}\right) \mu_{\mathrm{j}}^{0}(\mathrm{~s} \ln )=-\mathrm{v} \, \mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{m}_{\mathrm{j}}^{\mathrm{eq}}\left(\mathrm{s} \ln ; \mathrm{x}_{2}\right) / \mathrm{m}_{\mathrm{j}}^{\mathrm{eq}}(\mathrm{aq})\right]\]Thus the ratio \(\left[\mathrm{m}_{\mathrm{j}}^{\mathrm{eq}}\left(\mathrm{s} \ln ; \mathrm{x}_{2}\right) / \mathrm{m}_{\mathrm{j}}^{\mathrm{eq}}(\mathrm{aq})\right]\) is effectively the ratio of solubilities of salt \(j\) in the mixed aqueous solutions and aqueous solution. If the solubility of the salt increases with increase in \(x_{2}\), \(\Delta\left(\mathrm{aq} \rightarrow \mathrm{x}_{2}\right) \mu_{\mathrm{j}}^{0}(\mathrm{~s} \ln )\) is negative. In other words, the salt in aqueous solutions is stabilised by adding the co-solvent.Granted that solubility data lead to an estimate for \(\Delta\left(\mathrm{aq} \rightarrow \mathrm{x}_{2}\right) \mu_{\mathrm{j}}^{0}(\mathrm{~s} \ln )\), this quantity involves contributions from both cations and anions. For a salt containing two ionic substances \[\Delta\left(\mathrm{aq} \rightarrow \mathrm{x}_{2}\right) \mu_{\mathrm{j}}^{0}(\mathrm{~s} \ln )=\Delta\left(\mathrm{aq} \rightarrow \mathrm{x}_{2}\right) \mu_{+}^{0}(\mathrm{~s} \ln )+\Delta\left(\mathrm{aq} \rightarrow \mathrm{x}_{2}\right) \mu_{-}^{0}(\mathrm{~s} \ln )\]The background to this type of analysis centres on classic studes into the electrical conductivities of salt solutions. For a given salt in a solvent (at fixed \(\mathrm{T}\) and \(\mathrm{p}\)), the molar conductivity approaches a limiting value with decrease in concentration; \(\operatorname{limit}\left(c_{j} \rightarrow 0\right) \Lambda_{j}=\Lambda_{j}^{0}\). The limiting molar conductivity of a salt solution \({\Lambda^{0}}_{j}\) containing a 1:1 salt can be written as the sum of limiting ionic conductivities \({\lambda_{i}}^{0}\) of anions and cations. \[\Lambda_{\mathrm{j}}^{0}=\lambda_{+}^{0}+\lambda_{-}^{0}\]The transport number of an ion \(\mathrm{t}_{j}\) measures the ratio \(\lambda_{\mathrm{j}}^{0} / \Lambda\). Both \(\mathrm{t}_{j}\) and \(\Lambda\) can be measured and hence \(\lambda_{\mathrm{j}}^{0}\) calculated in the limit of infinite dilution characterizes ion \(j\) in a given solvent at defined \(\mathrm{T}\) and \(\mathrm{p}\). Discrimination between anions and cations arises from their electrical charges and hence the direction of migration of ions in an electric field. Nevertheless the task of measuring both \(\mathrm{t}_{j}\) and \(\Lambda\) is not trivial and some simple working hypothesis is often sought. The argument is advanced that the molar conductivities are equal in magnitude for two ions having similar size and solvation characteristics. This ‘extrathermodynamic’ assumption has been applied to a range of ‘onium salts includingThis ‘big ion – big ion’ assumption is carried over to the analysis of thermodynamic properties where we lack the discrimination between cations and anions based on their mobilities in an applied electric potentials. gradient. Then for example the change in solubility of one such salt in aqueous solution on adding a cosolvent ( e.g. ethanol) can be understood in terms of equal transfer thermodynamic potentials. \[(1 / 2) \, \Delta\left(\mathrm{aq} \rightarrow \mathrm{x}_{2}\right) \mu^{0}(\text { big cation big anion; } \mathrm{s} \ln )=\Delta\left(\mathrm{aq} \rightarrow \mathrm{x}_{2}\right) \mu^{0}(\text { big cation; } \mathrm{s} \ln )=\Delta\left(\mathrm{aq} \rightarrow \mathrm{x}_{2}\right) \mu^{0}(\text { big anion; } \mathrm{l} \ln )\]For example having obtained \(\Delta\left(\mathrm{aq} \rightarrow \mathrm{x}_{2}\right) \mu^{0}(\text { big cation; } \mathrm{ln})\), the difference in solubilities of the corresponding salt iodide is used to obtain the transfer parameter for iodide ions in the two solvents. \[\begin{aligned} \Delta(\mathrm{aq} \rightarrow&\left.\mathrm{x}_{2}\right) \mu^{0}\left(\mathrm{I}^{-} ; \mathrm{s} \ln \right)=\\ & \Delta\left(\mathrm{aq} \rightarrow \mathrm{x}_{2}\right) \mu^{0}(\text { big cation iodide; } \mathrm{s} \ln )-\Delta\left(\mathrm{aq} \rightarrow \mathrm{x}_{2}\right) \mu^{0}(\text { big cation } \mathrm{s} \ln ) \end{aligned}\]Considerable information is available in the chemical literature concerning ionic transfer parameters, particularly for solutes in binary aqueous mixtures at \(298.2 \mathrm{~K}\) and ambient pressure.8-21 Unfortunately there is no agreed composition scale for transfer parameters. Information includes transfer parameters based on concentration, molality and mole fractions scales for the solutes. The situation is further complicated by the fact that different scales are used to express composition of liquid mixtures. Common scales include mass%, mole fraction and vol%. Conversion between these scales is a tedious. Some examples of the required equations are presented in an Appendix to this Topic.FootnotesWe write down two equations for the same quantity, the chemical potential of solute \(j\). For the chemical potential of solute \(j\) in an ideal aqueous solution at ambient pressure ( i.e. close to the standard pressure, \(\mathrm{p}^{0}\)), \[\mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}\left(\mathrm{aq} ; \mathrm{m}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right]\]Here \(\mathrm{m}_{j}\) is the molality of solute \(j\); \(\mathrm{m}^{0} = 1 \mathrm{~mol kg}^{-1}\), the reference molality. However we may decide to express the composition of the solution in terms of the mole fraction of solute. If the properties of the solute are ideal, the chemical potential of solute \(j\), \(\mu_{j}(\mathrm{aq})\) is related to the mole fraction of solute \(x_{j}\). \[\mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}\left(\mathrm{aq} ; \mathrm{x}_{\mathrm{j}}=1\right)+\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{x}_{\mathrm{j}}\right]\]Equations (i) and (ii) describe the same property, \(\mu_{j}(\mathrm{aq})\). The property \(\mu_{j}^{0}\left(\mathrm{aq} ; x_{j}=1\right)\) is interesting because it describes the chemical potential of solute \(j\) in aqueous solution where the mole fraction of solute is unity; it is clearly an ‘extrapolated’ property of the solute.If \(\mathrm{n}_{j}\) is the amount of solute in a solution prepared using \(10^{2} \mathrm{~kg}\) of water, we can combine equations (i) and (ii); \(\mathrm{x}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} /\left[\left(10^{2} / \mathrm{M}_{1}\right)+\mathrm{n}_{\mathrm{j}}\right]\) where for a dilute solution \(\left(10^{2} / \mathrm{M}_{1}\right)>>\mathrm{n}_{\mathrm{j}}\); \(\mathrm{M}_{1}\) is the molar mass of water. \[\mu_{\mathrm{j}}^{0}\left(\mathrm{aq} ; \mathrm{m}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{n}_{\mathrm{j}} / 10^{2} \, \mathrm{m}^{0}\right]=\mu_{\mathrm{j}}^{0}\left(\mathrm{aq} ; \mathrm{x}_{\mathrm{j}}=1\right)+\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{n}_{\mathrm{j}} \, \mathrm{M}_{\mathrm{l}} / 10^{2}\right]\]Or, \[\mu_{\mathrm{j}}^{0}\left(\mathrm{aq} ; \mathrm{m}^{0}\right)-\mu_{\mathrm{j}}^{0}\left(\mathrm{aq} ; \mathrm{x}_{\mathrm{j}}=1\right)=\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{m}^{0} \, \mathrm{M}_{1}\right]\]We note that \(\left[\mathrm{m}^{0} \, \mathrm{M}_{1}\right]=\left[\mathrm{mol} \mathrm{kg}^{-1}\right] \,\left[\mathrm{kg} \mathrm{mol}^{-1}\right]=\)If \(\mathrm{n}_{j}\) is the amount of solute \(j\) in \(10^{2} \mathrm{~kg}\) of a solvent mixture, the chemical potential of solute \(j\) is given by equation (iv) \[\mu_{\mathrm{j}}(\operatorname{mix})=\mu_{\mathrm{j}}^{0}\left(\operatorname{mix} ; \mathrm{m}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{n}_{\mathrm{j}} / 10^{2} \, \mathrm{m}^{0}\right]\]We note that \(\left[\mathrm{n}_{\mathrm{j}} / 10^{2} \, \mathrm{m}^{0}\right]=\left[\mathrm{mol} / \mathrm{kg} \, \mathrm{mol} \, \mathrm{kg}^{-1}\right]=\). If the binary solvent mixture comprises \(w_{2} \%\) of the non-aqueous component, for a dilute solution of solute \(j\), the mole fraction of solute \(x_{j}\) is given by equation (vi) where \(\mathrm{M}_{2}\) is the molar mass of the cosolvent. \[\mathrm{x}_{\mathrm{j}}=\frac{\mathrm{n}_{\mathrm{j}}}{\left\{\left[\left(10^{2}-\mathrm{w}_{2} \%\right] / \mathrm{M}_{1}\right\}+\left\{\mathrm{w}_{2} \% / \mathrm{M}_{2}\right\}\right.}\]Using the mole fraction scale for solute \(j\), the chemical potential of solute \(j\) in the mixture, composition \(w_{2} \%\) is given by equation (vii). \[\mu_{\mathrm{j}}(\mathrm{mix})=\mu_{\mathrm{j}}^{0}\left(\mathrm{mix} ; \mathrm{x}_{\mathrm{j}}=1\right)+\mathrm{R} \, \mathrm{T} \, \ln \left[\frac{\mathrm{n}_{\mathrm{j}}}{\left\{\left[\left(10^{2}-\mathrm{w}_{2} \%\right] / \mathrm{M}_{1}\right\}+\left\{\mathrm{w}_{2} \% / \mathrm{M}_{2}\right\}\right.}\right]\]Equations (v) and (vii) describe the same property, the chemical potential of solute \(j\) in a mixed solvent system. Hence, \[\begin{aligned} &\mu_{\mathrm{j}}^{0}\left(\operatorname{mix} ; \mathrm{m}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{n}_{\mathrm{j}} / 10^{2} \, \mathrm{m}^{0}\right] \\ &=\mu_{\mathrm{j}}^{0}\left(\operatorname{mix} ; \mathrm{x}_{\mathrm{j}}=1\right)+\mathrm{R} \, \mathrm{T} \, \ln \left[\frac{\mathrm{n}_{\mathrm{j}}}{\left\{\left[\left(10^{2}-\mathrm{w}_{2} \%\right] / \mathrm{M}_{1}\right\}+\left\{\mathrm{w}_{2} \% / \mathrm{M}_{2}\right\}\right.}\right] \end{aligned}\]Or, \[\begin{aligned} &\mu_{\mathrm{j}}^{0}\left(\operatorname{mix} ; \mathrm{m}^{0}\right) \\ &=\mu_{\mathrm{j}}^{0}\left(\operatorname{mix} ; \mathrm{x}_{\mathrm{j}}=1\right)+\mathrm{R} \, \mathrm{T} \, \ln \left[\frac{10^{2} \, \mathrm{m}^{0}}{\left\{\left[\left(10^{2}-\mathrm{w}_{2} \%\right] / \mathrm{M}_{1}\right\}+\left\{\mathrm{w}_{2} \% / \mathrm{M}_{2}\right\}\right.}\right] \end{aligned}\]It is convenient at this point to comment on the difference in reference chemical potentials of solute \(j\) in aqueous solutions and a solvent mixture. Thus from equation (iv). \[\mu_{\mathrm{j}}^{0}\left(\mathrm{aq} ; \mathrm{m}^{0}\right)-\mu_{\mathrm{j}}^{0}\left(\mathrm{aq} ; \mathrm{x}_{\mathrm{j}}=\mathrm{l}\right)=\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{m}^{0} \, \mathrm{M}_{1}\right]\]And from equation (ix) \[\begin{aligned} \mu_{\mathrm{j}}^{0}\left(\operatorname{mix} ; \mathrm{m}^{0}\right)-\mu_{\mathrm{j}}^{0}\left(\operatorname{mix} ; \mathrm{x}_{\mathrm{j}}\right.&=1) \\ &=\mathrm{R} \, \mathrm{T} \, \ln \left[\frac{10^{2} \, \mathrm{m}^{0}}{\left\{\left[\left(10^{2}-\mathrm{w}_{2} \%\right] / \mathrm{M}_{1}\right\}+\left\{\mathrm{w}_{2} \% / \mathrm{M}_{2}\right\}\right.}\right] \end{aligned}\]The difference between equations (x) and (xi) yields an equation relating transfer parameters for solute \(j\) on the two composition scales. \[\begin{array}{r} \mu_{\mathrm{j}}^{0}\left(\operatorname{mix} ; \mathrm{m}^{0}\right)-\mu_{\mathrm{j}}^{0}\left(\mathrm{aq} ; \mathrm{m}^{0}\right)=\mu_{\mathrm{j}}^{0}\left(\mathrm{mix} ; \mathrm{x}_{\mathrm{j}}=1\right)-\mu_{\mathrm{j}}^{0}\left(\mathrm{aq} ; \mathrm{x}_{1}=1\right) \\ -\mathrm{R} \, \mathrm{T} \, \ln \left[\frac{\left\{10^{2}-\mathrm{w}_{2} \%\right\}+\left\{\mathrm{w}_{2} \% \, \mathrm{M}_{1} / \mathrm{M}_{2}\right\}}{10^{2}}\right] \end{array}\]Hence \[\begin{aligned} \Delta(\mathrm{aq} \rightarrow \mathrm{mix}) & \mu_{\mathrm{j}}^{0}(\mathrm{~m}-\mathrm{scale})=\Delta(\mathrm{aq} \rightarrow \mathrm{mix}) \mu_{\mathrm{j}}^{0}(\mathrm{x}-\text { scale }) \\ &-\mathrm{R} \, \mathrm{T} \, \ln \left[\frac{\left\{10^{2}-\mathrm{w}_{2} \%\right\}+\left\{\mathrm{w}_{2} \% \, \mathrm{M}_{1} / \mathrm{M}_{2}\right\}}{10^{2}}\right] \end{aligned}\]Or, \[\begin{aligned} &\Delta(\mathrm{aq} \rightarrow \mathrm{mix}) \mu_{\mathrm{j}}^{0}(\mathrm{~m}-\mathrm{scale})=\Delta(\mathrm{aq} \rightarrow \mathrm{mix}) \mu_{\mathrm{j}}^{0}(\mathrm{x}-\text { scale }) \\ &\quad-\mathrm{R} \, \mathrm{T} \, \ln \left[1-\left(\mathrm{w}_{2} \% / 10^{2}\right)+\left(\mathrm{w}_{2} \% / 10^{2}\right) \, \mathrm{M}_{\mathrm{l}} / \mathrm{M}_{2}\right] \end{aligned}\]Or, \[\begin{array}{r} \Delta(\mathrm{aq} \rightarrow \operatorname{mix}) \mu_{\mathrm{j}}^{0}(\mathrm{~m}-\text { scale })=\Delta(\mathrm{aq} \rightarrow \mathrm{mix}) \mu_{\mathrm{j}}^{0}(\mathrm{x}-\text { scale }) \\ -\mathrm{R} \, \mathrm{T} \, \ln \left\{1-\left[1-\left(\mathrm{M}_{1} / \mathrm{M}_{2}\right)\right] \,\left(\mathrm{w}_{2} \% / 10^{2}\right)\right\} \end{array}\]If solute \(j\) is a salt which is completely dissociated into \(ν\) ions in both aqueous solution and in the mixed solvent system, \[\begin{aligned} \Delta(\mathrm{aq} \rightarrow \mathrm{mix}) \mu_{\mathrm{j}}^{0}(\mathrm{~m}-\mathrm{scale})=\Delta(\mathrm{aq} \rightarrow \mathrm{mix}) \mu_{\mathrm{j}}^{0}(\mathrm{x}-\mathrm{scale}) \\ -\mathrm{v} \, \mathrm{R} \, \mathrm{T} \, \ln \left\{1-\left[1-\left(\mathrm{M}_{1} / \mathrm{M}_{2}\right)\right] \,\left(\mathrm{w}_{2} \% / 10^{2}\right)\right\} \end{aligned}\]Thus for each ionic substance contributing to the transfer property for the salt, \[\begin{aligned} \Delta(\mathrm{aq}&\rightarrow \mathrm{mix}) \mu_{\mathrm{j}}^{0}(\mathrm{~s} \ln )=\\ v_{+} \, \Delta(\mathrm{aq} \rightarrow \operatorname{mix}) \mu_{+}^{0}(\mathrm{~s} \ln )+v_{-} \, \Delta(\mathrm{aq} \rightarrow \mathrm{mix}) \mu_{-}^{0}(\mathrm{~s} \ln ) \end{aligned}\]Equations (xv) and (xvi) show that the difference between the transfer chemical potentials on the x- and m- scales is independent of temperature. The difference is based on the mass of the solvent components in the mixture. Consequently the transfer enthalpies on the two scales are equal. \[\Delta(\mathrm{aq} \rightarrow \mathrm{mix}) \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{~m}-\text { scale })=\Delta(\mathrm{aq} \rightarrow \mathrm{mix}) \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{x}-\text { scale })\]Therefore the difference in the transfer chemical potentials can be traced to differences in the transfer entropies. At constant pressure, \[\begin{aligned} &-\Delta(\mathrm{aq} \rightarrow \text { mix }) \mathrm{S}_{\mathrm{j}}^{0}(\mathrm{~m}-\text { scale })=\mathrm{d} \Delta(\mathrm{aq} \rightarrow \operatorname{mix}) \mu_{\mathrm{j}}^{0} / \mathrm{dT} \\ &=-\Delta(\mathrm{aq} \rightarrow \operatorname{mix}) \mathrm{S}_{\mathrm{j}}^{0}(\mathrm{x}-\text { scale }) \\ &+\mathrm{v} \, \mathrm{R} \, \ln \left\{\left[1-\left[1-\left(\mathrm{M}_{1} / \mathrm{M}_{2}\right)\right] \,\left(\mathrm{w}_{2} \% / 10^{2}\right)\right\}\right\} \end{aligned}\]A similar argument notes that the masses of the solvents forming the mixed solvents are independent of pressure (at fixed temperature) Therefore the volumes of transfer on molality and mole fraction scales are equal. In summary (at fixed \(\mathrm{T}\) and \mathrm{p}\)), \[\begin{aligned} \Delta(\mathrm{aq} \rightarrow \operatorname{mix}) \mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{s} \ln ) &=\mathrm{H}_{\mathrm{j}}^{\infty}(\operatorname{mix})-\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq}) \\ &=-\mathrm{T}^{2} \,\left[\partial\left\{\Delta(\mathrm{aq} \rightarrow \operatorname{mix}) \mu_{\mathrm{j}}^{0}(\mathrm{sln} ; \mathrm{T}) / \mathrm{T}\right\} / \mathrm{dT}\right] \end{aligned}\]Further, for the isobaric partial molar heat capacities, \[\begin{aligned} \Delta(\mathrm{aq} \rightarrow \operatorname{mix}) \mathrm{C}_{\mathrm{pj}}^{\infty}(\mathrm{s} \ln ) &=\mathrm{C}_{\mathrm{pj}}^{\infty}(\mathrm{mix})-\mathrm{C}_{\mathrm{pj}}^{\infty}(\mathrm{aq}) \\ &=\left[\partial\left\{\Delta(\mathrm{aq} \rightarrow \mathrm{mix}) \mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{s} \ln ; \mathrm{T})\right\} / \partial \mathrm{T}\right]_{\mathrm{p}} \end{aligned}\]Also \[\Delta(\mathrm{aq} \rightarrow \operatorname{mix}) \mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{s} \ln )=\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{mix})-\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq})\]The procedures described above are repeated but now in a comparison of the molality and concentration scales.For a solute \(j\) (at fixed \(\mathrm{T}\) and \(\mathrm{p}\)) in a solution having ideal thermodynamic properties, the chemical potential of solute \(j\) is related to concentration of solute \(j\), \(\mathrm{c}_{j}\) which by convention is expressed in terms of amount of solute in \(1 \mathrm{~dm}^{3}\) of solution at defined \(\mathrm{T}\) and \(\mathrm{p}\); i.e. \(\mathrm{c}_{\mathrm{j}}=\left[\mathrm{mol} \mathrm{~dm} ^ {-3} \right]\). A reference concentration \(\mathrm{c}_{r}\) describes a solution where one \(\mathrm{dm}^{3}\) of solution contains one mole of solute. Because the volume of a liquid depends on both temperature and pressure, these variables must be specified. Thus \[\mu_{j}(a q)=\mu_{j}^{0}(c-s c a l e ; a q)+R \, T \, \ln \left[c_{j}(a q) / c_{r}\right]\]The units of both \(\mathrm{c}_{j}(\mathrm{aq})\) and \(\mathrm{c}_{r}\) are \(\left[\mathrm{mol dm}^{-3}\right]\).Hence using equations (i) and (xxiii), \[\begin{aligned} &\mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{~m} ; \mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{m}_{\mathrm{j}}(\mathrm{aq}) / \mathrm{m}^{0}\right] \\ &=\mu_{\mathrm{j}}^{0}(\mathrm{c}-\mathrm{scale} ; \mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{c}_{\mathrm{j}}(\mathrm{aq}) / \mathrm{c}_{\mathrm{r}}\right] \end{aligned}\]For a solution in \(10^{2} \mathrm{~kg}\) of solvent, \[\mathrm{m}_{\mathrm{j}}(\mathrm{aq})=\mathrm{n}_{\mathrm{j}} / 10^{2} \mathrm{~mol} \mathrm{~kg}{ }^{-1}\]For a dilute solution, density \[\rho(\mathrm{aq})=\rho_{1}^{*}(\ell)\]Volume of a dilute solution with mass \[10^{2} \mathrm{~kg}=10^{2} / \rho_{1}^{*}(\ell)\]Concentration, \[\mathrm{c}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} \, \rho_{\mathrm{l}}^{*}(\ell) / 10^{2}\]Therefore equation (xxiv) can be written in the following form. \[\begin{aligned} \mu_{\mathrm{j}}^{0}(\mathrm{~m} ; \mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{n}_{\mathrm{j}} / 10^{2} \, \mathrm{m}^{0}\right] \\ &=\mu_{\mathrm{j}}^{0}(\mathrm{c}-\mathrm{scale} ; \mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{n}_{\mathrm{j}} \, \rho_{1}^{*}(\ell) / \mathrm{c}_{\mathrm{r}}\right] \end{aligned}\]For the solution in a binary aqueous mixture, \[\begin{aligned} \mu_{\mathrm{j}}^{0}(\mathrm{~m} ; \mathrm{mix})+\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{n}_{\mathrm{j}} / 10^{2} \, \mathrm{m}^{0}\right] \\ &=\mu_{\mathrm{j}}^{0}(\mathrm{c}-\text { scale; mix })+\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{n}_{\mathrm{j}} \, \rho(\mathrm{mix}) / \mathrm{c}_{\mathrm{r}}\right] \end{aligned}\]Then, \[\begin{aligned} \Delta(\mathrm{aq} \rightarrow&\mathrm{mix}) \mu_{\mathrm{j}}^{0}(\mathrm{~m}) \\ &=\Delta(\mathrm{aq} \rightarrow \mathrm{mix}) \mu_{\mathrm{j}}^{0}(\mathrm{c}-\mathrm{scale})+\mathrm{R} \, \mathrm{T} \, \ln \left[\rho(\mathrm{mix}) / \rho_{\mathrm{1}}^{*}(\ell)\right] \end{aligned}\]In the event that solute is a salt which produces \(ν\) moles of ions for each mole of salt, \[\begin{aligned} \Delta(\mathrm{aq} \rightarrow&\mathrm{mix}) \mu_{\mathrm{j}}^{0}(\mathrm{~m}) \\ &=\Delta(\mathrm{aq} \rightarrow \mathrm{mix}) \mu_{\mathrm{j}}^{0}(\mathrm{c}-\text { scale })+\mathrm{v} \, \mathrm{R} \, \mathrm{T} \, \ln \left[\rho(\mathrm{mix}) / \rho_{1}^{*}(\ell)\right] \end{aligned}\]For each ionic substances, e.g. a cation \[\begin{aligned} \Delta(\mathrm{aq} \rightarrow&\operatorname{mix}) \mu_{+}^{0}(\mathrm{~m}) \\ &=\Delta(\mathrm{aq} \rightarrow \operatorname{mix}) \mu_{+}^{0}(\mathrm{c}-\mathrm{scale})+v \, \mathrm{R} \, \mathrm{T} \, \ln \left[\rho(\operatorname{mix}) / \rho_{1}^{*}(\ell)\right] \end{aligned}\]Because the densities of water and each mixture depends on temperature at fixed pressure, the transfer enthalpies on molality and concentration scales differ. Thus \[\begin{aligned} \Delta(\mathrm{aq} \rightarrow&\operatorname{mix}) \mathrm{H}_{+}^{\infty}(\mathrm{m}) \\ &=\Delta(\mathrm{aq} \rightarrow \operatorname{mix}) \mathrm{H}_{+}^{\infty}(\mathrm{c}-\mathrm{scale})-\mathrm{v} \, \mathrm{R} \, \mathrm{T}^{2} \,\left[\partial \ln \left[\rho(\mathrm{mix}) / \rho_{1}^{*}(\ell)\right] / \partial \mathrm{T}\right]_{\mathrm{p}} \end{aligned}\]This page titled 1.10.36: Gibbs Energies- Salt Solutions- Aqueous Mixtures is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.11.1: Gibbs-Duhem Equation

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.11%3A_Gibbs-Duhem_Equation/1.11.01%3A_Gibbs-Duhem_Equation

This equation is at the heart of chemical thermodynamics. The Gibbs energy of a closed system can be expressed as follows. \[\mathrm{G}=\mathrm{G}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{\mathrm{i}}\right]\]Here \(\mathrm{n}_{\mathrm{i}}\) represents the amounts of all chemical substances in the system. Then for a system containing two chemical substances, 1 and 2, \[\mathrm{G}=\mathrm{G}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{2}\right]\]If we prepare a system containing \(\mathrm{k} \, \mathrm{n}_{1}\) and \(\mathrm{k} \, \mathrm{n}_{2}\) moles of the two chemical substances [cf. Euler’s Theorem], the Gibbs energy increases by a factor, \(\mathrm{k}\). Hence, \[\mathrm{G}=\mathrm{n}_{1} \, \mu_{1}+\mathrm{n}_{2} \, \mu_{2}\]In general, \[\mathrm{G}=\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{n}_{\mathrm{j}} \, \mu_{\mathrm{j}}\]Equation (c) is differentiated. \[\mathrm{dG}=\mathrm{n}_{1} \, \mathrm{d} \mu_{1}+\mu_{1} \, \mathrm{dn} \mathrm{n}_{1}+\mathrm{n}_{2} \, \mathrm{d} \mu_{2}+\mu_{2} \, \mathrm{dn}_{2}\]But, \[\mathrm{dG}=-\mathrm{S} \, \mathrm{dT}+\mathrm{V} \, \mathrm{dp}+\mu_{1} \, \mathrm{dn}_{1}+\mu_{2} \, \mathrm{dn}_{2}\]Therefore, combining equations (c) and (f), \[-S \, d T+V \, d p-n_{1} \, d \mu_{1}-n_{2} \, d \mu_{2}=0\]Therefore, for a system held at fixed \(\mathrm{T}\) and \(\mathrm{p}\), \[\mathrm{n}_{1} \, \mathrm{d} \mu_{1}+\mathrm{n}_{2} \, \mathrm{d} \mu_{2}=0\]Equation (h) expresses the ‘communication’ between the two chemical substances in the system. In some senses equation (h) is a Happy Family Equation. If the chemical potential of substance 1, \(\mu_{1}\) increases [i.e. \(\mathrm{d} \mu_{1}> 0\)], the chemical potential of chemical substance 2 decreases, [i.e. \(\mathrm{d} \mu_{2}<0\)], in order to hold the condition expressed by equation (h).Footnotes Similarly \(\mathrm{U}=\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{n}_{\mathrm{j}} \, \mu_{\mathrm{j}}+\mathrm{T} \, \mathrm{S}-\mathrm{p} \, \mathrm{V}\) \(\mathrm{H}=\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{n}_{\mathrm{j}} \, \mu_{\mathrm{j}}+\mathrm{T} \, \mathrm{S}\) and \(F=\sum_{j=1}^{j=i} n_{j} \, \mu_{j}-p \, V\) Generally, \(\mathrm{dG}=\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}}\left[\mathrm{n}_{\mathrm{j}} \, \mathrm{d} \mu_{\mathrm{j}}+\mu_{\mathrm{j}} \, \mathrm{dn} \mathrm{n}_{\mathrm{j}}\right]\) Generally \(\mathrm{dG}=-\mathrm{S} \, \mathrm{dT}+\mathrm{V} \, \mathrm{dp}+\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mu_{\mathrm{j}} \, \mathrm{dn}_{\mathrm{j}}\( Generally \(\sum_{j=1}^{j=i} n_{j} \, d \mu_{j}=0\) If one member of a family is sad for some reason, other members of the family say something along the lines, ‘cheer up --it is not as bad as all that’. If another member of the family becomes over-excited, the family says something along the lines, ‘calm down’. Similar things happen to chemical substances in a closed system at fixed \(\mathrm{T}\) and \(\mathrm{p}\).This page titled 1.11.1: Gibbs-Duhem Equation is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.11.2: Gibbs-Duhem Equation- Salt Solutions- Osmotic and Activity Coefficients

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.11%3A_Gibbs-Duhem_Equation/1.11.02%3A_Gibbs-Duhem_Equation-_Salt_Solutions-_Osmotic_and_Activity_Coefficients

We consider an aqueous solution prepared using \(1 \mathrm{~kg}\) of solvent, water, at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\) (\(\cong p^{0}\)). For an aqueous salt solution, the chemical potential \(\mu_{j}(\mathrm{aq})\) for salt \(j\) at molality \(\mathrm{m}_{j}\) is given by equation (a) where \(\gamma_{\pm}\) is the mean ionic activity coefficient of the salt. \[\mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{v} \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{Q} \, \mathrm{m}_{\mathrm{j}} \, \gamma_{\pm} / \mathrm{m}^{0}\right)\]By definition, at all \(\mathrm{T}\) and \(\mathrm{p}\), \[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{\pm}=1\]For the solvent, water, \[\mu_{1}(\mathrm{aq})=\mu_{1}^{\star}(\ell)-\mathrm{v} \, \mathrm{R} \, \mathrm{T} \, \phi \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\]By definition, at all \(\mathrm{T}\) and \(\mathrm{p}\), \[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi=1.0\]Chemical potentials \(\mu_{j}(\mathrm{aq})\) and \(\mu_{1}(\mathrm{aq})\) are linked by the Gibbs-Duhem equation. This, \[\left(1 / M_{1}\right) \, d \mu_{1}(a q)+v \, m_{j} \, d \mu_{j}(a q)=0\]\[\begin{aligned} &\left(1 / \mathrm{M}_{1}\right) \, \mathrm{d}\left[\mu_{1}^{*}(\ell)-\mathrm{v} \, \mathrm{R} \, \mathrm{T} \, \phi \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\right] \\ &\quad+\mathrm{v} \, \mathrm{m}_{\mathrm{j}} \, \mathrm{d}\left[\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{v} \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{Q} \, \mathrm{m}_{\mathrm{j}} \, \gamma_{\pm} / \mathrm{m}^{0}\right]=0\right. \end{aligned}\]\[\begin{aligned} &-v \, R \, T \, d\left[\phi \, m_{j}\right] \\ &+v \, m_{j} \, v \, R \, T \, d\left[\ln (Q)+\ln \left(m_{j}\right)+\ln \left(\gamma_{\pm}\right)-\ln \left(m^{0}\right)\right]=0 \end{aligned}\]\[-\mathrm{d}\left[\phi \, \mathrm{m}_{\mathrm{j}}\right]+\mathrm{v} \, \mathrm{m}_{\mathrm{j}} \,\left\{\mathrm{d} \ln \left(\mathrm{m}_{\mathrm{j}}\right)+\mathrm{d} \ln \left(\gamma_{\pm}\right)\right\}=0\]\[-\phi \, d m_{j}-m_{j} \, d \phi+v \, m_{j} \, d \ln \left(m_{j}\right)+v \, m_{j} \, d \ln \left(\gamma_{\pm}\right)=0\]Then, \[-\phi \, d m_{j}-m_{j} \, d \phi+v \, d m_{j}+v \, m_{j} \, d \ln \left(\gamma_{\pm}\right)=0\]\[v \, m_{j} \, d \ln \left(\gamma_{\pm}\right)=\phi \, d m_{j}-v \, d m_{j}+m_{j} \, d \phi\]Or, \[\mathrm{d} \ln \left(\gamma_{\pm}\right)=(\phi-v) \, \frac{\mathrm{dm}_{\mathrm{j}}}{\mathrm{v} \, \mathrm{m}_{\mathrm{j}}} + \frac{\mathrm{d} \phi}{\mathrm{v}}\]For a solute where one mole of pure solute forms one mole of solute in solution, \[\mathrm{d} \ln \left(\gamma\right)=(\phi-1) \, \frac{\mathrm{dm}_{\mathrm{j}}}{\mathrm{m}_{\mathrm{j}}} + \mathrm{d} \phi\]Then, \[\ln (\gamma)=(\phi-1) \,+\int_{0}^{\mathrm{m}(\mathrm{j})}(\phi-1) \, \mathrm{d} \ln \left(\mathrm{m}_{\mathrm{j}}\right)\]This page titled 1.11.2: Gibbs-Duhem Equation- Salt Solutions- Osmotic and Activity Coefficients is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.11.3: Gibbs-Duhem Equation- Solvent and Solutes- Aqueous Solutions

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.11%3A_Gibbs-Duhem_Equation/1.11.03%3A_Gibbs-Duhem_Equation-_Solvent_and_Solutes-_Aqueous_Solutions

A given aqueous solution at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\) (which is close to the standard pressure) is prepared using water (\(1 \mathrm{~kg}\)) and \(\mathrm{m}_{\mathrm{j}}\) moles of solute-\(\mathrm{j}\). The chemical potential of solute–j is given by equation (a) where \(\mu_{\mathrm{j}}^{0}(\mathrm{aq})\) is the chemical potential of solute-\(\mathrm{j}\) in an aqueous solutions where \(\mathrm{m}_{\mathrm{j}} = 1 \mathrm{~mol kg}^{-1}\) and the thermodynamic properties of the solution are ideal. \[\mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)\]At all \(\mathrm{T}\) and \(\mathrm{p}\), \[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{\mathrm{j}}=1\]The chemical potential of solvent, water \(\mu_{1}(\mathrm{aq})\) is given by equation (c) where \(\mu_{1}^{*}(\ell)\) is the chemical potential of water(\(\ell\)) at the same \(\mathrm{T}\) and \(\mathrm{p}\); \(\phi\) is the molal osmotic coefficient. \[\mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\ell)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\]At all \(\mathrm{T}\) and \(\mathrm{p}\), \[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi=1\]For a solution at fixed \(\mathrm{T}\) and \(\mathrm{p}\), the Gibbs-Duhem equation relates \(\mu_{1}(\mathrm{aq})\) and \(\mu_{\mathrm{j}}(\mathrm{aq})\) using equation (e). \[\left(1 / \mathrm{M}_{1}\right) \, \mathrm{d} \mu_{1}(\mathrm{aq})+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \mu_{\mathrm{j}}(\mathrm{aq})=0\]Therefore \[\begin{aligned} &\left(1 / \mathrm{M}_{1}\right) \, \mathrm{d}\left[\mu_{1}^{*}(\ell)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\right] \\ &+\mathrm{m}_{\mathrm{j}} \, \mathrm{d}\left[\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)\right]=0 \end{aligned}\]Thus, \[\mathrm{d}\left(-\phi \, \mathrm{m}_{\mathrm{j}}\right)+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}}\right)=0\]\[-\phi \, d m_{j}-m_{j} \, d \phi+m_{j} \, d \ln \left(m_{j}\right)+m_{j} \, d \ln \left(\gamma_{j}\right)=0\]From equation (h) (dividing by \(\mathrm{m}_{\mathrm{j}}\)) \[-\phi \, d \ln \left(m_{j}\right)-d \phi+d \ln \left(m_{j}\right)+d \ln \left(\gamma_{j}\right)=0\]\[\mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right)=\mathrm{d} \phi-(1-\phi) \, \mathrm{d} \ln \left(\mathrm{m}_{\mathrm{j}}\right)\]The latter equation is integrated between the limits ‘\(\mathrm{m}_{\mathrm{j}} = 0\)’ and \(\mathrm{m}_{\mathrm{j}}\); equivalent to limits ‘\(\phi =1\)’ and \(\phi\). \[\int_{0}^{m(j)} d \ln \left(\gamma_{j}\right)=\int_{\phi=1}^{\phi} \mathrm{d} \phi-\int_{0}^{m(j)}(1-\phi) \, d \ln \left(m_{j}\right)\]Then, \[-\ln \left(\gamma_{\mathrm{j}}\right)=(1-\phi)+\int_{0}^{\mathrm{m}(\mathrm{j})}(1-\phi) \, \mathrm{d} \ln \left(\mathrm{m}_{\mathrm{j}}\right)\]From equation (g) \[\mathrm{d}\left[\phi \, \mathrm{m}_{\mathrm{j}}\right]=\mathrm{m}_{\mathrm{j}} \, \mathrm{d}\left[\ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)\right]+\mathrm{m}_{\mathrm{j}} \, \mathrm{d}\left[\ln \left(\gamma_{\mathrm{j}}\right)\right]\]or, \[\mathrm{d}\left[\phi \, \mathrm{m}_{\mathrm{j}}\right]=\mathrm{m}^{0} \, \mathrm{d}\left[\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right]+\mathrm{m}_{\mathrm{j}} \, \mathrm{d}\left[\ln \left(\gamma_{\mathrm{j}}\right)\right]\]Following integration from ‘\(\mathrm{m}_{\mathrm{j}} = 0\)’ to \(\mathrm{m}_{\mathrm{j}}\), \[\phi \, \mathrm{m}_{\mathrm{j}}=\mathrm{m}_{\mathrm{j}}+\int_{0}^{\mathrm{m}_{\mathrm{j}}} \mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right)\]or, \[\phi=1+\left(1 / \mathrm{m}_{\mathrm{j}}\right) \, \int_{0}^{\mathrm{m}_{\mathrm{j}}} \mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right)\]Hence, \[\phi-1=\left(1 / \mathrm{m}_{\mathrm{j}}\right) \, \int_{0}^{\mathrm{m}_{\mathrm{j}}} \mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\mathrm{j}}\right)\]In other words, \((\phi-1)\) is related to the integral of \(\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \gamma_{\mathrm{j}}\) between the limits ‘\(\mathrm{m}_{\mathrm{j}} = 0\)’ and \(\mathrm{m}_{\mathrm{j}}\). Equation (q) marks the limit of the thermodynamics analysis. However we explore the significance of the equation by adopting an equation relating \(\phi\) and \(\mathrm{m}_{\mathrm{j}}\). Equation (r) signals one assumption in which \(\mathrm{r}\) is a parameter characteristic of the solution under examination. Thus \[\phi-1=\alpha \,\left(\mathrm{m}_{\mathrm{j}}\right)^{\mathrm{r}}\]Then \[-\ln \left(\gamma_{\mathrm{j}}\right)=(1-\phi) \,(1+\mathrm{r}) / \mathrm{r}\]Or, \[(1-\phi)=[r /(1+r)] \,\left\{-\ln \left(\gamma_{j}\right)\right\}\]From equation (m), \[\phi=1-\alpha \,\left(\mathrm{m}_{\mathrm{j}}\right)^{\mathrm{r}}\]If for example, \(\alpha > 1\), then \(\phi < 1\) for all solutions. According to equation (c), \[\mu_{1}(\mathrm{aq})-\mu_{1}^{*}(\ell)=-\left[1-\alpha \,\left(\mathrm{m}_{\mathrm{j}}\right)^{\mathrm{t}}\right] \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\]If therefore \(\phi>1, \mu_{1}(\mathrm{aq})<\mu_{1}^{*}(\ell)\); relative to the chemical potential of the pure solvent, the solvent in the solution is stabilised. For the solute according to equation (n), \[-\ln \left(\gamma_{\mathrm{j}}\right)=[(1+\mathrm{r}) / \mathrm{r}] \,\left[\alpha \,\left(\mathrm{m}_{\mathrm{j}}\right)^{\mathrm{r}}\right]\]Or, [Bjerrum’s Equation] \[-\ln \left(\gamma_{\mathrm{j}}\right)=[(1+\mathrm{r}) / \mathrm{r}] \,[1-\phi]\]Footnotes From equations (q) and (s), \[-\ln \left(\gamma_{\mathrm{j}}\right)=\alpha \,\left(\mathrm{m}_{\mathrm{j}}\right)^{\mathrm{r}}+\alpha \, \int_{0}^{\mathrm{m}(\mathrm{j})}\left(\mathrm{m}_{\mathrm{j}}\right)^{\mathrm{r}-1} \, d \mathrm{~m}_{\mathrm{j}}\]Then, \[\begin{aligned} -\ln \left(\gamma_{\mathrm{j}}\right) &=\alpha \,\left(\mathrm{m}_{\mathrm{j}}\right)^{\mathrm{r}}+\alpha \,\left(\mathrm{m}_{\mathrm{j}}\right)^{\mathrm{r}} / \mathrm{r} \\ &=\alpha \,\left(\mathrm{m}_{\mathrm{j}}\right)^{\mathrm{r}} \,[1+(1 / \mathrm{r})] \end{aligned}\]Hence, \(-\ln \left(\gamma_{\mathrm{j}}\right)=\alpha \,\left(\mathrm{m}_{\mathrm{j}}\right)^{\mathrm{r}} \,[1+\mathrm{r}] / \mathrm{r}\)This page titled 1.11.3: Gibbs-Duhem Equation- Solvent and Solutes- Aqueous Solutions is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.11.4: Gibbs-Duhem Equation- Aqueous Salt Solutions- Salt and Solvent- Debye-Huckel Limiting Law

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.11%3A_Gibbs-Duhem_Equation/1.11.04%3A_Gibbs-Duhem_Equation-_Aqueous_Salt_Solutions-_Salt_and_Solvent-_Debye-Huckel_Limiting_Law

A given aqueous salt solution at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\) (which is close to the standard pressure) is prepared using water (\(1 \mathrm{~kg}\)) and \(\mathrm{m}_{\mathrm{j}}\) moles of a 1:1 sa }\) and the thermodynamic properties of the solution are ideal. \[\mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+2 \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\pm} / \mathrm{m}^{0}\right)\]At all \(\mathrm{T}\) and \(\mathrm{p}\), \[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \gamma_{\pm}=1\]The chemical potential of solvent, water \(\mu_{1}(\mathrm{aq})\) is related to salt molality \(\mathrm{m}_{\mathrm{j}}\) by equation (c) where \(\mu_{1}^{*}(\ell)\) is the chemical potential of water(\(\ell\)) at the same \(\mathrm{T}\) and \(\mathrm{p}\) and \(\phi\) is the molal osmotic coefficient. \[\mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\ell)-2 \, \phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\]At all \(\mathrm{T}\) and \(\mathrm{p}\), \[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi=1\)For a solution at fixed \(\mathrm{T}\) and \(\mathrm{p}\), the Gibbs-Duhem equation relates \(\mu_{1}(\mathrm{aq})\) and \(\mu_{j}(\mathrm{aq})\) using equation (e). \[\left(1 / M_{1}\right) \, d \mu_{1}(a q)+m_{j} \, d \mu_{j}(a q)=0\]Therefore \[\begin{aligned} &\left(1 / \mathrm{M}_{1}\right) \, \mathrm{d}\left[\mu_{1}^{*}(\ell)-2 \, \phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\right] \\ &+\mathrm{m}_{\mathrm{j}} \, \mathrm{d}\left[\mu_{\mathrm{j}}^{0}(\mathrm{aq})+2 \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\pm} / \mathrm{m}^{0}\right)\right]=0 \end{aligned}\]\[\mathrm{d}\left(-\phi \, \mathrm{m}_{\mathrm{j}}\right)+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\pm}\right)=0\]At ‘\(\mathrm{m}_{\mathrm{j}} =0\)’, \(\phi\) is unity. Therefore integration of equation (g) yields equation (h) \[\phi=1+\frac{1}{m_{j}} \, \int_{0}^{m(j)} m_{j} \, d \ln \left(\gamma_{\pm}\right)\]We have an equation for \(\phi\) in terms of \(\mathrm{m}_{\mathrm{j}}\) and \(\gamma_{\pm}\). Equation (h) signals the limit of the thermodynamic analysis. To make progress we need an equation for \(\gamma_{\pm}\) in terms of \(\mathrm{m}_{\mathrm{j}}\). The Debye-Huckel Limiting Law provides such an equation having the form shown in equation (i) where \(\mathrm{S}_{\gamma}\) is positive and a function of temperature, pressure and relative permittivity of the solvent. Thus \[\ln \left(\gamma_{\pm}\right)=-S_{\gamma} \,\left(m_{j} / m^{0}\right)^{1 / 2}\]From equation (h), \[\phi=1-\frac{S_{\gamma}}{\left(m^{0}\right)^{1 / 2} \, m_{j}} \, \int_{0}^{m(j)} m_{j} \, d\left(m_{j}\right)^{1 / 2}\]Hence, \[\phi=1-\left(\mathrm{S}_{\gamma} / 3\right) \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}\)We examine the impact of equations (i) and (k) on the chemical potentials of solute and solvent. From equation (c), \[\mu_{1}(\mathrm{aq})-\mu_{1}^{*}(\ell)=-2 \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left[1-\left(\mathrm{S}_{\gamma} / 3\right) \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}\right]\]For a salt solution having thermodynamic properties which are ideal, \[\mu_{1}(\mathrm{aq} ; \mathrm{id})-\mu_{1}^{*}(\ell)=-2 \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\]Then, \[\mu_{1}(\mathrm{aq})-\mu_{1}(\mathrm{aq} ; \mathrm{id})=2 \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \,\left(\mathrm{S}_{\gamma} / 3\right) \,\left(\mathrm{m}^{0}\right)^{-1 / 2} \,\left(\mathrm{m}_{\mathrm{j}}\right)^{3 / 2}\]But \(\mathrm{S}_{\gamma}\) is positive. Hence \(\left[\mu_{1}(\mathrm{aq})-\mu_{1}(\mathrm{aq} ; \mathrm{id})\right]\) is positive so that in terms of the DHLL the chemical potential of water is raised above that in the corresponding solution having ideal thermodynamic properties.For the solute, equation (a) requires that \(\mu_{j}(\mathrm{aq})\) is given by equation (o). \[\mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+2 \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)+2 \, \mathrm{R} \, \mathrm{T} \, \ln \left(\gamma_{\pm}\right)\]Or, \[\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{id})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+2 \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)\]Then using equation (i), \[\mu_{\mathrm{j}}(\mathrm{aq})-\mu_{\mathrm{j}}(\mathrm{aq} ; \mathrm{id})=-2 \, \mathrm{R} \, \mathrm{T} \, \mathrm{S}_{\mathrm{\gamma}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}\]According therefore to the DHLL, salt j in a real solution is stabilised relative to that in an ideal solution. In other words according to the DHLL the salt is stabilised whereas the solvent is destabilised, the impact of ion-ion interactions on the Gibbs energy of a solution is moderated.Footnotes \(-\phi \, \mathrm{dm}_{\mathrm{j}}-\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \phi+\mathrm{m}_{\mathrm{j}} \, \mathrm{d}\left[\ln \left(\mathrm{m}_{\mathrm{j}}\right)\right]+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\pm}\right)=0\) \(-\phi \, \mathrm{dm}_{\mathrm{j}}-\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \phi+\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}_{\mathrm{j}}\right) \, \mathrm{dm} \mathrm{m}_{\mathrm{j}}+\mathrm{m}_{\mathrm{j}} \, \mathrm{d} \ln \left(\gamma_{\pm}\right)=0\)Then, \(-(\phi-1) \, d m_{j}-m_{j} \, d \phi+m_{j} \, d \ln \left(\gamma_{\pm}\right)=0\)Hence, \(-\int_{0}^{m(j)}(\phi-1) \, d m_{j}-\int_{0}^{m(j)} m_{j} \, d \phi=-\int_{0}^{m(j)} m_{j} \, d \ln \left(\gamma_{\pm}\right)\)Or, \(\phi=1+\frac{1}{m_{j}} \, \int_{0}^{m(j)} m_{j} \, d \ln \left(\gamma_{\pm}\right)\) Put \(\left(m_{j}\right)^{1 / 2}=x\); \(\int_{0}^{m(j)} x^{2} \, d x=x^{3} / 3=\left(m_{j}\right)^{3 / 2} / 3\)This page titled 1.11.4: Gibbs-Duhem Equation- Aqueous Salt Solutions- Salt and Solvent- Debye-Huckel Limiting Law is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.12.1: Heat Capacities- Isobaric- Solutions

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.12%3A_Expansions/1.12.01%3A_Heat_Capacities-_Isobaric-_Solutions

From the definition of enthalpy \(\mathrm{H}\), an infinitesimal small change in enthalpy is related to the corresponding change in thermodynamic energy \(\mathrm{dU}\) by equation (a). \[\mathrm{dH}=\mathrm{dU}+\mathrm{p} \, \mathrm{dV}+\mathrm{V} \, \mathrm{dp}\]If only ‘\(\mathrm{p}-\mathrm{V}\)’ work is involved, \[\mathrm{dU}=\mathrm{q}-\mathrm{p} \, \mathrm{dV}\]Then \[\mathrm{dH}=\mathrm{q}+\mathrm{V} \, \mathrm{dp}\]But in general terms, \[\mathrm{q}=\mathrm{C} \, \mathrm{dT}\]Here \(\mathrm{C}\) is the heat capacity of the system, an extensive variable. Hence for a change at constant pressure, \[\mathrm{dH}=\mathrm{C}_{\mathrm{p}} \, \mathrm{dT}\]Isobaric heat capacity is related to the change in enthalpy accompanying a change in temperature. \[\mathrm{C}_{p}=(\partial \mathrm{H} / \partial \mathrm{T})_{\mathrm{p}}\]\(\mathrm{C}_{\mathrm{p}}\) is an extensive variable; \(\mathrm{C}_{\mathrm{pm}}\) is the corresponding molar property. We develop the above analysis in a slightly different way in order to make an important point. We explore the relationship between the dependence of (\(\mathrm{G}/\mathrm{T}\)) on temperature atA calculus operation yields the following equation. \[\left[\frac{\partial(\mathrm{G} / \mathrm{T})}{\partial \mathrm{T}}\right]_{\mathrm{p}, \mathrm{A}}=\left[\frac{\partial(\mathrm{G} / \mathrm{T})}{\partial \mathrm{T}}\right]_{\mathrm{p}, \xi}-\frac{1}{\mathrm{~T}} \,\left[\frac{\partial \xi}{\partial \mathrm{A}}\right]_{\mathrm{T}, \mathrm{p}} \,\left[\frac{\partial \mathrm{A}}{\partial \mathrm{T}}\right]_{\mathrm{p}, \xi} \,\left[\frac{\partial \mathrm{G}}{\partial \xi}\right]_{\mathrm{T}, \mathrm{p}}\]But at equilibrium, \[A=-\left[\frac{\partial G}{\partial \xi}\right]_{\mathrm{T}, \mathrm{p}}=0\]Using the Gibbs-Helmholtz Equation, \[\mathrm{H}(\mathrm{A}=0)=\mathrm{H}\left(\xi^{\mathrm{eq}}\right)\]This result is expected because enthalpy \(\mathrm{H}\) is a strong state variable, a function of state which does not need a description of a pathway. This is not the case for isobaric heat capacities. Using the same calculus operation, \[\left[\frac{\partial \mathrm{H}}{\partial \mathrm{T}}\right]_{\mathrm{p}, \mathrm{A}}=\left[\frac{\partial \mathrm{H}}{\partial \mathrm{T}}\right]_{\mathrm{p}, \xi}-\left[\frac{\partial \xi}{\partial \mathrm{A}}\right]_{\mathrm{T}, \mathrm{p}} \,\left[\frac{\partial \mathrm{A}}{\partial \mathrm{T}}\right]_{\mathrm{p}, \xi} \,\left[\frac{\partial \mathrm{H}}{\partial \xi}\right]_{\mathrm{T}, \mathrm{p}}\]We cannot assume that the triple product term is zero. Hence there are two limiting isobaric heat capacities; the equilibrium isobaric heat capacity, \(\mathrm{C}_{\mathrm{p}}(\mathrm{A}=0)\) and the ‘frozen’ isobaric heat capacity, \(\mathrm{C}_{\mathrm{p}\left(\xi_{\mathrm{eq}\right)\). \[\mathrm{C}_{\mathrm{p}}(\mathrm{A}=0)=\mathrm{C}_{\mathrm{p}}\left(\xi^{\mathrm{eq}}\right)-\left[\frac{\partial \xi}{\partial \mathrm{A}}\right]_{\mathrm{T}, \mathrm{p}} \,\left[\frac{\partial \mathrm{A}}{\partial \mathrm{T}}\right]_{\mathrm{p}, \xi} \,\left[\frac{\partial \mathrm{H}}{\partial \xi}\right]_{\mathrm{T}, \mathrm{p}}\]In other words, the isobaric heat capacity is not a strong function of state. The property is concerned with a pathway between states. The term \left[-\left[\frac{\partial \xi}{\partial \mathrm{A}}\right]_{\mathrm{T}, \mathrm{p}} \,\left[\frac{\partial \mathrm{A}}{\partial \mathrm{T}}\right]_{\mathrm{p}, \xi} \,\left[\frac{\partial \mathrm{H}}{\partial \xi}\right]_{\mathrm{T}, \mathrm{p}}\right]\) is the relaxational isobaric heat capacity. \(\mathrm{C}_{\mathrm{p}}(\mathrm{A}=0)\), the equilibrium heat capacity, signals that when heat q passes into a system, the composition - organization of the system changes in order that the Gibbs energy of the system remains at a minimum. In contrast \(\mathrm{C}_{\mathrm{p}}\left(\xi_{\mathrm{eq}}\right)\), the frozen heat capacity, signals that no changes occur in the composition – organization in the system such that the Gibbs energy is displaced from the original minimum. Moreover the equilibrium isobaric heat capacity is always larger than the frozen isobaric heat capacity. Indeed we can often treat the extensive equilibrium property \(\mathrm{C}_{\mathrm{p}}(\mathrm{A}=0)\) as a function of state.Certainly isobaric heat capacities differentiate water as a solvent from other associated liquids such as \(\mathrm{H}_{2}\mathrm{O}_{2}\), and \(\mathrm{N}_{2}\mathrm{H}_{4}\) and low melting fused salts such as ethylammonium nitrate. Interestingly, among liquids, water has one of the highest heat capacitances; i.e. heat capacities per unit volume. Therefore hypothermia is often life threatening for babies and old persons because in order to raise their temperature a large amount of thermal energy has to be passed into the body in order to raise their temperature. This is often difficult without damaging the skin and other body tissues--- a consequence of humans being effectively concentrated aqueous systems.The isobaric heat capacity of a solution prepared using \(\mathrm{n}_{1}\) moles of solvent (water) and \(\mathrm{n}_{j}\) moles of solute \(j\) at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\) is defined by equation. \[\mathrm{C}_{\mathrm{p}}=\mathrm{C}_{\mathrm{p}}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}\right]\]We assume the system is at thermodynamic equilibrium such that the affinity for spontaneous change is zero at a minimum in Gibbs energy. The isobaric heat capacity of the solution is related to the composition using equation (m). \[\mathrm{C}_{\mathrm{p}}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{C}_{\mathrm{pl}}(\mathrm{aq})+\mathrm{n}_{\mathrm{j}} \, \mathrm{C}_{\mathrm{pj}}(\mathrm{aq})\]Here \(\mathrm{C}_{\mathrm{p}1}(\mathrm{aq})\) and \(\mathrm{C}_{\mathrm{pj}}(\mathrm{aq})\) are the partial molar isobaric heat capacities enthalpies of solvent and solute respectively. Alternatively \(\mathrm{C}_{\mathrm{p}}(\mathrm{aq})\) is given by equation (n) where \(\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)\) and \(\phi\left(\mathrm{C}_{\mathrm{pj}}\right)\) are the molar isobaric heat capacity of the pure solvent and the apparent molar isobaric heat capacity of the solute \(j\) respectively. Thus \[\mathrm{C}_{\mathrm{p}}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{C}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{C}_{\mathrm{pj}}\right)\]Footnotes M. Allen, D. F. Evans and R. Lumry, J. Solution Chem.,1985,14,549. M. Hadded, M. Biquard, P. Letellier and R. Schaal, Can. J. Chem.,1985,63,565.See J. K. Grime, in Analytical Solution Calorimetry, ed. J. K.Grime, Wiley, New York, 1985, chapter 1.For isochoric and isobaric heat capacities of liquids see, D. Harrison and E. A. Moelwyn-Hughes, Proc. R. Soc. London, Ser.A,1957, 239, 230.This page titled 1.12.1: Heat Capacities- Isobaric- Solutions is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.12.2: Heat Capacity- Isobaric- Partial Molar- Solution

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Equilibrium isobaric heat capacities of solutions can be treated for most purposes as extensive variables. Thus for an aqueous solution prepared using \(\mathrm{n}_{1}\) moles of solvent (water) and \(\mathrm{n}_{j}\) moles of a simple solute \(j\) the isobaric (equilibrium) heat capacity of the solution \(\mathrm{C}_{\mathrm{p}}(\mathrm{aq})\) can be related to the composition of the solution using equations (a) and (b). \[\mathrm{C}_{\mathrm{p}}(\mathrm{aq} ; \mathrm{A}=0)=\mathrm{n}_{1} \, \mathrm{C}_{\mathrm{p} 1}(\mathrm{aq})+\mathrm{n}_{\mathrm{j}} \, \mathrm{C}_{\mathrm{pj}}(\mathrm{aq})\]where \[\mathrm{C}_{\mathrm{pl}}(\mathrm{aq})=\left(\frac{\partial \mathrm{C}_{\mathrm{p}}(\mathrm{aq} ; \mathrm{A}=0)}{\partial \mathrm{n}_{1}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{j})} \quad \text { and } \mathrm{C}_{\mathrm{pj}}(\mathrm{aq})=\left(\frac{\partial \mathrm{C}_{\mathrm{p}}(\mathrm{aq} ; \mathrm{A}=0)}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{l})}\]Similar equations are encountered in a discussion of the partial molar enthalpies but with reference to these properties we develop a number of strategies because it is not possible to determine the enthalpy of a solution. In the present case the outlook is much more favourable because it is possible to measure isobaric heat capacities of solutions. The fact that we can measure the temperature dependence of the equilibrium enthalpy of a solution but not the actual enthalpy is an interesting philosophical point. Nevertheless it is informative to develop the analysis starting from equations relating partial molar enthalpies and compositions of solutions.A given aqueous solution is prepared using \(\mathrm{n}_{1}\) moles of solvent (water) and \(\mathrm{n}_{j}\) moles of solute. The partial molar enthalpies are related to the composition of the solution by the following equations. Thus \[\mathrm{H}_{1}(\mathrm{aq})=\mathrm{H}_{1}^{*}(\ell)+\mathrm{R} \, \mathrm{T}^{2} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,(\partial \phi / \partial \mathrm{T})_{\mathrm{p}}\]and \[\mathrm{H}_{\mathrm{j}}(\mathrm{aq})=\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})-\mathrm{R} \, \mathrm{T}^{2} \,\left[\partial \ln \left(\gamma_{\mathrm{j}}\right) / \partial \mathrm{T}\right)_{\mathrm{p}}\]At all \(\mathrm{T}\) and \(\mathrm{p}\), \[\operatorname{limit}\left(m_{j} \rightarrow 0\right) \phi=1 \text { and } \gamma_{j}=1\]By definition, \[\mathrm{C}_{\mathrm{p} 1}(\mathrm{aq})=\left(\frac{\partial \mathrm{C}_{\mathrm{p}}}{\partial \mathrm{n}_{1}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{j})}=\left(\frac{\partial \mathrm{H}_{1}}{\partial \mathrm{T}}\right)_{\mathrm{p}}=\left(\frac{\partial^{2} \mathrm{H}}{\partial \mathrm{n}_{1} \, \partial \mathrm{T}}\right)_{\mathrm{p}, \mathrm{n}(\mathrm{j})}\]And, \[\mathrm{C}_{\mathrm{pj}}(\mathrm{aq})=\left(\frac{\partial \mathrm{C}_{\mathrm{p}}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}}=\left(\frac{\partial \mathrm{H}_{\mathrm{j}}}{\partial \mathrm{T}}\right)_{\mathrm{p}}=\left(\frac{\partial^{2} \mathrm{H}}{\partial \mathrm{n}_{\mathrm{j}} \, \partial \mathrm{T}}\right)_{\mathrm{p}, \mathrm{n}}\]The latter two equations trace the story from the enthalpy of the solution to partial molar isobaric heat capacities. Using equation (f) in conjunction with equation (c) we obtain an equation for dependence for \(\mathrm{C}_{\mathrm{p}1}(\mathrm{aq})\) on molality \(\mathrm{m}_{j}\). \[\mathrm{C}_{\mathrm{pl} 1}(\mathrm{aq})=\mathrm{C}_{\mathrm{pl} 1}^{*}(\ell)+2 \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,(\partial \phi / \partial \mathrm{T})_{\mathrm{p}}+\mathrm{R} \, \mathrm{T}^{2} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \,\left(\partial^{2} \phi / \partial \mathrm{T}^{2}\right)_{\mathrm{p}}\]Similarly using equations (d) and (g), we obtain an equation relating \(\mathrm{C}_{\mathrm{p}j}(\mathrm{aq})\) and molality \(\mathrm{m}_{j}\); the origin of the two minus signs is the Gibbs - Helmholtz Equation. \[\mathrm{C}_{\mathrm{pj}}(\mathrm{aq})=\mathrm{C}_{\mathrm{pj}}^{\infty}(\mathrm{aq})-2 \, \mathrm{R} \, \mathrm{T} \,\left[\partial \ln \left(\gamma_{\mathrm{j}}\right) / \partial \mathrm{T}\right]_{\mathrm{p}}-\mathrm{R} \, \mathrm{T}^{2} \,\left[\partial^{2} \ln \left(\gamma_{\mathrm{j}}\right) / \partial \mathrm{T}^{2}\right]_{\mathrm{p}}\]We consider a solution prepared using \(1 \mathrm{~kg}\) of water and \(\mathrm{m}_{j}\) moles of solute \(j\). \[\mathrm{C}_{\mathrm{p}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{C}_{\mathrm{pl}}(\mathrm{aq})+\mathrm{m}_{\mathrm{j}} \, \mathrm{C}_{\mathrm{pj}} \text { (aq) }\]If the thermodynamic properties of the solution are ideal, from the definitions of both practical osmotic coefficient \(\phi\) and activity coefficient \(\gamma_{j}\), the last two terms in equations (h) and (i) are zero. \[\mathrm{C}_{\mathrm{p}}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{C}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{m}_{\mathrm{j}} \, \mathrm{C}_{\mathrm{pj}}^{\infty}(\mathrm{aq})\]This is an interesting equation because two experiments yield \(\mathrm{C}_{\mathrm{p}}(\mathrm{aq} ; \mathrm{id})\) and \(\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)\). Hence, granted the ideal conditions, we obtain an estimate of \(\mathrm{C}_{\mathrm{pj}}^{\infty}(\mathrm{aq})\), limiting isobaric heat capacity of solute \(j\) in solution. Unfortunately the assumption concerning ideal properties of a solution is often unrealistic. Nevertheless equation (k) offers a reference against which we can examine the properties of real solutions.Footnotes An important technological development was the design of the Picker flow calorimeter; P. Picker, P.-A. Leduc, P. R. Philip and J. E. Desnoyers, J. Chem. Thermodyn.,1971, 3,631. For details of calibration of the Picker calorimeter; D. E.White and R. H. Ward, J. Solution Chem.,1982,11,223. For extension to measurement of thermal expansion coefficients; J. F. Alary, M. N. Simard, J. Dumont and C. Jolicoeur, J. Solution Chem.,1982,11,755. \(\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~kg}^{-1}\right]=\left[\mathrm{kg} \, \mathrm{mol}^{-1}\right]^{-1} \,\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right]+\left[\operatorname{mol~kg}{ }^{-1}\right] \,\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right]\)This page titled 1.12.2: Heat Capacity- Isobaric- Partial Molar- Solution is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.12.3: Heat Capacities- Isobaric and Isochoric

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When heat \(\mathrm{q}\) passes smoothly (reversibly) into a closed system from the surroundings, the temperature of the system increases (if there are no phase changes, e.g. liquid to vapour). The increase in temperature \(\Delta \mathrm{T}\) is related to heat \(\mathrm{q}\) using equation (a). \[\mathrm{q}=\mathrm{C} \, \Delta \mathrm{T}\]Heat capacity \(\mathrm{C}\) is an extensive property of a system whereas \(\Delta \mathrm{T}\) is the change in an intensive variable. For a given amount of heat, a more dramatic increase in temperature is produced the lower is the heat capacity \(\mathrm{C}\). Moreover as defined by equation (a) the heat capacity of a system is not a thermodynamic function of state because heat capacity describes a pathway accompanying a change in temperature. Hence, we define precisely the pathway taken by the system. Two important classes of heat capacities areIsochoric and isobaric heat capacities are related to the isobaric expansions \(\mathrm{E}_{\mathrm{p}}\) and isothermal compression \(\mathrm{KT}\) using equation (b). \[\mathrm{C}_{\mathrm{V}}=\mathrm{C}_{\mathrm{p}}-\mathrm{T} \,\left(\mathrm{E}_{\mathrm{p}}\right)^{2} / \mathrm{K}_{\mathrm{T}}\]Heat capacities and compressions are simply related. \[\mathrm{K}_{\mathrm{T}} / \mathrm{K}_{\mathrm{S}}=\mathrm{C}_{\mathrm{p}} / \mathrm{C}_{\mathrm{V}}\]Footnotes According to a calculus operation, the dependences of entropy on temperature at constant volume and constant pressure are related. \(\left(\frac{\partial \mathrm{S}}{\partial \mathrm{T}}\right)_{\mathrm{V}}=\left(\frac{\partial \mathrm{S}}{\partial \mathrm{T}}\right)_{\mathrm{p}}-\left(\frac{\partial \mathrm{S}}{\partial \mathrm{p}}\right)_{\mathrm{T}} \,\left(\frac{\partial \mathrm{p}}{\partial \mathrm{V}}\right)_{\mathrm{T}} \,\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{p}}\) A Maxwell equation requires that \(\left(\frac{\partial S}{\partial p}\right)_{T}=-\left(\frac{\partial V}{\partial T}\right)_{p}\) Hence, \[\left(\frac{\partial \mathrm{S}}{\partial \mathrm{T}}\right)_{\mathrm{V}}=\left(\frac{\partial \mathrm{S}}{\partial \mathrm{T}}\right)_{\mathrm{p}}+\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{p}} \,\left(\frac{\partial \mathrm{p}}{\partial \mathrm{V}}\right)_{\mathrm{T}} \,\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{p}}\]But the isobaric expansion, \(E_{p}=\left(\frac{\partial V}{\partial T}\right)_{p}\) And the isothermal compression, \(\mathrm{K}_{\mathrm{T}}=-\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{T}}\) From the Gibbs –Helmholtz equation, \(\mathrm{C}_{\mathrm{V}}=\mathrm{T} \,\left(\frac{\partial \mathrm{S}}{\partial \mathrm{T}}\right)_{\mathrm{V}}\) And \(C_{p}=T \,\left(\frac{\partial S}{\partial T}\right)_{p}\) Then \[C_{V}=C_{p}-T \,\left(E_{p}\right)^{2} / K_{T}\]The latter equation is correct under the condition of either ‘at constant affinity \(\mathrm{A}\)’ or ‘at constant composition’. The starting point is the following equation. \[\begin{array}{r} \left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{T}}=-\left(\frac{\partial \mathrm{T}}{\partial \mathrm{p}}\right)_{\mathrm{V}} \,\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{p}}=-\left(\frac{\partial \mathrm{T}}{\partial \mathrm{p}}\right)_{\mathrm{V}} \,\left(\frac{\partial \mathrm{V}}{\partial \mathrm{S}}\right)_{\mathrm{p}} \,\left(\frac{\partial \mathrm{S}}{\partial \mathrm{T}}\right)_{\mathrm{p}} \\ \left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{S}}=-\left(\frac{\partial \mathrm{S}}{\partial \mathrm{p}}\right)_{\mathrm{V}} \,\left(\frac{\partial \mathrm{V}}{\partial \mathrm{S}}\right)_{\mathrm{p}}=-\left(\frac{\partial \mathrm{T}}{\partial \mathrm{p}}\right)_{\mathrm{v}} \,\left(\frac{\partial \mathrm{V}}{\partial \mathrm{S}}\right)_{\mathrm{p}} \,\left(\frac{\partial \mathrm{S}}{\partial \mathrm{T}}\right)_{\mathrm{V}} \end{array}\]Then \((\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{T}} /(\partial \mathrm{V} / \partial \mathrm{p})_{\mathrm{S}}=(\partial \mathrm{S} / \partial \mathrm{T})_{\mathrm{p}} /(\partial \mathrm{S} / \partial \mathrm{T})_{\mathrm{V}}\) Hence, \[\mathrm{K}_{\mathrm{T}} / \mathrm{K}_{\mathrm{S}}=\mathrm{C}_{\mathrm{p}} / \mathrm{C}_{\mathrm{V}}\]This page titled 1.12.3: Heat Capacities- Isobaric and Isochoric is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.12.4: Heat Capacities- Isobaric- Equilibrium and Frozen

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An aqueous solution is prepared using \(\mathrm{n}_{1}\) moles of water(\(\ell\)) and \(\mathrm{n}_{j}\) moles of solute at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\). The system is at equilibrium where the affinity for spontaneous change is zero. Hence, \[\mathrm{H}^{\mathrm{eq}}=\mathrm{H}^{\mathrm{eq}}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}, \mathrm{A}=0, \xi^{\mathrm{eq}}\right]\]Heat capacities describe a pathway. There are two limiting pathways. The system can be displaced either to a nearby state along a pathway for which \(\xi\) is constant or along a pathway for which the affinity for spontaneous change is constant. The accompanying differential changes in enthalpies are unlikely to be the same. In fact they are related using a calculus procedure. \[\left(\frac{\partial \mathrm{H}}{\partial \mathrm{T}}\right)_{\mathrm{A}, \mathrm{p}}=\left(\frac{\partial \mathrm{H}}{\partial \mathrm{T}}\right)_{\xi, \mathrm{p}}-\left(\frac{\partial \mathrm{A}}{\partial \mathrm{T}}\right)_{\xi, \mathrm{p}} \,\left(\frac{\partial \xi}{\partial \mathrm{A}}\right)_{\mathrm{T}, \mathrm{p}} \,\left(\frac{\partial \mathrm{H}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}\]If the original state was an equilibrium state we write this equation in the following form which incorporates an equation for the dependence of affinity \(\mathrm{A}\) on temperature at equilibrium. \[\left(\frac{\partial \mathrm{H}}{\partial \mathrm{T}}\right)_{\mathrm{A}=0, \mathrm{p}}=\left(\frac{\partial \mathrm{H}}{\partial \mathrm{T}}\right)_{\xi \mathrm{eq}, \mathrm{p}}-\frac{1}{\mathrm{~T}} \,\left[\left(\frac{\partial \mathrm{H}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}^{\mathrm{eq}}\right]^{2} \,\left(\frac{\partial \xi}{\partial \mathrm{A}}\right)_{\mathrm{T}, \mathrm{p}}^{\mathrm{eq}}\]Then from the definition of isobaric heat capacity, \[\mathrm{C}_{\mathrm{p}}(\mathrm{A}=0)=\mathrm{C}_{\mathrm{p}}\left(\xi^{\mathrm{eq}}\right)-\frac{1}{\mathrm{~T}} \,\left[\left(\frac{\partial \mathrm{H}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}^{\mathrm{eq}}\right]^{2} \,\left(\frac{\partial \xi}{\partial \mathrm{A}}\right)_{\mathrm{T}, \mathrm{p}}^{\mathrm{eq}}\]Here \((\partial \mathrm{H} / \partial \xi)_{\mathrm{T}, \mathrm{p}}^{\mathrm{eq}}\), is the enthalpy of reaction. For a stable equilibrium state \((\partial \mathrm{A} / \partial \xi)_{\mathrm{T}, \mathrm{p}}^{\mathrm{eq}}\), is negative. Hence, \[\mathrm{C}_{\mathrm{p}}(\mathrm{A}=0)>\mathrm{C}_{\mathrm{p}}\left(\xi^{\mathrm{eq}}\right)\]Here \(\mathrm{C}_{\mathrm{p}}(\mathrm{A}=0)\) is the equilibrium heat capacity signalling that when heat q passes into the system the composition/organisation of the system changes in order that the Gibbs energy of the system remains at a minimum. In contrast \(\mathrm{C}_{\mathrm{p}}\left(\xi^{\mathrm{eq}}\right)\) is the frozen capacity signalling that no changes occur in the composition/organisation in the system such that the Gibbs energy of the system is displaced from the original minimum. Moreover, equation (e) shows that the equilibrium isobaric heat capacity is always larger than the frozen isobaric heat capacity. Indeed we can often treat the extensive equilibrium property \(\mathrm{C}_{\mathrm{p}}(\mathrm{A}=0)\) as a function of state (although it is not).Footnote \(\frac{1}{T} \,\left[\left(\frac{\partial \mathrm{H}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}^{\mathrm{eq}}\right]^{2} \,\left(\frac{\partial \xi}{\partial \mathrm{A}}\right)_{\mathrm{T}, \mathrm{p}}=\frac{1}{[\mathrm{~K}]} \,\left[\frac{\mathrm{J}}{\mathrm{mol}}\right]^{2} \,\left[\frac{\mathrm{mol}}{\mathrm{J} \mathrm{mol}^{-1}}\right]=\left[\mathrm{J} \mathrm{K}^{-1}\right]\)This page titled 1.12.4: Heat Capacities- Isobaric- Equilibrium and Frozen is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.12.5: Heat Capacity- Isobaric- Solutions- Excess

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A given solution is prepared using \(1 \mathrm{~kg}\) of solvent (water) and \(\mathrm{m}_{j}\) moles of solute \(j\). If the thermodynamic properties of this solution are ideal, the isobaric heat capacity can be expressed as follows. \[\mathrm{C}_{\mathrm{p}}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{C}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{m}_{\mathrm{j}} \, \mathrm{C}_{\mathrm{pj}}^{\infty}(\mathrm{aq})\]On the other hand for a real solution the isobaric heat capacity can be expressed in terms of the apparent molar heat capacity of the solute, \(\phi \left(\mathrm{C}_{\mathrm{pj}}\right)\). \[\mathrm{C}_{\mathrm{p}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{C}_{\mathrm{pl}}^{*}(\ell)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{C}_{\mathrm{pj}}\right)\]The difference between \(\mathrm{C}_{\mathrm{p}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)\) and \(\mathrm{C}_{\mathrm{p}}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)\) defines the relative isobaric heat capacity of the solution \(\mathrm{J}\), an excess property. \[\mathrm{J}(\mathrm{aq})=\mathrm{C}_{\mathrm{p}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)-\mathrm{C}_{\mathrm{p}}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)\]Thermodynamics does not define the magnitude or sign of \(\mathrm{J}(\mathrm{aq})\). However, from the definitions of ideal and real partial molar isobaric capacities of solvent and solute, the following condition must hold. \[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{J}(\mathrm{aq})=0\]Relative quantities can also be defined for solute and solvent. \[\mathrm{J}_{j}(\mathrm{aq})=\mathrm{C}_{\mathrm{pj}}(\mathrm{aq})-\mathrm{C}_{\mathrm{pj}}^{\infty}(\mathrm{aq})\]\[\mathrm{J}_{1}(\mathrm{aq})=\mathrm{C}_{\mathrm{p} 1}(\mathrm{aq})-\mathrm{C}_{\mathrm{p} 1}^{*}(\ell)\]Also, \[\phi\left(\mathrm{J}_{\mathrm{j}}\right)=\phi\left[\mathrm{C}_{\mathrm{pj}}(\mathrm{aq})\right]-\mathrm{C}_{\mathrm{pj}}^{\infty}(\mathrm{aq})\]Hence, \[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{J}_{\mathrm{j}}(\mathrm{aq})=\mathrm{J}_{1}(\mathrm{aq})=\phi\left(\mathrm{J}_{\mathrm{j}}\right)=0\]Equation (c) defines a property \(\mathrm{J}\) which is an excess isobaric heat capacity of a solution prepared using \(1 \mathrm{~kg}\) of water. Thus, \[\mathrm{C}_{\mathrm{p}}^{\mathrm{E}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\mathrm{J}(\mathrm{aq})=\mathrm{C}_{\mathrm{p}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)-\mathrm{C}_{\mathrm{p}}\left(\mathrm{aq} ; \mathrm{id} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)\]From equations (a) and (b), \[\mathrm{C}_{\mathrm{p}}^{\mathrm{E}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\mathrm{m}_{\mathrm{j}} \,\left[\phi\left(\mathrm{C}_{\mathrm{pj}}\right)-\mathrm{C}_{\mathrm{pj}}^{\infty}(\mathrm{aq})\right]\]From equation (g), \[\mathrm{C}_{\mathrm{p}}^{\mathrm{E}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{J}_{\mathrm{j}}\right)\]Thus \(\phi \left(\mathrm{J}_{j}\right)\) is the relative apparent molar isobaric heat capacity of the solute in a given real solution. Isobaric heat capacities of solutions and related partial molar isobaric heat capacities reflect in characteristic fashion the impact of added solutes on water water interactionsFootnote \(\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~kg}^{-1}\right]=\left[\mathrm{kg} \mathrm{mol}^{-1}\right]^{-1} \,\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right]+\left[\mathrm{mol} \mathrm{kg}^{-1}\right] \,\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right]\)This page titled 1.12.5: Heat Capacity- Isobaric- Solutions- Excess is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.12.6: Heat Capacities- Isobaric- Dependence on Temperature

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.12%3A_Expansions/1.12.06%3A_Heat_Capacities-_Isobaric-_Dependence_on_Temperature

Interesting cases emerge where an equilibrium isobaric heat capacity reflects a change in composition as a consequence of the system changing composition in order to hold the system at equilibrium. The development of sensitive scanning calorimeters stimulated research in this subject, particularly with respect to biochemical research; e.g. multilamellar systems.We consider the case where a solution is prepared using \(\mathrm{n}_{1}\) moles of water and \(\mathrm{n}_{\mathrm{x}}\) moles of solute \(\mathrm{X}\). In solution at temperature \(\mathrm{T}\) (and fixed pressure \(\mathrm{p}\)) the following chemical equilibrium is established.\[\mathrm{X}(\mathrm{aq}) \Longrightarrow \mathrm{Y}(\mathrm{aq})\]By definition \(\alpha=\xi / \mathrm{n}_{\mathrm{x}}^{0}\), the degree of reaction forming substance \(\mathrm{Y}\) at equilibrium. Then \(\mathrm{n}_{\mathrm{x}}^{\mathrm{eq}}=\mathrm{n}_{\mathrm{x}}^{0} \,(1-\alpha)\) and \(\mathrm{n}_{\mathrm{y}}^{\mathrm{eq}}=\alpha \, \mathrm{n}_{\mathrm{x}}^{0}\).If \(\mathrm{w}_{1}\) is the mass of solvent, water(\(\ell\)), the equilibrium molalities are \(\mathrm{m}_{\mathrm{x}}^{0} \,(1-\alpha)\) for chemical substance \(\mathrm{X}\) and \(\alpha \, \mathrm{m}_{\mathrm{x}}^{0}\) for chemical substance \(\mathrm{Y}\). For the purposes of the arguments advanced here we assume that the thermodynamic properties of the solution are ideal. The equilibrium composition of the closed system at defined temperature and pressure is described by the equilibrium constant \(\mathrm{K}^{0}\). Then, \[\mathrm{K}^{0}=\alpha /(1-\alpha)\]Hence the (dimensionless and intensive) degree of reaction, \[\alpha=\mathrm{K}^{0} /\left(1+\mathrm{K}^{0}\right)\]Because \(\mathrm{K}^{0}\) is dependent on temperature then so is the degree of reaction. The extent to which an increase in temperature favours or disfavours formation of more \(\mathrm{Y}(\mathrm{aq})\) depends on the sign of the enthalpy of reaction, \(\Delta_{r}\mathrm{H}^{0}\). Thus, \[\frac{\mathrm{d} \alpha}{\mathrm{dT}}=\frac{\mathrm{K}^{0}}{\left(1+\mathrm{K}^{0}\right)^{2}} \,\left[\frac{\mathrm{d} \ln \mathrm{K}^{0}}{\mathrm{dT}}\right]\]The analysis at this point is considerably simplified if we assume that the limiting enthalpy of reaction, \(\Delta_{r}\mathrm{H}^{\infty}(\mathrm{aq})\) for the chemical reaction is independent of temperature (at pressure \(\mathrm{p}\)). Hence using the van’t Hoff equation, \[\frac{\mathrm{d} \alpha}{\mathrm{dT}}=\frac{1}{\mathrm{R} \, \mathrm{T}^{2}} \, \frac{\mathrm{K}^{0}}{\left(1+\mathrm{K}^{0}\right)} \, \Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\mathrm{aq})\]Thus the shift in the composition of the solution depends on the sign of the limiting enthalpy of reaction. If \(\Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\mathrm{aq})<0\), an increase in temperature favours an increase in the amount of \(\mathrm{X}(\mathrm{aq})\) at the expense of \(\mathrm{Y}(\mathrm{aq})\). At temperature \(\mathrm{T}\), the enthalpy of the solution is given by equation (e) where \(\mathrm{H}_{1}^{*}(\ell)\) is the molar enthalpy of the solvent. \[\mathrm{H}(\mathrm{aq} ; \mathrm{A}=0)=\mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\ell)+\mathrm{n}_{\mathrm{x}}^{0} \,(1-\alpha) \, \mathrm{H}_{\mathrm{x}}^{\infty}(\mathrm{aq})+\mathrm{n}_{\mathrm{x}}^{0} \, \alpha \, \mathrm{H}_{\mathrm{y}}^{\infty}(\mathrm{aq})\]or, \[\mathrm{H}(\mathrm{aq} ; \mathrm{A}=0)=\mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\ell)+\mathrm{n}_{\mathrm{x}}^{0} \, \mathrm{H}_{\mathrm{x}}^{\infty}(\mathrm{aq})+\alpha \, \mathrm{n}_{\mathrm{x}}^{0} \,\left[\mathrm{H}_{\mathrm{y}}^{\infty}(\mathrm{aq})-\mathrm{H}_{\mathrm{x}}^{\infty}(\mathrm{aq})\right]\]or, \[\mathrm{H}(\mathrm{aq} ; \mathrm{A}=0)=\mathrm{n}_{1} \, \mathrm{H}_{1}^{*}(\ell)+\mathrm{n}_{\mathrm{x}}^{0} \, \mathrm{H}_{\mathrm{x}}^{\infty}(\mathrm{aq})+\alpha \, \mathrm{n}_{\mathrm{x}}^{0} \, \Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\mathrm{aq})\]We assume that \(\Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\mathrm{aq})\) is independent of temperature together with the amount \(\mathrm{n}_{1}\). Then, \[\mathrm{C}_{\mathrm{p}}(\mathrm{aq} ; \mathrm{A}=0)=\left(\frac{\partial \mathrm{H}(\mathrm{aq}: \mathrm{A}=0)}{\partial \mathrm{T}}\right)_{\mathrm{p}}\]Hence, \[\mathrm{C}_{\mathrm{p}}(\mathrm{aq} ; \mathrm{A}=0)=\left\{\mathrm{n}_{1} \, \mathrm{C}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{n}_{\mathrm{x}}^{0} \, \mathrm{C}_{\mathrm{px}}^{\infty}(\mathrm{aq})\right\}+\Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\mathrm{aq}) \, \mathrm{n}_{\mathrm{x}}^{0}(\mathrm{~d} \alpha / \mathrm{dT})\]The terms in the { } brackets are not (formally) dependent on temperature and constitute a frozen contribution to \(\mathrm{C}_{\mathrm{p}}(\mathrm{aq}, \mathrm{A}=0)\), \(\mathrm{C}_{\mathrm{p}}(\mathrm{aq}: \xi)\). Then equations (d) and (i) yield an equation for \(\mathrm{C}_{\mathrm{p}}(\mathrm{aq}: \mathrm{A}=0)\) in terms of \(\left[\Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\mathrm{aq})\right]^{2}\). \[\mathrm{C}_{\mathrm{p}}(\mathrm{aq} ; \mathrm{A}=0)=\mathrm{C}_{\mathrm{p}}(\xi: \mathrm{aq})+\left[\frac{\left[\Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\mathrm{aq})\right]^{2}}{\mathrm{RT}^{2}} \, \frac{\mathrm{K}^{0}}{\left[1+\mathrm{K}^{0}\right]^{2}}\right] \, \mathrm{n}_{\mathrm{x}}^{0}\]in terms of one mole of chemical substance \(\mathrm{X}\) (i.e. \(\mathrm{n}_{\mathrm{x}}^{0}=1 \mathrm{~mol}\)), \[\mathrm{C}_{\mathrm{p}}(\mathrm{aq} ; \mathrm{A}=0)=\mathrm{C}_{\mathrm{p}}(\xi ; \mathrm{aq})+\left[\frac{\left[\Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\mathrm{aq})\right]^{2}}{\mathrm{R} \, \mathrm{T}^{2}} \, \frac{\mathrm{K}^{0}}{\left[1+\mathrm{K}^{0}\right]^{2}}\right]\]According to equation (k) a large equilibrium heat capacity is favoured by a high \(\mathrm{C}_{\mathrm{p}}(\xi ; \mathrm{aq})\) and a large enthalpy of reaction. The term \(\left[\Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\mathrm{aq})\right]^{2}\) ensures that irrespective of whether the reaction (as written) is exothermic or endothermic, \(\mathrm{C}_{\mathrm{p}}(\mathrm{A}=0)-\mathrm{C}_{\mathrm{p}}(\xi)\) is positive. The dependence of \(\left[\mathrm{C}_{\mathrm{p}}(\mathrm{aq}: \mathrm{A}=0)-\mathrm{C}_{\mathrm{p}}(\xi: \mathrm{aq})\right]\) on temperature forms a bell-shaped plot covering the range of temperatures when all added substance X is completely in the form of \(\mathrm{X}\) or of \(\mathrm{Y}\). The maximum in the bell occurs near the temperature at which \(\mathrm{K}^{0}\) is unity. If \(\mathrm{K}^{0}\) is unity, at this temperature \(\Delta_{r}\mathrm{G}^{0}\) is zero. In other words, at this temperature the reference chemical potentials of \(\mathrm{X}\) and \(\mathrm{Y}\), \(\mu^{0}(\mathrm{X})\) and \(\mu^{0}(\mathrm{Y})\) respectively are equal. Clearly therefore the temperature at the maximum in \(\mathrm{C}_{p}(\mathrm{~A}=0)-\mathrm{C}_{p}(\xi)\) is characteristic of the two solutes. Equation (k) forms the basis of the technique of differential scanning calorimetry (DSC) as applied to the investigation of the thermal stability of biologically important macromolecules. In the text book case, a plot of isobaric heat capacity against temperature forms a bell-shaped curve, the maximum corresponding to temperature at which the equilibrium constant for an equilibrium having the simple form discussed above is unity. The area under the curve yields the enthalpy change characterising the transition between the two forms \(\mathrm{X}\) and \(\mathrm{Y}\) of a single substance.The possibility exists that the temperature dependences of\(\mu^{0}(\mathrm{X})\) and \(\mu^{0}(\mathrm{Y})\) are such that the two plots intersect at two temperatures producing two maxima in the plot of \(\mathrm{C}_{\mathrm{p}}(\mathrm{A}=0)-\mathrm{C}_{\mathrm{p}}(\xi)\) against temperature.The patterns recorded by DSC scans for a \(\mathrm{X} \leftrightarrows \mathrm{Y}\) system can be understood in terms of the separate dependences of \(\left(\mu_{\mathrm{X}}^{0} / \mathrm{T}\right)\) and \(\left(\mu_{\mathrm{Y}}^{0} / \mathrm{T}\right)\) on temperature, where \(\mu^{0}(\mathrm{X})\) and \(\mu^{0}(\mathrm{Y})\) are the standard chemical potentials of substances \(\mathrm{X}\) and \(\mathrm{Y}\). The maximum in the recorded heat capacity occurs where the plots of \(\left(\mu_{\mathrm{X}}^{0} / \mathrm{T}\right)\) and \(\left(\mu_{\mathrm{Y}}^{0} / \mathrm{T}\right)\) against temperature cross. If these curves have a more complicated shape there is the possibility that they will cross at two temperatures. In fact this observation raises the possibility of identifying hot and cold denaturation of proteins using DSC. Similar extrema in isobaric heat capacities are recorded for gel-to-liquid transitions in vesicles.In more complex systems, the overall DSC scan can indicate the presence of domains in a macromolecule which undergo structural changes when the temperature is raised.Analysis of extrema in heat capacities becomes somewhat more complicated when two or more equilibria are coupled.Footnotes V. V. Plotnikov, J. M. Brandts, L.-N. Lin and J. F. Brandts, Anal. Biochem., 1997, 250,237. J. M. Sturtevant, Ann. Rev. Phys.Chem.,1987,38,463. S. Mabrey and J. M. Sturtevant, Proc. Natl. Acad., Sci. USA, 1976, 73, 3862. From equation (b) \[\begin{aligned} &\frac{\mathrm{d} \alpha}{\mathrm{dT}}=\frac{\mathrm{d}}{\mathrm{dT}}\left[\mathrm{K}^{0} \,\left(1+\mathrm{K}^{0}\right)^{-1}\right]=\left[\frac{1}{1+\mathrm{K}^{0}}-\frac{\mathrm{K}^{0}}{\left(1+\mathrm{K}^{0}\right)^{2}}\right] \, \frac{\mathrm{dK}}{\mathrm{dT}} \\ &=\frac{1}{\left(1+\mathrm{K}^{0}\right)^{2}} \, \frac{\mathrm{dK}^{0}}{\mathrm{dT}}=\frac{\mathrm{K}^{0}}{\left(1+\mathrm{K}^{0}\right)^{2}} \, \frac{1}{\mathrm{~K}^{0}} \, \frac{\mathrm{dK}^{0}}{\mathrm{dT}}=\frac{\mathrm{K}^{0}}{\left(1+\mathrm{K}^{0}\right)^{2}} \,\left[\frac{\mathrm{d} \ln \mathrm{K}^{0}}{\mathrm{dT}}\right] \end{aligned}\] \(\frac{\mathrm{d} \alpha}{\mathrm{dT}}=\frac{1}{\left[\mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\right]} \, \frac{1}{[\mathrm{~K}]^{2}} \, \frac{}{} \,\left[\mathrm{J} \mathrm{mol}^{-1}\right]=\frac{1}{[\mathrm{~K}]}\) \(\frac{\left[\Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\mathrm{aq})\right]^{2}}{\mathrm{R} \, \mathrm{T}^{2}} \, \frac{\mathrm{K}^{0}}{\left[1+\mathrm{K}^{0}\right]^{2}}=\frac{\left[\mathrm{J} \mathrm{mol}^{-1}\right]^{2}}{\left.\left[\mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\right] \mathrm{K}\right]^{2}} \, \frac{}{^{2}}=\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right]\) With \(\mathrm{h}=\Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\mathrm{aq})\), the second term in equation (k) can be written as follows \(\mathrm{y}=\frac{\mathrm{h}^{2}}{\mathrm{R} \, \mathrm{T}^{2}} \, \frac{\mathrm{K}^{0}}{\left(1+\mathrm{K}^{0}\right)^{2}}\). The pattern formed by the dependence of \(\mathrm{y}\) on temperature is given by \[\frac{\mathrm{dy}}{\mathrm{dT}}=\frac{\mathrm{h}^{2}}{\mathrm{R} \, \mathrm{T}^{2}} \,\left[\frac{1}{\left(1+\mathrm{K}^{0}\right)^{2}}-\frac{2 \, \mathrm{K}^{0}}{\left(1+\mathrm{K}^{0}\right)^{3}}\right] \frac{\mathrm{dK}^{0}}{\mathrm{dT}}-\frac{2 \, \mathrm{h}^{2} \, \mathrm{K}^{0}}{\mathrm{R} \, \mathrm{T}^{3} \,\left(1+\mathrm{K}^{0}\right)^{2}}\]Or, \[\frac{\mathrm{dy}}{\mathrm{dT}}=\frac{\mathrm{h}^{2} \, \mathrm{K}^{0}}{\mathrm{R} \, \mathrm{T}^{2} \,\left(1+\mathrm{K}^{0}\right)^{2}} \,\left[\frac{1-\mathrm{K}^{0}}{\left(1+\mathrm{K}^{0}\right)} \, \frac{\mathrm{d} \ln \left(\mathrm{K}^{0}\right)}{\mathrm{dT}}-\frac{2}{\mathrm{~T}}\right]\]Since \(\mathrm{d} \ln \left(\mathrm{K}^{0}\right) / \mathrm{dT}=\mathrm{h} / \mathrm{R} \, \mathrm{T}^{2}\), then \[\frac{\mathrm{dy}}{\mathrm{dT}}=\frac{\mathrm{h}^{2} \, \mathrm{K}^{0}}{\mathrm{R} \, \mathrm{T}^{3} \,\left(1+\mathrm{K}^{0}\right)^{2}} \,\left[\frac{1-\mathrm{K}^{0}}{\left(1+\mathrm{K}^{0}\right)} \, \frac{\mathrm{h}}{\mathrm{R} \, \mathrm{T}}-2\right]\]Hence the condition for an extremum in \(\mathrm{y}\) as a function of \(\mathrm{T}\) is \(\frac{1-\mathrm{K}^{0}}{\left(1+\mathrm{K}^{0}\right)} \, \frac{\mathrm{h}}{\mathrm{R} \, \mathrm{T}}-2=0\) Or \(\frac{1-\mathrm{K}^{0}}{\left(1+\mathrm{K}^{0}\right)}=\frac{2 \, \mathrm{R} \, \mathrm{T}}{\mathrm{h}}\) Then \[\mathrm{K}^{0}=\frac{1-(2 \, \mathrm{R} \, \mathrm{T} / \mathrm{h})}{1+(2 \, \mathrm{R} \, \mathrm{T} / \mathrm{h})}\]By definition \(h=\Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\mathrm{aq})\). Therefore if the magnitude of \(\Delta_{\mathrm{r}} \mathrm{H}^{\infty}(\mathrm{aq})\) is much larger than \(2 \, \mathrm{~R} \, \mathrm{T}\), the top of the bell shaped curve is reached at a temperature where \(\mathrm{K}^{0}\) is unity. In the general case, at approximately this temperature \(\mathrm{y}\) is a maximum. M. J. Blandamer, J. Burgess and J. M. W. Scott, Ann. Rep. Prog. Chem., Sect. C, Phys. Chem.,1985,82,77. J. M. Sturtevant, Thermodynamic Data for Biochemistry and Biotechnlogy, ed. H. J. Hinz, Springer-Verlag, Berlin,1974, pp. 349- 376. Th. Ackermann, Angew. Chem. Int. Ed. Engl.,1989, 28, 981. M. J. Blandamer, B. Briggs, M. D. Butt, M. Waters, P. M. Cullis, J. B. F. N. Engberts, D. Hoekstra and R. K. Mohanty, Langmuir, 1994, 10, 3488. M. J. Blandamer, B. Briggs, P. M. Cullis, J. B. F. N. Engberts and D. Hoekstra, J. Chem. Soc. Faraday Trans., 1994, 90, 1905. C. O. Pabo, R. T. Sauer, J. M. Sturtevant and M. Ptashne, Proc. Natl. Acad. .Sci., USA, 1979, 76, 1608. M. J. Blandamer, B. Briggs, P. M. Cullis, A. P. Jackson, A. Maxwell and R. J. Reece, Biochemistry, 1994, 33, 7510. M. J. Blandamer, J. Burgess and J. M. W. Scott, J. Chem. Soc. Faraday Trans.1, 1984, 80, 2881. G. J. Mains, J. W. Larson and L. G. Hepler, J.Phys.Chem.,1985,88,1257.This page titled 1.12.6: Heat Capacities- Isobaric- 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.

1.12.7: Expansions- Isobaric and Isentropic

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.12%3A_Expansions/1.12.07%3A_Expansions-_Isobaric_and_Isentropic

The volume of a closed system at thermodynamic equilibrium containing two chemical substances is defined by equation (a). \[\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{2}, \mathrm{~A}=0, \xi^{\mathrm{eq}}\right]\]At temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\), the chemical composition / organisation \(\xi^{\mathrm{eq}}\) corresponds to the state where the affinity for spontaneous change is zero.The system is displaced by a change in temperature to a neighbouring state where the affinity for spontaneous change is also zero; the organisation/composition changes to \(\xi^{\mathrm{eq}}(\mathrm{new})\). \[\mathrm{V}(\text { new })=\mathrm{V}\left[\mathrm{T}(\text { new }), \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{2}, \mathrm{~A}=0, \xi^{\mathrm{eq}}(\text { new })\right]\]The differential dependence on temperature of the volume defined in equation. (a) is the equilibrium isobaric thermal expansion, \(\mathrm{E}_{\mathrm{p}}(\mathrm{A}=0)\). \[\mathrm{E}_{\mathrm{p}}(\mathrm{A}=0)=\left(\frac{\partial \mathrm{V}}{\partial T}\right)_{\mathrm{p}, \mathrm{A}=0}\]The chemical composition/organisation changes to hold the affinity for spontaneous change at zero. Indeed the perturbation in the form of a change in temperature might have to be extremely slow so that the change in organisation/chemical composition keeps in step with the change in temperature.The isobaric expansion \(\mathrm{E}_{\mathrm{p}}(\mathrm{aq})\) for an aqueous solution containing solute \(j\) is related to the partial molar isobaric expansions of solute and solvent; equation (d). \[\mathrm{E}_{\mathrm{p}}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{E}_{\mathrm{pl}}(\mathrm{aq})+\mathrm{n}_{\mathrm{j}} \, \mathrm{E}_{\mathrm{pj}}(\mathrm{aq})\]Alternatively using the concept of an apparent molar property, we define an (equilibrium) apparent molar isobaric expansion for solute \(j\), \(\phi\left(E_{p j}\right)\). \[\mathrm{E}_{\mathrm{p}}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{E}_{\mathrm{pl} 1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right)\]\[\phi\left(\mathrm{E}_{\mathrm{pj}}\right)=\left[\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right) / \partial \mathrm{T}\right]_{\mathrm{p}}\]Generally little interest has been shown in either partial molar or apparent molar isentropic expansions of solutes. Complications are encountered in understanding isentropic expansions without the redeeming feature of practical accessibility via an analogue of the Newton -Laplace equation. The isentropic expansions \(\mathrm{E}_{\mathrm{S}}(\mathrm{aq})\) is defined by equation (g). \[\mathrm{E}_{\mathrm{S}}=\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{S}(\mathrm{aq})}\]The constraint on the partial derivative refers to the entropy of the solution \(\mathrm{S}(\mathrm{aq})\). As we change the amount of solute nj for fixed temperature and fixed pressure and amount of solvent \(\mathrm{n}_{1}\), so both \(\mathrm{V}(\mathrm{aq})\) and \(\mathrm{S}(\mathrm{aq})\) change yielding a new isentropic thermal expansion, \(\mathrm{E}_{\mathrm{S}}(\mathrm{aq})\) at a new entropy \(\mathrm{S}(\mathrm{aq})\). For a series of solutions having different molalities of solute, comparison of \(\mathrm{E}_{\mathrm{S}}(\mathrm{aq})\) is not straightforward because \({\mathrm{S}(\mathrm{aq})\) is itself a function of solution composition. Further comparison cannot be readily drawn with the isentropic thermal expansion of the solvent, \(\mathrm{E}_{\mathrm{SI}}^{*}(\ell)\); equation (h). \[\mathrm{E}_{\mathrm{S} 1}^{*}(\ell)=\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{T}}\right) \quad \text { at constant } \quad \mathrm{S}_{1}^{*}(\ell)\]\(\mathrm{E}_{\mathrm{S}}(\mathrm{aq})\) is a non-Gibbsian property. Consequently familiar thermodynamic relationships involving partial molar properties are not valid in the case of partial molar isentropic thermal expansions which are non-Lewisian properties. \(\left(\frac{\partial \mathrm{V}_{\mathrm{j}}}{\partial \mathrm{T}}\right)_{\mathrm{s}(\mathrm{aq})}\) is a semi-partial property.Footnote For a system at equilibrium where \(\mathrm{A} = 0\), \(\frac{\partial^{2} G}{\partial T \, \partial p}=\frac{\partial^{2} G}{\partial p \, \partial T}\)Therefore, \(\mathrm{E}_{\mathrm{p}}(\mathrm{A}=0)=\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \mathrm{A}=0}=-\left(\frac{\partial \mathrm{S}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{A}=0}\)This page titled 1.12.7: Expansions- Isobaric and Isentropic is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.12.8: Expansions- Solutions- Isobaric- Partial and Apparent Molar

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.12%3A_Expansions/1.12.08%3A_Expansions-_Solutions-_Isobaric-_Partial_and_Apparent_Molar

The volume of a given aqueous solution containing \(\mathrm{n}_{1}\) moles of water and \(\mathrm{n}_{j}\) moles of solute \(j\) is related to the composition by equation (a). \[\mathrm{V}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{V}_{1}(\mathrm{aq})+\mathrm{n}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}(\mathrm{aq})\]\(\mathrm{V}_{1}(\mathrm{aq})\) and \(\mathrm{V}_{j}(\mathrm{aq})\) are the partial molar volumes of water and solute \(j\) respectively. The (equilibrium) isobaric thermal expansion of the solution (at fixed pressure) \(\mathrm{E}_{\mathrm{p}}\) characterises the differential dependence of \(\mathrm{V}(\mathrm{aq})\) on temperature. \[\mathrm{E}_{\mathrm{p}}(\mathrm{aq})=[\partial \mathrm{V}(\mathrm{aq}) / \partial \mathrm{T}]_{\mathrm{p}, \mathrm{A}=0}\]\(\mathrm{E}_{\mathrm{p}}(\mathrm{aq})\) is an extensive property of the solution. Two partial molar isobaric thermal expansions are defined, characteristic of solute and solvent. \[\mathrm{E}_{\mathrm{p} 1}(\mathrm{aq})=\left(\partial \mathrm{V}_{1}(\mathrm{aq}) / \partial \mathrm{T}\right)_{\mathrm{p}}\]\[\mathrm{E}_{\mathrm{pj}}(\mathrm{aq})=\left(\partial \mathrm{V}_{\mathrm{j}}(\mathrm{aq}) / \partial \mathrm{T}\right)_{\mathrm{p}}\]From equation (a), \[\mathrm{E}_{\mathrm{p}}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{E}_{\mathrm{p} 1}(\mathrm{aq})+\mathrm{n}_{\mathrm{j}} \, \mathrm{E}_{\mathrm{pj}}(\mathrm{aq})\]In the treatment of volumetric properties of solutions we define an apparent molar volume of the solute, \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)\). By analogy we rewrite equation (e) in a form which defines the apparent molar isobaric expansion of the solute, \(\phi\left(\mathrm{E}_{\mathrm{j}}\right)\). Thus, \[\mathrm{E}_{\mathrm{p}}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{E}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right)\]Here, \[\phi\left(\mathrm{E}_{\mathrm{pj}}\right)=\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{T}}\right)_{\mathrm{p}}\]For the pure solvent, \[\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)=\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{T}}\right)_{\mathrm{p}}\]Footnotes \(\mathrm{E}_{\mathrm{p}}\) is an extensive property; the larger the volume \(\mathrm{V}\) the larger the change in volume for a given increase in temperature. \(E_{p}=\left[m^{3} K^{-1}\right] \quad E_{p 1}=\left[\mathrm{m}^{3} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\right] \quad E_{p j}=\left[\mathrm{m}^{3} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\right]\) \(\phi\left(\mathrm{E}_{\mathrm{j}}\right)=\left[\mathrm{m}^{3} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\right]\)This page titled 1.12.8: Expansions- Solutions- Isobaric- Partial and Apparent Molar is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.12.9: Expansions- Apparent Molar Isobaric- Composition Dependence

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.12%3A_Expansions/1.12.09%3A_Expansions-_Apparent_Molar_Isobaric-_Composition_Dependence

For many aqueous solutions at ambient temperature and pressure the dependence of apparent molar isobaric expansions for solute \(j\) \(\phi\left(E_{p j}\right)\) on molality \(\mathrm{m}_{j}\) is accounted for using an equation having the following general form. [The reason for choosing the molality scale is that \(\mathrm{m}_{j}\) is independent of \(\mathrm{T}\) and \(\mathrm{p}\) whereas concentration \(\mathrm{c}_{j}\) is not. \[\phi\left(E_{p j}\right)=a_{1}+a_{2} \,\left(m_{j} / m^{0}\right)+a_{3} \,\left(m_{j} / m^{0}\right)^{2} \ldots . .\]At low solute molalities the linear term is dominant. Granted therefore that equation (a) accounts for the observed pattern, we need to explore the analysis a little further. There are advantages in linking \(\phi\left(E_{p j}\right)\) and the partial molar property \(\mathrm{E}_{p j}(\mathrm{aq})\).For an aqueous solution prepared using \(1 \mathrm{~kg}\) of water and \(\mathrm{m}_{j}\) moles of solute \(j\) at fixed \(\mathrm{T}\) and \(\mathrm{p}\), \[\mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\]Hence, \[\mathrm{E}_{\mathrm{p}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{E}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right)\]\[\left(\frac{\partial \mathrm{E}_{\mathrm{p}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)}{\partial \mathrm{m}_{\mathrm{j}}}\right)=\mathrm{m}_{\mathrm{j}} \,\left(\frac{\partial \phi\left(\mathrm{E}_{\mathrm{pj}}\right)}{\partial \mathrm{m}_{\mathrm{j}}}\right)+\phi\left(\mathrm{E}_{\mathrm{pj}}\right)\]But \[\left(\frac{\partial \mathrm{E}_{\mathrm{p}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right.}{\partial \mathrm{m}_{\mathrm{j}}}\right)=\mathrm{E}_{\mathrm{pj}}(\mathrm{aq})\]\[\mathrm{E}_{\mathrm{pj}}(\mathrm{aq})=\mathrm{m}_{\mathrm{j}} \,\left(\frac{\partial \phi\left(\mathrm{E}_{\mathrm{pj}}\right)}{\partial \mathrm{m}_{\mathrm{j}}}\right)+\phi\left(\mathrm{E}_{\mathrm{pj}}\right)\]Hence the partial molar isobaric expansions for solute \(j\) can be calculated using the apparent molar isobaric expansions and its dependence on molality. Further if equation (a) accounts for the dependence of \(\phi\left(\mathrm{E}_{\mathrm{pj}}\right)\) on \(\mathrm{m}_{j}\), then \[\mathrm{E}_{\mathrm{pj}}=\mathrm{a}_{1}+2 \, \mathrm{a}_{2} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)+3 \, \mathrm{a}_{3} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2} \ldots \ldots\]Therefore, using equations (a) and (g), \[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi\left(\mathrm{E}_{\mathrm{pj}}\right)=\phi\left(\mathrm{E}_{\mathrm{pj}}\right)^{\infty}=\mathrm{E}_{\mathrm{pj}}^{\infty}(\mathrm{aq})\]In the next stage of the analysis we develop an argument starting with an equation for the chemical potential of solute \(j\) in solution. \[\mu_{j}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{m}_{\mathrm{j}} \, \gamma_{\mathrm{j}} / \mathrm{m}^{0}\right)+\int_{\mathrm{p}^{0}}^{p} \mathrm{~V}_{\mathrm{j}}^{\infty}(\mathrm{aq} ; \mathrm{T}) \, \mathrm{dp}\]Then with \(\mathrm{V}_{\mathrm{j}}(\mathrm{aq})=\left(\frac{\partial \mu_{\mathrm{j}}(\mathrm{aq})}{\partial \mathrm{p}}\right)_{\mathrm{T}}\), \[\mathrm{V}_{\mathrm{j}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \,\left[\partial \ln \left(\gamma_{\mathrm{j}}\right) / \partial \mathrm{p}\right]_{\mathrm{T}}\]With \(\mathrm{E}_{\mathrm{pj}}(\mathrm{aq})=\left(\frac{\partial \mathrm{V}_{\mathrm{j}}(\mathrm{aq})}{\partial \mathrm{T}}\right)_{\mathrm{p}}\), \[\mathrm{E}_{\mathrm{pj}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mathrm{E}_{\mathrm{pj}}^{\infty}(\mathrm{aq})+\mathrm{R} \,\left(\frac{\partial \ln \left(\gamma_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right)_{\mathrm{T}}+\mathrm{R} \, \mathrm{T} \,\left(\frac{\partial^{2} \ln \left(\gamma_{\mathrm{j}}\right)}{\partial \mathrm{p} \, \partial \mathrm{T}}\right)\]In the case of dilute solutions we might assert that \(\ln \left(\gamma_{\mathrm{j}}\right)\) is a linear function of molality \(\mathrm{m}_{j}\). Thus, \[\ln \left(\gamma_{\mathrm{j}}\right)=\mathrm{S}_{\gamma} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)\]By definition, \(\mathrm{S}_{\mathrm{V}}=\left(\frac{\partial \mathrm{S}_{\gamma}}{\partial \mathrm{p}}\right)_{\mathrm{T}}\) and \(\mathrm{S}_{\mathrm{Ep}}=\mathrm{S}_{\mathrm{V}}+\mathrm{T} \,\left(\frac{\partial \mathrm{S}_{\mathrm{V}}}{\partial \mathrm{T}}\right)_{\mathrm{p}}\) Then \[\mathrm{E}_{\mathrm{pj}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mathrm{E}_{\mathrm{pj}}^{\infty}(\mathrm{aq})+\mathrm{R} \, \mathrm{S}_{\mathrm{Ep}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)\]Thus we identify the basis of the parameter a2 in equation (a).Footnotes \(\ln \left(\gamma_{\mathrm{j}}\right)= \,\left[\mathrm{mol} \mathrm{kg}^{-1}\right] \,\left[\mathrm{mol} \mathrm{kg}{ }^{-1}\right]^{-1}\) \(\mathrm{S}_{\mathrm{V}}= /\left[\mathrm{N} \mathrm{m}^{-2}\right]=\left[\mathrm{N} \mathrm{m}^{-2}\right]^{-1}\) \(\mathrm{S}_{\mathrm{Ep}}=\left[\mathrm{N} \mathrm{m}^{-2}\right]^{-1}+[\mathrm{K}] \,\left[\mathrm{N} \mathrm{m}^{-2}\right]^{-1} \,\left[\mathrm{K}^{-1}=\left[\mathrm{N} \mathrm{m}^{-2}\right]^{-1}\right.\) \[\begin{aligned} \mathrm{E}_{\mathrm{pj}}(\mathrm{aq}) &=\mathrm{E}_{\mathrm{pj}}^{\infty}(\mathrm{aq})+\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,\left[\mathrm{N} \mathrm{m}^{-2}\right]^{-1} \,\left[\mathrm{mol} \mathrm{kg}^{-1}\right] \,\left[\mathrm{mol} \mathrm{kg}^{-1}\right]^{-1} \\ &=\mathrm{E}_{\mathrm{pj}}^{\infty}(\mathrm{aq})+\left[\mathrm{N} \mathrm{m} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,\left[\mathrm{N} \mathrm{m}^{-2}\right]^{-1} \\ &=\mathrm{E}_{\mathrm{pj}}^{\infty}(\mathrm{aq})+\left[\mathrm{m}^{3} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\right] \end{aligned}\]This page titled 1.12.9: Expansions- Apparent Molar Isobaric- Composition Dependence is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.12.10: Expansions- Solutions Apparent Molar Isobaric Expansions- Determination

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.12%3A_Expansions/1.12.10%3A_Expansions-_Solutions_Apparent_Molar_Isobaric_Expansions-_Determination

The volume of an aqueous solution \(\mathrm{V}(\mathrm{aq})\) is related to the amounts of solvent and solute through the molar volume of water \(\mathrm{V}_{1}^{*}(\ell)\) and the apparent molar volume of solute \(\phi \left(\mathrm{V}_{j}\right) at the same temperature and pressure; equation (a). \[\mathrm{V}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\]The isobaric temperature dependence of the apparent molar volume of solute \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)\) yields the apparent molar (isobaric) expansion of solute \(j\), \(\phi\left(\mathrm{E}_{j}\right)\). \[\phi\left(E_{p j}\right)=\left(\frac{\partial \phi\left(V_{j}\right)}{\partial T}\right)_{p}\]Equation (a) (as in most treatments of volumetric properties) is the starting equation for the development of equations which relate apparent molar isobaric expansions of a solute \(j\) to the measured isobaric expansibilities of solvent and solution. The following four equivalent equations are frequently quoted. A method is also available for direct determination of \(\mathrm{E}_{\mathrm{pj}}(\mathrm{aq})\) from density data determined as functions of \(\mathrm{T}\) and \(\mathrm{m}_{j}\).Molality Scale \[\phi\left(\mathrm{E}_{\mathrm{pj}}\right)=\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\alpha_{\mathrm{p}}(\mathrm{aq})-\alpha_{\mathrm{pl}}^{*}(\ell)\right]+\alpha_{\mathrm{p}}(\mathrm{aq}) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\]\[\begin{gathered} \phi\left(E_{p j}\right)=\left[m_{j} \, \rho(a q) \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\alpha_{p}(a q) \, \rho_{1}^{*}(\ell)-\alpha_{p 1}^{*}(\ell) \, \rho(a q)\right] \\ +\alpha_{p}(a q) \, M_{j} \,[\rho(a q)]^{-1} \end{gathered}\]Concentration Scale [4 - 7] \[\phi\left(\mathrm{E}_{\mathrm{pj}}\right)=\left[\frac{1}{\mathrm{c}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)}\right] \,\left[\alpha_{\mathrm{p}}(\mathrm{aq}) \, \rho_{1}^{*}(\ell)-\rho(\mathrm{aq}) \, \alpha_{\mathrm{pl}}^{*}(\ell)\right]+\alpha_{\mathrm{p} 1}^{*}(\ell) \, \mathrm{M}_{\mathrm{j}} / \rho_{1}^{*}(\ell)\]\[\phi\left(\mathrm{E}_{\mathrm{pj}}\right)=\left[\mathrm{c}_{\mathrm{j}}\right]^{-1} \,\left[\alpha_{\mathrm{p}}(\mathrm{aq})-\alpha_{\mathrm{pl}}^{*}(\ell)\right]+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \alpha_{\mathrm{p} 1}^{*}(\ell)\]The four equations (c) - (f) are thermodynamically correct, no assumptions being made in their derivation.The partial molar isobaric expansion \(\mathrm{E}_{\mathrm{pj}}(\mathrm{aq})\) is obtained using equation (g). \[E_{p j}(a q)=\phi\left(E_{p j}\right)+m_{j} \,\left(\frac{\partial \phi\left(E_{p j}\right)}{\partial m_{j}}\right)_{p}\]Footnotes From equation (a) with respect to the dependence of \(\mathrm{V}(\mathrm{aq}\) on temperature at constant \(\mathrm{p}\) and at “\(\mathrm{A} = 0\)”. \[(\partial \mathrm{V}(\mathrm{aq}) / \partial \mathrm{T})_{\mathrm{p}}=\mathrm{n}_{1} \,\left(\partial \mathrm{V}_{1}^{*}(\ell) / \partial \mathrm{T}\right)_{\mathrm{p}}+\mathrm{n}_{\mathrm{j}} \,\left(\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right) / \partial \mathrm{T}\right)_{\mathrm{p}}\]Using equation (b), \((\partial \mathrm{V}(\mathrm{aq}) / \partial \mathrm{T})_{\mathrm{p}}=\mathrm{n}_{1} \,\left(\partial \mathrm{V}_{1}^{*}(\ell) / \partial \mathrm{T}\right)_{\mathrm{p}}+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right)\) Hence, \[\left(\frac{1}{V(a q)}\right) \,\left(\frac{\partial \mathrm{V}(\mathrm{aq})}{\partial \mathrm{T}}\right)_{\mathrm{p}}=\mathrm{n}_{1} \,\left(\frac{\mathrm{V}_{1}^{*}(\ell)}{\mathrm{V}_{1}^{*}(\ell)}\right) \, \frac{1}{\mathrm{~V}(\mathrm{aq})} \,\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{T}}\right)_{\mathrm{p}}+\left(\frac{\mathrm{n}_{\mathrm{j}}}{\mathrm{V}(\mathrm{aq})}\right) \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right)\]Thus, \(\alpha_{\mathrm{p}}(\mathrm{aq})=\mathrm{n}_{1} \,\left(\frac{\mathrm{V}_{1}^{*}(\ell)}{\mathrm{V}(\mathrm{aq})}\right) \, \alpha_{\mathrm{p} 1}^{*}(\ell)+\frac{\mathrm{n}_{\mathrm{j}}}{\mathrm{V}(\mathrm{aq})} \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right)\) or, \(\mathrm{V}(\mathrm{aq}) \, \alpha_{\mathrm{p}}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell) \, \alpha_{\mathrm{p} 1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right)\)We again use equation (a) for \(\mathrm{V}(\mathrm{aq}\), \[\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right)=\left[\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\right] \, \alpha_{\mathrm{p}}(\mathrm{aq})-\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell) \, \alpha_{\mathrm{pl}}^{*}(\ell)\] But, \(\mathrm{V}_{1}^{*}(\ell)=\mathrm{M}_{1} / \rho_{1}^{*}(\ell)\) where \(\mathrm{M}_{1}\) is the molar mass of the solvent water. \[\phi\left(E_{p j}\right)=\frac{n_{1} \, M_{1}}{n_{j} \, \rho_{1}^{*}(\ell)} \, \alpha_{p}(\mathrm{aq})-\frac{n_{1} \, M_{1}}{n_{j} \, \rho_{1}^{*}(\ell)} \, \alpha_{1}^{*}(\ell)+\alpha_{p}(\mathrm{aq}) \, \phi\left(V_{j}\right)\]But \(\mathrm{m}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} / \mathrm{n}_{\mathrm{l}} \, \mathrm{M}_{\mathrm{l}}\). \[\phi\left(\mathrm{E}_{\mathrm{pj}}\right)=\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\alpha_{\mathrm{p}}(\mathrm{aq})-\alpha_{\mathrm{p} 1}^{*}(\ell)\right]+\alpha_{\mathrm{p}}(\mathrm{aq}) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\]Hence we obtain equation (c). With reference to equation (c), \[\begin{aligned} &{\left[\frac{1}{\mathrm{~m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)}\right] \,\left[\alpha_{\mathrm{p}}(\mathrm{aq})-\alpha_{\mathrm{p} 1}^{*}(\ell)\right]=\left[\frac{\mathrm{kg}}{\mathrm{mol}}\right] \,\left[\frac{\mathrm{m}^{3}}{\mathrm{~kg}}\right] \,\left[\mathrm{K}^{-1}\right]=\left[\mathrm{m}^{3} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\right]} \\ &\alpha_{\mathrm{p}}(\mathrm{aq}) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)=\left[\mathrm{K}^{-1}\right] \,\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right] \end{aligned}\] From \(\phi\left(\mathrm{E}_{\mathrm{pj}}\right)=\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\alpha_{\mathrm{p}}(\mathrm{aq})-\alpha_{\mathrm{p} 1}^{*}(\ell)\right]+\alpha_{\mathrm{p}}(\mathrm{aq}) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\) and, \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)=\frac{\mathrm{V}(\mathrm{aq})-\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)}{\mathrm{n}_{\mathrm{j}}}\)\[\phi\left(E_{p j}\right)=\left[m_{j} \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\alpha_{p}(a q)-\alpha_{p l}^{*}(\ell)\right]+\alpha_{p}(a q) \,\left[\frac{\mathrm{V}(\mathrm{aq})-\mathrm{n}_{1} \, V_{1}^{*}(\ell)}{\mathrm{n}_{\mathrm{j}}}\right]\]\[\phi\left(E_{p j}\right)=\left[m_{j} \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\alpha_{p}(\mathrm{aq})-\alpha_{p 1}^{*}(\ell)\right]+\alpha_{p}(\mathrm{aq}) \,\left[\frac{1}{\mathrm{c}_{j}}-\frac{\mathrm{n}_{1} \, \mathrm{M}_{1}}{\rho_{1}^{*}(\ell) \, \mathrm{n}_{\mathrm{j}}}\right]\]But \(\frac{1}{c_{j}}=\frac{M_{j}}{\rho(a q)}+\frac{1}{m_{j} \, \rho(a q)}\) \[\begin{aligned} &\phi\left(E_{\mathrm{pj}}\right)=\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\alpha_{\mathrm{p}}(\mathrm{aq})-\alpha_{\mathrm{pl}}^{*}(\ell)\right]+\left[\frac{\alpha_{\mathrm{p}}(\mathrm{aq})}{\mathrm{m}_{\mathrm{j}} \, \rho(\mathrm{aq})}+\frac{\alpha_{\mathrm{p}}(\mathrm{aq}) \, \mathrm{M}_{\mathrm{j}}}{\rho(\mathrm{aq})}-\frac{\alpha_{\mathrm{p}}(\mathrm{aq})}{\rho_{1}^{*}(\ell) \, \mathrm{m}_{\mathrm{j}}}\right] \\ &\phi\left(\mathrm{E}_{\mathrm{pj}}\right)=\frac{\alpha_{\mathrm{p}}(\mathrm{aq})}{\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)}-\frac{\alpha_{\mathrm{pl}}^{*}(\ell)}{\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)}+\frac{\alpha_{\mathrm{p}}(\mathrm{aq})}{\mathrm{m}_{\mathrm{j}} \, \rho(\mathrm{aq})}+\frac{\alpha_{\mathrm{p}}(\mathrm{aq}) \, \mathrm{M}_{\mathrm{j}}}{\rho(\mathrm{aq})}-\frac{\alpha_{\mathrm{p}}(\mathrm{aq})}{\rho_{1}^{*}(\ell) \, \mathrm{m}_{\mathrm{j}}} \end{aligned}\]Hence we obtain equation (d). \[\begin{gathered} \phi\left(E_{p j}\right)=\left[m_{j} \, \rho(a q) \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\alpha_{p}(a q) \, \rho_{1}^{*}(\ell)-\alpha_{p 1}^{*}(\ell) \, \rho(a q)\right] \\ +\alpha_{p}(a q) \, M_{j} \,[\rho(a q)]^{-1} \end{gathered}\] From \((\partial \mathrm{V}(\mathrm{aq}) / \partial \mathrm{T})_{\mathrm{p}}=\mathrm{n}_{1} \,\left(\partial \mathrm{V}_{1}^{*}(\ell) / \partial \mathrm{T}\right)_{\mathrm{p}}+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{E}_{\mathrm{pij}}\right)\) \[\alpha_{\mathrm{p}}(\mathrm{aq}) \, \mathrm{V}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell) \, \alpha_{\mathrm{p} 1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{E}_{\mathrm{p} j \mathrm{j}}\right)\]Or, \(\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right)=\mathrm{V}(\mathrm{aq}) \, \alpha_{\mathrm{p}}(\mathrm{aq})-\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell) \, \alpha_{\mathrm{p} 1}^{*}(\ell)\)But, \(\rho(\mathrm{aq})=\left(\mathrm{n}_{1} \, \mathrm{M}_{1}+\mathrm{n}_{\mathrm{j}} \, \mathrm{M}_{\mathrm{j}}\right) / \mathrm{V}(\mathrm{aq})\) or, \(\mathrm{n}_{1}=\left(\mathrm{V}(\mathrm{aq}) \, \rho(\mathrm{aq})-\mathrm{n}_{\mathrm{j}} \, \mathrm{M}_{\mathrm{j}}\right) / \mathrm{M}_{1}\) \[\begin{aligned} &n_{j} \, \phi\left(E_{p j}\right)=\left[V(a q) \, \alpha_{p}(a q)\right]-\left[V(a q) \, \rho(a q)-n_{j} \, M_{j}\right] \, V_{1}^{*}(\ell) \, \alpha_{p l}^{*}(\ell) / M_{1} \\ &\phi\left(E_{p j}\right)=\left[\frac{V(a q) \, \alpha_{p}(a q)}{n_{j}}\right]-\left[\frac{V(a q) \, \rho(a q) \, V_{1}^{*}(\ell) \, \alpha_{1}^{*}(\ell)}{n_{j} \, M_{1}}\right]+\left[\frac{V_{1}^{*}(\ell) \, \alpha_{p 1}^{*}(\ell) \, M_{j}}{M_{1}}\right] \end{aligned}\]But concentration \(\mathrm{c}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} / \mathrm{V}(\mathrm{aq})\) and \(\rho_{1}^{*}=\mathrm{M}_{1} / \mathrm{V}_{1}^{*}(\ell)\). \[\phi\left(\mathrm{E}_{\mathrm{j}}\right)=\left[\frac{\alpha_{\mathrm{p}}(\mathrm{aq})}{\mathrm{c}_{\mathrm{j}}}\right]-\left[\frac{\rho(\mathrm{aq}) \, \alpha_{\mathrm{pl}}^{*}(\ell)}{\mathrm{c}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)}\right]+\left[\frac{\alpha_{\mathrm{p} 1}^{*}(\ell) \, \mathrm{M}_{\mathrm{j}}}{\rho_{1}^{*}(\ell)}\right]\]Hence we obtain equation (e). \[\phi\left(E_{p j}\right)=\left[\frac{1}{c_{j} \, \rho_{1}^{*}(\ell)}\right] \,\left[\alpha_{p}(\mathrm{aq}) \, \rho_{1}^{*}(\ell)-\rho(a q) \, \alpha_{p 1}^{*}(\ell)\right]+\alpha_{p 1}^{*}(\ell) \, M_{j} / \rho_{1}^{*}(\ell)\] With reference to equation (e), \[\begin{aligned} &{\left[\frac{1}{\mathrm{c}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)}\right] \,\left[\alpha_{\mathrm{p}}(\mathrm{aq}) \, \rho_{1}^{*}(\ell)-\rho(\mathrm{aq}) \, \alpha_{\mathrm{pl} 1}^{*}(\ell)\right]=\left[\frac{\mathrm{m}^{3}}{\mathrm{~mol}}\right] \,\left[\frac{\mathrm{m}^{3}}{\mathrm{~kg}}\right] \,\left[\mathrm{K}^{-1}\right] \,\left[\mathrm{kg} \mathrm{m}^{-3}\right]} \\ &=\left[\mathrm{m}^{3} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\right] \end{aligned}\] The volume of a solution, \(\mathrm{V}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{n}_{1} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\)Concentration \(\mathrm{c}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} / \mathrm{V}(\mathrm{aq})\) or, \(\mathrm{c}_{\mathrm{j}}=\frac{\mathrm{n}_{\mathrm{j}}}{\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)}\)But molality \(\mathrm{m}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} / \mathrm{n}_{\mathrm{l}} \, \mathrm{M}_{\mathrm{l}}\) \(\mathrm{c}_{\mathrm{j}}=\frac{\mathrm{m}_{\mathrm{j}} \, \mathrm{n}_{1} \, \mathrm{M}_{1}}{\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)}\) or, \(\frac{1}{\mathrm{c}_{\mathrm{j}}}=\frac{\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)}{\mathrm{m}_{\mathrm{j}} \, \mathrm{n}_{1} \, \mathrm{M}_{1}}+\frac{\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\mathrm{n}_{1} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}}\) or, \(\frac{1}{c_{j}}=\frac{1}{m_{j} \, \rho_{1}^{*}(\ell)}+\phi\left(V_{j}\right)\) or, \(\frac{1}{\mathrm{~m}_{\mathrm{j}}}=\frac{\rho_{1}^{*}(\ell)}{\mathrm{c}_{\mathrm{j}}}-\rho_{1}^{*}(\ell) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\)From equation (c). \[\phi\left(\mathrm{E}_{\mathrm{pj}}\right)=\left[\frac{1}{\rho_{1}^{*}(\ell)}\right] \,\left[\frac{\rho_{1}^{*}(\ell)}{\mathrm{c}_{\mathrm{j}}}-\rho_{1}^{*}(\ell) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\right] \,\left[\alpha_{\mathrm{p}}(\mathrm{aq})-\alpha_{\mathrm{p} 1}^{*}(\ell)\right]+\alpha_{\mathrm{p}}(\mathrm{aq}) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\]Or, \[\begin{aligned} \phi\left(\mathrm{E}_{\mathrm{pj}}\right)=\left[\frac{1}{\mathrm{c}_{\mathrm{j}}}\right] \,\left[\alpha_{\mathrm{p}}(\mathrm{aq})\right.&\left.-\alpha_{\mathrm{pl}}^{*}(\ell)\right]-\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \alpha_{\mathrm{p}}(\mathrm{aq})+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \alpha_{\mathrm{pl}}^{*}(\ell) \\ &+\alpha_{\mathrm{p}}(\mathrm{aq}) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \end{aligned}\]We obtain equation (f) \[\phi\left(\mathrm{E}_{\mathrm{pj}}\right)=\left[\mathrm{c}_{\mathrm{j}}\right]^{-1} \,\left[\alpha_{\mathrm{p}}(\mathrm{aq})-\alpha_{\mathrm{pl}}^{*}(\ell)\right]+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \alpha_{\mathrm{pl}}^{*}(\ell)\] With reference to equation (f) \[\left[\frac{1}{\mathrm{c}_{\mathrm{j}}}\right] \,\left[\alpha_{\mathrm{p}}(\mathrm{aq})-\alpha_{\mathrm{pl}}^{*}(\ell)\right]=\left[\frac{\mathrm{m}^{3}}{\mathrm{~mol}}\right] \,\left[\mathrm{K}^{-1}\right]=\left[\mathrm{m}^{3} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\right]\] From \(\mathrm{E}_{\mathrm{p}}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{E}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{E}_{\mathrm{p} j}\right)\) Then, \(\left(\frac{\partial E_{p}}{\partial n_{j}}\right)_{T, p, n_{j}}=n_{j} \,\left[\frac{\partial \phi\left(E_{p j}\right)}{\partial n_{j}}\right]+\phi\left(E_{p j}\right)\)Or, \(\mathrm{E}_{\mathrm{pj}}(\mathrm{aq})=\mathrm{m}_{\mathrm{j}} \,\left[\frac{\partial \phi\left(\mathrm{E}_{\mathrm{pj}}\right)}{\partial \mathrm{m}_{\mathrm{j}}}\right]+\phi\left(\mathrm{E}_{\mathrm{pj}}\right)\) M. J. Blandamer and H. Hoiland, Phys.Chem.Chem.Phys.,1999,1,1873. This method starts out with the measured dependence of the density \(\rho(\mathrm{aq})\) on temperature and molality at fixed pressure about density \(\rho_{1}^{*}(\ell, \theta)\), at temperature \(\theta\) at same pressure. For example the data might be fitted to an equation having the following form yielding the b-coefficients. \[\begin{aligned} &\rho\left(\mathrm{m}_{\mathrm{j}}, \mathrm{T}\right)=\rho_{1}^{*}(\ell, \theta)+\mathrm{b}_{2} \,(\mathrm{T}-\theta) \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right) / \theta+\mathrm{b}_{3} \,(\mathrm{T}-\theta) \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2} / \theta \\ &+\mathrm{b}_{4} \,(\mathrm{T}-\theta)^{2} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right) / \theta^{2}+\mathrm{b}_{5} \,(\mathrm{T}-\theta)^{2} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2} / \theta^{2} \\ &\left(\frac{\partial \rho\left(\mathrm{aq} ; \mathrm{m}_{\mathrm{j}} ; \mathrm{T}\right)}{\partial \mathrm{T}}\right)_{\mathrm{m}_{\mathrm{j}}}=\mathrm{b}_{2} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right) / \theta+\mathrm{b}_{3} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2} / \theta \end{aligned}\]\[+2 \, b_{4} \,(\mathrm{T}-\theta) \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right) / \theta^{2}+2 \, \mathrm{b}_{5} \,(\mathrm{T}-\theta) \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right) / \theta^{2}\]and, \[\begin{aligned} &\left(\frac{\partial \rho\left(\mathrm{aq} ; \mathrm{m}_{\mathrm{j}} ; \mathrm{T}\right)}{\partial\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)}\right)_{\mathrm{T}}=\mathrm{b}_{2} \,(\mathrm{T}-\theta) / \theta+2 \, \mathrm{b}_{3} \,(\mathrm{T}-\theta) \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right) / \theta \\ &+\mathrm{b}_{4} \,(\mathrm{T}-\theta)^{2} / \theta^{2}+2 \, \mathrm{b}_{5} \,(\mathrm{T}-\theta)^{2} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right) / \theta^{2} \end{aligned}\]The density \(\rho(\mathrm{aq})\) of an aqueous solution molality \(\mathrm{m}_{j}\) prepared using \(1 \mathrm{~kg}\) of water is given by the following equation. \[\rho(\mathrm{aq})=\left[1+\mathrm{M}_{\mathrm{j}} \, \mathrm{m}_{\mathrm{j}}\right] / \mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)\]\[\mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left[1+\mathrm{M}_{\mathrm{j}} \, \mathrm{m}_{\mathrm{j}}\right] / \rho(\mathrm{aq})\]Also, \[\mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{V}_{1}(\mathrm{aq})+\mathrm{m}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}(\mathrm{aq})\]\[\left(\frac{\partial \mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)}{\partial \mathrm{m}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}}=\frac{\mathrm{M}_{\mathrm{j}}}{\rho(\mathrm{aq})}-\frac{1+\mathrm{M}_{\mathrm{j}} \, \mathrm{m}_{\mathrm{j}}}{[\rho(\mathrm{aq})]^{2}} \,\left(\frac{\partial \rho(\mathrm{aq})}{\partial \mathrm{m}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}}\]But, \[\mathrm{V}_{\mathrm{j}}(\mathrm{aq})=\left(\frac{\partial \mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)}{\partial \mathrm{m}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}}\]And, \[E_{p j}(a q)=\left(\frac{\partial V_{j}(a q)}{\partial T}\right)_{p, m_{j}}\]\[\mathrm{E}_{\mathrm{pj}}(\mathrm{aq})=\frac{\partial}{\partial \mathrm{T}} \,\left\{\frac{\mathrm{M}_{\mathrm{j}}}{\rho(\mathrm{aq})}-\left(\frac{1}{\rho(\mathrm{aq})}\right)^{2} \,\left(\frac{\partial \rho(\mathrm{aq})}{\partial \mathrm{m}_{\mathrm{j}}}\right) \,\left[1+\mathrm{M}_{\mathrm{j}} \, \mathrm{m}_{\mathrm{j}}\right]\right\}\]\[\begin{aligned} &\mathrm{E}_{\mathrm{pj}}(\mathrm{aq})=-\frac{\mathrm{M}_{\mathrm{j}}}{(\rho(\mathrm{aq}))^{2}} \,\left(\frac{\partial \rho(\mathrm{aq})}{\partial \mathrm{T}}\right)_{\mathrm{m}_{\mathrm{j}}}+\frac{2}{\left(\rho(\mathrm{aq})^{3}\right)} \,\left(\frac{\partial \rho(\mathrm{aq})}{\partial \mathrm{T}}\right)_{\mathrm{m}_{\mathrm{j}}} \,\left(\frac{\partial \rho(\mathrm{aq})}{\partial \mathrm{m}_{\mathrm{j}}}\right) \,\left[1+\mathrm{M}_{\mathrm{j}} \, \mathrm{m}_{\mathrm{j}}\right] \\ &-\frac{1}{(\rho(\mathrm{aq}))^{2}} \, \frac{\partial}{\partial \mathrm{T}} \,\left(\frac{\partial \rho(\mathrm{aq})}{\partial \mathrm{m}_{\mathrm{j}}}\right) \,\left[1+\mathrm{M}_{\mathrm{j}} \, \mathrm{m}_{\mathrm{j}}\right] \end{aligned}\]Using equation (k) in conjunction with equations (a) - (c), partial molar isobaric expansion \(\mathrm{E}_{\mathrm{pj}}(\mathrm{aq})\) is calculated from the density and its dependence on both temperature and molality of solute. In another development \(\mathrm{E}_{\mathrm{pj}}\) is related to \(\alpha_{\mathrm{p}}\) and its dependence on molality of solute. By definition, \[\alpha_{p}(\mathrm{aq})=-\frac{1}{V(\mathrm{aq})} \,\left(\frac{\partial \mathrm{V}(\mathrm{aq})}{\partial \mathrm{T}}\right)_{\mathrm{p}, \mathrm{m}(\mathrm{j})}\]Or, \[\alpha_{p}(a q)=-\frac{1}{\rho(a q)} \,\left(\frac{\partial \rho(a q)}{\partial T}\right)_{p, m(j)}\]At temperature \(\mathrm{T}\) and molality \(\mathrm{m}_{j}\), \[\left(\frac{\partial \rho(\mathrm{aq})}{\partial \mathrm{T}}\right)_{\mathrm{p}, \mathrm{m}(\mathrm{j})}=-\alpha_{\mathrm{p}}(\mathrm{aq}) \, \rho(\mathrm{aq})\]Using equation (n) \[\begin{aligned} &E_{p j}(a q)=\frac{M_{j} \, \alpha_{p}}{\rho(a q)}-\frac{2}{(\rho(a q))^{2}} \, \alpha_{p} \,\left(\frac{\partial \rho(a q)}{\partial m_{j}}\right) \,\left[1+M_{j} \, m_{j}\right] \\ &-\frac{1}{(\rho(a q))^{2}} \, \frac{\partial}{\partial m_{j}} \,\left(\frac{\partial \rho}{\partial T}\right) \,\left[1+M_{j} \, m_{j}\right] \end{aligned}\]But from equation (n) \[\frac{\partial}{\partial m_{j}} \,\left(\frac{\partial \rho}{\partial T}\right)=-\alpha_{p} \, \frac{\partial \rho(\mathrm{aq})}{\partial m_{j}}-\rho(\mathrm{aq}) \,\left(\frac{\partial \alpha_{p}}{\partial m_{j}}\right)\]Therefore, \[\begin{aligned} &\mathrm{E}_{\mathrm{pj}}=\frac{\mathrm{M}_{\mathrm{j}} \, \alpha_{\mathrm{p}}}{\rho(\mathrm{aq})}-\frac{2}{(\rho(\mathrm{aq}))^{2}} \, \alpha_{\mathrm{p}} \,\left(\frac{\partial \rho(\mathrm{aq})}{\partial \mathrm{m}_{\mathrm{j}}}\right) \,\left[1+\mathrm{M}_{\mathrm{j}} \, \mathrm{m}_{\mathrm{j}}\right] \\ &+\frac{1}{(\rho(\mathrm{aq}))^{2}} \, \alpha_{\mathrm{p}} \,\left(\frac{\partial \rho(\mathrm{aq})}{\partial \mathrm{m}_{\mathrm{j}}}\right) \,\left[1+\mathrm{M}_{\mathrm{j}} \, \mathrm{m}_{\mathrm{j}}\right]+\frac{1}{\rho(\mathrm{aq})} \,\left(\frac{\partial \alpha_{\mathrm{p}}}{\partial \mathrm{m}_{\mathrm{j}}}\right) \,\left[1+\mathrm{M}_{\mathrm{j}} \, \mathrm{m}_{\mathrm{j}}\right] \end{aligned}\]or, (with reordering of terms) \[\mathrm{E}_{\mathrm{p} j}=-\frac{\left[1+\mathrm{M}_{\mathrm{j}} \, \mathrm{m}_{\mathrm{j}}\right]}{(\rho(\mathrm{aq}))^{2}} \, \alpha_{\mathrm{p}} \,\left(\frac{\partial \rho(\mathrm{aq})}{\partial \mathrm{m}_{\mathrm{j}}}\right)+\frac{1+\mathrm{M}_{\mathrm{j}} \, \mathrm{m}_{\mathrm{j}}}{\rho(\mathrm{aq})} \,\left(\frac{\partial \alpha_{\mathrm{p}}}{\partial \mathrm{m}_{\mathrm{j}}}\right)+\frac{\mathrm{M}_{\mathrm{j}} \, \alpha_{\mathrm{p}}}{\rho(\mathrm{aq})}\]Using equation (g) for \(\mathrm{V}_{\mathrm{j}}(\mathrm{aq})\) \[\mathrm{E}_{\mathrm{pj}}=\mathrm{V}_{\mathrm{j}}(\mathrm{aq}) \, \alpha_{\mathrm{p}}+\frac{\left[1+\mathrm{M}_{\mathrm{j}} \, \mathrm{m}_{\mathrm{j}}\right]}{\rho(\mathrm{aq})} \,\left(\frac{\partial \alpha_{\mathrm{p}}}{\partial \mathrm{m}_{\mathrm{j}}}\right)\]The partial molar isobaric expansion \(\mathrm{E}_{\mathrm{pj}}\) is calculated from isobaric expansibility and its dependence on molality of solute.This page titled 1.12.10: Expansions- Solutions Apparent Molar Isobaric Expansions- Determination is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.12.11: Expansions- Isobaric- Apparent Molar- Neutral Solutes

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.12%3A_Expansions/1.12.11%3A_Expansions-_Isobaric-_Apparent_Molar-_Neutral_Solutes

For many aqueous solutions at ambient pressure and temperature, the dependence of \(\phi\left(\mathrm{E}_{\mathrm{pj}}\right)\) on molality of a neutral solute \(j\), \(\mathrm{m}_{j}\) is accounted for by an equation having the following general form. [The reason for choosing the molality scale is again the fact that \(\mathrm{m}_{j}\) is independent of \(\mathrm{T}\) and \(\mathrm{p}\) but concentration \(\mathrm{c}_{j}\) is not.] \[\phi\left(E_{p j}\right)=a_{1}+a_{2} \,\left(m_{j} / m^{0}\right)+a_{3} \,\left(m_{j} / m^{0}\right)^{2}+a_{4} \,\left(m_{j} / m^{0}\right)^{3}+\ldots\]At low molalities, the linear term is dominant. Granted therefore that equation(a) accounts for the observed pattern, we need a quantitative description which accounts for this pattern. There are advantages in linking directly the apparent property \(\phi\left(\mathrm{E}_{\mathrm{pj}}\right)\) and the partial molar property \(\mathrm{E}_{\mathrm{pj}}(\mathrm{aq})\).For an aqueous solution at fixed temperature and pressure, \[\mathrm{E}_{\mathrm{pj}}(\mathrm{aq})=\mathrm{m}_{\mathrm{j}} \,\left(\frac{\partial \phi\left(\mathrm{E}_{\mathrm{pj}}\right)}{\partial \mathrm{m}_{\mathrm{j}}}\right)+\phi\left(\mathrm{E}_{\mathrm{pj}}\right)\]Hence the partial molar isobaric expansion of solute \(j\) can be calculated from the apparent molar isobaric expansion and its dependence on molality, \(\mathrm{m}_{j}\). Hence if equation (a) satisfactorily describes the observed dependence of \(\phi\left(E_{p j}\right)\) on \(\mathrm{m}_{j}\), \[\mathrm{E}_{\mathrm{pj}}(\mathrm{aq})=\mathrm{a}_{1}+2 \, \mathrm{a}_{2} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)+3 \, \mathrm{a}_{3} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2}+\ldots\]Therefore, \[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \mathrm{E}_{\mathrm{pj}}(\mathrm{aq})=\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi \mathrm{E}_{\mathrm{pj}}(\mathrm{aq})=\mathrm{E}_{\mathrm{pj}}^{\infty}(\mathrm{aq})=\phi\left(\mathrm{E}_{\mathrm{pj}}\right)^{\infty}\]Consequently the parameter \(\mathrm{a}_{1}\) in equations (a) and (b) is the limiting partial molar isobaric expansion of solute \(j\). For dilute solutions, equation (c) takes the following simple form. \[\mathrm{E}_{\mathrm{pj}}(\mathrm{aq})=\mathrm{E}_{\mathrm{pj}}^{\infty}(\mathrm{aq})+\mathrm{R} \, \mathrm{S}_{\mathrm{Ep}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)\]In these terms we can identify the basis of the parameter \(\mathrm{a}_{2}\) in equations (a) and (c). Desrosiers et al used a quadratic (cf. equation (a)) to express the dependence of \(\phi\left(\mathrm{E}_{\mathrm{pj}}\right)\) at \(298 \mathrm{~K}\) on molality of urea in aqueous solutions; \(\phi\left(\mathrm{E}_{\mathrm{pj}}\right)=0.07 \mathrm{~cm}^{3} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\), coefficient \(\mathrm{a}_{2}\) being positive and coefficient \(\mathrm{a}_{3}\) being negative.The majority of published information concerns the dependence on temperature of \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}\). A survey based on a dilatometric study of 15 non-electrolytes in aqueous solution indicates that \(\left[\mathrm{d} \phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty} / \mathrm{dT}\right]\) is less than \(\left[\mathrm{dV}_{\mathrm{j}}^{*}(\ell) / \mathrm{dT}\right]\) for the pure liquid substance \(j\); the second derivative \(\left[\mathrm{d}^{2} \phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty} / \mathrm{dT}^{2}\right]\) is positive. However for hydrophilic solutes \(\left[\mathrm{d} \phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty} / \mathrm{dT}\right]\) is larger than \(\left[\mathrm{dV}_{\mathrm{j}}^{*}(\ell) / \mathrm{dT}\right]\) and \(\left[\mathrm{d}^{2} \phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty} / \mathrm{dT}^{2}\right]\) is negative. A similar pattern is observed for sucrose and urea for which \(\left[\mathrm{d}^{2} \phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty} / \mathrm{dT}^{2}\right]\) is negative. Indeed Hepler classified solutes in aqueous solutions as either structure-breaking (negative) or structure forming (positive) on the basis of the sign for \(\left[\mathrm{d}^{2} \phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty} / \mathrm{dT}^{2}\right]\). The dependence of \(\phi\left(V_{j}\right)^{\infty}\) on temperature for both glycine and alanine in \(\mathrm{NaCl}(\mathrm{aq})\) is small, For monosacchrides(aq) \(\mathrm{E}_{\mathrm{pj}}^{\infty}(\mathrm{aq})\) is positive.Footnotes N. Desrosiers, G. Perron, J. G. Mathieson, B. E. Conway and J. E. Desnoyers, J Solution Chem.,1974,3,789. J. I. Neal and D. A. I. Goring, J. Phys. Chem.,1970,74,658. J. Sengster, T.-T. Ling and F. Lenzi, J Solution Chem,1976,5,575. L.G. Hepler, Can J.Chem.,1969,47,4613. B. S. Lark, K.Balat and S. Singh, Indian J Chem., Sect A,25,534. S. Paljk, K. Balat and S. Singh, J. Chem. Eng Data, 1990,35.41.This page titled 1.12.11: Expansions- Isobaric- Apparent Molar- Neutral Solutes is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.12.12: Expansions- Isobaric- Salt Solutions- Apparent Molar

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.12%3A_Expansions/1.12.12%3A_Expansions-_Isobaric-_Salt_Solutions-_Apparent_Molar

In general terms the dependence of apparent molar isobaric expansions for salt \(j\) on the composition of a given solution can be described using the following empirical equation. \[\phi\left(E_{p j}\right)=a_{1}+a_{2} \,\left(m_{j} / m^{0}\right)^{1 / 2}+a_{3} \,\left(m_{j} / m^{0}\right)\]The presence of a term in \(\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}\) is not unexpected in the case of salt solutions. Moreover for dilute solutions the term in \(\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}\) is dominant. Hence \[\left[\partial \phi\left(E_{p j}\right) / \partial m_{j}\right]=(1 / 2) \, a_{2} \,\left(m_{j} \, m^{0}\right)^{-1 / 2}+a_{3} \,\left(1 / m^{0}\right)\]Then, \[\mathrm{E}_{\mathrm{pj}}(\mathrm{aq})=\mathrm{a}_{1}+(3 / 2) \, \mathrm{a}_{2} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}+2 \, \mathrm{a}_{3} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)\]Therefore parameter \(\mathrm{a}_{1}\) is the limiting partial molar and apparent molar isobaric expansion of solute \(j\) in solution. An explanation of the term in \(\left(m_{j} / m^{0}\right)^{1 / 2}\) based on the Debye-Huckel Limiting Law (DHLL).In general terms the chemical potential of salt \(j\) in aqueous solution at fixed \(\mathrm{T}\) and \(\mathrm{p}\) is related to molality \(\mathrm{m}_{j}\) using equation (d). \[\mu_{j}(a q)=\mu_{j}^{0}(a q)+v \, R \, T \, \ln \left(Q \, m_{j} \, \gamma_{\pm} / m^{0}\right)+\int_{p^{0}}^{p} V_{j}^{\infty}(a q) \, d p\]\[\mathrm{V}_{\mathrm{j}}(\mathrm{aq})=\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq})+\mathrm{v} \, \mathrm{R} \, \mathrm{T} \,\left(\partial \ln \gamma_{\pm} / \partial \mathrm{p}\right)_{\mathrm{T}}\]Therefore \[\left.\mathrm{E}_{\mathrm{pj}}(\mathrm{aq})=\mathrm{E}_{\mathrm{pj}}^{\infty}(\mathrm{aq})+\mathrm{v} \, \mathrm{R} \,\left\{\left[\mathrm{d} \ln \left(\gamma_{\pm}\right) / \mathrm{dp}\right)\right]_{\mathrm{T}}+\mathrm{T} \,\left[\mathrm{d}^{2} \ln \left(\gamma_{\pm}\right) / \mathrm{dp} \, \mathrm{dT}\right]\right\}\]According to the DHLL, \[\ln \left(\gamma_{\pm}\right)=-S_{\gamma} \,\left(m_{j} / m^{0}\right)^{1 / 2}\]By definition \[\mathrm{S}_{\mathrm{V}}=\left\lfloor\partial \mathrm{S}_{\gamma} / \partial \mathrm{p}\right\rfloor_{\mathrm{T}}\]Then, \[\mathrm{T} \,\left[\partial \ln \left(\gamma_{\pm}\right) / \partial \mathrm{p}\right]_{\mathrm{T}}=-\mathrm{T} \, \mathrm{S}_{\mathrm{v}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}\]Hence we write \[\mathrm{E}_{\mathrm{pj}}(\mathrm{aq})=\mathrm{E}_{\mathrm{pj}}^{\infty}(\mathrm{aq})-\mathrm{v}_{\mathrm{j}} \, \mathrm{R} \, \mathrm{S}_{\mathrm{Ep}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}\]where, \[\mathrm{S}_{\mathrm{Ep}}=\mathrm{S}_{\mathrm{V}}+\mathrm{T} \,\left[\partial \mathrm{S}_{\mathrm{V}} / \partial \mathrm{T}\right)_{\mathrm{p}}\]Therefore a linear dependence of \(\mathrm{E}_{\mathrm{p j}}(\mathrm{aq})\) on \(\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{1 / 2}\) for dilute solutions is predicted by the DHLL. Hence for dilute solutions \[\phi\left(E_{p j}\right)=\phi\left(E_{p j}\right)^{\infty}-(2 / 3) \, v \, R \, S_{E p} \,\left(m_{j} / m^{0}\right)^{1 / 2}\]For tetra-alkylammonium iodides(aq) \(\left[\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty} / \partial \mathrm{T}\right]_{\mathrm{p}}\) is positive, the magnitude increasing on going from \(\mathrm{Me}_{4}\mathrm{N}^{+}\) to \(\mathrm{Bu}_{4}\mathrm{N}^{+}\) [4,5,].Apparent molar isobaric expansions for divalent metal chlorides(aq) lead to estimates of ionic molar isobaric expansions based on \(\mathrm{E}_{p}^{\infty}\left(\mathrm{Cl}^{-} ; \mathrm{aq}\right)\) set at \(+0.046 \mathrm{~cm}^{3} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}\). The ionic estimates show a linear dependence on \(\left(\mathrm{r}_{\mathrm{j}}\right)^{-1}\), a pattern predicted by the Born equation. \(\phi\left(\mathrm{E}_{\mathrm{pj}}\right)^{\infty}\) for \(\mathrm{NaBPh}_{4}\) decreases gradually over the range 0 to 60 Celsius.\(\phi\left(E_{p j}\right)^{\infty}\) for \(\mathrm{NaF}(\mathrm{aq})\), \(\mathrm{Na}_{2}\mathrm{SO}_{4}(\mathrm{aq})\) and \(\mathrm{KCl}(\mathrm{aq})\) is positive.Footnotes \[\begin{aligned} &\mathrm{S}_{\gamma}= \quad \mathrm{S}_{\mathrm{V}}=\left[\mathrm{N} \mathrm{m}^{-2}\right]^{-1} \quad \mathrm{~T} \, \mathrm{S}_{\mathrm{V}}=[\mathrm{K}] \,\left[\mathrm{N} \mathrm{m}^{-2}\right]^{-1} \\ &\left\{\left[\partial \ln \left(\gamma_{\pm}\right) / \partial \mathrm{p}\right]_{\mathrm{T}}+\mathrm{T} \,\left[\partial^{2} \ln \left(\gamma_{\pm}\right) / \partial \mathrm{p} \, \partial \mathrm{T}\right]\right\} \\ &=\left\{\left[\mathrm{N} \mathrm{m}^{-2}\right]^{-1}+[\mathrm{K}] \, \,\left[\mathrm{N} \mathrm{m}^{-2}\right]^{-1}[\mathrm{~K}]^{-1}\right\}=\left[\mathrm{N} \mathrm{m}^{-2}\right]^{-1} \end{aligned}\] \[\begin{aligned} &\left[\mathrm{m}^{3} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}\right]=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}\right]- \,\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,\left[\mathrm{N} \mathrm{m}^{-2}\right]^{-1} \\ &=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}\right]-\left[\mathrm{m}^{3} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}\right] \end{aligned}\] \(\mathrm{S}_{\mathrm{Ep}}=\left[\mathrm{N} \mathrm{m}^{-2}\right]^{-1}+[\mathrm{K}] \,\left[\mathrm{N} \mathrm{m}^{-2}\right]^{-1} \,\left[\mathrm{K}^{-1}\right]=\left[\mathrm{N} \mathrm{m}^{-2}\right]^{-1}\) R. Gopal and M. A. Siddiqi, J.Phys.Chem.,1968,72,1814. F. Franks and H. T. Smith, Trans. Faraday Soc.,1967,63,2586. F. J. Millero and W. Drost –Hansen, J. Phys.Chem.,1968,72,1758. F. J. Millero, J. Phys. Chem., 1968, 72, 4589. F. J. Millero, J. Chem. Eng. Data, 1970,15,562. F. J. Millero and J. H. Knox, J. Chem. Eng. Data, 1973,18,407.This page titled 1.12.12: Expansions- Isobaric- Salt Solutions- Apparent Molar is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.12.13: Expansions- Isobaric- Binary Liquid Mixtures

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.12%3A_Expansions/1.12.13%3A_Expansions-_Isobaric-_Binary_Liquid_Mixtures

The isobaric (equilibrium) expansion of a liquid, volume \(\mathrm{V}\), is defined by equation (a). \[\mathrm{E}_{\mathrm{p}}=\left(\frac{\partial V}{\partial T}\right)_{p}\]Both \(\mathrm{E}_{\mathrm{p}}\) and \(\mathrm{V}\) are extensive properties of a mixture. Therefore it is convenient to refer to the molar property, \(\mathrm{E}_{\mathrm{pm}}(\operatorname{mix})\). Thus \[\mathrm{E}_{\mathrm{pm}}(\mathrm{mix})=\left(\frac{\partial \mathrm{V}_{\mathrm{m}}(\mathrm{mix})}{\partial \mathrm{T}}\right)_{\mathrm{p}}\]At fixed \(\mathrm{T}\) and \(\mathrm{p}\), \(\mathrm{V}_{\mathrm{m}}(\mathrm{mix})\) for a binary liquid mixture is related to the partial molar volumes of the two components. \[\mathrm{V}_{\mathrm{m}}(\operatorname{mix})=\mathrm{x}_{1} \, \mathrm{V}_{1}(\operatorname{mix})+\mathrm{x}_{2} \, \mathrm{V}_{2}(\mathrm{mix})\]From equation (b) \[\mathrm{E}_{\mathrm{pm}}(\mathrm{mix})=\mathrm{x}_{1} \,\left(\frac{\partial \mathrm{V}_{1}(\mathrm{mix})}{\partial \mathrm{T}}\right)_{\mathrm{p}}+\mathrm{x}_{2} \,\left(\frac{\partial \mathrm{V}_{2}(\mathrm{mix})}{\partial \mathrm{T}}\right)_{\mathrm{p}}\]For a binary mixture having molar volume \(\mathrm{V}_{\mathrm{m}}(\mathrm{mix})\) and density \(\rho(\mathrm{mix})\), \[\rho(\operatorname{mix})=\left(\mathrm{x}_{1} \, \mathrm{M}_{1}+\mathrm{x}_{2} \, \mathrm{M}_{2}\right) / \mathrm{V}_{\mathrm{m}}(\mathrm{mix})\]Here \(\mathrm{M}_{1}\) and \(\mathrm{M}_{2}\) are the molar masses of liquids 1 and 2 respectively. \[\mathrm{V}_{\mathrm{m}}(\mathrm{mix})=\left(\mathrm{x}_{1} \, \mathrm{M}_{1}+\mathrm{x}_{2} \, \mathrm{M}_{2}\right) / \rho(\mathrm{mix})\]Hence, \[\begin{aligned} {\left[\partial \mathrm{V}_{\mathrm{m}}(\operatorname{mix}) / \partial \mathrm{T}\right]_{\mathrm{p}} } &=\\ &-\left[\left(\mathrm{x}_{1} \, \mathrm{M}_{1}+\mathrm{x}_{2} \, \mathrm{M}_{2}\right) / \rho(\operatorname{mix})\right] \,[\partial \ln \{\rho(\operatorname{mix})\} / \partial \mathrm{T}]_{\mathrm{p}} \end{aligned}\]\(\\mathrm{E}_{\mathrm{pm}}(\mathrm{mix})\) is obtained for a given mixture from the isobaric dependence of density on temperature. There is merit in considering equations for \(\mathrm{E}_{\mathrm{pm}}(\operatorname{mix} ; \mathrm{id})\) of a binary mixture having ideal thermodynamic properties and hence for the related excess molar expansion \(\mathrm{E}_{\mathrm{pm}}^{\mathrm{E}}\). With, \[\mathrm{E}_{\mathrm{pm}}(\operatorname{mix} ; \mathrm{id})=\mathrm{x}_{1} \, \mathrm{E}_{1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{E}_{2}^{*}(\ell)\]\[\mathrm{E}_{\mathrm{pm}}^{\mathrm{E}}=\mathrm{E}_{\mathrm{pm}}(\mathrm{mix})-\mathrm{E}_{\mathrm{pm}}(\mathrm{mix} ; \mathrm{id})\]\(\mathrm{E}_{\mathrm{pm}}(\operatorname{mix} ; \mathrm{id})\) is the mole fraction weighted sum of the isobaric expansions of the pure liquid components at the same \(\mathrm{T}\) and \(\mathrm{p}\). The isobaric expansibility of an ideal binary liquid mixture \(\alpha_{p}(\operatorname{mix} ; \mathrm{id})\) is given by equation (j). \[\alpha_{\mathrm{p}}(\operatorname{mix} ; \mathrm{id})=\frac{\mathrm{x}_{1} \, \mathrm{E}_{\mathrm{p} 1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{E}_{\mathrm{p} 2}^{*}(\ell)}{\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}(\ell)}\]Or, \[\alpha_{\mathrm{p}}(\operatorname{mix} ; \text { id })=\frac{\mathrm{x}_{1} \, \mathrm{E}_{\mathrm{p} 1}^{*}(\ell)}{\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}(\ell)}+\frac{\mathrm{x}_{2} \, \mathrm{E}_{\mathrm{p} 2}^{*}(\ell)}{\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}(\ell)}\]Hence, \[\alpha_{\mathrm{p}}(\mathrm{mix} ; \mathrm{id})=\frac{\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell) \, \alpha_{\mathrm{p} 1}^{*}(\ell)}{\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}(\ell)}+\frac{\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}(\ell) \, \mathrm{E}_{\mathrm{p} 2}^{*}(\ell)}{\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}(\ell)}\]Hence, expansibility \(\alpha_{p}(\operatorname{mix} ; 1 \mathrm{~d})\) can be expressed in terms of the volume fractions of the corresponding ideal binary liquid mixture. \[\alpha_{p}(\operatorname{mix} ; \text { id })=\phi_{1}(\operatorname{mix} ; \text { id }) \, \alpha_{p 1}^{*}(\ell)+\phi_{2}(\operatorname{mix} ; \text { id }) \, \alpha_{p 2}^{*}(\ell)\]The excess (equilibrium) isobaric expansivity \(\alpha_{p}^{E}(\operatorname{mix})\) is given by mix equation (n). \[\alpha_{\mathrm{p}}^{\mathrm{E}}(\mathrm{mix})=\frac{1}{\mathrm{~V}_{\mathrm{m}}(\mathrm{mix})} \,\left[\left(\frac{\partial \mathrm{V}_{\mathrm{m}}^{\mathrm{E}}(\mathrm{mix})}{\partial \mathrm{T}}\right)_{\mathrm{p}}-\mathrm{V}_{\mathrm{m}}^{\mathrm{E}} \, \alpha_{\mathrm{p}}(\mathrm{mix} ; \mathrm{id})\right]\]From another standpoint the thermal expansion of a binary liquid mixture is analysed in terms of the differential dependence of rational activity coefficients on temperature and pressure. For liquid component 1 at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\), \[\mu_{1}(\operatorname{mix})=\mu_{1}^{0}(\ell)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1} \, \mathrm{f}_{1}\right)+\int_{\mathrm{p}}^{\mathrm{p}} \mathrm{V}_{1}^{*}(\ell) \, \mathrm{dp}\]Then \[\mathrm{V}_{1}(\mathrm{mix})=\mathrm{V}_{1}^{*}(\ell)+\mathrm{R} \, \mathrm{T} \,\left[\partial \ln \left(\mathrm{f}_{1}\right) / \partial \mathrm{p}\right]_{\mathrm{T}}\]At temperature \(\mathrm{T}\), \[\mathrm{E}_{\mathrm{p}_{1}}(\operatorname{mix})=\mathrm{E}_{\mathrm{p} 1}(\operatorname{mix} ; \mathrm{id})+\mathrm{R} \,\left[\partial \ln \left(\mathrm{f}_{1}\right) / \partial \mathrm{p}\right]_{\mathrm{T}}+\mathrm{R} \, \mathrm{T} \,\left[\frac{\partial}{\partial \mathrm{T}}\left(\frac{\partial \ln \left(\mathrm{f}_{1}\right)}{\partial \mathrm{p}}\right)_{\mathrm{T}}\right]_{\mathrm{p}}\]\[\begin{aligned} &\mathrm{E}_{\mathrm{p} 2}(\operatorname{mix})= \\ &\quad \mathrm{E}_{\mathrm{p} 2}(\mathrm{mix} ; \mathrm{id})+\mathrm{R} \,\left[\partial \ln \left(\mathrm{f}_{2}\right) / \partial \mathrm{p}\right]_{\mathrm{T}}+\mathrm{R} \, \mathrm{T} \,\left[\frac{\partial}{\partial \mathrm{T}}\left(\frac{\partial \ln \left(\mathrm{f}_{2}\right)}{\partial \mathrm{p}}\right)_{\mathrm{T}}\right]_{\mathrm{p}} \end{aligned}\]Two equations follow for the excess partial molar isobaric expansions of the components of the mixture. \[\mathrm{E}_{\mathrm{p} 1}^{\mathrm{E}}(\mathrm{mix})=\mathrm{R} \,\left[\partial \ln \left(\mathrm{f}_{1}\right) / \partial \mathrm{p}\right]_{\mathrm{T}}+\mathrm{R} \, \mathrm{T} \,\left[\frac{\partial}{\partial \mathrm{T}}\left(\frac{\partial \ln \left(\mathrm{f}_{1}\right)}{\partial \mathrm{p}}\right)_{\mathrm{T}}\right]_{\mathrm{p}}\]\[E_{\mathrm{p} 2}^{\mathrm{E}}(\mathrm{mix})=\mathrm{R} \,\left[\partial \ln \left(\mathrm{f}_{2}\right) / \partial \mathrm{p}_{\mathrm{T}}+\mathrm{R} \, \mathrm{T} \,\left[\frac{\partial}{\partial \mathrm{T}}\left(\frac{\partial \ln \left(\mathrm{f}_{2}\right)}{\partial \mathrm{p}}\right)_{\mathrm{T}}\right]_{\mathrm{p}}\right.\]Therefore for the mixture, \[\mathrm{E}_{\mathrm{pm}}^{\mathrm{E}}(\mathrm{mix})=\mathrm{x}_{1} \, \mathrm{E}_{\mathrm{p} 1}^{\mathrm{E}}(\mathrm{mix})+\mathrm{x}_{2} \, \mathrm{E}_{\mathrm{p} 2}^{\mathrm{E}}(\mathrm{mix})\]Footnotes For a binary liquid mixture at defined \(\mathrm{T}\) and \(\mathrm{p}\), \[\mathrm{V}_{\mathrm{m}}(\operatorname{mix})=\mathrm{V}_{\mathrm{m}}(\mathrm{mix} ; \mathrm{id})+\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}\]\[\alpha_{p}(\operatorname{mix})=\frac{1}{V_{m}(\operatorname{mix})} \, \frac{\partial}{\partial T}\left[V_{m}(\text { mix } ; 1 \mathrm{~d})+V_{m}^{E}\right]\]Or, \(\alpha_{p}(\operatorname{mix})=\frac{1}{V_{m}(\operatorname{mix})} \, \frac{\partial \mathrm{V}_{\mathrm{m}}(\mathrm{mix} ; \mathrm{id})}{\partial \mathrm{T}}+\frac{1}{\mathrm{~V}_{\mathrm{m}}(\mathrm{mix})} \, \frac{\partial \mathrm{V}_{\mathrm{m}}^{\mathrm{E}}}{\partial \mathrm{T}}\)But, \(\alpha_{p}(\operatorname{mix} ; \mathrm{id})=\frac{1}{\mathrm{~V}_{\mathrm{m}}(\mathrm{mix} ; \mathrm{id})} \, \frac{\partial \mathrm{V}_{\mathrm{m}}(\mathrm{mix} ; \mathrm{id})}{\partial \mathrm{T}}\)By definition, \[\begin{aligned} &\alpha_{p}^{E}=\alpha_{p}(\operatorname{mix})-\alpha_{p}(\operatorname{mix} ; \text { id })\\ &\alpha_{p}^{E}(\operatorname{mix})=\left[\frac{1}{V_{m}(\operatorname{mix})}-\frac{1}{V_{m}(\operatorname{mix} ; i \mathrm{~d})}\right] \, \frac{\partial V_{m}(\operatorname{mix} ; \mathrm{id})}{\partial \mathrm{T}}+\frac{1}{\mathrm{~V}_{\mathrm{m}}(\mathrm{mix})} \, \frac{\partial \mathrm{V}_{\mathrm{m}}^{\mathrm{E}}}{\partial \mathrm{T}}\\ &\alpha_{p}^{\mathrm{E}}(\operatorname{mix})=\left[\frac{\mathrm{V}_{\mathrm{m}}(\mathrm{mix} ; \mathrm{id})-\mathrm{V}_{\mathrm{m}}(\mathrm{mix})}{\mathrm{V}_{\mathrm{m}}(\mathrm{mix}) \, \mathrm{V}_{\mathrm{m}}(\mathrm{mix} ; \mathrm{id})}\right] \, \frac{\partial \mathrm{V}_{\mathrm{m}}(\mathrm{mix} ; \mathrm{id})}{\partial \mathrm{T}}+\frac{1}{\mathrm{~V}_{\mathrm{m}}(\operatorname{mix})} \, \frac{\partial \mathrm{V}_{\mathrm{m}}^{\mathrm{E}}}{\partial \mathrm{T}}\\ &\alpha_{\mathrm{p}}^{\mathrm{E}}(\operatorname{mix})=-\left[\frac{\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}}{\mathrm{V}_{\mathrm{m}}(\mathrm{mix})}\right] \, \alpha_{\mathrm{p}}(\mathrm{mix} ; \mathrm{id})+\frac{1}{\mathrm{~V}_{\mathrm{m}}(\mathrm{mix})} \, \frac{\partial \mathrm{V}_{\mathrm{m}}^{\mathrm{E}}}{\partial \mathrm{T}} \end{aligned}\]Hence, \[\alpha_{p}^{E}(\operatorname{mix})=\frac{1}{V_{m}(\operatorname{mix})} \,\left[\left(\frac{\partial V_{m}^{E}}{\partial T}\right)_{p}-V_{m}^{E} \, \alpha_{p}(\operatorname{mix} ; \text { id })\right]\]This page titled 1.12.13: Expansions- Isobaric- Binary Liquid Mixtures is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.12.14: Expansibilities- Isobaric- Binary Liquid Mixtures

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.12%3A_Expansions/1.12.14%3A_Expansibilities-_Isobaric-_Binary_Liquid_Mixtures

A given binary liquid mixture is prepared using liquids 1 and 2 at defined \(\mathrm{T}\) and \(\mathrm{p}\). The molar volume of this mixture is given by equation (a). In the event that thermodynamic properties of the mixture are ideal, the molar volume is given by equation (a). \[\mathrm{V}_{\mathrm{m}}(\operatorname{mix} ; \mathrm{id})=\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}(\ell)\]At fixed pressure, \[\left(\frac{\partial \mathrm{V}_{\mathrm{m}}(\operatorname{mix} ; \mathrm{id})}{\partial \mathrm{T}}\right)_{\mathrm{p}}=\mathrm{x}_{1} \,\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{T}}\right)_{\mathrm{p}}+\mathrm{x}_{2} \,\left(\frac{\partial \mathrm{V}_{2}^{*}(\ell)}{\partial \mathrm{T}}\right)_{\mathrm{p}}\]\[\begin{aligned} &\frac{\mathrm{V}_{\mathrm{m}}(\operatorname{mix} ; \mathrm{id})}{\mathrm{V}_{\mathrm{m}}(\operatorname{mix} ; \mathrm{id})} \,\left(\frac{\partial \mathrm{V}_{\mathrm{m}}(\operatorname{mix} ; \mathrm{id})}{\partial \mathrm{T}}\right)_{\mathrm{p}}= \\ &\mathrm{x}_{1} \, \frac{\mathrm{V}_{1}^{*}(\ell)}{\mathrm{V}_{1}^{*}(\ell)} \,\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{T}}\right)_{\mathrm{p}}+\mathrm{x}_{2} \, \frac{\mathrm{V}_{2}^{*}(\ell)}{\mathrm{V}_{2}^{*}(\ell)} \,\left(\frac{\partial \mathrm{V}_{2}^{*}(\ell)}{\partial \mathrm{T}}\right)_{\mathrm{p}} \end{aligned}\]Hence, \[\mathrm{V}_{\mathrm{m}}(\operatorname{mix} ; \mathrm{id}) \, \alpha_{\mathrm{p}}(\operatorname{mix} ; \mathrm{id})=\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell) \, \alpha_{\mathrm{p} 1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}(\ell) \, \alpha_{\mathrm{p} 2}^{*}(\ell)\]But \[\phi_{1}(\operatorname{mix} ; \mathrm{id})=\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell) / \mathrm{V}_{\mathrm{m}}(\operatorname{mix} ; \mathrm{id})\]And, \[\phi_{2}(\operatorname{mix} ; \mathrm{id})=\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}(\ell) / \mathrm{V}_{\mathrm{m}}(\operatorname{mix} ; \mathrm{id})\]Hence \[\alpha_{p}(\operatorname{mix} ; \text { id })=\phi_{1}(\operatorname{mix} ; \text { id }) \, \alpha_{p 1}^{*}(\ell)+\phi_{2}(\operatorname{mix} ; \text { id }) \, \alpha_{p 2}^{*}(\ell)\]For a real binary liquid mixture, \[\mathrm{V}_{\mathrm{m}}(\operatorname{mix})=\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}(\ell)+\mathrm{V}_{\mathrm{m}}^{\mathrm{E}}(\operatorname{mix})\]At fixed pressure, \[\left(\frac{\partial \mathrm{V}_{\mathrm{m}}(\mathrm{mix})}{\partial \mathrm{T}}\right)_{\mathrm{p}}=\mathrm{x}_{1} \,\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{T}}\right)_{\mathrm{p}}+\mathrm{x}_{2} \,\left(\frac{\partial \mathrm{V}_{2}^{*}(\ell)}{\partial \mathrm{T}}\right)_{\mathrm{p}}+\left(\frac{\partial \mathrm{V}_{\mathrm{m}}^{\mathrm{E}}}{\partial \mathrm{T}}\right)_{\mathrm{p}}\]Or, \[\begin{aligned} &\frac{\mathrm{V}_{\mathrm{m}}(\mathrm{mix})}{\mathrm{V}_{\mathrm{m}}(\mathrm{mix})} \,\left(\frac{\partial \mathrm{V}_{\mathrm{m}}(\mathrm{mix})}{\partial \mathrm{T}}\right)_{\mathrm{p}}= \\ &\mathrm{x}_{1} \, \frac{\mathrm{V}_{1}^{*}(\ell)}{\mathrm{V}_{1}^{*}(\ell)} \,\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{T}}\right)_{\mathrm{p}}+\mathrm{x}_{2} \, \frac{\mathrm{V}_{2}^{*}(\ell)}{\mathrm{V}_{2}^{*}(\ell)} \,\left(\frac{\partial \mathrm{V}_{2}^{*}(\ell)}{\partial \mathrm{T}}\right)_{\mathrm{p}}+\left(\frac{\partial \mathrm{V}_{\mathrm{m}}^{\mathrm{E}}}{\partial \mathrm{T}}\right)_{\mathrm{p}} \end{aligned}\]\[\begin{aligned} &\mathrm{V}_{\mathrm{m}}(\mathrm{mix}) \, \alpha_{\mathrm{p}}(\mathrm{mix})= \\ &\mathrm{x}_{1} \, \mathrm{V}_{1}^{*}(\ell) \, \alpha_{\mathrm{p} 1}^{*}(\ell)+\mathrm{x}_{2} \, \mathrm{V}_{2}^{*}(\ell) \, \alpha_{\mathrm{p} 2}^{*}(\ell)+\left(\frac{\partial \mathrm{V}_{\mathrm{m}}^{\mathrm{E}}}{\partial \mathrm{T}}\right)_{\mathrm{p}} \end{aligned}\]Or, \[\begin{aligned} &\alpha_{p}(\operatorname{mix})= \\ &\frac{1}{V_{\mathrm{m}}(\operatorname{mix})} \,\left[x_{1} \, V_{1}^{*}(\ell) \, \alpha_{p 1}^{*}(\ell)+x_{2} \, V_{2}^{*}(\ell) \, \alpha_{p 2}^{*}(\ell)+\left(\frac{\partial V_{m}^{E}}{\partial T}\right)_{p}\right] \end{aligned}\]We may also define an excess property using equation (k) but it is important to note that \(\alpha_{\mathrm{p}}^{\mathrm{E}}\) is not a simple second derivative of the excess molar Gibbs energy, \(\mathrm{G}_{\mathrm{m}}^{\mathrm{E}}\). \[\alpha_{p}^{E}(\operatorname{mix})=\alpha_{p}(\operatorname{mix})-\alpha_{p}(\operatorname{mix} ; \text { id })\]We start out using an alternative expression for \(\alpha_{p}(\operatorname{mix})\). \[\alpha_{p}(\operatorname{mix})=\frac{1}{V_{m}(\operatorname{mix})} \,\left[V_{m}(\operatorname{mix} ; \mathrm{id}) \, \alpha_{p}(\operatorname{mix} ; \mathrm{id})+\left(\frac{\partial \mathrm{V}_{\mathrm{m}}^{E}}{\partial T}\right)_{\mathrm{p}}\right]\]\[\begin{aligned} &\alpha_{p}^{E}(\operatorname{mix})= \\ &\frac{1}{V_{m}(\operatorname{mix})} \,\left[V_{m}(\operatorname{mix} ; \text { id }) \, \alpha_{p}(\operatorname{mix} ; \mathrm{id})+\left(\frac{\partial V_{m}^{E}}{\partial T}\right)_{p}\right]-\alpha_{p}(\operatorname{mix} ; \text { id }) \end{aligned}\]\[\begin{aligned} &\alpha_{\mathrm{p}}^{\mathrm{E}}(\operatorname{mix})= \\ &\frac{1}{\mathrm{~V}_{\mathrm{m}}(\operatorname{mix})} \,\left[\left(\frac{\partial \mathrm{V}_{\mathrm{m}}^{\mathrm{E}}}{\partial \mathrm{T}}\right)_{\mathrm{p}}+\left[\mathrm{V}_{\mathrm{m}}(\operatorname{mix} ; \mathrm{id})-\mathrm{V}_{\mathrm{m}}(\operatorname{mix})\right] \, \alpha_{\mathrm{p}}(\operatorname{mix} ; \mathrm{id}]\right. \end{aligned}\]Hence, \[\alpha_{p}^{E}(\operatorname{mix})=\frac{1}{V_{m}(\operatorname{mix})} \,\left[\left(\frac{\partial V_{m}^{E}}{\partial T}\right)_{p}-V_{m}^{E}(\operatorname{mix}) \, \alpha_{p}(\text { mix } ; \text { id }]\right.\]Footnotes \[\begin{aligned} &{\alpha_{\mathrm{p}}^{\mathrm{E}}(\operatorname{mix})=\left[\mathrm{K}^{-1}\right]} \\ &{\left[\left(\frac{\partial \mathrm{V}_{\mathrm{m}}^{\mathrm{E}}}{\partial \mathrm{T}}\right)_{\mathrm{p}}-\mathrm{V}_{\mathrm{m}}^{\mathrm{E}} \, \alpha_{\mathrm{p}}(\mathrm{mix} ; \mathrm{id})\right]=\frac{\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]}{[\mathrm{K}]}-\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right] \,\left[\mathrm{K}^{-1}\right]=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}\right]} \\ &\frac{1}{V_{m}(m i x)} \,\left[\left(\frac{\partial V_{m}^{E}}{\partial T}\right)_{p}-V_{m}^{E} \, \alpha_{p}(\text { mix } ; i d)\right] \\ &\quad \quad =\frac{1}{\left[\mathrm{~m}^{3} \mathrm{~mol}^{-1}\right]} \,\left[\mathrm{m}^{3} \mathrm{~mol}^{-1} \quad \mathrm{~K}^{-1}\right]=\left[\mathrm{K}^{-1}\right] \end{aligned}\]This page titled 1.12.14: Expansibilities- Isobaric- Binary Liquid Mixtures is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.12.15: Expansions- Isentropic- Solutions

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.12%3A_Expansions/1.12.15%3A_Expansions-_Isentropic-_Solutions

The Gibbs energy of a closed system at thermodynamic equilibrium containing two chemical substances is defined by equation (a) where the molecular composition/organisation is signalled by \(\xi^{\mathrm{eq}}\). \[\mathrm{G}=\mathrm{G}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{2}, \mathrm{~A}=0, \xi^{\mathrm{eq}}\right]\]\[\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{2}, \mathrm{~A}=0, \xi^{\mathrm{eq}}\right]\]\[\mathrm{S}=\mathrm{S}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{2}, \mathrm{~A}=0, \xi^{\mathrm{eq}}\right]\]A common feature is the use of the two intensive variables, temperature and pressure, in the definition of extensive properties \(\mathrm{G}\), \(\mathrm{V}\) and \(\mathrm{S}\). The properties \(\mathrm{G}\), \(\mathrm{V}\) and \(\mathrm{S}\) are Gibbsian.The system is perturbed by an increase in temperature along a path such that the affinity for spontaneous change remains zero and the entropy remains equal to that defined by equation (c). In principle we plot volume \(\mathrm{V}\) as a function of temperature at constant \(\mathrm{n}_{1}\), \(\mathrm{n}_{2}\), at '\(\mathrm{A}=0\)' and at a constant entropy equal to that defined by equation (c). The gradient of the plot at the point where the volume is defined by equation (b) yields the equilibrium isentropic expansion, \(\mathrm{E}_{\mathrm{S}}(\mathrm{A}=0)\); equation (d); isentropic = adiabatic and at equilibrium. \[\mathrm{E}_{\mathrm{S}}(\mathrm{A}=0)=(\partial \mathrm{V} / \partial \mathrm{T})_{\mathrm{S} . \mathrm{A}=0}\]\(\mathrm{E}_{\mathrm{S}}(\mathrm{A}=0)\) characterises the system defined by the Gibbsian set of independent variables \(\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{2}, \mathrm{~A}=0, \xi^{\mathrm{eq}}\right]\). As we change the amount of solute \(\mathrm{n}_{j}\) for a fixed temperature, pressure and amount of solvent \(\mathrm{n}_{1}\), so both \(\mathrm{V}(\mathrm{aq})\) and \(\mathrm{S}(\mathrm{aq})\) change yielding a new isentropic thermal expansion \(\mathrm{E}_{\mathrm{S}}(\mathrm{aq})\) at a new entropy \(\mathrm{S}(\mathrm{aq})\). For a series of solutions having different molalities, comparison of \(\mathrm{E}_{\mathrm{S}}(\mathrm{aq})\) is not straightforward because entropy \(\mathrm{S}(\mathrm{aq})\) is itself a function of solution composition. Further comparison cannot be readily drawn with the isentropic thermal expansion of the pure solvent \(\mathrm{E}_{\mathrm{Sl}}^{*}(\ell)\) equation (e). \[\mathrm{E}_{\mathrm{S} 1}^{*}(\ell ; \mathrm{A}=0)=\left(\partial \mathrm{V}_{1}^{*}(\ell) / \partial \mathrm{T}\right)_{\mathrm{A}=0} \text { at constant } \mathrm{S}_{1}^{*}(\ell)\]\(\mathrm{E}_{\mathrm{S}}(\mathrm{A}=0 ; \mathrm{aq})\) is a non-Gibbsian property. Consequently, familiar thermodynamic relationships involving partial molar properties are not valid in the case of partial molar isentropic (thermal) expansions which are non-Lewisian properties. \(\left[\partial \mathrm{V}_{\mathrm{j}}(\mathrm{aq}) / \partial \mathrm{T}\right]\) for solute-\(j\) in aqueous solution at constant \(\mathrm{S}(\mathrm{aq})\) is a semi-partial molar property.For an aqueous solution having entropy \(\mathrm{S}(\mathrm{aq})\), two partial molar isentropic expansions are defined for the solvent and solute. At \(\mathrm{S}(\mathrm{aq})\) characterised by \(\mathrm{T}\), \(\mathrm{p}\), \(\mathrm{n}_{1}\) and \(\mathrm{n}_{j}\), \[\mathrm{E}_{\mathrm{Sl}}(\mathrm{aq} ; \mathrm{def})=\left[\partial \mathrm{E}_{\mathrm{S}}(\mathrm{aq}) / \partial \mathrm{n}_{1}\right] \text { at fixed } \mathrm{T}, \mathrm{p} and \mathrm{n}_{j}\]and \[\mathrm{E}_{\mathrm{Sj}}(\mathrm{aq} ; \mathrm{def})=\left[\partial \mathrm{E}_{\mathrm{S}}(\mathrm{aq}) / \partial \mathrm{n}_{\mathrm{j}}\right] \text { at fixed } \mathrm{T}, \mathrm{p} \text { and } \mathrm{n}_{1}\]So that, \[\mathrm{E}_{\mathrm{S}}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{E}_{\mathrm{Sl}}(\mathrm{aq} ; \operatorname{def})+\mathrm{n}_{\mathrm{j}} \, \mathrm{E}_{\mathrm{Sj}}(\mathrm{aq} ; \text { def })\]Equation (h) relates \(\mathrm{E}_{\mathrm{S}}(\mathrm{aq})\) to the partial molar intensive isentropic properties of both solvent and solute.A similar problem is encountered in defining an apparent molar isentropic expansion for solute-\(j\), \(\phi\left(\mathrm{E}_{\mathrm{Sj} \mathrm{j}}\right)\). We might assert that \(\phi\left(\mathrm{E}_{\mathrm{Sj} \mathrm{j}}\right)\) is defined by the isentropic differential dependence \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)\) on temperature. Alternatively, we use an equation by analogy to those used to relate, for example, \(\mathrm{V}(\mathrm{aq})\) to \(V_{1}^{*}(\ell)\) and \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)\). Equation (i) relates \(\mathrm{V}(\mathrm{aq})\) to the apparent molar volume of solute j, φ(Vj). \[\mathrm{V}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\]Differentiation of equation (i) at constant entropy again raises the problem that the molar entropy \(\mathrm{S}(\mathrm{aq})\) does not equal the molar entropy of the pure solvent, \(\mathrm{S}_{1}^{*}(\ell)\). However, by analogy with the definition of \(\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \mathrm{def}\right)\) we define a quantity \(\phi\left(\mathrm{E}_{\mathrm{Sj}} ; \text { def }\right)\) using equation (j). \[\mathrm{E}_{\mathrm{s}}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{E}_{\mathrm{Sl}}^{*}(\ell)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{E}_{\mathrm{Sj}} ; \operatorname{def}\right)\]\(\mathrm{E}_{\mathrm{Sl}}^{*}(\ell)\) is the molar intensive property of the solvent. The isentropic expansion of the solution at entropy \(\mathrm{S}(\mathrm{aq})\) is linked with that of the pure solvent at entropy \(\mathrm{S}_{1}^{*}(\ell)\). Further \[\phi\left(E_{\mathrm{s} j} ; \operatorname{def}\right)=\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\alpha_{\mathrm{s}}(\mathrm{aq})-\alpha_{\mathrm{s} 1}^{*}(\ell)\right]+\alpha_{\mathrm{s}}(\mathrm{aq}) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\]\[\phi\left(E_{\mathrm{sj}} ; \text { def }\right)=\left[\alpha_{\mathrm{s}}(\mathrm{aq})-\alpha_{\mathrm{s} 1}^{*}(\ell)\right] \,\left(\mathrm{c}_{\mathrm{j}}\right)^{-1}+\alpha_{\mathrm{s} 1}^{*} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\]Footnotes From \(\left(\frac{\partial^{2} \mathrm{U}}{\partial \mathrm{S} \, \partial \mathrm{V}}\right)=\left(\frac{\partial^{2} \mathrm{U}}{\partial \mathrm{V} \, \partial \mathrm{S}}\right), \quad\left(\frac{\partial \mathrm{T}}{\partial \mathrm{V}}\right)_{\mathrm{s}}=-\left(\frac{\partial \mathrm{p}}{\partial \mathrm{S}}\right)_{\mathrm{V}}\) We invert the latter equation. Hence, \(E_{S}=\left(\frac{\partial V}{\partial T}\right)_{s}=-\left(\frac{\partial S}{\partial p}\right)_{V}\)The isentropic dependence of volume on temperature equals (with reversed sign) the isochoric dependence of entropy on pressure. J. C. R. Reis, M. J. Blandamer, M. I. Davis and G. Douheret., Phys. Chem. Chem. Phys.,2001,3,1465. J. C. R. Reis, J. Chem. Soc. Faraday Trans. 2,1982,78,1595. M. J. Blandamer, M. I. Davis, G. Douheret and J. C. R. Reis., Chem. Soc. Rev., 2001,30,8. From \[\begin{aligned} &\phi\left(E_{\mathrm{S} j} ; \text { def }\right)=\frac{E_{\mathrm{S}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)-\left(1 / \mathrm{M}_{1}\right) \, \mathrm{E}_{\mathrm{S} 1}^{*}(\ell)}{\mathrm{m}_{\mathrm{j}}} \\ &\phi\left(\mathrm{E}_{\mathrm{Sj}} ; \text { def }\right)=\frac{\mathrm{E}_{\mathrm{S}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)}{\mathrm{m}_{\mathrm{j}}}-\frac{\mathrm{E}_{\mathrm{S} 1}^{*}(\ell)}{\mathrm{m}_{\mathrm{j}} \, \mathrm{M}_{1}} \\ &\phi\left(\mathrm{E}_{\mathrm{Sj}} ; \operatorname{def}\right)=\frac{\alpha_{\mathrm{s}}(\mathrm{aq}) \, \mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)}{\mathrm{m}_{\mathrm{j}}}-\frac{\alpha_{\mathrm{S} 1}^{*}(\ell) \, \mathrm{V}_{1}^{*}(\ell)}{\mathrm{m}_{\mathrm{j}} \, \mathrm{M}_{1}} \end{aligned}\]But \(\mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1} / \mathrm{kg}=1\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\)Then \(\phi\left(\mathrm{E}_{\mathrm{s} j} ; \mathrm{def}\right)=\frac{\alpha_{\mathrm{s}}(\mathrm{aq}) \,\left[\left(1 / \mathrm{M}_{1}\right) \mathrm{V}_{1}^{*}(\ell)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\right]}{\mathrm{m}_{\mathrm{j}}}-\frac{\alpha_{\mathrm{s} 1}^{*}(\ell) \, \mathrm{V}_{1}^{*}(\ell)}{\mathrm{m}_{\mathrm{j}} \, \mathrm{M}_{1}}\)Or, \(\phi\left(\mathrm{E}_{\mathrm{Sj}} ; \text { def }\right)=\frac{\mathrm{V}_{1}^{*}(\ell)}{\mathrm{m}_{\mathrm{j}} \, \mathrm{M}_{1}} \,\left[\alpha_{\mathrm{s}}(\mathrm{aq})-\alpha_{\mathrm{sl}}^{*}(\ell)\right]+\alpha_{\mathrm{s}}(\mathrm{aq}) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\)Hence, \(\phi\left(\mathrm{E}_{\mathrm{s} \mathrm{j}} ; \operatorname{def}\right)=\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\alpha_{\mathrm{s}}(\mathrm{aq})-\alpha_{\mathrm{s} 1}^{*}(\ell)\right]+\alpha_{\mathrm{s}}(\mathrm{aq}) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\)But \(\frac{1}{\mathrm{~m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)}=\frac{1}{\mathrm{c}_{\mathrm{j}}}-\phi\left(\mathrm{V}_{\mathrm{j}}\right)\)Then \phi\left(E_{s j} ; \operatorname{def}\right)=\left[\frac{1}{c_{j}}-\phi\left(V_{j}\right)\right] \,\left[\alpha_{s}(\mathrm{aq})-\alpha_{\mathrm{s}_{1}}^{*}(\ell)\right]+\alpha_{\mathrm{s}}(\mathrm{aq}) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\)Or, \(\begin{aligned} \phi\left(\mathrm{E}_{\mathrm{Sj}} ; \operatorname{def}\right)=\left[\mathrm{c}_{\mathrm{j}}\right]^{-1} \,\left[\alpha_{\mathrm{s}}(\mathrm{aq})-\alpha_{\mathrm{s} 1}^{*}(\ell)\right] \\ &-\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \alpha_{\mathrm{s}}(\mathrm{aq})+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \alpha_{\mathrm{s} 1}^{*}(\ell)+\alpha_{\mathrm{s}}(\mathrm{aq}) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \end{aligned}\)Hence, \[\begin{aligned} &\phi\left(\mathrm{E}_{\mathrm{sj}} ; \mathrm{def}\right)=\left[\mathrm{c}_{\mathrm{j}}\right]^{-1} \,\left[\alpha_{\mathrm{s}}(\mathrm{aq})-\alpha_{\mathrm{s} 1}^{*}(\ell)\right]+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \alpha_{\mathrm{s} 1}^{*}(\ell) \\ \phi\left(\mathrm{E}_{\mathrm{Sj}} ; \text { def }\right)=\left[\mathrm{m}^{3} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\right] \end{aligned}\]This page titled 1.12.15: Expansions- Isentropic- Solutions is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.12.16: Expansions- Solutions- Apparent Molar Isentropic and Isobaric

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.12%3A_Expansions/1.12.16%3A_Expansions-_Solutions-_Apparent_Molar_Isentropic_and_Isobaric

In the context of the properties of aqueous solutions the concept of apparent molar properties is important with respect to the analysis of experimental results; e.g. apparent molar volume for solute \(j\) \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)\) calculated from the densities of a given solution and solvent at fixed \(\mathrm{T}\) and \(\mathrm{p}\). Similarly apparent molar isobaric expansions \(\phi\left(\mathrm{E}_{\mathrm{pj}}\right)\) characterise the dependence of \phi\left(\mathrm{V}_{j}\right) on temperature at fixed pressure. Nevertheless problems emerge when we turn attention to comparable isentropic properties. The way ahead involves definition of apparent molar isentropic expansions \(\phi\left(\mathrm{E}_{\mathrm{sj}} ; \text { def }\right)\) and apparent molar isentropic compressions \(\phi\left(\mathrm{K}_{\mathrm{sj}} ; \text { def }\right)\). These two properties are related; equation(a). \[\begin{aligned} &\phi\left(\mathrm{E}_{\mathrm{sj}} ; \mathrm{def}\right)= \\ &-\frac{\alpha_{\mathrm{S} 1}^{*}(\ell)}{\alpha_{\mathrm{p}}(\mathrm{aq})} \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right)+\frac{\alpha_{\mathrm{s}}(\mathrm{aq})}{\kappa_{\mathrm{s}}(\mathrm{aq})} \, \phi\left(\mathrm{K}_{\mathrm{s}}\right)+\frac{\alpha_{\mathrm{s}}(\mathrm{aq}) \, \kappa_{\mathrm{S} 1}^{*}(\ell)}{\kappa_{\mathrm{s}}(\mathrm{aq}) \, \sigma(\mathrm{aq})} \, \phi\left(\mathrm{C}_{\mathrm{pj}}\right) \\ &\quad+\left[\alpha_{\mathrm{p} 1}^{*}(\ell) \,\left(1+\frac{\alpha_{\mathrm{pl} 1}^{*}(\ell)}{\alpha_{\mathrm{p}}(\mathrm{aq})}\right)-\frac{\alpha_{\mathrm{s}}(\mathrm{aq}) \, \kappa_{\mathrm{S} 1}^{*}(\ell)}{\kappa_{\mathrm{s}}(\mathrm{aq})} \,\left(1+\frac{\sigma_{1}^{*}(\ell)}{\sigma(\mathrm{aq})}\right)\right] \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \end{aligned}\]Equation (b) relates the corresponding properties at infinite dilution. \[\frac{\phi\left(\mathrm{E}_{\mathrm{Sj}} ; \mathrm{def}\right)^{\infty}}{\alpha_{\mathrm{S} 1}^{*}(\ell)}=-\frac{\phi\left(\mathrm{E}_{\mathrm{pj}}\right)^{\infty}}{\alpha_{\mathrm{p} 1}^{*}(\ell)}+\frac{\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \mathrm{def}\right)^{\infty}}{\kappa_{\mathrm{S} 1}^{*}(\ell)}+\frac{\phi\left(\mathrm{C}_{\mathrm{pj}}\right)^{\infty}}{\sigma_{1}^{*}(\ell)}\]‘Semi’ apparent molar isentropic expansions and compressions are related using equation (c) \[\frac{1}{\alpha_{\mathrm{S}}(\mathrm{aq})} \,\left[\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{T}}\right]_{\mathrm{S}(\mathrm{aq})}=-\frac{1}{\kappa_{\mathrm{S}}(\mathrm{aq})} \,\left[\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right]_{\mathrm{S}(\mathrm{aq})}\]Equations (a) , (b) and (c) illustrate the power of thermodynamics in drawing together and relating the several properties of a solution.Footnotes In the following we simplify the algebra by omitting the descriptors (aq) and (\(\ell\)). The starting point is the following equation. \[\alpha_{\mathrm{s}}-\alpha_{\mathrm{s}}^{*}=\frac{\alpha_{\mathrm{s}}}{\kappa_{\mathrm{s}} \, \sigma} \,\left(\kappa_{\mathrm{s}} \, \sigma-\kappa_{\mathrm{s}}^{*} \, \sigma^{*}\right)-\frac{\alpha_{\mathrm{s}}^{*}}{\alpha_{\mathrm{p}}} \,\left(\alpha_{\mathrm{p}}-\alpha_{\mathrm{p}}^{*}\right)\]The latter equation is effectively an identity. From equation (a), \[\alpha_{\mathrm{s}}-\alpha_{\mathrm{s}}^{*}=\alpha_{\mathrm{s}}-\frac{\alpha_{\mathrm{s}}}{\kappa_{\mathrm{s}} \, \sigma} \, \kappa_{\mathrm{s}}^{*} \, \sigma^{*}-\alpha_{\mathrm{s}}^{*}+\frac{\alpha_{\mathrm{s}}^{*}}{\alpha_{\mathrm{p}}} \, \alpha_{\mathrm{p}}^{*}\]From equation (b), \(\alpha_{\mathrm{s}}-\alpha_{\mathrm{s}}^{*}=\alpha_{\mathrm{s}}-\alpha_{\mathrm{s}}^{*}+\frac{\kappa_{\mathrm{s}} \, \sigma}{\mathrm{T} \, \alpha_{\mathrm{p}} \, \kappa_{\mathrm{s}} \, \sigma} \, \kappa_{\mathrm{s}}^{*} \, \sigma^{*}-\frac{\kappa_{\mathrm{S}}^{*} \, \sigma^{*}}{\alpha_{\mathrm{p}} \, \mathrm{T}}\) or, \(\alpha_{\mathrm{s}}-\alpha_{\mathrm{s}}^{*}=\alpha_{\mathrm{s}}-\alpha_{\mathrm{s}}^{*}\)But as an identity, \[\kappa_{\mathrm{S}} \, \sigma-\kappa_{\mathrm{S}}^{*} \, \sigma^{*}=\sigma \,\left(\kappa_{\mathrm{S}}-\kappa_{\mathrm{S}}^{*}\right)+\kappa_{\mathrm{S}}^{*} \,\left(\sigma-\sigma^{*}\right)\]From equations (a) and (c). \(\alpha_{\mathrm{s}}-\alpha_{\mathrm{s}}^{*}=\frac{\alpha_{\mathrm{s}}}{\kappa_{\mathrm{s}}} \,\left(\kappa_{\mathrm{s}}-\kappa_{\mathrm{s}}^{*}\right)+\frac{\alpha_{\mathrm{s}} \, \kappa_{\mathrm{s}}^{*}}{\kappa_{\mathrm{s}} \, \sigma} \,\left(\sigma-\sigma^{*}\right)-\frac{\alpha_{\mathrm{s}}^{*}}{\alpha_{\mathrm{p}}}\left(\alpha_{\mathrm{p}}-\alpha_{\mathrm{p}}^{*}\right)\)But, \[\begin{aligned} &\phi\left(\mathrm{E}_{\mathrm{s} j} ; \text { def }\right)=\left[\mathrm{c}_{\mathrm{j}}\right]^{-1} \,\left[\alpha_{\mathrm{s}}-\alpha_{\mathrm{s}}^{*}\right]+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \alpha_{\mathrm{s}}^{*} \\ &\phi\left(\mathrm{E}_{\mathrm{S}_{\mathrm{j}}} ; \operatorname{def}\right)=\frac{\alpha_{\mathrm{s}}}{\kappa_{\mathrm{s}}} \, \frac{1}{\mathrm{c}_{\mathrm{j}}} \,\left(\kappa_{\mathrm{s}}-\kappa_{\mathrm{s}}^{*}\right)+\frac{\alpha_{\mathrm{s}} \, \kappa_{\mathrm{s}}^{*}}{\kappa_{\mathrm{s}} \, \sigma} \, \frac{1}{\mathrm{c}_{\mathrm{j}}} \,\left(\sigma-\sigma^{*}\right) \\ &-\frac{\alpha_{\mathrm{s}}^{*}}{\alpha_{\mathrm{p}}} \, \frac{1}{\mathrm{c}_{\mathrm{j}}} \,\left(\alpha_{\mathrm{p}}-\alpha_{\mathrm{p}}^{*}\right)+\alpha_{\mathrm{s}}^{*} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \end{aligned}\]But for the isobaric heat capacities \(\phi\left(\mathrm{C}_{\mathrm{pj}}\right)=\left[\mathrm{c}_{\mathrm{j}}\right]^{-1} \,\left[\sigma-\sigma^{*}\right]+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \sigma^{*}\)Also, \[\begin{array}{r} \phi\left(\mathrm{K}_{\mathrm{s} j} ; \text { def }\right)=\left[\mathrm{c}_{\mathrm{j}}\right]^{-1} \,\left[\kappa_{\mathrm{s}}-\kappa_{\mathrm{s}}^{*}\right]+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \kappa_{\mathrm{s}}^{*} \\ \phi\left(\mathrm{E}_{\mathrm{Sj}} ; \text { def }\right)=\frac{\alpha_{\mathrm{s}}}{\kappa_{\mathrm{S}}} \,\left[\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \text { def }\right)-\kappa_{\mathrm{s}}^{*} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\right] \end{array}\]Hence, \[\begin{aligned} &+\frac{\alpha_{\mathrm{s}} \, \kappa_{\mathrm{s}}^{*}}{\kappa_{\mathrm{s}} \, \sigma} \,\left[\phi\left(\mathrm{C}_{\mathrm{pj}}\right)-\sigma^{*} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\right] \\ &-\frac{\alpha_{\mathrm{s}}^{*}}{\alpha_{\mathrm{p}}} \,\left[\phi\left(\mathrm{E}_{\mathrm{pj}}\right)-\alpha_{\mathrm{p}}^{*} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\right]+\alpha_{\mathrm{s}}^{*} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \end{aligned}\]With a little reorganisation, \[\begin{aligned} \phi\left(\mathrm{E}_{\mathrm{Sj}} ; \operatorname{def}\right)=-& \frac{\alpha_{\mathrm{S}}^{*}}{\alpha_{\mathrm{p}}} \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right)+\frac{\alpha_{\mathrm{S}}}{\kappa_{\mathrm{S}}} \, \phi\left(\mathrm{K}_{\mathrm{Sj}} ; \operatorname{def}\right)+\frac{\alpha_{\mathrm{s}} \, \kappa_{\mathrm{s}}^{*}}{\kappa_{\mathrm{S}} \, \sigma} \, \phi\left(\mathrm{C}_{\mathrm{pj}}\right) \\ &+\left[\alpha_{\mathrm{s}}^{*} \,\left(1+\frac{\alpha_{\mathrm{p}}^{*}}{\alpha_{\mathrm{p}}}\right)-\frac{\alpha_{\mathrm{s}} \, \kappa_{\mathrm{s}}^{*}}{\kappa_{\mathrm{S}}}\left(1+\frac{\sigma^{*}}{\sigma}\right)\right] \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \end{aligned}\]Hence in the limit of infinite dilution, \[\frac{\phi\left(E_{\mathrm{S} j} ; \mathrm{def}\right)^{\infty}}{\alpha_{\mathrm{s}}^{*}}=-\frac{\phi\left(\mathrm{E}_{\mathrm{pj}}\right)^{\infty}}{\alpha_{\mathrm{p}}^{*}}+\frac{\phi\left(\mathrm{K}_{\mathrm{s} j} ; \mathrm{def}\right)^{\infty}}{\kappa_{\mathrm{s}}^{*}}+\frac{\phi\left(\mathrm{C}_{\mathrm{pj}}\right)^{\infty}}{\sigma^{*}}\] \[\begin{aligned} &{\frac{\phi\left(\mathrm{E}_{\mathrm{Sj}} ; \mathrm{def}\right)^{\infty}}{\alpha_{\mathrm{s}}^{*}}=\frac{\left[\mathrm{m}^{3} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}\right]}{\left[\mathrm{K}^{-1}\right]}=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]} \\ &\frac{\phi\left(\mathrm{E}_{\mathrm{pj}}\right)^{\infty}}{\alpha_{\mathrm{p}}^{*}}=\frac{\left[\mathrm{m}^{3} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}\right]}{\left[\mathrm{K}^{-1}\right]}=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right] \\ &\frac{\phi\left(\mathrm{K}_{\mathrm{Sj}} ; \mathrm{def}\right)^{\infty}}{\kappa_{\mathrm{S}}^{*}}=\frac{\left[\mathrm{m}^{3} \mathrm{~mol}^{-1} / \mathrm{N} \mathrm{m}^{-2}\right]}{\left[\mathrm{N} \mathrm{m}^{-2}\right]^{-1}}=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right] \\ &\frac{\phi\left(\mathrm{C}_{\mathrm{pj}}\right)^{\infty}}{\sigma^{*}}=\frac{\left[\mathrm{J} \mathrm{mol}^{-1}\right]}{\left[\mathrm{J} \mathrm{m}^{-3}\right]}=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right] \end{aligned}\]This page titled 1.12.16: Expansions- Solutions- Apparent Molar Isentropic and Isobaric is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.12.17: Expansions- Isentropic- Solutions- Apparent and Partial Molar

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.12%3A_Expansions/1.12.17%3A_Expansions-_Isentropic-_Solutions-_Apparent_and_Partial_Molar

A given solution is prepared using \(\mathrm{n}_{1}\) moles of solvent (water) and \(\mathrm{n}_{j}\) moles of solute \(j\). The volume of the system is defined by equation (a). \[\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}\right]\]We consider the case where the closed system is at equilibrium and hence where the affinity for spontaneous change is zero. The entropy of the system (at equilibrium) is defined by the same set of independent variables. Thus \[\mathrm{S}=\mathrm{S}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}\right]\]The system is perturbed at constant pressure by a change in temperature. The path followed by the system is such that the affinity for spontaneous change remains at zero (i.e. at equilibrium) and that the entropy of the system \(\mathrm{S}(\mathrm{aq})\) remains constant at that given by equation (b).The equilibrium isentropic expansion of the system is defined by equation (c). \[\mathrm{E}_{\mathrm{s}}(\mathrm{A}=0)=\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{S}(\mathrm{aq}), \mathrm{A}=0}\]\(\mathrm{E}_{\mathrm{S}}(\mathrm{A}=0)\) is an extensive property of the system. Nevertheless it is convenient to consider an intensive property. For example, \(\mathrm{E}_{\mathrm{S}}\left(\mathrm{aq} ; \mathrm{A}=0 ; \mathrm{w}_{1}=1 \mathrm{~kg} ; \mathrm{m}_{\mathrm{j}}\right)\) is the equilibrium isentropic expansion of a solution molality \(\mathrm{m}_{j}\) prepared using \(1 \mathrm{~kg}\) of water(\(\ell\)).For a system comprising pure solvent at defined \(\mathrm{T}\) and \(\mathrm{p}\) we define a molar (equilibrium) isentropic expansion, \(\mathrm{E}_{\mathrm{S}}^{*}(\ell)\); equation (d). \[\mathrm{E}_{\mathrm{S} 1}^{*}(\ell ; \mathrm{A}=0)=\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{T}}\right)_{\mathrm{s}_{1}^{*}(\ell), \mathrm{A}=0}\]The volume of a solution, molality \(\mathrm{m}_{j}\), prepared using \(1 \mathrm{~kg}\) of water(\(\ell\)) is related to the composition using either equations (e) or (f). \[\mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\]\[\mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{V}_{1}(\mathrm{aq})+\mathrm{m}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}(\mathrm{aq})\]A key problem emerges. We note that the conditions on the partial differential in equation (c) relate to the entropy of the aqueous solution. The latter condition is not the same as that invoked in equation (d) which refers to the molar entropy of the pure solvent. We could of course differentiate equation (e) with respect to temperature at fixed entropy S(aq). However we would encounter a term \(\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{T}}\right)_{\mathrm{s}(\mathrm{aq})}\). This is a complicated derivative where we might have hoped for a term \(\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{T}}\right)_{\mathrm{s}_{1}^{*}(\ell)}\). The way forward is to accept the problem and define a property, by analogy with the corresponding isobaric property, a property \(\phi\left(\mathrm{E}_{\mathrm{S} j} ; \text { def }\right)\) which has the appearance of proper thermodynamic apparent property. Then, \[\mathrm{E}_{\mathrm{S}}\left(\mathrm{aq} ; \mathrm{A}=0 ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{E}_{\mathrm{Sl}}^{*}(\ell ; \mathrm{A}=0)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{E}_{\mathrm{Sj}} ; \mathrm{def}\right)\]There is a subtle problem with respect to equation (f) which can be differentiated with respect to \(\mathrm{T}\) at constant \(\mathrm{S}(\mathrm{aq})\) as defined by equation (b). Then \[\mathrm{E}_{\mathrm{s}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \,\left(\frac{\partial \mathrm{V}_{1}(\mathrm{aq})}{\partial \mathrm{T}}\right)_{\mathrm{S}(\mathrm{aq})}+\mathrm{m}_{\mathrm{j}} \,\left(\frac{\partial \mathrm{V}_{\mathrm{j}}(\mathrm{aq})}{\partial \mathrm{T}}\right)_{\mathrm{S}(\mathrm{aq})}\]Partial molar isentropic expansions \(\mathrm{E}_{\mathrm{S}1}(\mathrm{aq})\) and \(\mathrm{E}_{\mathrm{S}j}(\mathrm{aq})\) are defined by the following equations. \[\mathrm{E}_{\mathrm{Sl}}(\mathrm{aq})=\left(\frac{\partial \mathrm{E}_{\mathrm{s}}(\mathrm{aq})}{\partial \mathrm{n}_{1}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{j})}\]\[\mathrm{E}_{\mathrm{Sj}}(\mathrm{aq})=\left(\frac{\partial \mathrm{E}_{\mathrm{j}}(\mathrm{aq})}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}}\]But \(\mathrm{E}_{\mathrm{S}1}\) and \(\mathrm{E}_{\mathrm{S}j}\) are non-Lewisian partial molar properties. Hence \[\mathrm{E}_{\mathrm{Sl}}(\mathrm{aq}) \neq\left(\frac{\partial \mathrm{V}_{1}(\mathrm{aq})}{\partial \mathrm{T}}\right)_{\mathrm{S}(\mathrm{aq})}\]\[\mathrm{E}_{\mathrm{S} j}(\mathrm{aq})=\left(\frac{\partial \mathrm{V}_{\mathrm{j}}(\mathrm{aq})}{\partial \mathrm{T}}\right)_{\mathrm{s}(\mathrm{aq})}\]Then, \[\mathrm{E}_{\mathrm{S}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{E}_{\mathrm{S} 1}(\mathrm{aq})+\mathrm{m}_{\mathrm{j}} \, \mathrm{E}_{\mathrm{Sj}}(\mathrm{aq})\]In practical terms equation (n) follows from equation (g), \[\phi\left(\mathrm{E}_{\mathrm{Sj}} ; \operatorname{def}\right)=\left(1 / \mathrm{m}_{\mathrm{j}}\right) \,\left[\mathrm{E}_{\mathrm{S}}\left(\mathrm{aq} ; \mathrm{A}=0 ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)-\left(1 / \mathrm{M}_{1}\right) \, \mathrm{E}_{\mathrm{S} 1}^{*}(\ell ; \mathrm{A}=0)\right]\]Two practical equations follow from equation (n) allowing \(\phi\left(\mathrm{E}_{\mathrm{Sj}} ; \operatorname{def}\right)\) to be calculated from the isentropic expansibilities of solutions and solvent, both volume intensive variables. \[\phi\left(\mathrm{E}_{\mathrm{s} j} ; \operatorname{def}\right)=\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\alpha_{\mathrm{s}}(\mathrm{aq})-\alpha_{\mathrm{sl}}^{*}(\ell)\right]+\alpha_{\mathrm{s}}(\mathrm{aq}) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\]\[\phi\left(E_{\mathrm{Sj}} ; \text { def }\right)=\left[c_{j}\right]^{-1} \,\left[\alpha_{\mathrm{s}}(\mathrm{aq})-\alpha_{\mathrm{s} 1}^{*}(\ell)\right]+\alpha_{\mathrm{S} 1}^{*}(\ell) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\]where \[\alpha_{\mathrm{s}}(\mathrm{aq})=\frac{1}{\mathrm{~V}(\mathrm{aq})} \,\left(\frac{\partial \mathrm{V}(\mathrm{aq})}{\partial \mathrm{T}}\right)_{\mathrm{s}(\mathrm{aq})}\]\[\alpha_{\mathrm{S} 1}^{*}(\ell)=\frac{1}{\mathrm{~V}_{1}^{*}(\ell)} \,\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{T}}\right)_{\mathrm{s}_{1}^{*}(\ell)}\]Footnotes From equation (n), \[\phi\left(\mathrm{E}_{\mathrm{Sj}} ; \operatorname{def}\right)=\left(1 / \mathrm{m}_{\mathrm{j}}\right) \,\left[\mathrm{E}_{\mathrm{s}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)-\left(1 / \mathrm{M}_{1}\right) \, \mathrm{E}_{\mathrm{sl}}^{*}(\ell)\right]\]We use equation (m) for a solution prepared using \(1 \mathrm{~kg}\) of water. \[\mathrm{E}_{\mathrm{s}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{E}_{\mathrm{S} 1}(\mathrm{aq})+\mathrm{m}_{\mathrm{j}} \, \mathrm{E}_{\mathrm{Sj}}(\mathrm{aq})\]Then \[\phi\left(\mathrm{E}_{\mathrm{S} j} ; \mathrm{def}\right)=\left(1 / \mathrm{m}_{\mathrm{j}}\right) \,\left[\left(1 / \mathrm{M}_{1}\right) \, \mathrm{E}_{\mathrm{S} 1}(\mathrm{aq})+\mathrm{m}_{\mathrm{j}} \, \mathrm{E}_{\mathrm{Sj}}(\mathrm{aq})-\left(1 / \mathrm{M}_{1}\right) \, \mathrm{E}_{\mathrm{Sl} 1}^{*}(\ell)\right]\]Or, \[\phi\left(\mathrm{E}_{\mathrm{s} j} ; \operatorname{def}\right)=\mathrm{E}_{\mathrm{S} j}(\mathrm{aq})+\left(1 / \mathrm{M}_{1}\right) \,\left(1 / \mathrm{m}_{\mathrm{j}}\right) \,\left[\mathrm{E}_{\mathrm{Sl}}(\mathrm{aq})-\mathrm{E}_{\mathrm{s} 1}^{*}(\ell)\right]\]Hence using equation (m), \[\phi\left(\mathrm{E}_{\mathrm{Sj}} ; \mathrm{def}\right)=\left(1 / \mathrm{m}_{\mathrm{j}}\right) \, \mathrm{E}_{\mathrm{s}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)-\left(1 / \mathrm{M}_{1}\right) \,\left(1 / \mathrm{m}_{\mathrm{j}}\right) \, \mathrm{E}_{\mathrm{Sl}}^{*}(\ell)\]Using equations (q) and (r), \[\phi\left(\mathrm{E}_{\mathrm{s}} ; \mathrm{def}\right)=\left(1 / \mathrm{m}_{\mathrm{j}}\right) \, \alpha_{\mathrm{s}}(\mathrm{aq}) \, \mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)-\left(1 / \mathrm{M}_{1}\right) \,\left(1 / \mathrm{m}_{\mathrm{j}}\right) \, \alpha_{\mathrm{s} 1}^{*}(\ell) \, \mathrm{V}_{1}^{*}(\ell)\]Or, \[\begin{aligned} \phi\left(\mathrm{E}_{\mathrm{s} j} ; \text { def }\right)=&\left(1 / \mathrm{m}_{\mathrm{j}}\right) \, \alpha_{\mathrm{s}}(\mathrm{aq}) \,\left[\left(1 / \mathrm{M}_{1}\right) \mathrm{V}_{1}^{*}(\ell)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\right] \\ &-\left(1 / \mathrm{M}_{1}\right) \,\left(1 / \mathrm{m}_{\mathrm{j}}\right) \, \alpha_{\mathrm{s} 1}^{*}(\mathrm{l}) \, \mathrm{V}_{1}^{*}(\ell) \end{aligned}\]Or \[\phi\left(E_{\mathrm{Sj}} ; \operatorname{def}\right)=\left(\mathrm{V}_{1}^{*}(\ell) / \mathrm{m}_{\mathrm{j}} \, \mathrm{M}_{1}\right) \,\left[\alpha_{\mathrm{s}}(\mathrm{aq})-\alpha_{\mathrm{s} 1}^{*}(\ell)\right]+\alpha_{\mathrm{s}}(\mathrm{aq}) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\]Or \[\phi\left(E_{\mathrm{s} j} ; \operatorname{def}\right)=\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1} \,\left[\alpha_{\mathrm{s}}(\mathrm{aq})-\alpha_{\mathrm{s} 1}^{*}(\ell)\right]+\alpha_{\mathrm{s}}(\mathrm{aq}) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\]Also \(\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\ell)\right]^{-1}=\left[1 / \mathrm{c}_{\mathrm{j}}\right]-\phi\left(\mathrm{V}_{\mathrm{j}}\right)\) Then, \[\phi\left(\mathrm{E}_{\mathrm{sj}} ; \mathrm{def}\right)=\left[\left(1 / \mathrm{c}_{\mathrm{j}}\right)-\phi\left(\mathrm{V}_{\mathrm{j}}\right)\right] \,\left[\alpha_{\mathrm{s}}(\mathrm{aq})-\alpha_{\mathrm{sl}}^{*}(\ell)\right]+\alpha_{\mathrm{s}}(\mathrm{aq}) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\]Or, \[\begin{aligned} \phi\left(\mathrm{E}_{\mathrm{Sj}} ; \mathrm{def}\right)=\left(1 / \mathrm{c}_{\mathrm{j}}\right) \,\left[\alpha_{\mathrm{s}}(\mathrm{aq})-\alpha_{\mathrm{s} 1}^{*}(\ell)\right] \\ &-\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \alpha_{\mathrm{s}}(\mathrm{aq})+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \alpha_{\mathrm{s} 1}^{*}(\ell)+\alpha_{\mathrm{s}}(\mathrm{aq}) \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \end{aligned}\]Hence, \[\phi\left(\mathrm{E}_{\mathrm{Sj}} ; \mathrm{def}\right)=\left(1 / \mathrm{c}_{\mathrm{j}}\right) \,\left[\alpha_{\mathrm{s}}(\mathrm{aq})-\alpha_{\mathrm{S} 1}^{*}(\ell)\right]+\phi\left(\mathrm{V}_{\mathrm{j}}\right) \, \alpha_{\mathrm{s} 1}^{*}(\ell)\]For further details see---- J.C.R.Reis, G. Douheret, M.I.Davis, I.J.Fjellanger and H.Hoiland, Phys. Chem. Chem. Phys., 2008,10, 561.This page titled 1.12.17: Expansions- Isentropic- Solutions- Apparent and Partial Molar is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.12.18: Expansions- Solutions- Isentropic Dependence of Apparent Molar Volume of Solute on Temperature and Pressure

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.12%3A_Expansions/1.12.18%3A_Expansions-_Solutions-_Isentropic_Dependence_of_Apparent_Molar_Volume_of_Solute_on_Temperature_and_Pressure

The starting point is the following calculus operation. \[\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{T}}\right)_{\mathrm{s}}=\left(\frac{\partial \mathrm{p}}{\partial \mathrm{T}}\right)_{\mathrm{s}} \,\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right)_{\mathrm{s}}\]Also \[\left(\frac{\partial \phi\left(V_{\mathrm{j}}\right)}{\partial T}\right)_{\mathrm{s}}=\frac{\mathrm{C}_{\mathrm{p}}}{\mathrm{T}} \,\left(\frac{\partial \mathrm{T}}{\partial \mathrm{V}}\right)_{\mathrm{p}} \,\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right)_{\mathrm{s}}\]Or, \[\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{T}}\right)_{\mathrm{s}}=\frac{\mathrm{C}_{\mathrm{p}}}{\mathrm{T}} \, \frac{\mathrm{V}}{\mathrm{V}} \,\left(\frac{\partial \mathrm{T}}{\partial \mathrm{V}}\right)_{\mathrm{p}} \,\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right)_{\mathrm{s}}\]Hence, \[\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial T}\right)_{\mathrm{s}}=\frac{\sigma}{\mathrm{T} \, \alpha_{\mathrm{p}}} \,\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right)_{\mathrm{s}}\]But \(\frac{\sigma}{\mathrm{T}}=\frac{\left[\alpha_{p}\right]^{2}}{\delta}\) Then \[\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{T}}\right)_{\mathrm{s}}=\frac{\alpha_{\mathrm{p}}}{\delta} \,\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right)_{\mathrm{s}}\]Also, \[\alpha_{\mathrm{s}}=-\kappa_{\mathrm{s}} \, \sigma / \mathrm{T} \, \alpha_{\mathrm{p}}\]Then \(\frac{\alpha_{\mathrm{s}}}{\kappa_{\mathrm{s}}}=-\frac{\alpha_{\mathrm{p}}}{\delta}\) Hence \[\frac{\left[\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right) / \partial \mathrm{T}\right]_{\mathrm{s}}}{\alpha_{\mathrm{s}}}=-\frac{\left[\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right) / \partial \mathrm{p}\right]_{\mathrm{s}}}{\kappa_{\mathrm{s}}}\]This page titled 1.12.18: Expansions- Solutions- Isentropic Dependence of Apparent Molar Volume of Solute 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.

1.12.19: Expansions- Solutions- Isentropic Dependence of Partial Molar Volume on Temperature

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.12%3A_Expansions/1.12.19%3A_Expansions-_Solutions-_Isentropic_Dependence_of_Partial_Molar_Volume_on_Temperature

We switch the condition on a derivative expressing the dependence of partial molar volume on temperature. \[\left(\frac{\partial \mathrm{V}_{\mathrm{j}}}{\partial \mathrm{T}}\right)_{\mathrm{s}}=\left(\frac{\partial \mathrm{V}_{\mathrm{j}}}{\partial \mathrm{T}}\right)_{\mathrm{p}}+\left(\frac{\partial \mathrm{V}_{\mathrm{j}}}{\partial \mathrm{p}}\right)_{\mathrm{T}}\left(\frac{\partial \mathrm{p}}{\partial \mathrm{T}}\right)_{\mathrm{s}}\]Then \[\left(\frac{\partial V_{j}}{\partial T}\right)_{\mathrm{s}}=E_{p j}-\left(\frac{\partial V_{j}}{\partial p}\right)_{T} \, \frac{(\partial S / \partial T)_{p}}{(\partial S / \partial p)_{T}}\]Or, \[\left(\frac{\partial \mathrm{V}_{\mathrm{j}}}{\partial \mathrm{T}}\right)_{\mathrm{S}}=\mathrm{E}_{\mathrm{pj}}+\frac{(\partial \mathrm{S} / \partial \mathrm{T})_{\mathrm{p}}}{(\partial \mathrm{S} / \partial \mathrm{p})_{\mathrm{T}}} \, \mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\]But \[\frac{(\partial S / \partial T)_{p}}{(\partial S / \partial p)_{T}}=-\frac{C_{p}}{T \,(\partial V / \partial T)_{p}}=-\frac{C_{p}}{T \, V \, \alpha_{p}}\]Or, \[\frac{(\partial \mathrm{S} / \partial \mathrm{T})_{\mathrm{p}}}{(\partial \mathrm{S} / \partial \mathrm{p})_{\mathrm{T}}}=-\frac{\sigma}{\mathrm{T} \, \alpha_{\mathrm{p}}}\]Hence, \[\left(\frac{\partial \mathrm{V}_{\mathrm{j}}}{\partial \mathrm{T}}\right)_{\mathrm{s}}=\mathrm{E}_{\mathrm{pj}}-\frac{\sigma}{\mathrm{T} \, \alpha_{\mathrm{p}}} \, \mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\]This page titled 1.12.19: Expansions- Solutions- Isentropic Dependence of Partial Molar Volume on Temperature is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.12.20: Expansions- Isentropic- Liquid Mixtures

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.12%3A_Expansions/1.12.20%3A_Expansions-_Isentropic-_Liquid_Mixtures

A given binary liquid mixture has mole fraction \(x_{1}\left[=1-x_{2}\right]\) at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\). The system is at equilibrium at a minimum in Gibbs energy where the affinity for spontaneous change is zero. The molar volume and molar entropy of the mixtures are given by equations (a) and (b). \[\mathrm{V}_{\mathrm{m}}=\mathrm{V}_{\mathrm{m}}\left[\mathrm{T}, \mathrm{p}, \mathrm{x}_{1}\right]\]\[\mathrm{S}_{\mathrm{m}}=\mathrm{S}_{\mathrm{m}}\left[\mathrm{T}, \mathrm{p}, \mathrm{x}_{1}\right]\]These two equations describe the properties of the system in the \(\mathrm{T}-\mathrm{p}\)-composition domain; i.e. a Gibbsian description. We consider two dependences of the volume on temperature under the constraint that the affinity for spontaneous change remains at zero; i.e equilibrium expansions.The isobaric expansion is defined by equation (c). \[\mathrm{E}_{\mathrm{p}}(\operatorname{mix})=\left(\frac{\partial \mathrm{V}(\operatorname{mix})}{\partial \mathrm{T}}\right)_{\mathrm{p}}\]The isentropic expansion is defined by equation (d) \[\mathrm{E}_{\mathrm{S}}(\operatorname{mix})=\left(\frac{\partial \mathrm{V}(\operatorname{mix})}{\partial \mathrm{T}}\right)_{\mathrm{S}}\]In the latter case the system tracks a path with increase in temperature where the affinity for spontaneous change remains at zero and the entropy remains the same at that defined by equation (b). [\(\mathrm{NB} \mathrm{~E}_{p}(\operatorname{mix})\) and \(E_{S}(\operatorname{mix})\) as defined by equations (c) and (d) are extensive properties.] The two expansions are related through the (equilibrium) isobaric heat capacity \(\mathrm{C}_{\mathrm{p}} (\operatorname{mix})\)and the (equilibrium) isothermal compression \(\mathrm{K}_{\mathrm{T}}(\operatorname{mix})\). Thus \[\mathrm{E}_{\mathrm{S}}(\operatorname{mix})=\mathrm{E}_{\mathrm{p}}(\operatorname{mix})-\frac{\mathrm{C}_{\mathrm{p}}(\operatorname{mix}) \, \mathrm{K}_{\mathrm{T}}(\operatorname{mix})}{\mathrm{T} \, \mathrm{E}_{\mathrm{p}}(\operatorname{mix})}\]In the context of the property \(\mathrm{E}_{\mathrm{p}(\operatorname{mix})\), the entropy of the system changes with an increase in temperature at constant pressure. But by definition the entropy does not change for an isentropic expansion, \(\mathrm{E}_{\mathrm{S}(\operatorname{mix})\).For a binary liquid mixture having ideal thermodynamic properties, \[E_{S}(\operatorname{mix} ; i d)=E_{p}(\operatorname{mix} ; i d)-\frac{C_{p}(\operatorname{mix} ; i d) \, K_{T}(\operatorname{mix} ; i d)}{T \, E_{p}(\operatorname{mix} ; i d)}\]In this comparison we note that \(\mathrm{E}_{\mathrm{p}(\operatorname{mix})\) and \(\mathrm{E}_{\mathrm{p}}(\operatorname{mix} ; \mathrm{id})\) refer to the same pressure but the entropies referred to in \(\mathrm{E}_{\mathrm{S}}(\operatorname{mix})\) and \(\mathrm{E}_{\mathrm{S}}(\operatorname{mix} ; \mathrm{id})\) are not the same. The same contrast arises when we set out the two equations describing expansions of the pure liquids. \[\mathrm{E}_{\mathrm{S} 1}^{*}(\ell)=\mathrm{E}_{\mathrm{p} 1}^{*}(\ell)-\frac{\mathrm{C}_{\mathrm{p} 1}^{*}(\ell) \, \mathrm{K}_{\mathrm{T} 1}^{*}(\ell)}{\mathrm{T} \, \mathrm{E}_{\mathrm{p} 1}^{*}(\ell)}\]\[\mathrm{E}_{\mathrm{S} 2}^{*}(\ell)=\mathrm{E}_{\mathrm{p} 2}^{*}(\ell)-\frac{\mathrm{C}_{\mathrm{p} 2}^{*}(\ell) \, \mathrm{K}_{\mathrm{T} 2}^{*}(\ell)}{\mathrm{T} \, \mathrm{E}_{\mathrm{p} 2}^{*}(\ell)}\]The subject is complicated by the galaxy of entropies implied by the phrase ‘at constant entropy’.Footnote Using a calculus operation, \[\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{S}}=\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{p}}-\left(\frac{\partial \mathrm{S}}{\partial \mathrm{T}}\right)_{\mathrm{p}} \,\left(\frac{\partial \mathrm{p}}{\partial \mathrm{S}}\right)_{\mathrm{T}} \,\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{T}}\]We note two Maxwell equations. From \(\mathrm{U}=\mathrm{U}[\mathrm{S}, \mathrm{V}], \quad \partial^{2} \mathrm{U} / \partial \mathrm{S} \, \partial \mathrm{V}=\partial^{2} \mathrm{U} / \partial \mathrm{V} \, \partial \mathrm{S}\) Then \[\left(\frac{\partial \mathrm{T}}{\partial \mathrm{V}}\right)_{\mathrm{s}}=-\left(\frac{\partial \mathrm{p}}{\partial \mathrm{S}}\right)_{\mathrm{V}}\]We invert the latter equation. Hence \[\begin{aligned} \mathrm{E}_{\mathrm{S}}=&\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{s}}=-\left(\frac{\partial \mathrm{S}}{\partial \mathrm{p}}\right)_{\mathrm{V}}=\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{s}} \,\left(\frac{\partial \mathrm{S}}{\partial \mathrm{V}}\right)_{\mathrm{p}} \\ &=-\mathrm{K}_{\mathrm{S}} \,\left(\frac{\partial \mathrm{S}}{\partial \mathrm{T}}\right)_{\mathrm{p}} \,\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{p}}=-\mathrm{K}_{\mathrm{s}} \, \mathrm{C}_{\mathrm{p}} / \mathrm{T} \, \mathrm{E}_{\mathrm{p}} \end{aligned}\]Similarly \[\partial^{2} \mathrm{G} / \partial \mathrm{T} \, \partial \mathrm{p}=\partial^{2} \mathrm{G} / \partial \mathrm{p} \, \partial \mathrm{T}\]Then, \[E_{p}=\left(\frac{\partial V}{\partial T}\right)_{p}=-\left(\frac{\partial S}{\partial p}\right)_{T}\]Also at equilibrium, \(\mathrm{S}=-\left(\frac{\partial \mathrm{G}}{\partial \mathrm{T}}\right)_{\mathrm{p}}\)But \(\mathrm{G}=\mathrm{H}-\mathrm{T} \, \mathrm{S}\). Then \(\mathrm{H}=\mathrm{G}-\mathrm{T} \,\left(\frac{\partial \mathrm{G}}{\partial \mathrm{T}}\right)_{\mathrm{p}}\) \[\frac{\partial \mathrm{H}}{\partial \mathrm{T}}=\frac{\partial \mathrm{G}}{\partial \mathrm{T}}-\mathrm{T} \,\left(\frac{\partial^{2} \mathrm{G}}{\partial \mathrm{T}^{2}}\right)-\frac{\partial \mathrm{G}}{\partial \mathrm{T}}\]Further, \[\left(\frac{\partial \mathrm{H}}{\partial \mathrm{T}}\right)_{\mathrm{p}}=\mathrm{C}_{\mathrm{p}}=-\mathrm{T} \,\left(\frac{\partial^{2} \mathrm{G}}{\partial \mathrm{T}^{2}}\right)_{\mathrm{p}}=\mathrm{T} \,\left(\frac{\partial \mathrm{S}}{\partial \mathrm{T}}\right)_{\mathrm{p}}\]Based on equation (a), \(\mathrm{E}_{\mathrm{S}}=\mathrm{E}_{\mathrm{p}}-\mathrm{C}_{\mathrm{p}} \, \mathrm{K}_{\mathrm{T}} / \mathrm{T} \, \mathrm{E}_{\mathrm{p}}\)This page titled 1.12.20: Expansions- Isentropic- Liquid Mixtures is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.12.21: Expansions- Solutions- Partial Molar Isobaric and Isentropic

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.12%3A_Expansions/1.12.21%3A_Expansions-_Solutions-_Partial_Molar_Isobaric_and_Isentropic

The starting point is equation (a). \[\mathrm{E}_{\mathrm{p}}=\mathrm{E}_{\mathrm{S}}+\frac{\mathrm{K}_{\mathrm{T}} \, \mathrm{C}_{\mathrm{p}}}{\mathrm{T} \, \mathrm{E}_{\mathrm{p}}}\]We differentiate this equation with respect to the amount of solute \(\mathrm{n}_{j}\) at fixed \(\mathrm{T}\), \(\mathrm{p}\) and amount of solvent \(\mathrm{n}_{1}\). \[\mathrm{E}_{\mathrm{pj}}=\mathrm{E}_{\mathrm{Sj}}+\frac{1}{\mathrm{~T}} \,\left[\frac{\mathrm{K}_{\mathrm{T}}}{\mathrm{E}_{\mathrm{p}}} \, \mathrm{C}_{\mathrm{pj}}+\frac{\mathrm{C}_{\mathrm{p}}}{\mathrm{E}_{\mathrm{p}}} \, \mathrm{K}_{\mathrm{Tj}}-\frac{\mathrm{K}_{\mathrm{T}} \, \mathrm{C}_{\mathrm{p}}}{\left(\mathrm{E}_{\mathrm{p}}\right)^{2}} \, \mathrm{E}_{\mathrm{pj}}\right]\]Or, \[\mathrm{E}_{\mathrm{pj}}=\mathrm{E}_{\mathrm{Sj}}+\frac{1}{\mathrm{~T}} \, \frac{\mathrm{K}_{\mathrm{T}} \, \mathrm{C}_{\mathrm{p}}}{\mathrm{E}_{\mathrm{p}}} \,\left[\frac{\mathrm{C}_{\mathrm{p} j}}{\mathrm{C}_{\mathrm{p}}}+\frac{\mathrm{K}_{\mathrm{T} \mathrm{J}}}{\mathrm{K}_{\mathrm{T}}}-\frac{\mathrm{E}_{\mathrm{p} j}}{\mathrm{E}_{\mathrm{p}}}\right]\]We convert to volume intensive variables. \[\mathrm{E}_{\mathrm{pj}}=\mathrm{E}_{\mathrm{S} \mathrm{j}}+\frac{1}{\mathrm{~T}} \, \frac{\kappa_{\mathrm{T}} \, \mathrm{C}_{\mathrm{p}}}{\alpha_{\mathrm{p}} \, \mathrm{V}} \,\left[\frac{\mathrm{V} \, \mathrm{C}_{\mathrm{pj}}}{\mathrm{C}_{\mathrm{p}}}+\frac{\mathrm{V} \, \mathrm{K}_{\mathrm{T} j}}{\mathrm{~K}_{\mathrm{T}}}-\frac{\mathrm{V} \, \mathrm{E}_{\mathrm{p} j}}{\mathrm{E}_{\mathrm{p}}}\right]\]Or, \[E_{p j}=E_{S j}+\frac{1}{T} \, \frac{K_{T} \, \sigma}{\alpha_{p}} \,\left[\frac{C_{p j}}{\sigma}+\frac{K_{T_{j}}}{K_{T}}-\frac{E_{p j}}{\alpha_{p}}\right]\]But \[\varepsilon=K_{\mathrm{T}} \, \sigma / \mathrm{T} \, \alpha_{\mathrm{p}}\]Then, \[\frac{\mathrm{E}_{\mathrm{pj}}-\mathrm{E}_{\mathrm{S} j}}{\varepsilon}=-\frac{\mathrm{E}_{\mathrm{pj}}}{\alpha_{\mathrm{p}}}+\frac{\mathrm{K}_{\mathrm{T} j}}{\kappa_{\mathrm{T}}}+\frac{\mathrm{C}_{\mathrm{pj}}}{\sigma}\]Hence for an aqueous solution in the limit of infinite dilution, \[\frac{\mathrm{E}_{\mathrm{pj}}^{\infty}(\mathrm{aq})-\mathrm{E}_{\mathrm{sj}}^{\infty}(\mathrm{aq})}{\varepsilon_{1}^{*}(\ell)}=-\frac{\mathrm{E}_{\mathrm{pj}}^{\infty}(\mathrm{aq})}{\alpha_{\mathrm{p} 1}^{*}(\ell)}+\frac{\mathrm{K}_{\mathrm{T} j}^{\infty}(\mathrm{aq})}{\kappa_{\mathrm{T} 1}^{*}(\ell)}+\frac{\mathrm{C}_{\mathrm{pj}}^{\infty}}{\sigma_{1}^{*}(\ell)}\]We start with the equation, \[\mathrm{E}_{\mathrm{s}}=-\frac{\mathrm{K}_{\mathrm{s}} \, \mathrm{C}_{\mathrm{p}}}{\mathrm{T} \, \mathrm{E}_{\mathrm{p}}}\]The latter equation is differentiated with respect to the amount of solute \(\mathrm{n}_{j}\) in a solution at fixed \(\mathrm{T}\), fixed \(\mathrm{p}\) and fixed amount of solvent, \(\mathrm{n}_{1}\). \[\begin{aligned} \left(\frac{\partial E_{S}}{\partial n_{j}}\right)_{T, p, n}=-\frac{C_{p}}{T \, E_{p}} \,\left(\frac{\partial K_{s}}{\partial n_{j}}\right)_{T, p, n}-\frac{K_{s}}{T \, E_{p}} \,\left(\frac{\partial C_{p}}{\partial n_{j}}\right)_{T, p, n} \\ &+\frac{K_{s} \, C_{p}}{T \,\left(E_{p}\right)^{2}} \,\left(\frac{\partial E_{p}}{\partial n_{j}}\right)_{T, p, n} \end{aligned}\]Or, \[E_{S_{j}}=-\frac{C_{p}}{T \, E_{p}} \, \frac{K_{s}}{K_{s}} \, K_{s_{j}}-\frac{K_{s}}{T \, E_{p}} \, \frac{C_{p}}{C_{p}} \, C_{p j}+\frac{K_{s} \, C_{p}}{T \,\left(E_{p}\right)^{2}} \, E_{p j}\]We rewrite the latter equation in terms of volume intensive variables. \[\mathrm{E}_{\mathrm{Sj}}=-\frac{1}{\mathrm{~T}} \, \frac{\sigma}{\alpha_{\mathrm{p}}} \, \frac{\kappa_{\mathrm{s}}}{\kappa_{\mathrm{s}}} \, \mathrm{K}_{\mathrm{Sj}}-\frac{1}{\mathrm{~T}} \, \frac{\kappa_{\mathrm{S}}}{\alpha_{\mathrm{p}}} \, \frac{\sigma}{\sigma} \, \mathrm{C}_{\mathrm{pj}}+\frac{\kappa_{\mathrm{s}} \, \sigma}{\mathrm{T} \,\left(\alpha_{\mathrm{p}}\right)^{2}} \, \mathrm{E}_{\mathrm{pj}}\]But \[\alpha_{\mathrm{s}}=-\frac{\kappa_{\mathrm{s}} \, \sigma}{\mathrm{T} \, \alpha_{\mathrm{p}}}\]Then \[E_{\mathrm{S}_{\mathrm{j}}}=\alpha_{\mathrm{s}} \, \frac{\mathrm{K}_{\mathrm{s} \mathrm{j}}}{\kappa_{\mathrm{s}}}+\alpha_{\mathrm{s}} \, \frac{\mathrm{C}_{\mathrm{p} j}}{\sigma}-\alpha_{\mathrm{s}} \, \frac{\mathrm{E}_{\mathrm{pj}}}{\alpha_{\mathrm{p}}}\]Therefore (with a change of order) \[\frac{E_{\mathrm{s} j}}{\alpha_{p}}=-\frac{E_{p j}}{\alpha_{p}}+\frac{K_{S j}}{K_{s}}+\frac{C_{p j}}{\sigma}\]This page titled 1.12.21: Expansions- Solutions- Partial Molar Isobaric and Isentropic is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.12.22: Expansions and Compressions- Solutions- Isentropic Dependence of Volume on Temperature and Pressure

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.12%3A_Expansions/1.12.22%3A_Expansions_and_Compressions-_Solutions-_Isentropic_Dependence_of_Volume_on_Temperature_and_Pressure

The starting point is the calculus operation for a double differential. \[\frac{\partial^{2} U}{\partial S \, \partial V}=\frac{\partial^{2} U}{\partial V \, \partial S}\]Then, \(\left(\frac{\partial T}{\partial V}\right)_{s}=-\left(\frac{\partial p}{\partial S}\right)_{v}\) Or, \[\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{s}}=-\left(\frac{\partial \mathrm{S}}{\partial \mathrm{p}}\right)_{\mathrm{V}}\]But, \[\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{s}}=\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{s}} \,\left(\frac{\partial \mathrm{S}}{\partial \mathrm{V}}\right)_{\mathrm{p}}\]Also we note that \[\left(\frac{\partial \mathrm{S}}{\partial \mathrm{T}}\right)_{\mathrm{p}}=\left(\frac{\partial \mathrm{S}}{\partial \mathrm{V}}\right)_{\mathrm{p}} \,\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{p}}\]Then, \[\left(\frac{\partial V}{\partial T}\right)_{s}=\left(\frac{\partial V}{\partial p}\right)_{s} \,\left(\frac{\partial S}{\partial T}\right)_{p} \,\left(\frac{\partial T}{\partial V}\right)_{p}\]However from the Gibbs - Helmholtz Equation, \(\left(\frac{\partial S}{\partial T}\right)_{p}=\frac{C_{p}}{T}\)Then \(\left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{s}}=\left(\frac{\partial \mathrm{V}}{\partial \mathrm{p}}\right)_{\mathrm{s}} \,\left(\frac{\partial \mathrm{T}}{\partial \mathrm{V}}\right)_{\mathrm{p}} \, \frac{\mathrm{C}_{\mathrm{p}}}{\mathrm{T}}\) Or, \[\mathrm{E}_{\mathrm{s}}=-\frac{\mathrm{K}_{\mathrm{s}} \, \mathrm{C}_{\mathrm{p}}}{\mathrm{T} \, \mathrm{E}_{\mathrm{p}}}\]We divide both sides of equation (f) by volume \(\mathrm{V}\). Hence \[\alpha_{\mathrm{s}}=-\kappa_{\mathrm{s}} \, \sigma / \mathrm{T} \, \alpha_{\mathrm{p}}\]This page titled 1.12.22: Expansions and Compressions- Solutions- Isentropic Dependence of Volume 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.

1.12.23: Expansions- The Difference

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.12%3A_Expansions/1.12.23%3A_Expansions-_The_Difference

For a solution, \[\varepsilon=\alpha_{p}-\alpha_{s}=\kappa_{T} \, \sigma / T \, \alpha_{p}\]In order to simplify the algebra, we omit (aq) and (\(\ell\)) when describing the properties of an aqueous solution and the pure liquid respectively. Superscript '*' identifies the pure solvent. \[\varepsilon^{*}=\alpha_{\mathrm{p}}^{*}-\alpha_{\mathrm{S}}^{*}=\kappa_{\mathrm{T}}^{*} \, \sigma^{*} / \mathrm{T} \, \alpha_{\mathrm{p}}^{*}\]Hence, \[\varepsilon-\varepsilon^{*}=\frac{\varepsilon}{\kappa_{\mathrm{T}} \, \sigma} \,\left[\kappa_{\mathrm{T}} \, \sigma-\kappa_{\mathrm{T}}^{*} \, \sigma^{*}\right]-\frac{\varepsilon^{*}}{\alpha_{\mathrm{p}}} \,\left[\alpha_{\mathrm{p}}-\alpha_{\mathrm{p}}^{*}\right]\]The latter equation is effectively an identity. According to equation (c) \[\varepsilon-\varepsilon=\varepsilon-\frac{\varepsilon}{\kappa_{\mathrm{T}} \, \sigma} \, \kappa_{\mathrm{T}}^{*} \, \sigma^{*}-\varepsilon+\frac{\varepsilon}{\alpha_{\mathrm{p}}} \, \alpha_{\mathrm{p}}^{*}\]We use equations (a) and (b) in the second and fourth terms on the right hand side of the latter equation. \[\varepsilon-\varepsilon^{*}=\varepsilon-\frac{\varepsilon}{\kappa_{\mathrm{T}} \, \sigma} \, \varepsilon^{*} \, \mathrm{T} \, \alpha_{\mathrm{p}}^{*}-\varepsilon^{*}+\frac{\varepsilon^{*} \, \alpha_{\mathrm{p}}^{*} \, \mathrm{T} \, \varepsilon}{\kappa_{\mathrm{T}} \, \sigma}\]Or \(\varepsilon-\varepsilon^{*}=\varepsilon-\varepsilon^{*}\) Further, as an identity, \[\kappa_{\mathrm{T}} \, \sigma-\kappa_{\mathrm{T}}^{*} \, \sigma^{*}=\sigma \,\left(\kappa_{\mathrm{T}}-\kappa_{\mathrm{T}}^{*}\right)+\kappa_{\mathrm{T}}^{*} \,\left(\sigma-\sigma^{*}\right)\]From equation (c), \[\varepsilon-\varepsilon^{*}=\frac{\varepsilon}{\kappa_{\mathrm{T}}} \,\left[\kappa_{\mathrm{T}}-\kappa_{\mathrm{T}}^{*}\right]+\frac{\varepsilon \, \kappa_{\mathrm{T}}^{*}}{\kappa_{\mathrm{T}} \, \sigma} \,\left(\sigma-\sigma^{*}\right)-\frac{\varepsilon^{*}}{\alpha_{\mathrm{p}}} \,\left[\alpha_{\mathrm{p}}-\alpha_{\mathrm{p}}^{*}\right]\]But \[\phi\left(\mathrm{E}_{\mathrm{sj}} ; \mathrm{def}\right)=\frac{\alpha_{\mathrm{s}}-\alpha_{\mathrm{s}}^{*}}{\mathrm{c}_{\mathrm{j}}}+\alpha_{\mathrm{s}}^{*} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\]The analogue for \(\phi\left(E_{p j}\right)\) is the following equation. \(\phi\left(\mathrm{E}_{\mathrm{pj}}\right)=\frac{\alpha_{\mathrm{p}}-\alpha_{\mathrm{p}}^{*}}{\mathrm{c}_{\mathrm{j}}}+\alpha_{\mathrm{p}}^{*} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\) Hence \[\phi\left(E_{p j}\right)-\phi\left(E_{S_{j}} ; \operatorname{def}\right)=\frac{\varepsilon-\varepsilon^{*}}{c_{j}}+\varepsilon^{*} \, \phi\left(V_{j}\right)\]From equation (g), dividing by \(\mathrm{c}_{j}\), \[\begin{gathered} \frac{\varepsilon-\varepsilon}{c_{j}}=\frac{\varepsilon}{\kappa_{T}} \, \frac{1}{c_{j}} \,\left[\kappa_{T}-\kappa_{T}^{*}\right]+\frac{\varepsilon \, \kappa_{T}^{*}}{\kappa_{T} \, \sigma} \, \frac{1}{c_{j}} \,\left[\sigma-\sigma^{*}\right] \\ -\frac{\varepsilon^{*}}{\alpha_{p}} \, \frac{1}{c_{j}} \,\left[\alpha_{p}-\alpha_{p}^{*}\right] \end{gathered}\]But from equation (i) \[\frac{\varepsilon-\varepsilon^{*}}{c_{j}}=\phi\left(E_{p j}\right)-\phi\left(E_{S j} ; \operatorname{def}\right)-\varepsilon^{*} \, \phi\left(V_{j}\right)\]Equations having similar form for \(\left(\kappa_{\mathrm{T}}-\kappa_{\mathrm{T}}^{*}\right),\left(\sigma-\sigma^{*}\right)\) and \(\left(\alpha_{p}-\alpha_{p}^{*}\right)\) are readily generated. Hence \[\begin{aligned} \phi\left(\mathrm{E}_{\mathrm{pj}}\right)-\phi\left(\mathrm{E}_{\mathrm{Sj}} ; \mathrm{def}\right) &=\frac{\varepsilon}{\kappa_{\mathrm{T}}} \,\left[\phi\left(\mathrm{K}_{\mathrm{Tj}}\right)-\kappa_{\mathrm{T}}^{*} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\right]+\frac{\varepsilon \, \kappa_{\mathrm{T}}^{*}}{\kappa_{\mathrm{T}} \, \sigma} \,\left[\phi\left(\mathrm{C}_{\mathrm{pj}}\right)-\sigma^{*} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\right] \\ &-\frac{\varepsilon^{*}}{\alpha_{\mathrm{p}}} \,\left[\phi\left(\mathrm{E}_{\mathrm{pj}}\right)-\alpha_{\mathrm{p}}^{*} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\right]+\varepsilon^{*} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \end{aligned}\]Therefore \[\begin{aligned} \phi\left(E_{\mathrm{pj}}\right)-\phi\left(E_{\mathrm{Sj}} ; \operatorname{def}\right) &=-\frac{\varepsilon}{\alpha_{\mathrm{p}}} \, \phi\left(\mathrm{E}_{\mathrm{pj}}\right)+\frac{\varepsilon}{\kappa_{\mathrm{T}}} \, \phi\left(\mathrm{K}_{\mathrm{Tj}}\right)+\frac{\varepsilon \, \kappa_{\mathrm{T}}^{*}}{\kappa_{\mathrm{T}} \, \sigma} \, \phi\left(\mathrm{C}_{\mathrm{pj}}\right) \\ &+\left[\varepsilon * \,\left(1+\frac{\alpha_{\mathrm{p}}^{*}}{\alpha_{\mathrm{p}}}\right)-\frac{\varepsilon \, \kappa_{\mathrm{T}}^{*}}{\kappa_{\mathrm{T}}} \,\left(1+\frac{\sigma^{*}}{\sigma}\right)\right] \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \quad(\mathrm{m}) \end{aligned}\]In the limit of infinite dilution, \[\frac{\phi\left(E_{\mathrm{pj}}\right)^{\infty}-\phi\left(\mathrm{E}_{\mathrm{Sj}} ; \mathrm{def}\right)^{\infty}}{\varepsilon_{1}^{*}(\ell)}=-\frac{\phi\left(\mathrm{E}_{\mathrm{pj}}\right)^{\infty}}{\alpha_{\mathrm{p} 1}^{*}(\ell)}+\frac{\phi\left(\mathrm{K}_{\mathrm{T}}\right)^{\infty}}{\kappa_{\mathrm{Tl}}^{*}(\ell)}+\frac{\phi\left(\mathrm{C}_{\mathrm{pj}}\right)^{\infty}}{\sigma_{1}^{*}(\ell)}\]This page titled 1.12.23: Expansions- The Difference is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.12.24: Expansions- Equations

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.12%3A_Expansions/1.12.24%3A_Expansions-_Equations

We simplify the algebra by omitting the descriptors (aq) and (\(\ell\)) in the following equations. The starting point is the following equation. \[\alpha_{S}-\alpha_{S}^{*}=\frac{\alpha_{S}}{\kappa_{S} \, \sigma} \,\left(\kappa_{S} \, \sigma-\kappa_{S}^{*} \, \sigma^{*}\right)-\frac{\alpha_{S}^{*}}{\alpha_{p}} \,\left(\alpha_{p}-\alpha_{p}^{*}\right)\]The latter equation is effectively an identity. Thus from equation (a) \[\alpha_{S}-\alpha_{S}^{*}=\alpha_{S}-\frac{\alpha_{S}}{\kappa_{S} \, \sigma} \, \kappa_{S}^{*} \, \sigma^{*}-\alpha_{S}^{*}+\frac{\alpha_{S}^{*}}{\alpha_{p}} \, \alpha_{p}^{*}\]But \(\alpha_{\mathrm{s}}=-\kappa_{\mathrm{s}} \, \sigma / \mathrm{T} \, \alpha_{\mathrm{p}}\) and \(\alpha_{\mathrm{p}}^{*} / \alpha_{\mathrm{s}}^{*}=-\kappa_{\mathrm{s}}^{*} \, \sigma^{*} / \mathrm{T}\)Then from (b), \(\alpha_{\mathrm{s}}-\alpha_{\mathrm{s}}^{*}=\alpha_{\mathrm{s}}-\alpha_{\mathrm{s}}^{*}+\frac{\kappa_{\mathrm{s}} \, \sigma}{\mathrm{T} \, \alpha_{\mathrm{p}} \, \kappa_{\mathrm{s}} \, \sigma} \, \kappa_{\mathrm{s}}^{*} \, \sigma^{*}-\frac{\kappa_{\mathrm{s}}^{*} \, \sigma^{*}}{\alpha_{\mathrm{p}} \, \mathrm{T}}\) or \(\alpha_{\mathrm{s}}-\alpha_{\mathrm{s}}^{*}=\alpha_{\mathrm{s}}-\alpha_{\mathrm{s}}^{*}\)But as an identity, \[\kappa_{\mathrm{S}} \, \sigma-\kappa_{\mathrm{S}}^{*} \, \sigma^{*}=\sigma \,\left(\kappa_{\mathrm{S}}-\kappa_{\mathrm{S}}^{*}\right)+\kappa_{\mathrm{S}}^{*} \,\left(\sigma-\sigma^{*}\right)\]Then from equations (a) and (c), \[\alpha_{\mathrm{s}}-\alpha_{\mathrm{s}}^{*}=\frac{\alpha_{\mathrm{s}}}{\kappa_{\mathrm{s}}} \,\left(\kappa_{\mathrm{s}}-\kappa_{\mathrm{s}}^{*}\right)+\frac{\alpha_{\mathrm{s}} \, \kappa_{\mathrm{s}}^{*}}{\kappa_{\mathrm{s}} \, \sigma} \,\left(\sigma-\sigma^{*}\right)-\frac{\alpha_{\mathrm{s}}^{*}}{\alpha_{\mathrm{p}}} \,\left(\alpha_{\mathrm{p}}-\alpha_{\mathrm{p}}^{*}\right)\]But, \[\phi\left(E_{\mathrm{S}} ; \operatorname{def}\right)=\left[\mathrm{c}_{\mathrm{j}}\right]^{-1} \,\left(\alpha_{\mathrm{s}}-\alpha_{\mathrm{s}}^{*}\right)+\alpha_{\mathrm{s}}^{*} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\]Hence \[\begin{aligned} \phi\left(E_{\mathrm{s} j} ; \operatorname{def}\right)=& \frac{\alpha_{\mathrm{s}}}{\kappa_{\mathrm{s}}} \,\left[\mathrm{c}_{\mathrm{j}}\right]^{-1} \,\left(\kappa_{\mathrm{s}}-\kappa_{\mathrm{s}}^{*}\right)+\frac{\alpha_{\mathrm{s}} \, \kappa_{\mathrm{s}}^{*}}{\kappa_{\mathrm{s}} \, \sigma} \,\left[\mathrm{c}_{\mathrm{j}}\right]^{-1} \,\left(\sigma-\sigma^{*}\right) \\ &-\frac{\alpha_{\mathrm{s}}^{*}}{\alpha_{\mathrm{p}}} \,\left[\mathrm{c}_{\mathrm{j}}\right]^{-1} \,\left(\alpha_{\mathrm{p}}-\alpha_{\mathrm{p}}^{*}\right)+\alpha_{\mathrm{s}}^{*} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \end{aligned}\]For isobaric heat capacities, \[\phi\left(\mathrm{C}_{\mathrm{pj}}\right)=\left[\mathrm{c}_{\mathrm{j}}\right]^{-1} \,\left(\sigma-\sigma^{*}\right)+\sigma^{*} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\]Also \[\phi\left(K_{\mathrm{S}_{j}} ; \operatorname{def}\right)=\left[\mathrm{c}_{\mathrm{j}}\right]^{-1} \,\left(\kappa_{\mathrm{s}}-\kappa_{\mathrm{s}}^{*}\right)+\kappa_{\mathrm{s}}^{*} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\]Hence \[\begin{aligned} \phi\left(\mathrm{E}_{\mathrm{s} j} ; \operatorname{def}\right)=& \frac{\alpha_{\mathrm{s}}}{\kappa_{\mathrm{s}}} \,\left[\phi\left(\mathrm{K}_{\mathrm{s} j} ; \operatorname{def}\right)-\kappa_{\mathrm{s}}^{*} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\right]+\frac{\alpha_{\mathrm{s}} \, \kappa_{\mathrm{s}}^{*}}{\kappa_{\mathrm{s}} \, \sigma} \,\left[\phi\left(\mathrm{C}_{\mathrm{pj}}\right)-\sigma^{*} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\right] \\ &-\frac{\alpha_{\mathrm{s}}^{*}}{\alpha_{\mathrm{p}}} \,\left[\phi\left(\mathrm{E}_{\mathrm{pj}}\right)-\alpha_{\mathrm{p}}^{*} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\right]+\alpha_{\mathrm{s}}^{*} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \end{aligned}\]Then with a little reorganisation, \[\begin{aligned} \phi\left(\mathrm{E}_{\mathrm{s} j} ; \operatorname{def}\right)=&-\frac{\alpha_{\mathrm{s}}^{*}}{\alpha_{\mathrm{p}}} \, \phi\left(\mathrm{E}_{\mathrm{p} j}\right)+\frac{\alpha_{\mathrm{s}}}{\kappa_{\mathrm{s}}} \, \phi\left(\mathrm{K}_{\mathrm{s} j} ; \operatorname{def}\right)+\frac{\alpha_{\mathrm{s}} \, \kappa_{\mathrm{s}}^{*}}{\kappa_{\mathrm{s}} \, \sigma} \, \phi\left(\mathrm{C}_{\mathrm{pj}}\right) \\ &+\left[\alpha_{\mathrm{s}}^{*} \,\left(1+\frac{\alpha_{\mathrm{p}}^{*}}{\alpha_{\mathrm{p}}}\right)-\frac{\alpha_{\mathrm{s}} \, \kappa_{\mathrm{s}}^{*}}{\kappa_{\mathrm{s}}} \,\left(1+\frac{\sigma^{*}}{\sigma}\right)\right] \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \end{aligned}\]Hence, in the limit of infinite dilution, \(\frac{\phi\left(\mathrm{E}_{\mathrm{Sj}} ; \operatorname{def}\right)^{\infty}}{\alpha_{\mathrm{S} 1}^{*}(\ell)}=-\frac{\phi\left(\mathrm{E}_{\mathrm{pj}}\right)^{\infty}}{\alpha_{\mathrm{p} 1}^{*}(\ell)}+\frac{\phi\left(\mathrm{K}_{\mathrm{s} j} ; \mathrm{def}\right)^{\infty}}{\kappa_{\mathrm{s} 1}^{*}(\ell)}+\frac{\phi\left(\mathrm{C}_{\mathrm{pj}}\right)^{\infty}}{\sigma_{1}^{*}(\ell)}\)This page titled 1.12.24: Expansions- Equations is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.13.1: Equilibrium and Frozen Properties

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.13%3A_Equilibrium/1.13.01%3A_Equilibrium_and_Frozen_Properties

The Gibbs energy \(\mathrm{G}\) of a given closed system is characterised by the independent variables temperature \(\mathrm{T}\), pressure \(\mathrm{p}\) and composition \(\xi\). \[\mathrm{G}=\mathrm{G}[\mathrm{T}, \mathrm{p}, \xi] \label{a}\]In the state defined by Equation \ref{a} the affinity for spontaneous change is \(\mathrm{A}\). Starting with the system in the state defined by equation (a) it is possible to change the pressure (at fixed temperature) and thereby perturb the system to neighbouring states where the affinity \(\mathrm{A}\) is the same. The differential dependence of \(\mathrm{G}\) on pressure along this path is given by the partial differential \((\partial \mathrm{G} / \partial \mathrm{p})_{\mathrm{T}, \mathrm{A}}\). Returning to the state defined by Equation \ref{a} we envisage a perturbation by a change in pressure (at fixed temperature) along a path such that the extent of chemical reaction \(\xi\) remains constant; the corresponding differential dependence of \(\mathrm{G}\) is given by \((\partial \mathrm{G} / \partial \mathrm{p})_{\mathrm{T}, \xi}\).The two partial derivatives are related by equation (b) for a system at constant temperature.\[\left[\frac{\partial \mathrm{G}}{\partial \mathrm{p}}\right]_{\mathrm{A}}=\left[\frac{\partial \mathrm{G}}{\partial \mathrm{p}}\right]_{\xi}-\left[\frac{\partial \mathrm{A}}{\partial \mathrm{p}}\right]_{\xi} \,\left[\frac{\partial \xi}{\partial \mathrm{A}}\right]_{\mathrm{p}} \,\left[\frac{\partial \mathrm{G}}{\partial \xi}\right]_{\mathrm{p}} \label{b}\]The important result which emerges from this equation concerns the properties of a system at chemical equilibrium where the affinity for spontaneous change is zero, the rate of change \(\mathrm{d} \xi / \mathrm{dt}\) is zero, the Gibbs energy is a minimum and, significantly, \((\partial G / \partial \xi)_{\mathrm{T}, \mathrm{p}}\) is zero. Hence \[V=\left[\frac{\partial G}{\partial p}\right]_{T, A=0}=\left[\frac{\partial G}{\partial p}\right]_{T, \xi(e q)}\]Thus we confirm that the volume \(\mathrm{V}\) of a system is a strong state variable, the dependence of \(\mathrm{G}\) on pressure (at constant \(\mathrm{T}\)) at constant ‘\(\mathrm{A}=0\)’ and at constant composition, \(\xi^{\mathrm{eq}}\) are identical. However if we turn our attention on to expansibilities and compressibilities we find that it is important to distinguish between two sets of properties, equilibrium and frozen.This page titled 1.13.1: Equilibrium and Frozen Properties is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.13.2: Equilibrium- Isochoric and Isobaric Paramenters

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.13%3A_Equilibrium/1.13.02%3A_Equilibrium-_Isochoric_and_Isobaric_Paramenters

In a description of a given closed system we define two extensive state variables, the Gibbs energy \(\mathrm{G}\) and the Helmholtz energy \(\mathrm{F}\). \[\mathrm{G}=\mathrm{U}+\mathrm{p} \, \mathrm{V}-\mathrm{T} \, \mathrm{S}\]\[\mathrm{F}=\mathrm{U}-\mathrm{T} \, \mathrm{S}\]Hence, \[\mathrm{G}=\mathrm{F}+\mathrm{p} \, \mathrm{V}\]The latter interesting equation links two practical thermodynamic potentials;The dependence of \(\mathrm{G}\) on extent of reaction at constant temperature and pressure is related to the differential dependence of \(\mathrm{F}\) on \(\xi\) at fixed temperature and pressure. \[\left(\frac{\partial \mathrm{G}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}=\left(\frac{\partial \mathrm{F}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}+\mathrm{p} \,\left(\frac{\partial \mathrm{V}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}\]At equilibrium where \(\mathrm{A} = 0\), \(\xi = \xi^{\mathrm{eq}\) and the Gibbs energy is a minimum [i.e. \((\partial \mathrm{G} / \partial \xi)_{\mathrm{T}, \mathrm{p}}=0\)], \[\left(\frac{\partial F}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{A}=0}=\mathrm{p} \,\left(\frac{\partial \mathrm{V}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{A}=0}\]In other words the differential dependence of the Helmholtz energy on extent of reaction at equilibrium (at constant \(\mathrm{T}\) and \(\mathrm{p}\)) is related to the volume of reaction. We rewrite equation (c); \[\mathrm{F}=\mathrm{G}-\mathrm{p} \, \mathrm{V}\]At constant temperature and volume, \[\left(\frac{\partial \mathrm{F}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{V}}=\left(\frac{\partial \mathrm{G}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{V}}-\mathrm{V} \,\left(\frac{\partial \mathrm{p}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{V}}\]At equilibrium (at constant \(\mathrm{T}\) and \(\mathrm{V}\)) where the Helmholtz energy \(\mathrm{F}\) is a minimum, clearly the Gibbs energy is not at a minimum. The dependence of both \(\mathrm{G}\) and \(\mathrm{F}\) on temperature at equilibrium can be expressed using two Gibbs - Helmholtz equations. Thus, \[\left[\frac{\partial(\Delta \mathrm{G} / \mathrm{T})}{\partial(1 / \mathrm{T})}\right]_{\mathrm{p}, \mathrm{A}=0}^{\mathrm{eq}}=\Delta \mathrm{H}^{\mathrm{eq}}\]\[\left[\frac{\partial(\Delta \mathrm{F} / \mathrm{T})}{\partial(1 / \mathrm{T})}\right]_{\mathrm{V}, \mathrm{A}=0}^{\mathrm{eq}}=\Delta \mathrm{U}^{\mathrm{eq}}\]From a practical standpoint, determination of \(\Delta\mathrm{H}^{\mathrm{eq}}\) is reasonably straightforward because over a range of temperatures the isobaric condition is readily satisfied. Thus we probe this differential dependence at a series of defined temperatures at fixed pressure; i.e. over the range \(\mathrm{T}-\delta \mathrm{T}\) to \(\mathrm{T}+\delta \mathrm{T}\) about \(\mathrm{T}\) for a number of temperatures.The condition ‘at constant volume’ presents problems. In principle we change the pressure to hold \(\mathrm{V}\) constant over a range of temperatures. Then we probe the differential dependence of \((\Delta \mathrm{F} / \mathrm{T})\) at a series of fixed temperatures; e.g. over the range \(\mathrm{T}-\Delta \mathrm{T}\) to \(\mathrm{T}-\Delta \mathrm{T}\) about a given temperature T. If the range of temperatures is large, there is a high probability that very high pressures will be required to hold the global isochoric condition.Another approach probes the dependence of \((\Delta \mathrm{F} / \mathrm{T})\) on temperature at a series of temperatures where volume \(\mathrm{V}\) is held constant by changing the pressure over the range \[\mathrm{T}_{\mathrm{i}}-\delta \mathrm{T}\) to \[\mathrm{T}_{\mathrm{i}}-\delta \mathrm{T}\] about \(\mathrm{T}_{\mathrm{i}}\). Volume \(\mathrm{V}_{\mathrm{i}}\) is constant over a small range of temperature. Here the isochoric condition is local to temperature \(\mathrm{T}\); thus \(\Delta\mathrm{U}\) is obtained at \(\mathrm{T}_{\mathrm{i}}\) and \(\mathrm{V}_{\mathrm{i}}\). Under these circumstances, comparison of derived \(\Delta \mathrm{U}\) - quantities as a function of temperature is not straightforward.Interestingly the solvent water presents pairs of temperatures either side of the TMD where molar volume of water is the same at, for example, ambient pressure. It might be possible to explore this feature by assuming that the volumes of two very dilute solutions are also identical at matched pairs of temperatures.This page titled 1.13.2: Equilibrium- Isochoric and Isobaric Paramenters is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.13.3: Equilibirium- Solid-Liquid

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.13%3A_Equilibrium/1.13.03%3A_Equilibirium-_Solid-Liquid

A given homogeneous liquid system comprises two chemical substances \(\mathrm{i}\) and \(\mathrm{j}\) at known \(\mathrm{T}\) and \(\mathrm{p}\). The temperature and/or pressure are changed. Consequently chemical substance \(\mathrm{j}\) spontaneously separates out as a solid phase but substance \(\mathrm{i}\) does not. Hence the liquid becomes richer in chemical substance \(\mathrm{i}\).The starting point of the analysis is the following equation for the affinity for spontaneous transfer of substance \(\mathrm{j}\) from phase II to phase I. \[\begin{aligned} \delta\left(\frac{A_{j}}{T}\right)=& \frac{\left[\Delta_{\text {trans }} H_{j}^{0}(T, p)\right]}{T^{2}} \, \delta T \\ &-\frac{\left[\Delta_{\text {trans }} V_{j}^{0}(T, p)\right]}{T} \, \delta p+R \, \delta \ln \left[\frac{x_{j}(I) \, f_{j}(I)}{x_{j}(\text { II }) \, f_{j}(I I)}\right] \end{aligned}\]For two equilibrium states such that \(\delta\left(\mathrm{A}_{\mathrm{j}} / \mathrm{T}\right)\) is zero for the transfer of chemical substance \(\mathrm{j}\) from phase II to phase I, \[\mathrm{R} \, \delta \ln \left[\frac{\mathrm{x}_{\mathrm{j}}(\mathrm{II}) \, \mathrm{f}_{\mathrm{j}}(\mathrm{II})}{\mathrm{x}_{\mathrm{j}}(\mathrm{I}) \, \mathrm{f}_{\mathrm{j}}(\mathrm{I})}\right]=\frac{\left[\Delta_{\text {trans }} \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{~T}, \mathrm{p})\right]}{\mathrm{T}^{2}} \, \delta \mathrm{T}-\frac{\left[\Delta_{\text {trans }} \mathrm{V}_{\mathrm{j}}^{0}(\mathrm{~T}, \mathrm{p})\right]}{\mathrm{T}} \, \delta \mathrm{p}\]In this application, chemical substance \(\mathrm{i}\) cannot exist in phase I. Then the equilibrium states are determined by substance \(\mathrm{j}\). Further we consider the case where state I corresponds to pure \(\mathrm{j}\) such that \(x_{j}(I) \, f_{j}(I)\) is unity at reference temperature \(\mathrm{T}_{\text{ref}}\) and reference pressure pref. We integrate equation (b) between these two states. \[\begin{aligned} &\ln \left[\mathrm{x}_{\mathrm{j}}(\mathrm{II}) \, \mathrm{f}_{\mathrm{j}}(\mathrm{II})\right]= \\ &\qquad \int_{\mathrm{T}(\mathrm{ref})}^{\mathrm{T}} \frac{\left[\Delta_{\text {trans }} \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{~T}, \mathrm{p})\right]}{\mathrm{R} \, \mathrm{T}^{2}} \, \mathrm{dT}-\int_{\mathrm{p}(\mathrm{ref})}^{\mathrm{p}} \frac{\left[\Delta_{\text {trans }} \mathrm{V}_{\mathrm{j}}^{0}(\mathrm{~T}, \mathrm{p})\right]}{\mathrm{R} \, \mathrm{T}} \, \mathrm{dp} \end{aligned}\]In the event that the pressure is constant, \[\ln \left[\mathrm{x}_{\mathrm{j}}(\mathrm{II}) \, \mathrm{f}_{\mathrm{j}}(\mathrm{II})\right]=\int_{\mathrm{T}(\mathrm{ref})}^{\mathrm{T}} \frac{\left[\Delta_{\text {trans }} \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{~T}, \mathrm{p})\right]}{\mathrm{R} \, \mathrm{T}^{2}} \, \mathrm{dT}\]Footnote By definition, for the transfer of one mole of chemical substance j from phase II to phase I, \(A_{j}=-\left[\mu_{j}(\mathrm{I})-\mu_{j}(\mathrm{II})\right] ; \mathrm{Or}, \mathrm{A}_{\mathrm{j}}=\mu_{\mathrm{j}}(\mathrm{II})-\mu_{\mathrm{j}}(\mathrm{I})\)This page titled 1.13.3: Equilibirium- Solid-Liquid is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.13.4: Equilibrium- Liquid-Solid- Schroeder - van Laar Equation

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.13%3A_Equilibrium/1.13.04%3A_Equilibrium-_Liquid-Solid-_Schroeder_-_van_Laar_Equation

A given homogeneous binary liquid system (at pressure \(\mathrm{p}\)) contains two chemical substances \(\mathrm{i}\) and \(\mathrm{j}\) at temperature \(\mathrm{T}\). The liquid system is cooled and only substance \(\mathrm{j}\) separates out as the pure solid substance \(\mathrm{j}\). Hence,\[\ln \left[\mathrm{x}_{\mathrm{j}}(\ell) \,\mathrm{f}_{\mathrm{j}}(\ell)\right]=\int_{\mathrm{T}_{\mathrm{j}}^{0}}^{\mathrm{T}} \frac{\left[\Delta_{\text {trans }} \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{~T}, \mathrm{p})\right]}{\mathrm{R} \,\mathrm{T}^{2}} \,\mathrm{dT} \label{a}\]Here \(x_{j}(\ell)\) is the mole fraction composition of the liquid; \(f_{j}(\ell)\) is the rational activity coefficient of substance \(j\) in the liquid mixture at mole fraction \(x_{j}(\ell)\) and temperature \(\mathrm{T}\). \(\mathrm{T}_{\mathrm{j}}^{0}\) is the melting point of pure \(j\) substance \(j\) at pressure \(\mathrm{p}\); i.e., both liquid and solid phases are pure chemical substance \(j\).In the event that \(\Delta_{\text {trans }} \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{~T}, \mathrm{p})\) is independent of temperature [i.e. \(\Delta_{\text {trans }} C_{p j}^{0}(T, p)\) is zero] Equation \ref{a} is integrated to yield Equation \ref{b}.\[-\ln \left[\mathrm{x}_{\mathrm{j}}(\ell) \,\mathrm{f}_{\mathrm{j}}(\ell)\right]=\dfrac{\Delta_{\text {fus}} \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{~T}, \mathrm{p})}{\mathrm{R}} \, \left(\dfrac{1}{\mathrm{~T}}-\frac{1}{\mathrm{~T}_{\mathrm{j}}^{0}}\right) \label{b}\]The phenomenon under consideration is fusion so that \(\Delta_{\text {fus }} \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{~T}, \mathrm{p})\) is the enthalpy of fusion of chemical substance \(j\) at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\). In the event that the thermodynamic properties of the liquid-solid system are ideal, Equation \ref{b} simplifies to Equation \ref{c}.\[-\ln \left[\mathrm{x}_{\mathrm{j}}(\ell)\right]=\frac{\Delta_{\mathrm{f}} \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{~T}, \mathrm{p})}{\mathrm{R}} \left(\frac{1}{\mathrm{~T}}-\frac{1}{\mathrm{~T}_{\mathrm{j}}^{0}}\right) \label{c}\]Equation \ref{c} is the Schroeder- van Laar Equation.Footnote I. Prigogine and R Defay, Chemical Thermodynamics, transl. D. H. Everett, Longmans Greeen, London, 1953.This page titled 1.13.4: Equilibrium- Liquid-Solid- Schroeder - van Laar Equation is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.13.5: Equilibrium - Eutectics

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.13%3A_Equilibrium/1.13.05%3A_Equilibrium-_Eutectics

A given homogeneous binary liquid system (at pressure \(\mathrm{p}\)) contains two chemical substances \(\mathrm{i}\) and \(\mathrm{j}\) at temperature \(\mathrm{T}\). The liquid system is cooled and only substance \(\mathrm{j}\) separates out as the pure solid substance \(\mathrm{j}\) leaving the liquid richer in chemical substance \(\mathrm{i}\). The mole fraction composition of the liquid is given by the Schroeder-van Laar equation written in the following form.\[-\ln \left[\mathrm{x}_{\mathrm{j}}(\ell) \mathrm{f}_{\mathrm{j}}(\ell)\right]=\frac{\left[\Delta_{\text {fus }} \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{~T})\right]}{\mathrm{R}} \left[\frac{1}{\mathrm{~T}}-\frac{1}{\mathrm{~T}_{\text {fus; } \mathrm{j}}^{0}}\right] \label{a}\]In many cases as the mole fraction composition of substance \(\mathrm{i}\) in the liquid increases the equilibrium temperature \(\mathrm{T}\) decreases until at the eutectic temperature \(\mathrm{T}_{\mathrm{e}}\) and mole fraction \(\left(\mathbf{X}_{\mathrm{j}}\right)_{\mathrm{e}}\) the system comprises a solid, the eutectic mixture. In the event that the thermodynamic properties of the system can be described as ideal, Equation \ref{a} simplifies to Equation \ref{b} where it is assumed that \(\mathrm{f}_{\mathrm{j}}(\ell)\) is unity at all temperatures. Then\[-\ln \left[\mathrm{x}_{\mathrm{j}}(\ell)\right]=\frac{\left[\Delta_{\mathrm{fus}} \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{~T})\right]}{\mathrm{R}} \left[\frac{1}{\mathrm{~T}}-\frac{1}{\mathrm{~T}_{\text {fus; } ; \mathrm{j}}^{0}}\right] \label{b}\]For the other component \(\mathrm{i}\), a corresponding plot is obtained when on cooling the liquid mixture only pure solid \(\mathrm{i}\) separates out. \[-\ln \left[\mathrm{x}_{\mathrm{i}}(\ell) \mathrm{f}_{\mathrm{i}}(\ell)\right]=\frac{\left[\Delta_{\mathrm{fus}} \mathrm{H}_{\mathrm{i}}^{0}(\mathrm{~T})\right]}{\mathrm{R}} \left[\frac{1}{\mathrm{~T}}-\frac{1}{\mathrm{~T}_{\text {fus }, \mathrm{i}}^{0}}\right]\]If the thermodynamic properties of the system are ideal then the analogue of equation (b) is equation (d). \[-\ln \left[\mathrm{x}_{\mathrm{i}}(\ell)\right]=\frac{\left[\Delta_{\text {fus }} \mathrm{H}_{\mathrm{i}}^{0}(\mathrm{~T})\right]}{\mathrm{R}} \left[\frac{1}{\mathrm{~T}}-\frac{1}{\mathrm{~T}_{\text {fuss } ; \mathrm{i}}^{0}}\right]\]The two curves described by equations (a) and (c) meet at the eutectic temperature. \(\mathrm{T}_{\mathrm{e}}\). Granted that the thermodynamic properties of the system are ideal, the following two equations follow from equations (b) and (d). \[-\ln \left[\mathrm{x}_{\mathrm{j}}^{\mathrm{e}}(\ell)\right]=\frac{\left[\Delta_{\mathrm{fus}} \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{~T})\right]}{\mathrm{R}} \left[\frac{1}{\mathrm{~T}_{\mathrm{e}}}-\frac{1}{\mathrm{~T}_{\text {fus } ; j}^{0}}\right]\]\[-\ln \left[\mathrm{x}_{\mathrm{i}}^{\mathrm{e}}(\ell)\right]=-\ln \left[1-\mathrm{x}_{\mathrm{j}}^{\mathrm{e}}\right]=\frac{\left[\Delta_{\text {fus }} \mathrm{H}_{\mathrm{i}}^{0}(\mathrm{~T})\right]}{\mathrm{R}} \left[\frac{1}{\mathrm{~T}_{\mathrm{e}}}-\frac{1}{\mathrm{~T}_{\text {fuss; } \mathrm{i}}^{0}}\right]\]In the event that \[\frac{\left[\Delta_{\text {fus }} \mathrm{H}_{\mathrm{j}}^{0}(\mathrm{~T})\right]}{\mathrm{R}} \left[\frac{1}{\mathrm{~T}_{\mathrm{e}}}-\frac{1}{\mathrm{~T}_{\text {fuss } ; \mathrm{j}}^{0}}\right]=\frac{\left[\Delta_{\text {fus }} \mathrm{H}_{\mathrm{i}}^{0}(\mathrm{~T})\right]}{\mathrm{R}} \left[\frac{1}{\mathrm{~T}_{\mathrm{e}}}-\frac{1}{\mathrm{~T}_{\text {fuss } ; \mathrm{i}}^{0}}\right]\]then \(x_{i}^{e}=x_{j}^{e}=0.5\). The impact of the non-ideal thermodynamic properties can be explored using equation (a) and (c) in conjunction with empirical equations relating, for example, \(\mathrm{f}_{\mathrm{j}}(\ell)\) and \(\mathrm{x}_{\mathrm{j}}(\ell)\); e.g. equation (g). \[\ln \left[\mathrm{f}_{\mathrm{j}}(\ell)\right]=\alpha \left[1-\mathrm{x}_{\mathrm{j}}(\ell)\right]^{2}\] I. Prigogine and R. Defay, Chemical Thermodynamics, tranls. D. H. Everett, Longmans Greeen, London, 1953.This page titled 1.13.5: Equilibrium - Eutectics is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.13.6: Equilibrium - Depression of Freezing Point of a Solvent by a Solute

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A given homogeneous liquid system (at pressure \(\mathrm{p}\)) comprises solvent \(\mathrm{i}\) and solute \(\mathrm{j}\) at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\). In the absence of solute \(\mathrm{j}\), the freezing point of the solvent is \(\mathrm{T}_{1}^{0}\). But in the presence of solute \(\mathrm{j}\) the freezing point is temperature \(\mathrm{T}\) where \(\mathrm{T} < \mathrm{T}_{1}^{0}\). The depression of freezing point \(\theta\left[=\mathrm{T}_{1}^{0}-\mathrm{T}\right]\) is recorded for a solution where the mole fraction of solvent is \(\mathrm{x}_{1}(\mathrm{sln})\). If the solution is dilute, we can assume that the thermodynamic properties of the solution are ideal. From the Schroeder-van Laar equation,\[-\ln \left[x_{1}(s \ln )\right]=\frac{\left[\Delta_{f} H_{1}^{0}(T)\right]}{R} \,\left[\frac{1}{T}-\frac{1}{T_{1}^{0}}\right]\]\[-\ln \left[\mathrm{x}_{1}(\mathrm{~s} \ln )\right]=\frac{\Delta_{\mathrm{f}} \mathrm{H}_{1}^{0}}{\mathrm{R}} \, \frac{\theta}{\left(\mathrm{T}_{1}^{0}-\theta\right) \, \mathrm{T}_{1}^{0}}\]If \[\mathrm{T}_{1}^{0}-\theta \cong \mathrm{T}_{1}^{0},-\ln \left[\mathrm{x}_{1}(\mathrm{~s} \ln )\right]=\frac{\Delta_{\mathrm{f}} \mathrm{H}_{1}^{0}}{\mathrm{R}} \, \frac{\theta}{\left(\mathrm{T}_{1}^{0}\right)^{2}}\]Or, \[\ln \left[\frac{1}{x_{1}(s \ln )}\right]=\frac{\Delta_{\mathrm{f}} H_{1}^{0}}{R} \, \frac{\theta}{\left(T_{1}^{0}\right)^{2}}\]Hence \[\theta=\left[\frac{\mathrm{R} \,\left(\mathrm{T}_{1}^{0}\right)^{2} \, \mathrm{M}_{1}}{\Delta_{\mathrm{f}} \mathrm{H}_{1}^{0}}\right] \, \mathrm{m}_{\mathrm{j}}\]The quantity enclosed in the […] brackets is characteristic of the solvent.Footnotes \(\theta=\mathrm{T}_{1}^{0}-\mathrm{T}\); \(\frac{1}{\mathrm{~T}}-\frac{1}{\mathrm{~T}_{1}^{0}}=\frac{\mathrm{T}_{1}^{0}-\mathrm{T}}{\mathrm{T} \, \mathrm{T}_{1}^{0}}=\frac{\mathrm{T}_{1}^{0}-\mathrm{T}}{\left(\mathrm{T}_{1}^{0}-\theta\right) \, \mathrm{T}_{1}^{0}}=\frac{\theta}{\left(\mathrm{T}_{1}^{0}-\theta\right) \, \mathrm{T}_{1}^{0}}\) \(\frac{1}{x_{1}}=\frac{1}{1-x_{j}}=\frac{1}{1-\left[n_{j} /\left(n_{1}+n_{j}\right)\right]}=\frac{n_{1}+n_{j}}{n_{1}+n_{j}-n_{j}}\) For a solution where the molality of solute \(j=m_{j}\) \(\mathrm{m}_{\mathrm{j}}=\frac{\mathrm{n}_{\mathrm{j}}}{\mathrm{n}_{1} \, \mathrm{M}_{1}}\) Then, \(\frac{1}{\mathrm{x}_{1}}=\frac{\mathrm{n}_{1}+\mathrm{n}_{1} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}}{\mathrm{n}_{1}}\) \(-\ln \left[\mathrm{x}_{1}(\mathrm{~s} \ln )\right]=-\ln \left[1+\mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}\right]\); \(-\ln \left[x_{1}(\operatorname{sln})\right]=-\ln \left[1-x_{j}(s \ln )\right] \approx x_{j}\) \(x_{j}=\frac{m_{j}}{\left(1 / M_{1}\right)+m_{j}} \approx m_{j} \, M_{1}\) see I Prigogine and R Defay, Chemical Thermodynamics, trans. D. H. Everett, Longmans Green, London, 1953.This page titled 1.13.6: Equilibrium - Depression of Freezing Point of a Solvent by a Solute is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.13.7: Equilibrium - Liquid-Solids - Hildebrand Rules

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Topics_in_Thermodynamics_of_Solutions_and_Liquid_Mixtures/01%3A_Modules/1.13%3A_Equilibrium/1.13.07%3A_Equilibrium-_Liquid-Solids-_Hildebrand_Rules

A given homogeneous liquid system (at pressure \(\mathrm{p}\)) contains two chemical substances \(\mathrm{i}\) and \(\mathrm{j}\) at temperature \(\mathrm{T}\). Chemical substance \(\mathrm{j}\) at temperature \(\mathrm{T}\) and \(\mathrm{p}\) is a liquid which being in vast excess in this system is the solvent. The system is cooled and pure solid substance \(\mathrm{i}\) separates out leaving the system less concentrated in the solute \(\mathrm{i}\). The solution is dilute and we assume that the thermodynamic properties of the solution are ideal. Then, from the Schroeder–van Laar Equation\[-\ln \left[\mathrm{x}_{\mathrm{i}}(\mathrm{s} \ln )\right]=\frac{\left[\Delta_{\mathrm{fius}} \mathrm{H}_{\mathrm{i}}^{0}(\mathrm{~T}, \mathrm{p})\right]}{\mathrm{R}} \,\left[\frac{1}{\mathrm{~T}}-\frac{1}{\mathrm{~T}_{\mathrm{i}}^{0}}\right] \label{a}\]\(\Delta_{\text {fus }} \mathrm{H}_{\mathrm{i}}^{0}\) is the molar enthalpy of fusion of chemical substance \(\mathrm{i}\), melting point \(\mathrm{T}_{\mathrm{i}}^{0}\). Mole fraction \(\mathrm{x}_{\mathrm{i}}(\mathrm{s} \ln )\) is the composition of the saturated solution at temperature \(\mathrm{T}\); i.e. the solubility of substance \(\mathrm{i}\). From Equation \ref{a},\[\ln \left[\frac{1}{\mathrm{x}_{\mathrm{i}}(\mathrm{s} \ln )}\right]^{\mathrm{eq}}=\frac{\left[\Delta_{\mathrm{fus}} \mathrm{H}_{\mathrm{i}}^{0}(\mathrm{~T}, \mathrm{p})\right]}{\mathrm{R}} \,\left[\frac{\mathrm{T}_{\mathrm{i}}^{0}-\mathrm{T}}{\mathrm{T} \, \mathrm{T}_{\mathrm{i}}^{0}}\right] \label{b}\]Equation \ref{b} forms the background to several generalisations concerning solubilities; i.e. Hildebrand Rules. We note that \(\Delta_{\text {fus }} \mathrm{H}_{\mathrm{i}}^{0}(\mathrm{~T}, \mathrm{p})\) and \(\mathrm{T}_{\mathrm{i}}^{0}\) characterise the solute.Footnote see I. Prigogine and R. Defay, Chemical Thermodynamics, transl. D. H. Everett, Longmans Greeen, London, 1953.This page titled 1.13.7: Equilibrium - Liquid-Solids - Hildebrand Rules is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis.

1.14.1: Adiabatic

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For a closed system undergoing a change in thermodynamic properties under the adiabatic constraint, heat does not cross between system and surroundings. The system is thermally insulated (i.e. adiabatically enclosed) from the surroundings. The First Law for closed systems has the following form.\[\Delta \mathrm{U}=\mathrm{q}+\mathrm{w}\]For changes under the adiabatic constraint, \(\mathrm{q}\) is zero. Then for adiabatic changes,\[\Delta \mathrm{U}=\mathrm{w} .\]Footnote ‘Adiabatic’ means impassable, from the Greek: ‘a = not’ + ‘dia = through’ + ‘bathos = deep’.This page titled 1.14.1: Adiabatic is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.14.2: Adsorption- Langmuir Adsorption Isotherm- One Adsorbate

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The theoretical basis of Adsorption Isotherms is customarily described in terms of a balance of rates of adsorption and desorption. Three important assumptions are made.In many interesting cases, small molecules (guest, adsorbate) bind to larger polymeric molecules (host, adsorbent) which provide a surface on which the smaller (guest) molecules are adsorbed. Adsorption data for such systems often follow a Langmuir pattern. Here we use a thermodynamic approach in descriptions of a thermodynamic equilibrium. Most texts describing Langmuir adsorption use a `kinetic model’, as indeed did Langmuir is his description of adsorption.We consider the case where water is the solvent and substance \(j\) is the adsorbate. In the absence of adsorbate \(j\), the surface of the adsorbent is covered with water. When adsorbate \(j\) is added to the system, the adsorption is described by the following equation. \[\mathrm{j}\left(\mathrm{aq} ; \mathrm{x}_{\mathrm{j}}\right)+\mathrm{H}_{2} \mathrm{O}\left(\mathrm{ad} ; \mathrm{x}_{1}^{\mathrm{ad}}\right) \rightarrow \mathrm{j}\left(\mathrm{ad} ; \mathrm{x}_{\mathrm{j}}^{\mathrm{ad}}\right)+\mathrm{H}_{2} \mathrm{O}\left(\mathrm{aq} ; \mathrm{x}_{1}\right)\]The latter equation describes a physical process rather than a ‘chemical’ reaction but the symbolism is common. Thus \(x_{j}\) is the mole fraction of solute \(j\) in the aqueous phase; \(\mathrm{x}_{\mathrm{1}}^{\mathrm{ad}}\) is the mole fraction of water in a thin solution adjacent to the surface of the adsorbate; \(x_{1}\) is the mole fraction of water in the aqueous solution.; \(\mathrm{x}_{\mathrm{j}}^{\mathrm{ad}}\) is the mole fraction of the substance \(j\) in the adsorbed layer. The process represented by equation (a) involves displacement of water from the ‘thin’ solution next to the adsorbent into the bulk solution. The reverse process describes the displacement of substance \(j\) from this layer by water(\(\lambda\)). At equilibrium, the two driving forces are balanced. We use a simple on/off model for the adsorption in a closed aqueous system at fixed temperature \(\mathrm{T}\) and fixed pressure \(p\) (\(\cong p^{0}\)). \[\begin{array}{lc} \text { Then, } \quad \mathrm{j}(\mathrm{aq}) \Leftarrow \,s \mathrm{j}(\mathrm{ad}) \\ \text { At } \mathrm{t}=0, \quad \mathrm{n}_{\mathrm{j}}^{0} \quad 0 \quad \mathrm{~mol} \\ \text { At } t=\infty \quad n_{j}^{0}-\xi \quad \xi \quad \mathrm{mol} \end{array}\]The latter condition refers to the equilibrium state; \(\xi\) is the extent of binding of guest solute \(j\) to the host adsorbate. To describe this equilibrium we need equations for the chemical potentials of \(j(\mathrm{aq})\) and \(j(\mathrm{ad}\)). The fraction of adsorbent surface covered by chemical substance \(j\) is defined as \(\theta\). If there are \(\mathrm{N}\) sites on the adsorbate for adsorption, the amount of sites occupied equals \(\mathrm{N} \, \theta\) and the amount of vacant sites equals \([\mathrm{N} \,(1-\theta)]\).The aim of the analysis is a plot showing the degree of occupancy of the surface of the adsorbent \(\theta\) as a function of the equilibrium composition of the system. If the experiment involves calorimetry, we require equations which describe the heat \(\mathrm{q}\) associated with injection of a small aliquot of a solution containing the adsorbate into a solution containing the adsorbent.The chemical potential of adsorbed chemical substance \(j\), \(\mu_{j}(\mathrm{ad})\) is related to \(\theta\) using a general equation. \[\mu_{\mathrm{j}}(\mathrm{ad})=\mu_{\mathrm{j}}^{0}(\mathrm{ad})+\mathrm{R} \, \mathrm{T} \, \ln [\mathrm{f}(\theta)]\]In order to make progress we need an explicit equation for \(\mathrm{f}(\theta)\); \(\mu_{\mathrm{j}}^{0}(\mathrm{ad})\) is the chemical potential of an ideal adsorbate on an ideal adsorbent where \(\theta =1/2\) at the same \(\mathrm{T}\) and \(p\). At equilibrium the chemical potential of solute \(j\) in solution equals the chemical potential of the adsorbate. We express the composition of the solution in terms of the concentration of chemical substance \(j\), \(\mathrm{c}_{j}\). \[\text { At equilibrium. } \quad \mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}(\mathrm{ad})\]\[\text { For solute, } \mathrm{j}(\mathrm{aq}), \quad \mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{c}_{\mathrm{j}}(\mathrm{aq}) \, \mathrm{y}_{\mathrm{j}}(\mathrm{aq}) / \mathrm{c}_{\mathrm{r}}\right]\]By definition, \(\mathrm{c}_{\mathrm{r}} = 1 \mathrm{~mol dm}^{−3}\); \(\gamma_{j}\) is the solute activity coefficient where, \[\lim \operatorname{it}\left(c_{j} \rightarrow 0\right) y_{j}(a q)=1 \quad \text { at all } T \text { and } p\]\(\mu_{\mathrm{j}}^{0}(\mathrm{aq})\) is the chemical potential of solute \(j\) in solution having unit concentration \(\mathrm{c}_{j}\), the thermodynamic properties of solute \(j\) being ideal. From equations (c) and (e) the equilibrium condition (d) requires, by definition, that \[\begin{aligned} &\Delta_{\mathrm{ad}} \mathrm{G}^{0}=\mu_{\mathrm{j}}^{0}(\mathrm{ad})-\mu_{\mathrm{j}}^{0}(\mathrm{aq}) \\ &=-\mathrm{R} \, \mathrm{T} \, \ln [\mathrm{f}(\theta)]+\mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{c}_{\mathrm{j}}(\mathrm{aq}) \, \mathrm{y}_{\mathrm{j}}(\mathrm{aq}) / \mathrm{c}_{\mathrm{r}}\right] \\ &=-\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{K}_{\mathrm{ad}}\right) \end{aligned\]The dimensionless property \(\mathrm{K}_{\mathrm{ad}\) is the equilibrium adsorption constant which depends on both \(\mathrm{T}\) and \(p\). \[\text { Then, } \mathrm{K}_{\mathrm{ad}}=\mathrm{f}(\theta) \, \mathrm{c}_{\mathrm{r}} / \mathrm{c}_{\mathrm{j}}(\mathrm{aq}) \, \mathrm{y}_{\mathrm{j}}(\mathrm{aq})\]We envisage a solution volume \(\mathrm{V}\) prepared using \(n_{j}{}^{0}\) moles of chemical substance \(j\). [The assumption is usually made that the volume of the system is the volume of the solution.] The equilibrium concentration of substance \(j\), \(\mathrm{c}_{\mathrm{j}}^{\mathrm{eq}}(\mathrm{aq})\) is given by equation (i) where \(\xi^{\mathrm{eq}\) is the equilibrium extent of adsorption \[\mathrm{c}_{\mathrm{j}}^{\mathrm{eq}}(\mathrm{aq})=\left(\mathrm{n}_{\mathrm{j}}^{0}-\xi^{\mathrm{eq}}\right) / \mathrm{V}\]\[\mathrm{K}_{\mathrm{ad}}=\left[\mathrm{f}(\theta) /\left(\mathrm{n}_{\mathrm{j}}^{0}-\xi\right)\right] \,\left[\mathrm{V} \, \mathrm{c}_{\mathrm{r}} / \mathrm{y}_{\mathrm{j}}(\mathrm{aq})\right]\]Equation (j) describes the composition of the system. Granted therefore the applicability of equation (i) we anticipate specific applications of this equation will, at minimum, yield two pieces of information.(i) Dependence of \(\theta\) on total concentration of \(j\) in the system, \(\mathrm{n}_{\mathrm{j}}^{0} / \mathrm{V}\).(ii) Dependence of percentage of chemical substance \(j\) bound on total concentration of \(j\) in the system, \(n_{j}^{0} / V\).If the system is prepared using \(n_{1}\) moles of water, the enthalpy of the system is given by equation (k). \[\mathrm{H}(\text { system })=\mathrm{n}_{1} \, \mathrm{H}_{1}(\mathrm{aq})+\left(\mathrm{n}_{\mathrm{j}}^{0}-\xi^{\mathrm{eq}}\right) \, \mathrm{H}_{\mathrm{j}}(\mathrm{aq})+\xi^{\mathrm{eq}} \, \mathrm{H}_{\mathrm{j}}(\mathrm{ad})\]The dependence of enthalpy on the extent of adsorption is given by equation (\(\lambda\)). \[\begin{aligned} &{[\partial \mathrm{H}(\text { system }) / \partial \xi \xi]=} \\ &\begin{aligned} \mathrm{n}_{1} \,\left[\partial \mathrm{H}_{1}(\mathrm{aq}) / \partial \xi\right]+\left(\mathrm{n}_{\mathrm{j}}^{0}-\xi^{\mathrm{eq}}\right) \,\left[\partial \mathrm{H}_{\mathrm{j}}(\mathrm{aq}) / \partial \xi\right]-\mathrm{H}_{\mathrm{j}}(\mathrm{aq}) \\ &+\mathrm{H}_{\mathrm{j}}(\mathrm{ad})+\xi^{\mathrm{eq}} \,\left[\partial \mathrm{H}_{\mathrm{j}}(\mathrm{ad}) / \partial \xi\right] \end{aligned} \end{aligned}\]As a working hypothesis we assume that the properties of substance \(j\) are ideal both in solution and as adsorbate. In other words there are no solute \(j \rightleftarrows\) solute \(j\) interactions in the aqueous solution, no bound \(j \rightleftarrows\) bound \(j\) interactions between adsorbed molecules and no solute \(j \rightleftarrows\) bound \(j\) interactions.In summary the adsorbed \(j\) molecules form monolayers and the adsorbed molecules are non-interacting with other adsorbed \(j\) molecules and with \(j\) molecules in solution.\[\text { Therefore, }[\partial \mathrm{H}(\text { system }) / \partial \xi \xi]=\mathrm{H}_{\mathrm{j}}(\mathrm{ad})^{0}-\mathrm{H}_{\mathrm{j}}(\mathrm{aq})^{\infty}=\Delta_{\mathrm{ad}} \mathrm{H}^{0}\]Here \(\mathrm{H}_{\mathrm{j}}(\mathrm{aq})^{\infty}\) is the limiting partial molar enthalpy of solute \(j\) meaning that in effect the solute molecules in solution are infinitely apart. \(\mathrm{H}_{\mathrm{j}}(\mathrm{ad})^{0}\) is the standard partial molar enthalpy of the adsorbate, implying that on the surface of the host adsorbent the adsorbate molecules are infinitely far apart; i.e. there are no adsorbate-adsorbate interactions. \(\Delta_{\mathrm{ad}} \mathrm{H}^{0}\) is the standard molar enthalpy for the adsorption of substance \(j\) from aqueous solution on to the adsorbate.In the next stage we require an equation for \(\mathrm{f}(\theta)\) in order to obtain an explicit equation for the chemical potential of adsorbed substance \(j\). We use the Langmuir model; \[f(\theta)=\theta /(1-\theta)\]\(\theta\) is the fraction of the surface covered by the adsorbate at equilibrium, the fraction \((1- \theta)\) being left bare; note that \(\theta\) is an intensive variable. \[\text { Then } \mu_{\mathrm{j}}(\mathrm{ad})=\mu_{\mathrm{j}}^{0}(\mathrm{ad})+\mathrm{R} \, \mathrm{T} \, \ln [\theta /(1-\theta)]\]The standard state for adsorbed j molecules corresponds to the situation where \(\theta=1 / 2\); i.e \([\theta /(1-\theta)]\) is unity. It is interesting to put some numbers to these variables. We set \(\chi=\ln [\theta /(1-\theta)]\). Then for \(\theta = 0.1\), \(\mathrm{R} \, \mathrm{T} \, \chi=-2.197 \, \mathrm{R} \, \mathrm{T}\). Hence when the surface is less than 50% covered the chemical potential of the adsorbate is less than in the adsorbed standard state. For \(\theta =1/2\), \(\mathrm{R} \, \mathrm{T} \, \chi=0\); at this stage the chemical potential of substance \(j\) in the adsorbed state equals that in the reference (standard) state. For \(\theta =0.9\), \(\mathrm{R} \, \mathrm{T} \, \chi=2.197 \, \mathrm{R} \, \mathrm{T}\). As the surface occupancy passes 0.5, the chemical potential of the adsorbate increases above that in the reference state. According to equation (i) for a system where solute \(–j\) and adsorbate \(–j\) have ideal thermodynamic properties, \[\mathrm{K}_{\mathrm{ad}}=[\theta /(1-\theta)] \,\left[\mathrm{V} \, \mathrm{c}_{\mathrm{r}} /\left(\mathrm{n}_{\mathrm{j}}^{0}-\xi\right)\right]\]With increase in \(\xi\) so \(\theta\) increases. Unfortunately we do not know how \(\theta\) and \(\xi\) are related. One procedure assumes that \(\theta\) is proportional to \(\xi_{\mathrm{eq}\), the constant of proportionality being \(\pi\). \[\mathrm{K}_{\mathrm{ad}}=\left[\pi \, \xi^{\mathrm{eq}} /\left(1-\pi \, \xi^{\mathrm{eq}}\right)\right] \,\left[\mathrm{V} \, \mathrm{c}_{\mathrm{r}} /\left(\mathrm{n}_{\mathrm{j}}^{0}-\xi^{\mathrm{eq}}\right)\right]\]\[\text { Hence, }\left(\mathrm{K}_{\mathrm{ad}} / \mathrm{V}\right) \,\left(1 / \mathrm{c}_{\mathrm{r}}\right) \,\left(\mathrm{n}_{\mathrm{j}}^{0}-\xi^{\mathrm{eq}}\right)=\pi \, \xi^{\mathrm{eq}} /\left(1-\pi \, \xi^{\mathrm{eq}}\right)\]By definition,\(\beta=\left[\mathrm{K}_{\mathrm{ad}} / \mathrm{V} \, \mathrm{c}_{\mathrm{r}}\right]^{-1}\) \[\text { Then, } \pi \,\left(\xi^{\mathrm{eq}}\right)^{2}-\xi^{\mathrm{eq}} \,\left[1+\pi \, \mathrm{n}_{\mathrm{j}}^{0}+\beta \, \pi\right]+\mathrm{n}_{\mathrm{j}}^{0}=0\]Equation (t) is a quadratic in the required variable \(\xi^{\mathrm{eq}\). With increase in amount of solute \(j\) in the system so the extent of adsorption increases. Recognising that \(\beta\) is taken as a constant, \[\left[\mathrm{d} \xi / \mathrm{dn}_{\mathrm{j}}^{0}\right]=\left[1-\pi \, \xi^{\mathrm{eq}}\right] /\left[\beta \, \pi+\pi \,\left(\mathrm{n}_{\mathrm{j}}^{0}-\xi^{\mathrm{eq}}\right)+\left(1-\pi \, \xi^{\mathrm{eq}}\right)\right]\]Equation (u) leads to an estimate of the dependence of \(\theta\) on the concentration of chemical substance j in the system. This analysis assumes no interactions between adsorbed molecules on the adsorbent. This assumption is probably too drastic. One approach which takes account of such interactions is the Freundlich Adsorption Isotherm. The chemical potentials of the adsorbed chemical substance \(j\) is related to \(\theta\) using equation (v). \[\mu_{\mathrm{j}}(\mathrm{ad})=\mu_{\mathrm{j}}^{0}(\mathrm{ad})+\mathrm{R} \, \mathrm{T} \, \ln [\theta /(1-\theta)]-\mathrm{R} \, \mathrm{T} \, \mathrm{a} \, \theta\]The parameter ‘a’ is an adsorbate-adsorbate interaction parameter. For systems where \(\mathrm{a} < 0\) (where \(\theta\) is always positive) repulsion between adsorbed molecules raises the chemical potential above that described by the Langmuir model and disfavours adsorption. For system where \(\mathrm{a} > 0\), attraction between adsorbed \(j\) molecules lowers their chemical potentials below the chemical potentials described by the Langmuir model; i.e. adsorption is enhanced above that required by the ideal model. But as for the Langmuir model, \(\theta\) is dependent on the extent of adsorption and thus almost certainly on the geometric properties of guest and host.If the thermodynamic properties of the solute \(j\) in solution are assumed to be ideal, equation (r ) is rewritten as follows. \[\mathrm{K}_{\mathrm{ad}}=\left[\pi \, \xi^{\mathrm{eq}} /\left(1-\pi \, \xi^{\mathrm{eq}}\right)\right] \, \exp \left(-\mathrm{a} \, \pi \, \xi^{\mathrm{eq}}\right) \,\left[\mathrm{V} \, \mathrm{c}_{\mathrm{r}} /\left(\mathrm{n}_{\mathrm{j}}^{0}-\xi^{\mathrm{eq}}\right)\right]\]Another approach writes \(\mathrm{f}(\theta)\) using a general equation having the following form. \[\mathrm{f}(\theta)=\left[\frac{\theta}{1-\theta}\right] \,\left[\frac{1}{\mathrm{n}^{\mathrm{n}}}\right] \,\left[\frac{\theta+\mathrm{n} \,(1-\theta)}{1-\theta}\right]^{\mathrm{n}-1}\]For the Langmuir adsorption isotherm, \(n\) is unity. Otherwise \(n\) is a positive integer; \(\mathrm{n}=1,2,3, \ldots\).Thus when \(n\) is set at 2, \[f(\theta)=\left[\frac{\theta}{1-\theta}\right] \,\left[\frac{1}{4}\right] \,\left[\frac{2-\theta}{1-\theta}\right]\]We comment on terminology. In enzyme chemistry, the term substrate refers to (in relative terms) small molecules which bind to an enzyme, a macromolecule. However in the subject describing adsorption of molecules on a surface, the molecules which are adsorbed are called the adsorbate. The macromolecular host is the adsorbent or substrate. In other words the meaning of the term ‘substrate’ differs in the two subjects.In general terms each enzyme has a unique site at which the adsorbate (substrate) is bound. However the term adsorbent implies that the ‘surface’ has a number of sites at which the adsorbate is adsorbed. The extent to which the sites are specific to a particular adsorbent is often less marked than in the case of enzymes. Nevertheless there is a common theme in which a substance \(j\) ‘free’ in solution loses translational freedom by coming in contact with a larger molecular system, being then held by that system.Footnotes I. Langmuir, J.Am.Chem.Soc.,1918,40,1361. D. H. Everett, Pure Appl. Chem.,1986, 58, 967. J. O’M. Bockris and S. U. M. Khan, Surface Electrochemistry, Plenum Press, New York, 1993. B. E. Conway, H. Angerstein-Kozlowska and H. P. Dhar, Electrochim.Acta, 1975, 19,189. M. J. Blandamer, B. Briggs, P. M. Cullis, K. D. Irlam, J. B. F. N. Engberts and J. Kevelam, J. Chem. Soc. Faraday Trans.,1998, 94,259. K. S. Pitzer, Thermodynamics, McGraw-Hill, New York, 3rd edn., 1995,chapter 23. H. Freundlich, Z.Phys.Chem.,1906,57,384.This page titled 1.14.2: Adsorption- Langmuir Adsorption Isotherm- One Adsorbate is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.14.3: Absorption Isotherms - Two Absorbates

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We consider the case where in addition to the adsorbent there are two adsorbates, chemical substances \(i\) and \(j\) in aqueous solution. In the simplest case the thermodynamic properties of the system are ideal. In other words for both solutes and adsorbates there are no \(i - i\), \(j - j\) and \(i - j\) interactions. Analysis of the adsorption using the Langmuir adsorption isotherm leads to two terms describing the surface coverage, \(\theta_{i}\) and \(\theta_{j}\) plus the total surface coverage, \(\theta_{i} + \theta_{j}\). The chemical potentials of the solutes in solution are described in terms of their concentrations (assuming pressure \(p\) is close to the standard pressure); \mathrm{c}_{\mathrm{r}}=1 \mathrm{~mol dm}^{-3}. \[\mu_{j}(a q)=\mu_{j}^{0}(a q)+R \, T \, \ln \left(c_{j} / c_{r}\right)\]\[\mu_{i}(a q)=\mu_{i}^{0}(a q)+R \, T \, \ln \left(c_{i} / c_{r}\right)\]The upper limit of the total surface occupancy is unity and so we expect as \(\left(\theta_{i} + \theta_{j} \right)\) approaches unity the sum of the chemical potentials \(\mu_{j}(\mathrm{ad})\) and \(\mu_{i}(\mathrm{ad})\) approaches \(+\infty\), thereby opposing any tendency for further solute to be adsorbed. If we assert that there are no substrate-substrate interactions on the surface, the chemical potential of adsorbate \(j\) can be formulated as follows. \[\mu_{j}(\mathrm{ad})=\mu_{j}^{0}(\mathrm{ad})+\mathrm{R} \, \mathrm{T} \, \ln \left[\theta_{\mathrm{j}} /\left(1-\theta_{\mathrm{i}}-\theta_{\mathrm{j}}\right)\right]\]The denominator \(\left(1-\theta_{i}-\theta_{j}\right)\) takes account of the fact that adsorbate \(i\) also occupies the surface. Thus as \(\left(\theta_{i} + \theta_{j} \right)\) approaches unity there are no more sites on the surface for adsorbate \(j\) (and adsorbate \(i\) ) to occupy. \[\text { Thus } \operatorname{limit}\left[\left(1-\theta_{\mathrm{i}}-\theta_{\mathrm{j}}\right) \rightarrow 0\right] \mu_{\mathrm{j}}(\mathrm{ad})=+\infty\]The equilibrium between chemical substance \(j\) as solute and adsorbate is described as follows. \[\text { Solute }-\mathrm{j}+\text { Polymer Surface } \Leftrightarrow \text { Adsorbate } \mathrm{j}\]\[\text { Prepared } \mathrm{n}_{\mathrm{j}}^{0} \quad \quad \quad 0 \mathrm{~mol}\]\[\text { Equilib. } \quad \mathrm{n}_{\mathrm{j}}^{0}-\xi_{\mathrm{j}} \quad \xi_{\mathrm{j}}+\xi_{\mathrm{i}} \quad \xi_{\mathrm{j}} \mathrm{mol}\]We express the fraction of surface coverage as proportional to the extent of adsorption via a proportionality constant \(\pi\) which is a function of the sizes of the solutes and geometric parameters describing the surface. \[\text { Then, } \quad \theta_{i}=\pi_{i} \, \xi_{i} \quad \text { and } \theta_{j}=\pi_{j} \, \xi_{j}\]\[\text { At equilibrium } \mu_{\mathrm{j}}^{\mathrm{eq}}(\mathrm{aq})=\mu_{\mathrm{j}}^{\mathrm{eq}} \text { (ad) and } \mu_{\mathrm{i}}^{\mathrm{eq}}(\mathrm{aq})=\mu_{\mathrm{i}}^{\mathrm{eq}}(\mathrm{ad})\]Hence for a system having volume \(\mathrm{V}\), \[\begin{aligned} \mu_{\mathrm{j}}^{0}(\mathrm{aq})+\mathrm{R} \, \mathrm{T} \, \ln \left[\left(\mathrm{n}_{\mathrm{j}}^{0}-\xi_{\mathrm{j}}\right) / \mathrm{V} \, \mathrm{c}_{\mathrm{r}}\right] \\ &=\mu_{\mathrm{j}}^{0}(\mathrm{ad})+\mathrm{R} \, \mathrm{T} \, \ln \left[\frac{\pi_{\mathrm{j}} \, \xi_{\mathrm{j}}}{1-\pi_{\mathrm{j}} \, \xi_{\mathrm{j}}-\pi_{\mathrm{i}} \, \xi_{\mathrm{i}}}\right] \end{aligned}\]\[\text { By definition, } \Delta_{\mathrm{ad}} \mathrm{G}_{\mathrm{j}}^{0}=-\mathrm{R} \, \mathrm{T} \, \ln \mathrm{K}_{\mathrm{j}}=\mu_{\mathrm{j}}^{0}(\mathrm{ad})-\mu_{\mathrm{j}}^{0}(\mathrm{aq})\]\[\text { Hence, } \mathrm{K}_{\mathrm{j}}=\left[\frac{\pi_{\mathrm{j}} \, \xi_{\mathrm{j}}}{1-\pi_{\mathrm{j}} \, \xi_{\mathrm{j}}-\pi_{\mathrm{i}} \, \xi_{\mathrm{i}}}\right] \, \frac{\mathrm{V} \, \mathrm{c}_{\mathrm{r}}}{\left(\mathrm{n}_{\mathrm{j}}^{0}-\xi_{\mathrm{j}}\right)}\]A similar equation is obtained for equilibrium constant \(\mathrm{K}_{i}\). Both equations are quadratics in the extent of adsorptionFootnotes M. J. Blandamer, B. Briggs, P. M. Cullis, K. D. Irlam, J. B. F. N. Engberts and J. Kevelam, J. Chem. Soc. Faraday Trans.,1998, 94, 259.This page titled 1.14.3: Absorption Isotherms - Two Absorbates is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.14.4: Apparent Molar Properties- Solutions- Background

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A given solution, volume \(\mathrm{V}\), is prepared using \(n_{1}\) moles of solvent (e.g. water) and \(n_{j}\) moles of solute, chemical substance \(j\).Thus\[ V(a q)=n_{1} \, V_{1}(a q)+n_{j} \, V_{j}(a q) \label{a}\]Here \(\mathrm{V}_{1}(\mathrm{aq})\) is the partial molar volume of the solvent and \(\mathrm{V}_{j}(\mathrm{aq})\) is the partial molar volume of the solute-\(j\). Experiment yields the density of this solution at defined \(\mathrm{T}\) and \(p\). In order to say something about this solution we would like to comment on the two partial molar volumes, \(\mathrm{V}_{1}(\mathrm{aq})\) and \(\mathrm{V}_{2}(\mathrm{aq})\). But we have only three known variables; the amounts of solvent and solute and the density. If we change the amount of, say, solute then \(\mathrm{V}(\mathrm{aq})\) together with the two partial molar volumes change. So we end up with more unknowns than known variables. Hence it would appear that no progress can be made. All is not lost. Equation (a) is rewritten in terms of the molar volume of the solvent, \(\mathrm{V}_{1}^{*}(\lambda)\) which is calculated from the density of the pure solvent and its molar mass. At a given \(\mathrm{T}\) and \(p\), density \(\rho_{1}^{*}(\lambda)=\mathrm{M}_{1} / \mathrm{V}_{1}^{*}(\lambda)\). We replace \(\mathrm{V}_{j}(\mathrm{aq})\) in Equation \ref{a} by the apparent molar volume, \(\phi\left(\mathrm{V}_{j}\right)\); Equation \ref{b}.\[\mathrm{V}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\lambda)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \label{b}\]Now we have only one unknown variable. But we anticipate that the apparent molar volume \(\phi \left(\mathrm{V}_{j} \right)\) depends on the composition of the solution, the solute, \(\mathrm{T}\) and \(p\).These comments concerning partial molar volumes establish a pattern which can be carried over to other partial molar properties. The following apparent molar properties of solutes are important; (i) apparent molar enthalpies \(\phi\left(\mathrm{H}_{\mathrm{j}}\right)\), (ii) apparent molar isobaric heat capacities \(\phi\left(\mathrm{C}_{p j}\right)\), (iii) apparent molar isothermal compressions \(\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)\), and (iv) apparent molar isobaric expansions \(\phi\left(\mathrm{E}_{\mathrm{pj}}\right)\). Apparent molar (defined) isentropic compressions \(\phi\left(K_{\mathrm{Sj}} ; \mathrm{def}\right)\), and apparent molar (defined) isentropic expansions \(\phi\left(K_{\mathrm{Sj}} ; \mathrm{def}\right)\) are also quoted but new complexities emerge.Lewis and Randall commented that ‘apparent molal quantities have little thermodynamic utility’, a statement repeated in the second but not in the third edition of this classic monograph. Suffice to say, their utility in the analysis of experimental results has been demonstrated by many authors.Apparent molar properties of solutes \(\phi\left(\mathrm{E}_{\mathrm{pj}}\right)\), \(\phi\left(K_{\mathrm{Sj}} ; \mathrm{def}\right)\), \(\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)\), \(\phi\left(K_{\mathrm{Sj}} ; \mathrm{def}\right)\) and \(\phi\left(\mathrm{C}_{p j}\right)\) are calculated using in turn the extensive properties of solutions, isobaric expansions \(\mathrm{E}_{p}\), isentropic expansions \(\mathrm{E}_{\mathrm{S}}\), isothermal compressions \(\mathrm{K}_{\mathrm{T}}\), isentropic compressions \(\mathrm{K}_{\mathrm{S}}\) and isobaric heat capacities \(\mathrm{C}_{p}\).Footnotes Equation (a) is interesting . We do not have to add the phrase ‘at constant temperature and pressure’ G. N. Lewis and M. Randall, Thermodynamics and The Free Energy of Chemical Substances, McGraw-Hill, New York, 1923. [The title on the front cover of the monograph is simply ‘Thermodynamics’.] G. N. Lewis and M. Randall, Thermodynamics, revised by K. S. Pitzer and L. Brewer, McGraw-Hill, New York, 2nd. edn. 1961, p. 108. K. S. Pitzer, Thermodynamics, McGraw-Hill, New York, 3d. edn., 1995.This page titled 1.14.4: Apparent Molar Properties- Solutions- Background is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.14.5: Apparent Molar Properties- Solutions- General

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Consider the general apparent molar property \(\phi \left( \mathrm{Q}_j \right) \) and the corresponding extensive property of a solutions, \(\mathrm{Q}\); e.g. \(\mathrm{C}_{p}\), \(\mathrm{E}_{p}\), \(\mathrm{E}_{\mathrm{S}}\), \(\mathrm{K}_{\mathrm{T}}\) and \(\mathrm{K}_{\mathrm{S}}\). The latter are all extensive properties of a given system. The corresponding volume intensive property \(q\) is given by the ratio \(\mathrm{Q} / \mathrm{V}\); c.f. \(q =\) isobaric expansivity \(\alpha_{p}\), isentropic expansivity \(\alpha_{S}\), isothermal compressibility \(\mathrm{K}_{\mathrm{T}}\), isentropic compressibility \(\mathrm{K}_{\mathrm{S}}\), and heat capacitance \(\sigma\) respectively. The following four general equations show how the volume intensive properties of the solution and solvent, \(q\) and \({q_{1}}^{*}\) respectively, form the basis for the calculation of apparent molar property \(\phi\left(Q_{j}\right)\). \[\phi\left(Q_{j}\right)=\left(q-q_{1}^{*}\right) \,\left(m_{j} \, \rho_{1}^{*}\right)^{-1}+q \, \phi\left(V_{j}\right)\]\[\phi\left(Q_{j}\right)=\left(q-q_{1}^{*}\right) \,\left(c_{j}\right)^{-1}+q_{1}^{*} \, \phi\left(V_{j}\right)\]\[\phi\left(Q_{j}\right)=\left(q \, \rho_{1}^{*}-q_{1}^{*} \, \rho\right) \,\left(m_{j} \, \rho \, \rho_{1}^{*}\right)^{-1}+q \, M_{j} \, \rho^{-1}\]\[\phi\left(Q_{j}\right)=\left(q \, \rho_{1}^{*}-q_{1}^{*} \, \rho\right) \,\left(c_{j} \, \rho_{1}^{*}\right)^{-1}+q_{1}^{*} \, M_{j} \,\left(\rho_{1}^{*}\right)^{-1}\]In these four equations, \(\rho\) is the density of the solution; \({\rho_{1}}^{*}\) is the density of the solvent at the same \(\mathrm{T}\) and \(p\). The theme running through these equations is the link between the apparent molar property \(\phi\left(Q_{j}\right)\) of a given solute and the measured property \(q\). Interestingly apparent molar enthalpies break the pattern in that the enthalpy of a solution cannot be measured. Nevertheless apparent molar enthalpies are used in the analysis of calorimetric results. There are no advantages in defining apparent chemical potentials and apparent molar entropies of solutes.Footnotes M. J. Blandamer, M. I. Davis, G.Douheret and J. C. R. Reis, Chem. Soc. Rev.,2001, 30,8. \(\phi\left(Q_{j}\right)\) is defined by the following equation with reference to the extensive variable \(\mathrm{Q}\) in terms of amounts of solvent and solute \(n_{1}\) and \(n_{j}\) respectively where \({\mathrm{Q}_{1}}^{*}\) is the molar property of the solvent at the same \(\mathrm{T}\) and \(p\). \[Q=n_{1} \, Q_{1}^{*}+n_{j} \, \phi\left(Q_{j}\right)\]We shift to volume intensive properties \(q\) and \({q_{1}}^{*}\). \[\mathrm{V} \, \mathrm{q}=\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\lambda) \, \mathrm{q}_{1}^{*}+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{Q}_{\mathrm{j}}\right)\]We express the volume using the following equation incorporating apparent molar volume \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)\) and molar volume of the solvent \(V_{1}^{*}(\lambda)\) at the same \(\mathrm{T}\) and \(p\). \[\mathrm{V}=\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\lambda)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)\]We solve equation (b) for \(\phi\left(\mathrm{V}_{\mathrm{j}}\right)\) using equation (c). \[\text{Hence, } \phi\left(\mathrm{Q}_{\mathrm{j}}\right)=\frac{\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\lambda) \, \mathrm{q}}{\mathrm{n}_{\mathrm{j}}}+\mathrm{q} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right)-\frac{\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\lambda) \, \mathrm{q}_{\mathrm{l}}^{*}}{\mathrm{n}_{\mathrm{j}}}\]But \(\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\lambda) / \mathrm{n}_{\mathrm{j}}=\left(\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}\right)^{-1}\).Equation (e) follows. \[\phi\left(Q_{j}\right)=\left(q-q_{1}^{*}\right) \,\left(m_{j} \, \rho_{1}^{*}\right)^{-1}+q \, \phi\left(V_{j}\right)\]Using the latter equation, molalities are converted to concentrations using equation (f). \[\left(\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}\right)^{-1}=\left(\mathrm{c}_{\mathrm{j}}\right)^{-1}-\phi\left(\mathrm{V}_{\mathrm{j}}\right)\]\[\text { Then } \phi\left(Q_{j}\right)=\left(q-q_{1}^{*}\right) \,\left(c_{j}\right)^{-1}-\phi\left(V_{j}\right) \,\left(q-q_{1}^{*}\right)+q \, \phi\left(V_{j}\right)\]\[\text {Or } \phi\left(Q_{j}\right)=\left(q-q_{1}^{*}\right) \,\left(c_{j}\right)^{-1}+q_{1}^{*} \, \phi\left(V_{j}\right)\]We return to equation (b) and express the volume using the following equation. \[\mathrm{V}=\left[\mathrm{n}_{1} \, \mathrm{M}_{1}+\mathrm{n}_{\mathrm{j}} \, \mathrm{M}_{\mathrm{j}}\right] / \rho\]\[\text { Then } \phi\left(Q_{j}\right)=\frac{n_{1} \, M_{1} \, q}{n_{j} \, \rho}+\frac{M_{j} \, q}{\rho}-\frac{n_{1} \, V_{1}^{*} \, q_{1}^{*}}{n_{j}}\]\[\text { Then } \phi\left(Q_{j}\right)=\frac{q}{m_{j} \, \rho}+\frac{M_{j} \, q}{\rho}-\frac{q_{1}^{*}}{m_{j} \, \rho_{1}^{*}}\]Or, \[\begin{aligned} &\phi\left(Q_{j}\right)= \\ &\left(q \, \rho_{1}^{*}-q_{1}^{*} \, \rho\right) \,\left(m_{j} \, \rho \, \rho_{1}^{*}\right)^{-1}+q \, M_{j} \, \rho^{-1} \end{aligned}\]To obtain an equation using concentrations, we use the following equation. \(1 / \mathrm{m}_{\mathrm{j}}=\rho / \mathrm{c}_{\mathrm{j}}-\mathrm{M}_{\mathrm{j}}\)Thus \[\phi\left(Q_{j}\right)=\frac{q \, \rho_{1}^{*}-q_{1}^{*} \, \rho}{c_{j} \, \rho_{1}^{*}}-\frac{M_{j} \,\left(q \, \rho_{1}^{*}-q_{1}^{*} \, \rho\right)}{\rho_{1}^{*} \, \rho}+\frac{q \, M_{j}}{\rho}\]Or, \[\phi\left(Q_{j}\right)=\left(q \, \rho_{1}^{*}-q_{1}^{*} \, \rho\right) \,\left(c_{j} \, \rho_{1}^{*}\right)^{-1}+q_{1}^{*} \, M_{j} \,\left(\rho_{1}^{*}\right)^{-1}\]Because the equations used for converting molalities to concentrations are exact, no approximations are involved. Therefore equations (e), (h),(\(\lambda\)) and m) are rigorously equivalent. \[\begin{aligned} \mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\lambda) / \mathrm{n}_{\mathrm{j}} &=\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\lambda) \, \rho_{1}^{*}(\lambda) / \mathrm{n}_{\mathrm{j}} \, \rho_{1}^{*}(\lambda) \\ &=\mathrm{w}_{1} / \mathrm{n}_{\mathrm{j}} \, \rho_{1}^{*}(\lambda)=\left[\mathrm{m}_{\mathrm{j}} \, \rho_{1}^{*}(\lambda)\right]^{-1} \end{aligned}\] From equations (b) and (i), \[\left(\mathrm{n}_{1} \, \mathrm{M}_{1}+\mathrm{n}_{\mathrm{j}} \, \mathrm{M}_{\mathrm{j}}\right) \, \mathrm{q} / \rho=\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\lambda) \, \mathrm{q}_{1}^{*}+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{Q}_{\mathrm{j}}\right)\] \(\mathrm{n}_{1} \, \mathrm{M}_{\mathrm{l}} / \mathrm{n}_{\mathrm{j}}=\left(\mathrm{m}_{\mathrm{j}}\right)^{-1}\) From equation (\(\lambda\)), \(\phi\left(Q_{j}\right)=\left\{\left[q \, \rho_{1}^{*}(\lambda)-q_{1}^{*} \, \rho\right] / \rho \, \rho_{1}^{*}(\lambda)\right\} \,\left[\left(\rho / c_{j}\right)-M_{j}\right]+\left[q \, M_{j} / \rho\right]\)This page titled 1.14.5: Apparent Molar Properties- Solutions- General is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

1.14.6: Axioms

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An axiom is a statement of principle which is generally accepted to be true. Axioms cannot be conveniently demonstrated otherwise there would be no need to use axioms. The Laws of Thermodynamics are axioms. However we can test and confirm by experiment many of the consequences following from the Laws of Thermodynamics. We can debate (if we so wish) if these laws were actually discovered in the sense that the laws were always there. Alternatively we might argue that these laws were formulated by brilliant scientists because prior to their formulation these laws did not exist.Footnote M. L. McGlashan, J.Chem.Educ.,1966,43,226.This page titled 1.14.6: Axioms is shared under a Public Domain license and was authored, remixed, and/or curated by Michael J Blandamer & Joao Carlos R Reis via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.