1.5.4: Chemical Potentials- Solutions- Composition
A given aqueous solution at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\) (both near ambient) was prepared using \(\mathrm{n}_{1}\) moles of water and \(\mathrm{n}_{j}\) moles of urea (i.e. chemical substance \(j\)). The Gibbs energy \(\mathrm{G}(\mathrm{aq})\), an extensive property (variable), is given by the sum of products of amounts of each chemical substance and chemical potentials [1].
\[\mathrm{G}(\mathrm{aq})=\mathrm{n}_{1} \, \mu_{1}(\mathrm{aq})+\mathrm{n}_{\mathrm{j}} \, \mu_{\mathrm{j}}(\mathrm{aq}) \label{a} \]
Equation \ref{a} is key although we cannot put number values to \(\mathrm{G}(\mathrm{aq})\), \(\mu_{1}(\mathrm{aq})\) and \(\mu_{j}(\mathrm{aq}\). The latter two quantities are, respectively, the chemical potentials of the solvent, water and solute \(j\) in the aqueous solution at the same temperature and pressure. Equation \ref{a} seems a strange starting point granted it contains three quantities which we can never know. Matters can only improve.
There is merit in turning attention to an intensive property describing the Gibbs energy of a solution prepared using \(1 \mathrm{~kg}\) of solvent, \(\mathrm{G}\left(\mathrm{aq} ; \mathrm{w}_{1}=1.0 \mathrm{~kg}\right)\). Therefore, we do not have to worry about the size of the flask containing the solution. The same descriptor applies to \(0.1 \mathrm{~cm}^{3}\) or \(10 \mathrm{m}^{3}\) of a given solution [2,3].
\[\text { By definition } \quad \mathrm{G}\left(\mathrm{aq} ; \mathrm{w}_{1}=1.0 \mathrm{~kg}\right)=\mathrm{G}(\mathrm{aq}) / \mathrm{w}_{1} \nonumber \]
\[\mathrm{G}\left(\mathrm{aq} ; \mathrm{w}_{1}=1.0 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mu_{1}(\mathrm{aq})+\mathrm{m}_{\mathrm{j}} \, \mu_{\mathrm{j}}(\mathrm{aq}) \nonumber \]
\(\mathrm{M}_{1}\) is the molar mass of solvent, water, and mj is the molality of solute \(j\). Again we cannot put number values to \(\mathrm{G}\left(\mathrm{aq} ; \mathrm{w}_{1}=1.0 \mathrm{~kg}\right)\), \(\mu_{1}(\mathrm{aq})\) and \(\mu_{\mathrm{j}}(\mathrm{aq})\). Faced with this situation, the well-established approach involves an examination of differences. With respect to \(\mu_{1}(a q)\), the properties of water in an aqueous solution are compared with the properties of water at the same temperature and pressure. In these terms, we compare \(\mu_{1}(\mathrm{aq}, \mathrm{T}, \mathrm{p})\) with \(\mu_{1}^{*}(\ell, T, p)\). The superscript * in the latter term indicates that the chemical substance is pure and the symbol '\(\ell\)' indicates that this substance is a liquid. Hence, comparison is drawn with the chemical potential of pure liquid water at the same \(\mathrm{T}\) and \(\mathrm{p}\). In one sense we regard the solute as a controlled impurity perturbing the properties of the solvent. [We use the subscript '1' to indicate chemical substance 1 which in the convention used here refers to the solvent; water in the case of aqueous solutions.]
In considering the properties of, for example, urea in this aqueous solution molality \(\mathrm{m}_{j}\), we need a reference state against which to compare the properties of urea in the real solution prepared by dissolving \(\mathrm{n}_{j}\) moles of urea in \(\mathrm{n}_{1}\) moles of water. There is little point in comparing the properties of solute, urea with those of solid urea, a hard crystalline solid. Instead, we identify a reference solution state.
In general terms chemists explore how the chemical potentials of solvent and solute in an aqueous solution are related to the composition of the solution. Equations which offer such relationships should satisfy two criteria [4]: in the limit of infinite dilution (i) the partial molar volumes \(\mathrm{V}_{\mathrm{j}}(\mathrm{aq})\) and \(\mathrm{V}_{1}(\mathrm{aq})\) are meaningful and (ii) the partial molar enthalpies \(\mathrm{H}_{\mathrm{j}}(\mathrm{aq}\) and \(\mathrm{H}_{1}(\mathrm{aq})\) are meaningful. In other words, these properties do not approach an asymptotic limit of either \(+ \infty\) or \(– \infty\) with increasing dilution. For this reason physical chemists usually favour expressing the composition of solutions in molalities.
In summary analysis of the properties of solutions and liquid mixtures is built around the somewhat abstract concept of the chemical potential introduced by J. Willard Gibbs and by Pierre Duhem. The task of showing chemists the significance and application of this concept was left to Lewis and Randall in their classic monograph [5] published in 1923 [6]. Chemical potentials are one example of a class of properties called partial molar which provide the key link between macroscopic thermodynamic descriptions of systems and molecular properties [7].
Footnotes
[1] \(\mathrm{G}(\mathrm{aq})=[\mathrm{mol}] \,\left[\mathrm{J} \mathrm{mol}{ }^{-1}\right]+[\mathrm{mol}] \,\left[\mathrm{J} \mathrm{mol}^{-1}\right]=[\mathrm{J}]\)
[2] \(\begin{aligned}
&\mathrm{G}\left(\mathrm{aq} ; \mathrm{w}_{1}=1.0 \mathrm{~kg}\right)= \\
&\quad\left[1 / \mathrm{kg} \mathrm{mol}^{-1}\right] \,\left[\mathrm{J} \mathrm{mol}^{-1}\right]+\left[\mathrm{mol} \mathrm{kg}^{-1}\right] \,\left[\mathrm{J} \mathrm{mol}^{-1}\right]=\left[\mathrm{J} \mathrm{kg}^{-1}\right]
\end{aligned}\)
[3] With reference to equations (a) and (c), we must avoid the temptation to write “at constant temperature and pressure”. This condition is implicit in the description of the system using the independent variables \(\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}\right]\) for the aqueous solution containing the solute with the added condition that \(\mathrm{T}\) and \(\mathrm{p}\) are intensive variables; i.e. the set of independent variables is Gibbsian. Nonetheless there is often merit in using a complete set of descriptions of a system even if we over-define the variable under discussion. In describing the Gibbs energy defined by equation (a), we might write \(\mathrm{G}(\mathrm{T} ; \mathrm{p} ; \mathrm{aq})\). Similarly for the system described by equation (b) it is often helpful to write \(\mathrm{G}\left(\mathrm{T} ; \mathrm{p} ; \mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg} ; \mathrm{m}_{\mathrm{j}}\right)\). In reviewing the properties of solutions our interest, unless otherwise stated, centres on solutions at equilibrium where the affinity \(\mathrm{A}\) is zero and the organisation characteristic of the equilibrium system, \(\xi^{e q}\). We may find it helpful to write \(\mathrm{G}\left(\mathrm{T} ; \mathrm{p} ; \mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg} ; \mathrm{m}_{\mathrm{j}} ; \mathrm{A}=0 ; \xi^{\mathrm{eq}}\right)\), replacing \(\mathrm{G}\) by \(\mathrm{H}\), \(\mathrm{S}\) and \(\mathrm{V}\) for the corresponding enthalpy, entropy and volume of this solution. Clearly this over-definition is somewhat silly. Nevertheless it is often preferable to over-define a system rather than under-define when mistakes can arise.
[4] J. E. Garrod and T. M. Herrington, J. Chem. Educ., 1969, 46 , 165.
[5] G. N. Lewis and M. L. Randall, Thermodynamics and the Free Energy of Chemical Substances, McGraw-Hill, New York, 1923.
[6] See also, G. N. Lewis, Proc. Am. Acad. Arts Sci.,1907, 43 ,259.
[7] L. Hepler, Thermochim. Acta, 1986, 100 , 171.