1.5.5: Chemical Potentials- Solutions- Partial Molar Properties
- Page ID
- 373371
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)A given solution comprises \(\mathrm{n}_{1}\) moles of solvent, liquid chemical substance 1, and \(\mathrm{n}_{\mathrm{j}}\) moles of solute, chemical substance \(\mathrm{j}\). We ask ---- What contributions are made by the solvent and by the solute to the volume of the solution at defined \(\mathrm{T}\) and \(\mathrm{p}\)? In fact we can only guess at these contributions. This is disappointing. The best that we can do is to probe the sensitivity of the volume of a given solution to the addition of small amounts of solute and of solvent. This approach leads to a set of properties called partial molar. The starting point is the Gibbs energy of a solution. We develop an argument which places the Gibbs energy at the centre from which all other thermodynamic variables develop.
A given closed system comprises \(\mathrm{n}_{1}\) moles of solvent (e.g. water) and \(\mathrm{n}_{\mathrm{j}}\) moles of a simple solute j (e.g. urea) at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\). The Gibbs energy of the solution is defined by equation (a).
\[\mathrm{G}=\mathrm{G}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{\mathrm{l}}, \mathrm{n}_{\mathrm{j}}\right] \label{a} \]
We introduce a partial derivative having the following form: \(\left(\frac{\partial G}{\partial n_{j}}\right)_{T, p, n_{1}}\). The latter partial differential describes the differential dependence of Gibbs energy \(\mathrm{G}\) on the amount of chemical substance \(\mathrm{j}\). By definition, the chemical potential of chemical substance \(\mathrm{j}\),
\[\mu_{\mathrm{j}}=\left(\frac{\partial \mathrm{G}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}_{\mathrm{l}}} \nonumber \]
We also envisage that displacement of the system by adding \(\delta n_{j}\) moles of chemical substance \(\mathrm{j}\) from the original state to a neighbouring state produces a change in Gibbs energy at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\). In one class of displacements the system moves along a path of constant affinity for spontaneous reaction \(\mathrm{A}\). In another displacement the system moves along a path at constant organisation/composition, \(\xi\); i.e. frozen. These two pathways are related by the following equation. For the system at fixed \(\mathrm{T}\), \(\mathrm{p}\) and \(\mathrm{n}_{1}\)
\[\left[\frac{\partial \mathrm{G}}{\partial \mathrm{n}_{\mathrm{j}}}\right]_{\mathrm{A}}=\left[\frac{\partial \mathrm{G}}{\partial \mathrm{n}_{\mathrm{j}}}\right]_{\xi}-\left[\frac{\partial \mathrm{A}}{\partial \mathrm{n}_{\mathrm{j}}}\right]_{\xi} \,\left[\frac{\partial \xi}{\partial \mathrm{A}}\right]_{\mathrm{n}(\mathrm{j})} \,\left[\frac{\partial \mathrm{G}}{\partial \xi}\right]_{\mathrm{n}(\mathrm{j})} \nonumber \]
The conditions, constant \(\mathrm{T}\) and \(\mathrm{p}\), refer to intensive variables. We direct attention to a closed system at equilibrium where ‘\(\mathrm{A} = 0\)’ and the composition \(\xi=\xi^{\mathrm{eq}}\). Moreover at equilibrium, \((\partial \mathrm{G} / \partial \xi)_{\mathrm{T}, \mathrm{p}}\) is zero. Therefore the chemical potential of chemical substance \(\mathrm{j}\) in a system at equilibrium is defined by the following equation. Hence from equation (c),
\[\mu_{\mathrm{j}}=\left(\frac{\partial \mathrm{G}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{~A}=0}=\left(\frac{\partial \mathrm{G}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}_{1},, \mathrm{G}^{c_{q}}} \nonumber \]
A similar argument in the context of chemical substance 1 shows that,
\[\mu_{1}=\left(\frac{\partial \mathrm{G}}{\partial \mathrm{n}_{1}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}_{\mathrm{j}}, \mathrm{A}=0}=\left(\frac{\partial \mathrm{G}}{\partial \mathrm{n}_{1}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}_{\mathrm{j}}, \zeta^{\mathrm{eq}}} \nonumber \]
Equations (d) and (e) are key results. Similarly for a closed system at equilibrium at fixed \(\mathrm{T}\) and fixed \(\mathrm{p}\) (at a minimum in \(\mathrm{G}\), \(\mathrm{A} = 0\), \(\xi=\xi^{\mathrm{eq}}\) ), for all \(i\)-substances,
\[V_{j}(A=0)=V_{j}\left(\xi^{e q}\right) \nonumber \]
\[\mathrm{S}_{\mathrm{j}}(\mathrm{A}=0)=\mathrm{S}_{\mathrm{j}}\left(\xi^{\mathrm{eq}}\right) \nonumber \]
\[\mathrm{H}_{\mathrm{j}}(\mathrm{A}=0)=\mathrm{H}_{\mathrm{j}}\left(\xi^{e q}\right) \nonumber \]
\[\mu_{j}(A=0)=\mu_{j}\left(\xi^{e q}\right) \nonumber \]
But in the case of, for example, isobaric expansions and isobaric heat capacities, \(\mathrm{E}_{\mathrm{pj}}(\mathrm{A}=0) \neq \mathrm{E}_{\mathrm{pj}}\left(\xi^{\mathrm{eq}}\right)\) and \(\mathrm{C}_{\mathrm{pj}}(\mathrm{A}=0) \neq \mathrm{C}_{\mathrm{pj}}\left(\xi^{e q}\right)\). The identifications, (f) to (i), arise because these variables are first derivatives of the Gibbs energy of a closed system at equilibrium where \((\partial \mathrm{G} / \partial \xi)\) at fixed \(\mathrm{T}\) and \(\mathrm{p}\) is zero.