1.10.22: Gibbs Energies- Salt Solutions- Solvent
- Page ID
- 384736
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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 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).
\[\mu_{1}(\mathrm{aq})=\mu_{1}^{*}(\ell)-2 \, \phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \nonumber \]
For the corresponding ideal solution, \(\phi = 1.0\) at all \(\mathrm{T}\) and \(\mathrm{p}\), Hence,
\[\mu_{1}(\mathrm{aq} ; \mathrm{id})=\mu_{1}^{*}(\ell)-2 \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \nonumber \]
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).
\[\mu_{1}(\mathrm{aq})=\mu_{1}^{*}\left(\ell ; \mathrm{p}^{0}\right)-\mathrm{v} \, \phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}+\int_{\mathrm{p}(0)}^{\mathrm{p}} \mathrm{V}_{1}^{*}(\ell) \, \mathrm{dp} \nonumber \]
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 [1,2].
\[\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})-\mathrm{v} \, \phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \nonumber \]
Footnotes
[1] For relevant Tables see;’ R. A. Robinson and R. H. Stokes, Electrolyte Solutions, 2nd edn.(revised), Butterworths, London, 1965, Appendix 8.
[2] The impact of salts on osmotic coefficients is illustrated by the properties of aqueous solutions containing alkylammonium salts.
- S. Lindenbaum, J. Phys.Chem.,1971, 75,3733; and references therein.
- G. E. Boyd, A. Schwartz and S. Lindenbaum, J. Phys.Chem.,1966, 70, 821; and references therein.