1.5.14: Chemical Potentials; Solute; Concentration Scale
- Page ID
- 373395
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Both molalities and mole fractions are based on the masses of solvent and solute in a solution. Hence neither the molality \(\mathrm{m}_{j}\) nor mole fraction \(\mathrm{x}_{j}\) of a non-reacting solute depend on temperature and pressure. In fact, when we differentiate the equations for \(\mu_{j}(\mathrm{aq} ; \mathrm{T})\) with respect to pressure we take advantage of the fact that \(\mathrm{m}_{j}\) and \(\mathrm{x}_{j}\) do not depend on pressure. In addition, molalities and mole fractions are precise; there is no ambiguity concerning the amount of solvent and solute in the solution.
However, when we describe the meaning and significance of the activity coefficient \(\gamma_{j}\) and the meaning of the term 'infinite dilution' we refer to the distance between solute molecules. In fact, in reviewing the properties of solutions, chemists are often more interested in the distance between solute molecules than in their masses. [The same can be said about the interest shown by humans in the behaviour of other human beings!] Therefore, chemists often use concentrations to express the composition of solutions.
The concentration of solute \(\mathrm{c}_{j}\) describes the amount of chemical substance \(j\) in a given volume of solution; \(\mathrm{c}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} / \mathrm{V}\). The common method for expressing \(\mathrm{c}_{j}\) uses the unit '\(\mathrm{mol} \mathrm{dm}\)' [1,2]. By definition [at temperature \(\mathrm{T}\) and pressure \(p\left(\cong p^{0}\right)\)]
\[\mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{c}-\mathrm{scale})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{c}_{\mathrm{j}} \, \mathrm{y}_{\mathrm{j}} / \mathrm{c}_{\mathrm{r}}\right) \nonumber \]
\(\mathrm{c}_{\mathrm{r}}\) is the reference concentration, \(1 \mathrm{~mol dm}^{-3}\); \(\mathrm{c}_{j}\) is expressed using the same unit; \(\mathrm{y}_{j}\) is the activity coefficient for the solute \(j\) on the concentration scale.
\[\text { By definition, } \quad \lim \operatorname{it}\left(c_{j} \rightarrow 0\right) y_{j}=1 \quad \text { (at all } T \text { and } p \text { ) } \nonumber \]
\(\mu_{\mathrm{j}}^{0}(\mathrm{aq} ; \mathrm{c}-\text { scale })\) is the chemical potential of solute \(j\) in an ideal \(\left(\mathrm{y}_{\mathrm{j}}=1.0\right)\) aqueous solution (at the same \(\mathrm{T}\) and \(\mathrm{p}\)) where the concentration of solute \(\mathrm{c}_{\mathrm{j}}=1.0 \mathrm{~mol} \mathrm{dm}^{-3}\)
Footnotes
[1] Using the base SI units, concentration is given by the ratio \(\left(n_{j} / V\right)\) where \(\mathrm{V}\) is expressed using the unit \(\mathrm{m}^{3}\); \(\mathrm{n}_{j}\) is the amount of chemical substance \(j\), the unit being the mole. Nevertheless in the present context, general practice uses the reference concentration \(1 \mathrm{~mol dm}^{-3}\); \(\mathrm{c}_{j}\) is expressed using the unit, \(\mathrm{mol dm}^{-3}\). This practice emerges from the fact that for dilute aqueous solutions at ambient \(\mathrm{T}\) and \(\mathrm{p}\), unit concentration of solute, \(1 \mathrm{~mol dm}^{-3}\) is almost exactly unit molality, \(1 \mathrm{~mol kg}^{-1}\).
[2] For comments on standard states see E. M. Arnett and D. R. McKelvey, in Solute-Solvent Interactions, ed. J. F. Coetzee, M. Dekker, New York, 1969, chapter 6.