1.5.14: Chemical Potentials; Solute; Concentration Scale
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.