1.14.49: Osmotic Coefficient
There is possible disadvantage in an approach using the mole fraction scale to express the composition of a solution. Granted
- that our interest is often in the properties of solutes in aqueous solutions,
- that the amount of solvent greatly exceeds the amount of solute in a solution, and
- that the sensitivity of equipment developed by chemists is sufficient to probe the properties of quite dilute solutions,
the mole fraction scale for the solvent is not the most convenient method for expressing the composition of a given solution [1-3]. Hence another equation relating \(\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})\) to the composition of a solution finds favor.
By definition, for a solution containing a single solute, chemical substance \(\mathrm{j}\) [4],
\[\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \nonumber \]
Or, in terms of the standard chemical potential for water at temperature \(\mathrm{T}\) and standard pressure \(\mathrm{p}^{0}\),
\[\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{1}^{*}\left(\ell ; \mathrm{T} ; \mathrm{p}^{0}\right)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}+\int_{\mathrm{p}^{0}}^{\mathrm{p}} \mathrm{V}_{1}^{*}(\ell ; \mathrm{T}) \, \mathrm{dp} \nonumber \]
\(\mathrm{M}_{1}\) is the molar mass of water; \(\phi\) is the practical osmotic coefficient which is characteristic of the solute, molality mj, temperature and pressure. By definition \(\phi\) is unity for ideal solutions at all temperatures and pressures.
\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi=1.0 \text { at all T and } \mathrm{p} \nonumber \]
Further for ideal solutions, the partial differentials \((\partial \phi / \partial T)_{p}, \left(\partial^{2} \phi / \partial T^{2}\right)_{p} \text { and } (\partial \phi / \partial \mathrm{p})_{\mathrm{T}}\) are zero. For an ideal solution [5],
\[\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p} ; \mathrm{id})=\mu_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})-\mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \nonumber \]
We rewrite equation (d) in the following form:
\[\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p} ; \mathrm{id})-\mu_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})=-\mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \nonumber \]
Hence with an increase in molality of solute in an ideal aqueous solution, the solvent is stabilized, being at a lower chemical potential than that for pure water. We contrast the chemical potentials of the solvent in real and ideal solutions using an excess chemical potential, \(\mu_{1}^{E}(a q ; T ; p)\);
\[\begin{aligned}
\mu_{1}^{\mathrm{E}}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p}) &=\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})-\mu_{1}(\mathrm{aq} ; \mathrm{id} ; \mathrm{T} ; \mathrm{p}) \\
&=(1-\phi) \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}
\end{aligned} \nonumber \]
The term \((1 - \phi)\) is often encountered because it expresses succinctly the impact of the solute on the properties of the solvent. At a given molality (and fixed temperature and pressure), \(\phi\) is characteristic of the solute.
In the case of a salt \(\mathrm{j}\) which on complete dissociation forms ν ions the analogue of equation (a) takes the following form.
\[\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p})=\mu_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})-\mathrm{v} \, \phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \nonumber \]
Footnotes
[1] N. Bjerrum, Z. Electrochem., 1907, 24 ,259.
[2] G. N. Lewis and M. Randall, Thermodynamics, revised by K. S. Pitzer and L. Brewer, McGraw-Hill, New York, 2nd edn., 1961, chapter 22.
[3] Mole fractions of solvent \(\mathrm{x}_{1}\) for aqueous solutions having gradually increasing molality of solute \(\mathrm{m}_{j}.
| (A) | \(\mathrm{m}_{\mathrm{j}} / \mathrm{mol} \mathrm{kg}^{-1}=10^{-3}\); | \(\mathrm{x}_{1}=0.999982\) | \(x_{j}=1.8 \times 10^{-5}\) |
| (B) | \(\mathrm{m}_{\mathrm{j}} / \mathrm{mol} \mathrm{kg}=10^{-2}\); | \(\mathrm{x}_{1}=0.99982\) | \(x_{\mathrm{j}}=1.8 \times 10^{-4}\) |
| (C) | \(\mathrm{m}_{\mathrm{j}} / \mathrm{mol} \mathrm{kg}=10^{-1}\); | \(\mathrm{x}_{1}=0.9982\) | \(x_{j}=1.8 \times 10^{-3}\) |
| (D) | \(\mathrm{m}_{\mathrm{j}} / \mathrm{mol} \mathrm{kg}{ }^{-1}=0.5\); | \(\mathrm{x}_{1}=0.9911\) | \(\mathrm{x}_{\mathrm{j}}=8.9 \times 10^{-3}\) |
| (E) | \(\mathrm{m}_{\mathrm{j}} / \mathrm{mol} \mathrm{kg}=1.0\) | \(\mathrm{x}_{1}=0.9823\) | \(\mathrm{x}_{\mathrm{j}}=1.77 \times 10^{-2}\) |
[4]
\[\left[\mathrm{J} \mathrm{mol}^{-1}\right]=\left[\mathrm{J} \mathrm{mol}^{-1}\right]-[1] \,\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}] \,\left[\mathrm{kg} \mathrm{mol}^{-1}\right] \,\left[\mathrm{mol} \mathrm{kg}^{-1}\right] \nonumber \]
The definitions of ideal solutions expressed in equations (i) and (ii) are not in conflict.
\[\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p} ; \mathrm{id})=\mu_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})-\mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}} \nonumber \]
\[\mu_{1}(\mathrm{aq} ; \mathrm{T} ; \mathrm{p} ; \mathrm{id})=\mu_{1}^{*}(\ell ; \mathrm{T} ; \mathrm{p})+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{x}_{1}\right) \nonumber \]
Thus for an ideal solution these equations require that, \(v-\mathrm{M}_{1} \, \mathrm{m}_{\mathrm{j}}=\ln \left(\mathrm{x}_{1}\right)\) But
\[\ln \left(\mathrm{x}_{1}\right)=\ln \left[\mathrm{M}_{1}^{-1} /\left(\mathrm{M}_{1}^{-1}+\mathrm{m}_{\mathrm{j}}\right)\right]=-\ln \left(1.0+\mathrm{m}_{\mathrm{j}} \, \mathrm{M}_{1}\right) \nonumber \]
Bearing in mind that \(\mathrm{M}_{1} = 0.018 \mathrm{~kg mol}^{-1}\), then for dilute solutions \(\ln \left(1.0+\mathrm{m}_{\mathrm{j}} \, \mathrm{M}_{1}\right)=\mathrm{m}_{\mathrm{j}} \, \mathrm{M}_{1}\).