# 16.14: Colligative Properties - Solubility of a Gas

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A similar analysis yields an equation for the solubility of a gas in a liquid solvent as a function of temperature at a fixed pressure, \(P^{\#}\). We refer to the gas component as \(A\) and the liquid component as \(B\). We assume that solvent \(B\) is nonvolatile, so that the gas phase with which it is in equilibrium is essentially pure gaseous solute \(A\). We again find that the properties of the solvent have no role in our model, and the solubility of gas \(A\) is the same in every solvent.

We assume that low concentrations of the solute obey Henry’s law and choose the solution-phase standard state for solute \(A\) to be the pure hypothetical liquid \(A\) whose vapor pressure is \({\textrm{ĸ}}_A\) at \(T\). From Section 16.4, we then have \({\tilde{a}}_{A,\mathrm{solution}}=y_A\), so that \(d{ \ln {\tilde{a}}_{A,\mathrm{solutio}\mathrm{n}}\ }=d{ \ln y_A\ }\) at any temperature. Substituting into the general equation

\[d{\mu }_A={\overline{V}}_AdP-\overline{S}_AdT+RT\left(d{ \ln {\tilde{a}}_A\ }\right)\] we have \[d{\mu }_{A,\mathrm{solution}}=-\overline{S}_{A,\mathrm{solution}}\ dT+RT\left(d{ \ln y_A\ }\right)\]

The pressure of gas-phase \(A\) is constant at \(P^{\#}\), and \(dP=0\). We choose the gas-phase activity standard state to be pure gas \(A\) at \(P^{\#}\) and \(T\). Since this makes the activity of the pure gas unity at any temperature, we have \(d{ \ln {\tilde{a}}_{B,gas}\ }=0\). Substituting, we have

\[d{\mu }_{A,\mathrm{gas}}=-\overline{S}_{A,\mathrm{gas}}\ dT\]

Any constant-pressure process that maintains equilibrium between gas-phase \(A\) and solution-phase \(A\) must involve the same change in the chemical potential of \(A\) in

each phase, so that \(d{\mu }_{A,\mathrm{gas}}=d{\mu }_{A,\mathrm{solution}}\), and

\[-\overline{S}_{A,\mathrm{gas}}\ dT=-\overline{S}_{A,\mathrm{solution}}\ dT+RT\left(d{ \ln y_A\ }\right)\]

so that \[d{ \ln y_A\ }=-\frac{\left(\overline{S}_{A,\mathrm{gas}}-\overline{S}_{A,\mathrm{solution}}\right)}{RT}\ dT\]

The difference \(\overline{S}_{A,\mathrm{gas}}-\overline{S}_{A,\mathrm{solution}}\) is the entropy change for the equilibrium—and hence reversible—process in which one mole of substance \(A\) originally in solution vaporizes into a gas phase consisting of essentially pure gas \(A\) while the system is at the constant pressure \(P^{\#}\). Let us designate the enthalpy change for this reversible process at \(P^{\#}\) and \(T\) as \(\Delta_{\mathrm{vap}}{\overline{H}}_{A,\mathrm{solution}}\). Then, we have

\[\overline{S}_{A,\mathrm{gas}}-\overline{S}_{A,\mathrm{solution}}=\frac{\Delta_{\mathrm{vap}}{\overline{H}}_{A,\mathrm{solution}}}{T}\]

so that

\[d{ \ln y_A\ }=-\frac{\Delta_{\mathrm{vap}}{\overline{H}}_{A,\mathrm{solution}}}{RT^2}\ dT\]

Since enthalpy changes are generally relatively insensitive to temperature, we expect that, at least over small ranges of \(y_A\) and \(T\), \(\Delta_{\mathrm{vap}}{\overline{H}}_{A,\mathrm{solution}}\) is approximately constant. Since the vaporization process takes \(A\) from a state in which it has some of the characteristics of a liquid into a gaseous state, we can be confident that \(\Delta_{\mathrm{vap}}{\overline{H}}_{A,\mathrm{solution}}>0\). This conclusion implies that

\[\frac{d{ \ln y_A\ }}{dT}<0\]

Thus, our thermodynamic model leads us to the conclusion that the solubility of gas \(A\) decreases as the temperature increases. That the solubilities of gases generally decrease with increasing temperature is a well-known experimental observation. It stands in contrast to the observation that the solubilities of liquid or solid—at \(P^{\#}\) and \(T\)—substances generally increase with increasing temperature. Our analysis of gas solubility provides a satisfying theoretical interpretation for an experimental observation which otherwise appears to be counterintuitive.

The meaning of \(\Delta_{\mathrm{vap}}{\overline{H}}_{A,\mathrm{solution}}\) is unambiguous. Our analysis enables us to measure it by experimentally measuring \(y_A\) as a function of \(T\). We can estimate \(\Delta_{\mathrm{vap}}{\overline{H}}_{A,\mathrm{solution}}\) from another perspective: When we consider the “solution” in which \(y_A=1\), the vaporization process is the vaporization of liquid \(A\) into a gas phase of pure \(A\) at \(P^{\#}\) and \(T\). Since we assume that \(A\) is stable as a gas at \(T\), the boiling point of pure liquid \(A\) must be less than \(T\) at \(P^{\#}\) and the vaporization \(A\) must be a spontaneous process at \(P^{\#}\) and \(T\). The enthalpy of vaporization datum which is most accessible for liquid \(A\) is that for the reversible vaporization at one atmosphere and the normal boiling point, \(T_B\), which we designate as \(\Delta_{\mathrm{vap}}H^o_A\). If we stipulate that \(P^{\#}\) is one atmosphere; assume that our solubility equation remains valid as \(y_A\) increases from \(y_A\approx 0\) to \(y_A=1\); and assume that the enthalpy of vaporization is approximately constant between the boiling point of \(A\) and \(T\), we have \(\Delta_{\mathrm{vap}}{\overline{H}}_{A,\mathrm{solution}}=\Delta_{\mathrm{vap}}H^o_A\). Then,

\[d{ \ln y_A\ }=-\frac{\Delta_{\mathrm{vap}}H^o_A}{RT^2}\ dT\]

and

\[\int^{y_A}_1{d{ \ln y_A\ }}=\int^T_{T_B}{-\frac{\Delta_{\mathrm{vap}}H^o_A}{RT^2}}\ dT \nonumber\]

so that \[{ \ln y_A\ }=\frac{\Delta_{\mathrm{vap}}H^o_A}{R}\left(\frac{1}{T}-\frac{1}{T_B}\right)\]

Check

Viewed critically, the accuracy of the approximation

\[\Delta_{\mathrm{vap}}{\overline{H}}_{A,\mathrm{solution}}\approx \Delta_{\mathrm{vap}}H^o_A \nonumber\]

is dubious. The assumptions we make to reach it are essentially equivalent to assuming that the cohesive forces in solution are about the same between \(A\) molecules as they are between \(A\) molecules and \(B\) molecules. We expect this approximation to be more accurate the more closely the solution exhibits ideal behavior. However, if solvent \(B\) is to satisfy our assumption that the solvent is nonvolatile, the cohesive interactions between \(B\) molecules must be greater than those between \(A\) molecules, and this not consistent with ideal-solution behavior.