# 16.14: Colligative Properties - Solubility of a Gas


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.

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