1.14.62: Solubilities of Gases in Liquids

Comparison of the solubilities of volatile chemical substance \(j\) in liquids \(\ell_{1}\) and \(\ell_{2}\) yields an estimate of the difference in reference chemical potentials, \(\Delta\left(\ell_{1} \rightarrow \ell_{2}\right) \mu_{\mathrm{j}}^{0}(\mathrm{~T})\).This is a classic subject [1-10] with two consequences.

1. A vast amount of information has been published, not all, unfortunately, of high quality.
2. Many terms and definitions have been developed. Determination of thermodynamic parameters characterizing gaseous solubilities is not straightforward. Account has to be taken of the fact that the properties of real gases are not perfect.

A closed system contains two phases, liquid and gaseous, at temperature \(\mathrm{T}\). The liquid is water; a sparingly soluble chemical substance \(j\) exists in both gas and liquid phases. A phase equilibrium is established for substance \(j\) in the two phases. In terms of the Phase Rule, there are two phases and two components. Hence there are two degrees of freedom. If the temperature and pressure are defined, the compositions of the two phases are fixed. In terms of chemical potentials with reference to substance \(j\) the following condition holds.

\[\mu_{\mathrm{j}}^{\mathrm{eq}}\left(\mathrm{aq} ; \mathrm{x}_{\mathrm{j}}, \mathrm{p}, \mathrm{T}\right)=\mu_{\mathrm{j}}^{\mathrm{eq}}\left(\mathrm{g} ; \mathrm{p}_{\mathrm{j}}, \mathrm{T}\right)\]

Here \(\mathrm{p}_{j}\) is the equilibrium partial pressure of substance \(j\) in the gas phase at pressure \(\mathrm{p}\) where pressure \(\mathrm{p}\) equals \(\left(\mathrm{p}_{j} + \mathrm{p}_{1}\right)^{\mathrm{eq}}\) where \(\mathrm{p}_{1}\) is the equilibrium partial pressure of water in the vapor phase. Equation (a) establishes the thermodynamic basis of the phenomenon discussed here. However historical and practical developments resulted in quite different approaches to the description of the solubilities of gases in liquids. The thermodynamic treatment is not straightforward if we recognize that the thermodynamic properties of the vapor (i.e. gas phase) and the solution are not ideal. When both the solubility and the partial pressure of the ‘solute’ in the gas phase are low, the assumption is often made that the thermodynamic properties of gas and solution are ideal. Then the analysis of solubility is reasonably straightforward [1-3]. In a sophisticated analysis, account must be taken of the intermolecular interactions in the vapor phase and solute-solute interactions in solution [4-10]. We review the basis of analyses where the thermodynamic properties of gas and liquid phases are ideal. Nevertheless equation (a) is the common starting point for the analysis.

Bunsen Coefficient, \(\alpha\)

By definition, the Bunsen Coefficient \(\alpha\) is the volume of a gas at \(273.15 \mathrm{~K}\) and standard pressure \(\mathrm{p}^{0}\) which dissolves in unit volume of a solvent when the partial pressure of the gas equals \(\mathrm{p}^{0}\). Experiment yields the volume \(\mathrm{V}_{j}(\mathrm{g})\) of gas \(j\) at temperature \(\mathrm{T}\) and partial pressure \(\mathrm{p}_{j}\) absorbed by volume \(\mathrm{V}_{\mathrm{s}}\) of solvent at temperature \(\mathrm{T}\). The volume of gas at \(273.15 \mathrm{~K}\) and standard pressure \(\mathrm{p}^{0}\), \(\mathrm{V}_{\mathrm{j}}\left(\mathrm{g} ; \mathrm{p}^{0} ; 273.15 \mathrm{~K}\right)\) is given by equation (b) assuming that gas \(j\) has the properties of a perfect gas.

\[\mathrm{V}_{\mathrm{j}}\left(\mathrm{g} ; \mathrm{p}^{0} ; 273.15 \mathrm{~K}\right)=\left[\mathrm{p}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}\left(\mathrm{g} ; \mathrm{T} ; \mathrm{p}_{\mathrm{j}}\right) \, 273.15 \mathrm{~K} / \mathrm{p}^{0} \, \mathrm{T}\right]\]

Hence experiment yields the ratio, \(\left[\mathrm{V}_{\mathrm{j}}\left(\mathrm{g} ; \mathrm{T} ; \mathrm{p}_{\mathrm{j}}\right) / \mathrm{V}_{\mathrm{s}}(\ell ; \mathrm{T} ; \mathrm{p})\right]\). By simple proportion we obtain the volume, \(\mathrm{V}_{\mathrm{j}}\left(\mathrm{g} ; \mathrm{p}^{0} ; 273.15 \mathrm{~K}\right)\) in the event that the gas \(j\) was at pressure \(\mathrm{p}^{0}\) above the liquid phase. Bunsen coefficient,

\[\alpha=\frac{\mathrm{V}_{\mathrm{j}}\left(\mathrm{g} ; \mathrm{T} ; \mathrm{p}_{\mathrm{j}}\right)}{\mathrm{V}_{\mathrm{s}}(\ell ; \mathrm{T} ; \mathrm{p})} \,\left(\frac{\mathrm{p}_{\mathrm{j}}}{\mathrm{p}^{0}}\right) \, \frac{273.15 \mathrm{~K}}{\mathrm{~T}}\]

At ambient pressure, if the partial pressure of the solvent is negligibly small,

\[\alpha=\frac{\mathrm{V}_{\mathrm{j}}(\mathrm{g} ; \mathrm{T} ; \mathrm{p})}{\mathrm{V}_{\mathrm{s}}(\ell ; \mathrm{T} ; \mathrm{p})} \,\left(\frac{273.15 \mathrm{~K}}{\mathrm{~T}}\right)\]

The assumption that substance \(j\) is a perfect gas can be debated but the correction is often less that 1%.

Oswald Coefficient

A given closed system comprises gaseous and liquid phases, at temperature \(\mathrm{T}\). The system is at equilibrium such that equation (a) holds. We assume that the thermodynamic properties of the solution and the gas phase are ideal and that an equilibrium exists for chemical substance \(j\) between the two phases. The solution is at pressure \(\mathrm{p}^{0}\), standard pressure which is close to ambient pressure.

\[\begin{aligned}
\mu_{\mathrm{j}}^{0}\left(\mathrm{~g} ; \mathrm{id} ; \mathrm{T} ; \mathrm{p}^{0}\right)+\mathrm{R} \, \mathrm{T} \, & \ln \left(\mathrm{p}_{\mathrm{e}}^{\mathrm{eq}} / \mathrm{p}^{0}\right) \\
&=\mu_{\mathrm{j}}^{0}\left(\mathrm{~s} \ln ; \mathrm{id} ; \mathrm{T} ; \mathrm{p}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{c}_{\mathrm{j}} / \mathrm{c}_{\mathrm{r}}\right)
\end{aligned}\]

For a perfect gas,

\[\mathrm{p}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}(\mathrm{g})=\mathrm{n}_{\mathrm{j}} \, \mathrm{R} \, \mathrm{T}\]

For a solution,

\[\mathrm{c}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} / \mathrm{V}(\mathrm{aq})\]

In these terms \(\mathrm{V}(\mathrm{aq})\) is the volume of solution which dissolves \(\mathrm{n}_{j}\) moles of chemical substance \(j\) from the gas phase.

\[\begin{aligned}
\mu_{\mathrm{j}}^{0}\left(\mathrm{~g} ; \mathrm{id} ; \mathrm{T} ; \mathrm{p}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\frac{\mathrm{n}_{\mathrm{j}} \, \mathrm{R} \, \mathrm{T}}{\mathrm{V}_{\mathrm{j}}(\mathrm{g})} \, \frac{1}{\mathrm{p}^{0}}\right) \\
&=\mu_{\mathrm{j}}^{0}\left(\mathrm{~s} \ln ; \mathrm{id} ; \mathrm{T} ; \mathrm{p}^{0}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\frac{\mathrm{n}_{\mathrm{j}}}{\mathrm{V}(\mathrm{aq})} \, \frac{1}{\mathrm{c}_{\mathrm{r}}}\right)
\end{aligned}\]

The Ostwald Coefficient \(\mathrm{L}\) is defined in terms of reference chemical potentials of substance \(j\) in solution and gas phase.

\[\Delta_{s \ln } \mathrm{G}^{0}\left(\mathrm{~T}, \mathrm{p}^{0}\right)=-\mathrm{R} \, \mathrm{T} \, \ln (\mathrm{L})=\mu_{\mathrm{j}}^{0}\left(\mathrm{~s} \ln ; \mathrm{id} ; \mathrm{T} ; \mathrm{p}^{0}\right)-\mu_{\mathrm{j}}^{0}\left(\mathrm{~g} ; \mathrm{id} ; \mathrm{T} ; \mathrm{p}^{0}\right)\]

From equation (h),

\[\Delta_{\mathrm{s} \ln } \mathrm{G}^{0}\left(\mathrm{~T}, \mathrm{p}^{0}\right)=\mathrm{R} \, \mathrm{T} \, \ln \left[\frac{\mathrm{n}_{\mathrm{j}} \, \mathrm{R} \, \mathrm{T}}{\mathrm{V}_{\mathrm{j}}(\mathrm{g})} \, \frac{\mathrm{V}(\mathrm{aq}) \, \mathrm{c}_{\mathrm{r}}}{\mathrm{n}_{\mathrm{j}} \, \mathrm{p}^{0}}\right]\]

\[\Delta_{s \ln } \mathrm{G}^{0}\left(\mathrm{~T}, \mathrm{p}^{0}\right)=\mathrm{R} \, \mathrm{T} \, \ln \left[\frac{\mathrm{V}(\mathrm{aq})}{\mathrm{V}_{\mathrm{j}}(\mathrm{g})} \, \frac{\mathrm{R} \, \mathrm{T} \, \mathrm{c}_{\mathrm{r}}}{\mathrm{p}^{0}}\right]\]

Because \(\mathrm{V}(\mathrm{aq})\) and \(\mathrm{V}_{j}(\mathrm{g})\) are expressed in the same units, the Oswald coefficient is dimensionless. The key assumption is that the thermodynamic properties of gas and solution phases are ideal.

Ostwald coefficients can be defined in several ways [4]. In the analysis set out above we refer to the volume of the solution containing solvent and solute \(j\). Another definition refers to the volume of pure liquid which dissolves a volume of gas \(\mathrm{V}_{j}\).

\[\mathrm{L}^{0}=\left[\mathrm{V}_{\mathrm{g}} / \mathrm{V}^{*}(\ell)\right]^{\mathrm{eq}}\]

A third definition refers to ratio of concentrations of substances \(j\) in liquid and gas phases.

\[\mathrm{L}_{\mathrm{c}}=\left[\mathrm{c}_{\mathrm{j}}^{\mathrm{L}} / \mathrm{c}_{\mathrm{j}}^{\mathrm{V}}\right]^{\mathrm{eq}}\]

In effect an Oswald coefficient describes an equilibrium distribution for a volatile solute between gas phase and solution. The Oswald coefficient is related to the (equilibrium) mole fraction of dissolved gas, \(x_{j}\) using equation (n) where \(\mathrm{p}_{j}\) is the partial pressure of chemical substance \(j\) and \(\mathrm{V}_{1}^{*}(\ell)\) is the molar volume of the solvent [11].

\[\mathrm{x}_{2}=\left\{\left[\mathrm{R} \, \mathrm{T} / \mathrm{L} \, \mathrm{p}_{\mathrm{j}} \, \mathrm{V}_{1}^{*}(\ell)\right]+1\right\}^{-1}\]

Henry’s Law Constant

In a given closed system at temperature \(\mathrm{T}\), gas and solution phases are in equilibrium. The thermodynamic properties of both phases are ideal. Then according to Henry’s Law, the partial pressure of volatile solute \(j\) is a linear function of the concentration \(\mathrm{c}_{j}\) at fixed temperature.

\[\mathrm{p}_{\mathrm{j}}(\operatorname{vap})=\mathrm{K}_{\mathrm{c}} \, \mathrm{c}_{\mathrm{j}} / \mathrm{c}_{\mathrm{r}}\]

\(\mathrm{K}_{\mathrm{c}}\) is the Henry’s Law constant (on the concentration scale), characteristic of solvent, solute and temperature. Similarly on the mole fraction scale,

\[\mathrm{p}_{\mathrm{j}}(\operatorname{vap})=\mathrm{K}_{\mathrm{x}} \, \mathrm{x}_{\mathrm{j}}\]

This subject, gas solubilities, is enormously important. We draw attention to some interesting reports concerning solubilities with particular reference to aqueous solutions and the environment [12].

Footnote

[1] R. Battino and H. L. Clever, Chem.Rev.,1966,66,395.

[2] E. Wilhelm and R. Battino, Chem.Rev.,1973,73,1.

[3] E. Wilhelm, R. Battino and R. J. Wilcock, Chem.Rev.,1977,77, 219.

[4] R. Battino, Fluid Phase Equilib.,1984,15,231.

[5] E. Wilhelm, Pure Appl.Chem.,1985,57,303.

[6] E. Wilhelm, Fluid Phase Equilib.,1986,27,233

[7] E. Wilhelm, Thermochim. Acta,1990,162,43.

[8] R.Fernandez-Prini and R. Crovetto, J. Phys. Chem. Ref.Data,1989,18,1231.

[9] R. Battino, T. R. Rettich and T. Tominaga, J. Phys. Chem. Ref. Data, 1984,13,563.

[10] T. R. Rettich, Y. P. Handa, R. Battino and E. Wilhelm, J. Phys. Chem.,1981,85,3230.

[11]

\[\begin{aligned}
&\frac{\mathrm{n}_{\mathrm{j}} \, \mathrm{R} \, \mathrm{T}}{\mathrm{V}_{\mathrm{j}}(\mathrm{g})} \, \frac{\mathrm{V}(\mathrm{aq}) \, \mathrm{c}_{\mathrm{r}}}{\mathrm{n}_{\mathrm{j}} \, \mathrm{p}^{0}}=\frac{[\mathrm{mol}] \,\left[\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}\right] \,[\mathrm{K}]}{\left[\mathrm{m}^{3}\right]} \, \frac{\left[\mathrm{m}^{3}\right] \,\left[\mathrm{mol} \mathrm{m}^{-3}\right]}{[\mathrm{mol}] \,\left[\mathrm{N} \mathrm{m}^{-2}\right]} \\
&=\frac{[\mathrm{N} \mathrm{m}]}{\left[\mathrm{m}^{3}\right]} \, \frac{1}{\left[\mathrm{~N} \mathrm{\textrm {m } ^ { - 2 } ]}\right.}=[1]
\end{aligned}\]

For a perfect gas \(j\),

\[\mathrm{p}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}(\mathrm{g})=\mathrm{n}_{\mathrm{j}} \, \mathrm{R} \, \mathrm{T}\]

\[V_{j}(g)=n_{j} \, R \, T / p_{j}\]

For \(\mathrm{n}_{1}\) moles of liquid 1, density \(\rho_{1}^{*}(\ell)\),

\[\mathrm{n}_{1} \, \mathrm{V}_{1}^{*}(\ell)=\mathrm{n}_{1} \, \mathrm{M}_{1} / \rho_{1}^{*}(\ell)\]

By definition, at temperature \(\mathrm{T}\),

\[\mathrm{L}=\mathrm{V}_{\mathrm{j}}(\mathrm{g}) / \mathrm{V}_{1}^{*}(\ell)\]

\[\mathrm{L}=\frac{\mathrm{n}_{\mathrm{j}} \, \mathrm{R} \, \mathrm{T}}{\mathrm{p}_{\mathrm{j}}} \, \frac{\rho_{\mathrm{l}}^{*}(\ell)}{\mathrm{n}_{1} \, \mathrm{M}_{\mathrm{l}}}\]

Hence,

\[\frac{\mathrm{n}_{1}}{\mathrm{n}_{\mathrm{j}}}=\frac{\mathrm{R} \, \mathrm{T}}{\mathrm{L} \, \mathrm{p}_{\mathrm{j}}} \, \frac{\rho_{1}^{*}(\ell)}{\mathrm{M}_{1}}=\frac{\mathrm{R} \, \mathrm{T}}{\mathrm{L} \, \mathrm{p}_{\mathrm{j}} \, \mathrm{V}_{1}^{*}(\ell)}\]

But mole fraction of solute \(j\) in solution,

\[\mathrm{x}_{\mathrm{j}}=\frac{\mathrm{n}_{\mathrm{j}}}{\mathrm{n}_{1}+\mathrm{n}_{\mathrm{j}}}=\frac{1}{\left(\mathrm{n}_{\mathrm{l}} / \mathrm{n}_{\mathrm{j}}\right)+1}\]

From equations (f) and (g),

\[\mathrm{x}_{\mathrm{j}}=\left[\left\{\mathrm{R} \, \mathrm{T} / \mathrm{L} \, \mathrm{p}_{\mathrm{j}} \, \mathrm{V}_{1}^{*}(\ell)\right\}+1\right]^{-1}\]

[12]

1. \(\mathrm{O}_{2}(\mathrm{aq})\); E. Douglas, J. Phys.Chem.,1964,68,169.
2. \(\mathrm{H}_{\mathrm{g}}(\mathrm{aq})\): E. Onat, J. Inorg. Nucl. Chem..,1974,36,2029.
3. \(\mathrm{H}_{2}\mathrm{S}(\mathrm{aq})\); E .C .Clarke and David N. Glew, Can. J. Chem., 1961,49,691.
4. \(\mathrm{N}_{2}(\mathrm{aq})\); T. R. Rettich, R. Battino and E. Wilhelm, J. Solution Chem.,1984,13,335.
5. Benzene(aq): D. S. Arnold, C. A. Plank, E. E. Erikson and F. P. Pike, Ind. Eng. Chem.,1958,3,253.
6. Polychlorinated Biphenyls(aq): W. Y. Shiu and D. Mackay, J. Phys. Chem. Ref. Data, 1986,15,911.
7. Cummenes; D. N. Glew and R. E. Robertson, J. Phys. Chem.,1956,60,332.
8. Fluorocarbons(aq); W.-Y. Wen and J. A. Muccitelli, J. Solution Chem.,1979,8,225.
9. Hydrocarbons; C. McAuliffe, J. Phys. Chem.,1966,70,1267.
10. \(\mathrm{O}_{2}(\mathrm{g})\) in water + alcohol mixtures; R.W.Cargill, J. Chem. Soc., Faraday Trans.,1996,72,2296.
11. Hydrocarbons in alcohol + water mixtures; R. W. Cargill and D. E Macphee, J. Chem. Res.(S),1986,2301.
12. \(\mathrm{N}_{2}(\mathrm{aq}); \mathrm{~H}_{2}(\mathrm{aq}); 298 - 640 \mathrm{~K}\); J. Alvarez, R. Crovetto and R. Fernandez-Prini, Ber. Bunsenges. Phys. Chem.,1988,92,935.
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14. Gases in ethylene glycol; R. Fernandez-Prini, R. Crovetto and N. Gentili, J. Chem. Thermodyn.,1987,19,1293.
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17. RH(aq); W.E. May, S.P. Wasik, M. E. Miller, Y.B. Tewari, J.M. Brown-Thomas and R. N. Goldberg, J. Chem. Eng. Data, 1983,28,197.
18. \(\mathrm{C}_{2}\mathrm{H}_{4}(\mathrm{aq} + \mathrm{~amine})\); E. Sada, H. Kumazawa and M. A. Butt, J. Chem. Eng. Data, 1977, 22,277.
19. A(aq); A. Ben-Naim, J. Phys. Chem.,1965,69,3245;1968,72,2998.
20. Viny chloride(aq); W. Hayduk and H. Laudie, J. Chem. Eng. Data, 1974,19,253.
21. Halogenated hydrocarbons(aq); A.L.Horvath, J. Chem. Documentation, 1972,12,163.
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23. \(\mathrm{CO}_{2}(\mathrm{aq} + \mathrm{~ROH})\);R. W. Cargill, J.Chem. Res(S),1982,230.