Empirical laws for dilute aqueous solutions of nonpolar gases

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Abstract
Analyses of precise measurements on dilute aqueous solutions of seven nonpolar gases have revealed several empirical laws that may suggest new theoretical approaches to the structure of water and act as critical tests for specific models. Relationships have been discovered among the solution parameters, the thermodynamic properties of the gases, and the molecular parameters of the gases. The properties of the solutions within the 0–50 °C temperature range of the measurements are illuminated by extrapolating the results to examine the way the systems would behave if they obeyed the same rules outside the experimental temperature range as they do within it. The empirical results are: (1) ln(1/k) =A3(T1/T−1) +A2(T1/T−1)2, where T1 is the absolute temperature at which the Henry coefficient k hypothetically would be unity. (2) A2=36.855 is a dimensionless constant universal to the seven gases. It has the thermodynamic significance of being equal to the hypothetical value at the temperature T1 of the difference in partial molal heat capacity at constant pressure for the gas between the usual standard states for the liquid and gaseous phases, divided by twice the gas constant R. (3) For the noble gases both T1 and A3 are linear functions of the square root of the force constant ε/kB of the gas, and also approximately linear with the fourth root of the gas polarizability. A3 times R is equal to the hypothetical value of the partial molal entropy of solution at T1. (4) For the noble gases, extrapolated graphs of ΔH̄° vs 1/T intersect at a nearly common temperature not very different from the critical temperature for water. A similar statement applies to ΔS̄° vs 1/T. An especially interesting corollary of these laws is that at any ’’scaled temperature’’ T/T1, ΔC̄°p has the same value for all the gases and it is inversely proportional to the square of the scaled temperature. Graphs of ΔS̄° vs (T1/T)2 are linear, with the same slope for all gases and with intercepts which vary smoothly from helium to xenon for the noble gases. The same statement applies to ΔH̄°/T1 vs T1/T. If Henry’s law and law (1) described the properties of the system, not only in the very dilute solutions where the measurements were made, but also at much lower temperatures and much higher dissolved gas concentrations, T1 would be approximately the normal boiling temperature of the hypothetical pure liquefied gas. The values of T1 from the solubility measurements have been found to be related smoothly to the actual boiling temperatures. Similarly, for the noble gases the calculated value of the partial molal enthalpy of solution at T1 is a smooth function of the actual heat of vaporization of the pure liquid. Frank and Evans found for nonpolar gases at a fixed temperature that plots of ΔS̄° vs ΔH̄° were straight lines. Although graphs of our results suggest a similar trend, the apparent relationship is misleading and the two variables are not in fact linearly related. Instead, the equations for the thermodynamic variables lead to (−ΔS̄°)−(−ΔS̄°)1= 1/2[(ΔC̄p°)−(ΔC̄p°)1]. Values for ΔH̄°, ΔS̄°, and ΔC̄p° have been compared with results obtained by other workers. The new values for ΔC̄p° may be the first reasonably reliable ones for that variable. The parameters in Table XV yield probably the best currently available values for the solubilities of helium, neon, krypton, and xenon in pure water. The precisions of these solubilities are approximately 0.1% to 0.2%. Henry’s law specifies the variation of gas solubility with pressure. The corresponding dependence of gas solubility on temperature is given by law (1). When combined they yield f= x exp[A3(1−T1/T)−A2(1−T1/T)2], which constitutes an equation of state for the dilute aqueous solution of nonpolar gases.