# Free energy of solvation and Henry's law constant

I am trying to calculate the free energy of solvation of $$\ce{CO2}$$ from its Henry's law constant. As given on Wikipedia, the dimensionless Henry's law constant is $$0.83$$. If I try to calculate $$\Delta G$$ from this, I got

\begin{align} \Delta G &= -RT\ln K \\ &= -\pu{8.3145 J mol-1 K-1}\times\pu{298.15 K}\times\ln 0.83 \\ &= \pu{0.46 kJ mol-1} \\ &= \pu{0.11 kcal mol-1}. \end{align}

However, I saw an article saying that the experimental value is $$\pu{0.24 kcal mol-1}$$. The article cited from the CRC Handbook of Chemistry and Physics, while Wikipedia cited from this compilation, which I think is also a reliable source.

So what did I do wrong in my calculation of $$\Delta G$$ from the Henry's law constant?

• It seems your calculation is fine. I find it a bit suspicious that the article from 2018 cites very old edition of CRC Handbook from 1995. Newer 97th edition from 2017 doesn't include free energies of solvation at all, only enthalpies of hydration. The difference in numbers might stem from different approaches of determination of the Gibbs free energy. – andselisk Feb 4 '19 at 11:56
• Also that article contains interesting passage on p. 17: "Thus, at QM/MM level of theory the value of hydration free energy of $\ce{CO2}$ amounts to 0.01±0.92 kcal/mol. This value is in good agreement with the experimental value of 0.24 kcal/mol". This confidence interval looks suspiciously broad and I don't see how it's in a "good agreement" at all. Probably I'm wrong and theoretical chemists who know better could comment on that. – andselisk Feb 4 '19 at 12:01
• I think solvation free energies having an uncertainty of 1 kcal/mol is not uncommon in theoretical calculations. The solvation free energy happens to be close to zero so that the uncertainty appears huge. – mck Feb 5 '19 at 4:17

The CRC Handbook of Chemistry and Physics [1] provides data for the solubility of $$CO_2$$ in $$H_2O$$ originally from reference [2]. The reported data are mole ratio solubilities $$\chi$$ of the gas at a partial pressure of one atmosphere (101.325 kPa), but can readily be converted into Henry's law solubility in other units:

$$H_{CP} = 1000 \delta_{H2O} \chi /M_{H2O} = 997 \times 6.15 \times 10^{-4}/18 = 0.034 mol/L atm$$

or in molal and bar units

$$H_{mP} = 1000\chi /1.013 (bar/atm)M_{H2O}=1000\times 6.15 \times 10^{-4}/(18 \times 1.013) = 0.034 mol/kg bar$$

Reference [2] is one of the original sources of the data tabulated in the Wikipedia, compiled by Rolf Sander in reference [3]. Other references in [3] provide similar values of $$H_{CP} \approx 0.034$$ mol/Latm. The value of $$H_{CP}$$ in the wikipedia is therefore consistent with the solubility reported in the CRC Handbook.

The article by Prasetyo and Hofer [4], referred to in the OP, cites the CRC Handbook as the source of the following reported experimental value (see Table 4): $$\Delta_{solv} G^o = 0.24 kcal/mol$$

However, from $$H_{mP} = 0.034$$ mol/kgbar (or $$H_{CP} = 0.034$$ mol/Latm) we obtain $$\Delta_{solv} G^o = -RTlog(H_{CP} \frac{P^o}{m^o})= 2.0 kcal/mol$$ which is far larger than the value reported in the article.

To attempt further progress, note that comparing standard free energies of solvation requires awareness of the associated reference states.

Prasetyo and Hofer indicate (pg 6474) that their choice of standard state is P° = 1 bar, m° = 1 mol/kg, and T = 298.15 K. The experimental values Prasetyo and Hofer compare to simulation results (see Table 4) refer presumably to this standard state. But this is approximately the same as the molar standard state associated with the data reported in the Wikipedia (P° = 1 atm, c° = 1 M, 298.15 K). Certainly the energies computed in the two scales don't differ significantly, as shown above. I therefore suspect that a different standard state was used in their calculation, or free energies for different (chemical) processes are being reported.

It is worth adding that the value $$\Delta_{solv} G^o = 0.24 kcal/mol$$ is encountered in other articles that also reference the 75th Edition of the CRC Handbook.

References

[1] The CRC Handbook of Chemistry and Physics, 85th Edition, section 8-87, SOLUBILITY OF SELECTED GASES IN WATER, L. H. Gevantman.

[2] R. Crovetto, Evaluation of Solubility Data for the System CO2-H2O, J. Phys. Chem. Ref. Data, 20, 575, 1991.

[3] R. Sander: Compilation of Henry's law constants (version 4.0) for water as solvent, Atmos. Chem. Phys., 15, 4399-4981 (2015).

[4] N. Prasetyo and T.S. Hofer J. Chem. Theory Comput. 2018, 14, 12, 6472-6483

• I wonder how you obtained 2.0 kcal/mol. I guess you calculated it by -RT ln H? But your H has units and it should not represent an equilibrium constant. That's why I used a dimensionless H in my calculation. – mck Feb 5 '19 at 4:15
• @mck H is reported with units so people understand the associated reference state, not because you are supposed to take the logarithm of the units. H is not unitless, but $HP^o/m^o$ or $HP^o/c^o$ are, depending on the definition of H. Since $P^o = 1, c^o=1$ and $m^o=1$ in the appropriate units, it all works out. See my changes to the answer. – Buck Thorn Feb 5 '19 at 15:03