How is hydration free energy of a solute measured experimentally?

Question

I am reading a paper (Shirts, M. R.; and Pitera, J. W. and Swope, W. C. and Pande, V. S. J. Chem. Phys. 2003, 119, 5740-5761). Near the beginning of the paper (page 5745), the authors state:

Experimental free energies of hydration for weakly soluble solutes are determined from concentration measurements made on two phase systems, where one phase consists of a vapor with a partial pressure $P_s$ of some solute molecule of type $s$, and the other phase consists of an aqueous solution with a number density concentration for solute molecules of $\rho_s^{\ell}$. When such a two phase system is at equilibrium with respect to transfer of molecules of type $s$ between the phases, the solvation free energy is given by $$\Delta G_{\text{solv}} = kT\ln(P_s / (\rho_s^{\ell} kT)).$$

What does this mean in practice? For example, suppose that I would like to measure the hydration free energy $\Delta G_{\text{solv}}$ of a small molecule (e.g., the amino acid arginine). That is to say, I would like to measure the solvation free energy of arginine in water. (Since arginine has a charged side chain at neutral pH, I would expect $\Delta G_{\text{solv}}$ to be negative, because arginine is likely stabilized by water, which is polar.)

Does the expression for $\Delta G_{\text{solv}}$ above imply that I just need an aqueous solution of arginine? Then would I need to somehow measure the partial vapor pressure $P_s$ of the arginine vapor which exists above the arginine aqueous solution? How is such a partial vapor pressure vapor measurement done in practice? Thanks for your time.

Answer 1

Your interpretation is correct. Theoretically, you will need only an aqueous solution and a fixed volume of gas, and then be able to measure equilibrium concentrations. Although, you may not observe very much vapor pressure without using elevated temperatures (as you noted, the zwitterion is very soluble).

Derivation

When a species $x$ is in equilibrium between two states, $$x_{state1} \rightleftharpoons x_{state2}$$ the equilibrium constant is given by the ratio of concentration of $x$ in each state: $$K = \frac {[x_{state2}]} {[x_{state1}]}.$$ The equilibrium constant relates to Gibbs free energy by the following equation: $$\Delta G = -RT \ln(K)=-RT\ln\left(\frac{x_{state2}}{x_{state1}}\right)=RT\ln\left(\frac{x_{state1}}{x_{state2}}\right).$$ Substituting for an aqueous/vapor phase system yields the cited equation. The literature uses Boltzmann's constant, $k$, in place of the gas constant $R$ because they are using units of molecules instead of moles. (The extra $kT$ term within the logarithm comes from the conversion of pressure to number density concentration using the ideal gas law: $\frac{n}{V}=\frac{P}{RT}$, again using molecular units instead of moles will require $k$ instead of $R$.)

Measurement

Determining the "best" method for gas phase measurement depends on what instrumentation you have available to you and how accurate you need/want to your results. Mass spectrometry would be ideal if it employs a soft ionization technique. If the molecular ion doesn't fraction, you could use the raw counts compared to an internal standard to calculate the concentration.

You don't necessarily need to measure the vapor pressure directly, either. If you have a method for measuring the aqueous concentration, you could calculate the vapor pressure based on decrease in solution concentration: $$\Delta G=RT \ln \left(\frac{[arg_{aq}]_i-[arg_{aq}]_f}{[arg_{aq}]_f}\right)=RT \ln \left(\frac{[arg_{aq}]_i}{[arg_{aq}]_f}-1\right)$$ where $[arg_{aq}]_i$ is the initial aqueous arginine concentration (total arginine) and $[arg_{aq}]_f$ is the final concentration (equilibrium amount). HPLC (or even UV-Vis if that is all you have available) would be a good method, but you would have to either make a calibration curve or use an internal standard (your buffer may work).

Notes

If you have access to J. Phys. Chem. and/or Biochemistry, this article cites articles (ref. 29, 82) that have experimental data for free energies of hydration for amino acids (all but arginine). They may be helpful to consult for methods.