Theory

Henry’s law states that the concentration of a solute in liquid \(C_s\)  is proportional to its partial pressure \(P\) above that liquid:

\[ C_s = H P \quad \quad (1) \]

Combining Eqn. (1) with the ideal gas law \( P = C_v RT\) where \(C_v\) is the vapor-phase solute concentration, we relate the Henry constant to the ratio of solute concentrations in the liquid and vapor:

\[ H R T = C_s/C_v = e^{-\beta \Delta G} \quad \quad (2) \]

Here \(\Delta G\) is the solvation free energy, i.e., the change in free energy to transfer a solute from vapor to liquid at fixed concentration.  The more soluble a solute, the more negative its solvation free energy, and the larger its Henry constant \(H\).

Figure 1. Molecular structures of methanol, ethanol, and 1-propanol.

In this tutorial, we use MD simulations to determine \(\Delta G\) for three alcohols in water: methanol, ethanol, and 1-propanol (Figure 1). The solubility of these alkyl alcohols is expected to decrease with increasing number of CH₂ groups, which are nonpolar and hence hydrophobic.

Solvation free energy

We determine the solvation free energy by computing the thermodynamic work to decouple a solute molecule from the solvent (thermodynamic integration):

\[ \Delta G = \int_0^1 d \lambda \, \left \langle \frac{d H(\lambda)}{d \lambda} \right \rangle \quad \quad (3) \]

Here \(H\) is the Hamiltonian, which depends on the coupling parameter \(\lambda\).  For \( \lambda=0\) (state A), the solute interacts fully with the solvent; at \( \lambda=1 \) (state B), the interactions are completely “turned off”.  The integrand \( \langle dH/d \lambda \rangle \) can be thought of as a generalized thermodynamic “force” (change of the average energy \( \langle dH \rangle \) with a small change in coupling \( d \lambda \).