Multi-particle interaction in a model of the hydrophobic interaction

The multi-particle potential of mean force between interstitial solute molecules in Ben-Naim's one-dimensional, many-state lattice model is calculated. The solution is a direct extension of an earlier calculation of the two-particle interaction by Kolomeisky and Widom (Faraday Dis. 112 (1999) 81). It is found that the many-particle interaction potential is a sum of pair potentials between neighboring particles only. An exact equation of state, expressing the activity in the temperature and the pressure, is derived. The resulting solubility of a gaseous hydrophobe, which is defined osmotically, is calculated and found to increase considerably with the gas density.