The atoms of an ideal monatomic gas of N particles confined in a box of volume V can be adsorbed onto the surface of the box (surface area A), where they are bound to the surface with binding energy ϵ, but they can move around on the surface like an ideal two-dimensional gas. Use the results of Problem 8.19 to write the (Helmholtz) free energy of this system as a function of N₃ and N₂, the numbers of gas atoms in the bulk and the surface respectively. Minimize the free energy (Problem 8.18) to find the ratio N₂/N₃ when the system reaches thermal equilibrium. What happens in the limits where T → 0, T → ∞, and ћ → 0?