In this problem we will consider a vapor fluid interface on top of a solid surface.  A schematic is shown below:

This is a zoomed-in picture of the edge of a liquid droplet sitting on a surface.  The change in the Helmholtz energy due to a change in the position of the vapor-liquid interface along the surface is:

Where  are the interaction energies (in N/m, analogous to the intermolecular potential but for macroscopic substances) between the three phases and are material-dependant constants.  L is the length of the interface in-and-out of the figure (in-and-out of the page) and θ is the contact angle


(a) Compute the work (W) required to move the liquid-vapor interface from x0 to a distance x at constant T, L and θ

(b) Find dU for this system for a given T (for constant T).

(c) You perform an experiment and find that the change in temperature as a function of distance (x) under adiabatic conditions is 10 Kelvin/mm.  Find how the contact angle would change as a function of applied heat under isothermal conditions at constant x (and constant L).