I have been more or less struggeling to understand an equation that is apparently used in almost all books covering crystals in any way. Basically every book that I have found explains the following:
In a system with a one-component vapour at the phase boundary vapour-solid, the difference in chemical potential between the two phases is given by:
$$\Delta\mu = \Delta\mu_v - \Delta\mu_c,$$
where $\Delta\mu_v$ and $\Delta\mu_s$ are the deviaitions of the chemical potentials of the vapor and solid states from their equilibrium positions, both being functions of pressure only ($T$ is being kept constant).
If those deviations are small, then we can write the $\Delta$s as partial derivatives which are integrated from $P_0$ (equilibrium pressure) to $P$ (actual pressure). By treating the vapor as an ideal gas, they finally arrive at:
$$\Delta\mu = kT\ln{\left(\frac{P}{P_0}\right)},$$
which explains how the change in chemical potential leading to nucleation is affected by the deviation of the pressure from the equilibrium point.
This all is quite easy to understand and fine with me, but now every book says the following: In case of solution crystallization the equation may be rewritten to give:
$$\Delta\mu = kT\ln{\left(\frac{C}{C_0}\right)},$$
where $C$ and $C_0$ are the actual and equilibrium (saturated) concentrations of the solute. However, no book that I have found (and I searched through a lot) actually derives or even explains why the pressures can be swapped by concentrations in case of solutions.
My explanation would be that thinking of pressure being linearly dependent on the amount of particles crossing unit area in unit time in the vapor and comparing that to the concentration which gives a linear dependence on the amount of particles crossing unit area in unit time in the solution, the two equations could be the same.
However, I don't know if that is correct, nor have I read this explanation anywhere but I need some proper source that can be quoted for the last equation or a derivation for the latter from first principals.
Could somebody please help me with that?
