Thermodynamics, Statistical Physics, and Quantum Mechanics

162

or

SOLUTIONS

where we only take intoaccount the first-orderterms in the solute. If we

alsoassume, which is usually thecase, that theosmotic pressure isalso

small, i.e., we obtain, from (S.4.56.11),

where and are the concentrations of the solutes:
Therefore, with the same accuracy, we arrive at the final

A different derivation of this formula may be found in Landau and Lifshitz,

Statistical Physics, Sect. 88.

Clausius–Clapeyron (Stony Brook)

a) We know that, at equilibrium, the chemical potentials of two phases

should be equal:

Here we write to emphasize the fact that the pressure depends

on the temperature. By taking the derivative of (S.4.57.1) with respect to

temperature, we obtain

Taking into account that and where s and

are the entropy and volume per particle, and substituting into (S.4.57.2),

where subscripts 1 and 2 refer to the two phases. On the other hand,

where is the latent heat per particle, so we can rewrite

(S.4.57.3) in the form

which is the Clausius–Clapeyron equation.

formula:

we have

4.57