Blundell Thermal Physics · Problem 28.4

Phase transitions

Physics — Statistical Mechanics · temperature-dependence-latent-heat

Problem

(a) Show that the temperature dependence of the latent heat of vaporization

L

is given by the following expression:

\frac{\mathrm{d}}{\mathrm{d}T}\left(\frac{L}{T}\right) = \frac{C_{pv} - C_{pL}}{T} + \left[\left(\frac{\partial S_v}{\partial p}\right)_T - \left(\frac{\partial S_L}{\partial p}\right)_T\right]\frac{\mathrm{d}p}{\mathrm{d}T}. \tag{28.68}

In this equation,

S_v

and

S_L

are the entropies of the vapour and liquid and

C_{pv}

and

C_{pL}

are the heat capacities of the vapour and liquid. Hence show that

L = L_0 + L_1 T

, where

L_0

and

L_1

are constants. (b) Show further that when the saturated vapour of an incompressible liquid is expanded adiabatically, some liquid condenses out if

C_{pL} + T\frac{\mathrm{d}}{\mathrm{d}T}\left(\frac{L}{T}\right) < 0, \tag{28.69}

where

C_{pL}

is the heat capacity of the liquid (which is assumed constant) and

L

is the latent heat of vaporization. (Hint: consider the gradient of the phase boundary in the

p

–

T

plane and the corresponding curve for adiabatic expansion.)