Problems and Solutions on Thermodynamics and Statistical Mechanics

120 Problems €4 Solutions on Thermodynamics d Statistical Mechanics

(d) Show that the heat capacity at constant volume of a Van der Waals
gas is a function of temperature alone (i.e., independent of V).

(MIT)

Solution:

= 0, we get

(a) As shown in Fig. 1.33, from (dp/dV)T=T, = 0 and (azp/dV2)r=r,

3a (Vc - b)3

T--

• v,4

so
a 8a
27b2’ ‘- 27bR

Vc = 3b,p, = - T--

(b) pcVciRTc = 318.

(c) In Fig. 1.33, the horizontal line CD is the modified isotherm. The

area of CAE is equal to that of EBD. The idea is that the common points,

i.e., C and D of the Van der Waals isotherm and the physical isotherm have

the same Gibbs free energy. Because of G = G(T,p), the equality of T’s

and p’s respectively will naturally cause the equality of G. In this way,

That is,

LE Vdp - Lc Vdp = /DB Vdp - LB Vdp, or AS~AE = ASEBD

For a Van der Waals gas, the equation of state gives

so that