Problems and Solutions on Thermodynamics and Statistical Mechanics

10 Problem €4 Solutiow on Thermdpamics El Statistical Mechanic8

From the Van der Waals equation

(p + a/Vz)(V - b) = RT ,

we obtain

Therefore

R

RTV3

a 2ab 2a(~ - b)2 *

c,=c+

p-,+, vv 1-

1011

A solid object has a density p, mass M, and coefficient of linear expan-

sion a. Show that at pressure p the heat capacities C, and C,, are related

bY

C, - C,, = 3aMp/p.

( Wisconsin)

Soh tion:

(%),-

From the first law of thermodynamics dQ = dU + pdV and

(g) ,, (for solid), we obtain

c,-c"=(g),-(g) =p$T. dV

U

1 dV
From the definition of coefficient of linear expansion a = asolid/3 = - -
3V dT'
we obtain
M

• = 3aV = 3a-.

dV

dT P

Substituting this in (*), we find

M

P

c, - c,, = 3a-p.