Problems and Solutions on Thermodynamics and Statistical Mechanics

but whose velocity distribution is arbitrary). If the gas discussed in the
previous parts were to be made ideal, what would be the restrictions on
the constants a through f?

(MIT)

Solution:

(a) We have dU(T, V) = C,dT +

~=-(g)~=-(g)~+T($$) V

Hence dU = (dT'I2V + eT2V + f T'12)dT - ( ;T'l2 - 2bT3 + cVV2) dV,

(b) Since U(T,V) is a state variable dU(T,V) is a total differential,
which requires

that is,

Hence a = 0, d = 0, e = 6b.

(c) Using the result in (b) we can write

dU(T, V) = d(2bT3V) + f T112dT - cV-2dV.

Hence

U(T, V) = 2bT3V + 2 fT3I2/3 + cV-' + const.

(d) Imagine that an ideal reflecting plane surface is placed in the gas.

The pressure exterted on it by atoms of velocity v is

The mean internal energy density of an ideal gas is just its mean kinetic

1- N

2 V

energy density, i.e.,

'L1= -mu2. -.