Problems and Solutions on Thermodynamics and Statistical Mechanics

354 Problema d Solutions on Thermodynamics d Statistical Mechanics

(c) Since the free energy F is an extensive quantity and

1

3

= -p = --u(T) ,

we have

u(T)V.

F=--^1

3

From thermodynamics we also have F = U - TS, where U = UV is the
internal energy, S is the entropy, and

s = - (g)

V

Hence

du dT

-=4r,

U

1

3

giving u = aT4,p = -aT4, where a is a constant.

2 166

A gas of interacting atoms has an equation of state and heat capacity

at constant volume given by the expressions

p(T, V) = aT'12 + bT3 + cV-~ ,

CV(T, V) = dT112V + eT2V + fT1I2 ,

where a through f are constants which are independent of T and V.

(a) Find the differential of the internal energy dU(T, V) in terms of dT

(b) Find the relationships among a through f due to the fact that

(c) Find U(T,V) as a function of T and V.

(d) Use kinetic arguments to derive a simple relation between p and U

for an ideal monatomic gas (a gas with no interactions between the atoms,

and dV.

U(T,V) is a state variable.