Problems and Solutions on Thermodynamics and Statistical Mechanics

328 Problema t3 Solutiona on Thermodynamica €4 Statistical Mechanics

J

(b) 1) U = --, M = *p, S = kln2;

3

2)U=---pH, M=p, S=O;

3. U = - - 3pH, M = 3p, 5' = 0;

2

J

2

4) U = 0, M = 0, S = 3kln2.

(c) When H = 0, the system has three energy levels (-g,O,g). At

J

T = 0 K, the system is at ground state; when 0 < kT << -, the energy of

the system is enhanced by

2

AE e- JPkT

1

T2

and

CH o( -e- JPkT

As temperature increases, E and CH increase rapidly. When the system

is near the state where all the energy levels are uniformly occupied, the

increase of energy becomes slower and CH drops. When kT >> J, AE o(

l/exp(J/kT) and CH o( 7. Finally the energy becomes constant and

CH = 0. The CH(T, H = O~VS T curve is sketched in Fig. 2.31.

1

C" I T, 0

1

kT

where P = -.

(e) M= -__ ia 1nQ = -[3e-BJ/2sinh(3PpH) 21.L

P aH Q

+(eDJl2 + 2) sinh(PpH)].

When kT >> pH or J, Q % 6, M w 4/?p2H.