Problems and Solutions on Thermodynamics and Statistical Mechanics

Statistical Physic8 217

and

5 mgL

e = -kT 2 - emgLlkT - 1 '

ae 5 k(rngL)2 emgLlkT

aT 2 (kT)2 (emgLlkT - 1)2

C" = - = -k-

5

Tk, for T +O ,

2k,

for T + 03.

Ell *---- -yL

Fig. 2.12

2051

Ideal monatomic gas is enclosed in cylinder of radius a and length L.

The cylinder rotates with angular velocity w about its symmetry axis and

the ideal gas is in equilibrium at temperature T in the coordinate system

rotating with the cylinder. Assume that the gas atoms have mass rn, have

no internal degrees of freedom, and obey classical statistics.

(a) What is the Hamiltonian in the rotating coordinates system?

(b) What is the partition function for the system?

(c) What is the average particle number density as a function of r?

(MIT)

Solution:

(a) The Hamiltonian for a single atom is

h'=-++d--mwr PI2 1 22 ,

2m 2

L

2

0, rIa,(z(<-,

m, otherwise.

4(W,Z) =