Problems and Solutions on Thermodynamics and Statistical Mechanics

Statistical Physic8 289

Solution:

The energy of a non-interacting degenerate electron gas is:

PP Ep2

E=8nVL zdp

where E is the energy of a single electron, pp is the Fermi momentum,

pF = (3N/8~V)'/~h.

(a) For the extreme relativistic case, E = cp, so we have energy

E=- 2scv^4

h3 'F '

which gives the equation of state

• _- 1E
  and pressure p = -
  T=O
  1
  3

pV= -E

(b) For a real electron,

where p is its momentum, giving

EM 2scV[p; + (mcp~)~]/h~.

The condition for the result of part (a) to be approximately valid is PF >>

mc. or

N as mc 3

->+) v3

Either N -+ 00 or V -+ 0 will satisfy this condition.

2114

Consider a box of volume V containing electron-positron pairs and

photons in equilibrium at a temperature T = l/kp. Assume that the

equilibrium is established by the reaction

7++e++e-.