Problems and Solutions on Thermodynamics and Statistical Mechanics

220 Problem8 d Solution8 on Thermodynamics d Statintical Mechanic8

Suppose that the number density of protons is 101'/cm3. Find the
chemical potentials for the electrons and positrons. Find the temperature
at which the positron density is l/cm3. Find the temperature at which it
is 101°/cm3.
(Prince ton)
Solution:
For kT/m,c2 << 1, nuclear reactions may be neglected. From charge
conservation, we have n- = np + n+, where n-, n+ are the number den-
sities of electrons and positrons respectively. For a non-relativistic non-
degenerate case, we have

n- =2 ( 2Tm,kT h2 ) 3/2 exp(p-;Tm'c2) ,

where p- and p+ are the chemical potentials of electrons and positrons

respectively. From the chemical equilibrium condition, we obtain p- =

-p+ = p. Hence

n+/n- = exp(-2p/kT).

For n+ = l/cm3, n- fic np = lo1' /cm3, we have exp(p/kT) = lo5 or

p/kT M 11.5. Substituting these results into the expression of n-, we have

T M 1.2~10~ K, sop w 1.6XlO-'erg. For n = 101'/cm3, exp(p/kT) = a,

p/kT NN 0.4. Substituting these results into the expression of n+, we get

T w 1.5 x lo8 K, p M 8.4 x lo-' erg.

2054

Consider a rigid lattice of distinguishable spin 1/2 atoms in a magnetic

field. The spins have two states, with energies -poH and +poH for spins up

(1) and down (l), respectively, relative to H. The system is at temperature

T.

(a) Determine the canonical partition function z for this system.

(b) Determine the total magnetic moment M = po(N+ - N-) of the

(c) Determine the entropy of the system.

system.

( wis co nsin)