Problems and Solutions on Thermodynamics and Statistical Mechanics

292 Problems €4 Solutions on Thermodynamics B Statistical Mechanics

Positron charge = 1.6~10-l~ coulomb; Bolteniann’s constant k = 1.38 x
erg/K].
(d) If the interaction cross sections in the electron-positron gas are
typically of order of magnitude of the Thompson cross section UT = 8mg/3
(classical electron radius 70 = 2.8 x cm), estimate the mean free
time between collisions of the particles. If the expansion rate in part (b) M
lo4 sec-’, is the assumption that the gas remains in equilibrium reasonable?
Why?
(SUIVY, Buffalo)

Solution:
(a) In the stated approximation, we have
P
kT
&=pc, -woo.

so

The average number density is

The average energy density is

(b) The quasi-static adiabatic expansion process satisfies the equation

d(pV) = -pdV. Neglecting the particle mass, we have p = p/3 (analogous

to a photon gas), then

dp - - - 4dV

P 3v)

giving

from which we obtain T cx V-’l3.

n cx V-I.

p cx v-4/3 ,

Hence the particle number density

(c) The average distance between particles r cx n-’/3. The ratio of the

Coulomb interaction energy per particle to the particle kinetic energy is

e2/r -N- e2n1I3 --N- e2 1

kT kT hc - 137 ‘