Problems and Solutions on Thermodynamics and Statistical Mechanics

352 Problems d Solutioru on Therdpamica 6' Statistical Mechanics

Solution:

Let n(w)dw denote the number of photons in the angular frequency
interval w - w + dw. Consider the pressure exerted on the walls by such
photons in the volume element dV at (r, 6, p) (Fig. 2.34). The probability
that they collide with an area dA of the wall is dA.cos 6/4ar2, each collision
contributing an impulse 2k cos 6 perpendicular to dA. Therefore, we have

Fig. 2.34.

dp, = - df

dAdt

dA. cos 6

ndw dV. ' 2k cos 6

• 
  
  
  
  • 4rr2
    dAdt
    dw
    4adt
    = -2nkcos2 6 sin Odrdddp ,
  
  
  
  

P= 1

dp, = 1 :kcdw = 1 *du 3.

r<cdt

uu

3 3v

Integrating we get p = - = -, where u is the energy density and U is

the total energy. From the thermodynamic equation

dU = TdS - pdV ,

and p = U/3V, we obtain dU = 3pdV + 3Vdp. Hence 4pdV + 3Vdp = TdS.

For an adiabatic process dS = 0, 4- + 3- = 0. Integrating we have

dV dp

VP

4

pv4I3 = const., 7 = ;.