Problems and Solutions on Thermodynamics and Statistical Mechanics

Thermodpam’cs 87

1088

Blackbody radiation.

(a) Derive the Maxwell relation

(b) From his electromagnetic theory Maxwell found that the pressure

p from an isotropic radiation field is equal to - the energy density u(T) :

where V is the volume of the cavity. Using the first

and second laws of thermodynamics together with the result obtained in
part (a) show that u obeys the equation

1

3

1

3 3v

p = -u(T) = -

1 du 1

3 dT 3

u = -T- - -U

(c) Solve this equation and obtain Stefan’s law relating u and T.

( was co nsin)

Solution:

(a) From the equation of thermodynamics dF = -SdT-pdV, we know

=-s,

(%)v (%) T =-p.

we get

Noting ~ a2F - - d2F

avaT aTav’

(b) The total energy of the radiation field is U(T,V) = u(T)V. Sub-

stituting it into the second law of thermodynamics:

(!g)T = T (g)T -P=T ($) V -P 7

T du 1

3 dT 3

we find u = -- - -u.

du

(c) The above formula can be rewritten as T- = 4u, whose solution

dT

is u = aT4, where a is the constant of integration. This is the famous

Stefan’s law of radiation for a black body.