Thermodynamics, Statistical Physics, and Quantum Mechanics

THERMODYNAMICS AND STATISTICAL PHYSICS 37

a) Show that the isothermal work done by the gas is

b)

where is the average number of photons in the mode.

Showthat the radiationpressure is equal to onethird of theenergy

density:

c) Show that for a nonrelativistic Fermi gas the pressure is

4.77 Blackbody Radiation and Early Universe (Stony

Brook)

The entropy of the blackbody radiation in the early universe does not

change if the expansion is so slow that the occupation of each photon mode

remains constant (or the other way around). To illustrate this consider the

following problem. A one-dimensional harmonic oscillator has an infinite

series of equally spaced energy states, with where is a positive

integer or zero and is the classical frequency of the oscillator.

a) Showthat for a harmonicoscillator thefreeenergy is

b)Find theentropyS. Establish the connectionbetweenentropy and

occupancy of themodes by showingthat for onemode offrequency

the entropy is a function of photon occupancy only:

4.78 Photon Gas (Stony Brook)

Consider a photon gas attemperatureTinside a container of volume V.
Derive the equation of state and compare it to that of the classical ideal
gas (which has the equation Also compute the energy of
the photon gas in terms of PV. You need not get all the numerical factors
in this derivation.