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Thermodynamics: Stats


Problems 2

Summary Problems 2

Problem :

Write down the Planck Distribution Function.

< s > =

Problem :

Looking at the Planck distribution function, describe what happens at the high and low frequency limits.

For τσ, the occupation of these low frequency states is very high, approaching . This poses no problem however because it is the density of photons per frequency space that is physically important, and so there are very few frequencies that have this high occupancy.

For τσ, the occupation of the high frequency states is near zero.

Problem :

Explain why there is a factor of 1/8 when we rewrite the sum as an integral in our derivation of the Stefan-Boltzmann law of radiation.

When we are summing over quantum states, only non-negative quantum numbers are allowed. Writing the integral alone sums over all 8 quadrants in n-space, and so we divide by 8 to get the right answer.

Problem :

Write down the Stefan-Boltzmann law of radiation.

= τ4

Problem :

Describe why it is reasonable to expect that the entropy of a photon gas would go as τ3.

We saw that the energy goes as τ4, and we can recall that the temperature can be defined as with appropriate variables held constant. The only way to satisfy such a requirement is to have σ go as τ3.