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Thermodynamics: Stats

Thermal Radiation



Planck Distribution

We want to learn about the spectrum of thermal radiation in a cavity. This doesn't sound familiar at first. We often call thermal radiation "blackbody radiation", but we won't go into the terminology too much here.

We know from our understanding of quantum mechanics that photons are quantized particles. Take any given mode, or frequency, of oscillation in a cavity. Then we say that each mode can be occupied only by an integral number of photons. In terms of energy, the allowable energies for the mode of frequency σ are = s σ . Here, we can think of s as being the number of photons in that mode.

For a given frequency, then, we can calculate the partition function:

Z = e -s σ/τ

This, however, is just an expansion of a closed form:

Z =

Now, we can find the average number of photons in a particular mode, or < s > , using the formula for average value of a property:

< s > = sP(s) =

Substituting in for Z and doing some unillustrative algebra yields the final result:

< s > =

The result is known as the Planck distribution function and gives us the average number of photons in the mode with frequency σ .

Stefan-Boltzmann Law

From the Planck distribution function, we can derive the energy density in the cavity. Convince yourself that the total energy in the cavity is given by:

U = < >

Each corresponds to the energy at a particular frequency σ n , and summing over all of the averages should yield the total energy. More explicitly:

U = < > = < s > σ =

Here, we can use the standard quantum method of letting the cavity be a cube and quantizing the frequencies to obtain σ n = nΠc/L if L is the length of a side of the cube.

We need one more trick to complete the derivation. The sum over positive n in 3 dimensions becomes 4Πn 2 dn . With those tools, we can plug through some more algebra to obtain:

= τ 4

The result is known as the Stefan-Boltzmann law of radiation. The significant aspect of the formula is that the energy density is proportional to the fourth power of the temperature.

We call a collection of photons a "photon gas". The term "gas" will not only refer to the traditional understanding of air, but will refer to any mobile conglomerate of a particle. You may hear of an "electron gas" or the like in your study of thermodynamics.

We can derive the entropy of the blackbody photons to be σ = ()3 .

The reason that this topic is included in thermodynamics is that it describes thermal radiation, and the results can be derived using some of the equations we have recently developed, along with some quantum mechanics.

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