Thermodynamics of Black-Body Radiation

Problem 1

Show that the photon gas in thermal equilibrium has zero chemical potential.

solution

Now let us consider in the next 4 problems some volume $V$, filled with black-body radiation of temperature $T$.

Problem 2

Find the number of photons with frequencies in the interval $[\omega ,\omega +d\omega]$.

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Problem 3

Find the total number of photons.

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Problem 4

What is this number for a gas oven at room temperature and at maximum heat?

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Problem 5

What is the energy of photons with frequencies in the interval $[ \omega ,\omega +d\omega]$?

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Problem 6

Calculate the free energy, entropy and total energy of black-body radiation.

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Problem 7

Calculate the thermal capacity of black-body radiation.

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Problem 8

Find the pressure of black-body radiation and construct its state equation.

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Problem 9

Find the adiabatic equation for the black-body radiation.

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Problem 10

Why CMB cannot be used to warm up food like in the microwave oven?

solution

The relic radiation, or CMB, corresponds to the black-body radiation with temperature $T_{CMB}=2.725\:K.$ According to the main principle of thermodynamics, heat cannot transfer from a less heated body to more heated one, and thus the body (food in our case), which initially had temperature $T_0>T_{CMB},$ will emit more energy in the environment then absorb back, until the equilibrium installs with the CMB radiation at temperature $T_{CMB}.$

Problem 11

The binding energy of electron in the hydrogen atom equals to $13.6\ eV$. What is the temperature of the Planck distribution with this average photon energy?

solution

The Planck distribution has maximum at frequency $\omega_m = 2.822 kT/\hbar.$ Then one obtains $ kT = 13.6/2.822 = 4.82\mbox{eV}$ and $T\approx 5.6 \cdot {10^4}\mbox{K}$.