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### Problem 1: chemical potential

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

In the next 4 problems we consider some volume $V$, filled with black-body radiation of temperature $T$.

### Problem 2: number density distribution

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

### Problem 3: total number

Find the total number of photons.

### Problem 4: gas oven

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

### Problem 5: energy distribution

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

### Problem 6: thermodynamic potentials

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

### Problem 7: thermal capacity

Calculate the thermal capacity of black-body radiation.

### Problem 8: pressure

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

### Problem 10: CMB as microwave

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

### Problem 11: Planck distribution

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?

### Problem 12: power-law

Find in power-law cosmology (see Chapter 3) time dependence of the CMB temperature $T(t)$.