# Thermodynamics of Black-Body Radiation

## Contents

- 1 Problem 1: chemical potential
- 2 Problem 2: number density distribution
- 3 Problem 3: total number
- 4 Problem 4: gas oven
- 5 Problem 5: energy distribution
- 6 Problem 6: thermodynamic potentials
- 7 Problem 7: thermal capacity
- 8 Problem 8: pressure
- 9 Problem 9: adiabatic equation
- 10 Problem 10: CMB as microwave
- 11 Problem 11: Planck distribution
- 12 Problem 12: power-law

### 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 9: adiabatic equation

Find the adiabatic equation for the black-body radiation.

### Problem 10: CMB as microwave

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

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: 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?

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 $T\approx 5.6 \cdot {10^4}\mbox{K}$.

### Problem 12: power-law

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

In power-law cosmology the scale factor $a(t)$and the CMB temperature $T(t)$ are related through the relation \[\frac{T_{0} }{T} =\frac{a}{a_{0} } =\left(\frac{t}{t_{0} } \right)^{\alpha } =\left(\frac{t}{t_{0} } \right)^{1/1+q} \] where $q$is the deceleration parameter. Consequently, \[T(t)=T_{0} \left(\frac{t}{t_{0} } \right)^{-\alpha } =T_{0} \left(\frac{t}{t_{0} } \right)^{-1/1+q} \]