# Thermodynamics of Non-Relativistic Gas

### Problem 1

Find the ratio of thermal capacities of matter in the form of monatomic gas and radiation.

### Problem 2

Expansion of the Universe tends to violate thermal equilibrium between the radiation ($T\propto a^{-1}$) and gas of non-relativistic particles ($T\propto a^{-2}$). Which of these two components determines the temperature of the Universe?

### Problem 3

Show that in the non-relativistic limit ($kT\ll mc^2$) $p\ll\rho$.

### Problem 4

Show that for a system of particles in thermal equilibrium the following holds $\frac{dp}{dT}=\frac1T(\rho+p).$

### Problem 5

Show that for a substance with the equation of state $p=w\rho$ the following holds $T\propto\rho^{\frac{w}{w+1}}.$

### Problem 6

Show that for a substance with the equation of state $p=w\rho$ the following holds $T\propto a^{-3w}.$

### Problem 7

Obtain a generalization of Stefan-Boltzmann law for cosmological fluid, described by generalized polytropic equation of state $p=w\rho+k\rho^{1+1/n},$ assuming that $1+w+k\rho^{1/n}>0$.

### Problem 8

Show that the velocity of sound for cosmological fluid described by generalized polytropic equation of state $$p=w\rho+k\rho^{1+1/n}$$, where $1+w+k\rho^{1/n}>0$, vanishes at the point where the temperature is extremum.