Thermodynamics of Non-Relativistic Gas

Problem 1

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

solution

$$\frac{c_m}{c_\gamma}=\frac{1.5nk}{4\alpha T^3},$$ where $n$ is the concentration of the atoms.

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?

solution

It is the radiation. Temperature of the matter relaxes to that of radiation, because the latter has much greater thermal capacity.

Problem 3

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

solution

Consider for example the case of non-relativistic gas. Then $p = nkT,\;\rho = mn,$ and therefore $$ p = \frac{kT}{m}\rho \ll 1. $$

Problem 4

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

solution

Entropy of the system of particles in thermal equilibrium is the function $S=S(V,T)$ such that \[dS=\frac1T(dE+pdV)=\frac1T[d(\rho V)+pdV]=\frac1T\left[V\frac{d\rho}{dT}dT+(p+\rho)dV\right].\] Then \[\frac{\partial S}{\partial V}=\frac1T(\rho+p),\ \frac{\partial S}{\partial V}=\frac{\partial S}{\partial T}=\frac{V}{T}\frac{d\rho}{dT}.\] The above given derivatives must satisfy the condition \[\frac{\partial}{\partial T}\left[\frac1T(\rho+p)\right]=\frac{\partial}{\partial V}\left(\frac{V}{T}\frac{d\rho}{dT}.\right)\] Finally one obtains \[\frac{dp}{dT}=\frac1T(\rho+p).\]

Problem 5

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

solution

\[\frac{dp}{dT}=\frac1T(\rho+p)\Rightarrow\frac{d\rho}{d T}=\frac{1+w}{w}\frac\rho T,\] \[\frac{w}{1+w}\frac{d\rho}{\rho}=\frac{dT}{T}\Rightarrow T\propto\rho^{\frac{w}{w+1}}.\]

Problem 6

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

solution

\[T\propto\rho^{\frac{w}{w+1}},\ \rho\propto a^{-3(1+w)}\Rightarrow T\propto a^{-3w}.\]

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 $1+w+k\rho^{1/n}>0$.

solution

For the polytropic equation of state the thermodynamical equation \[\frac{dp}{dT}=\frac1T(\rho+p)\] becomes \[(w+k(1+1/n)\rho^{1/n})\frac{d\rho}{dT}=\frac1T(w+1+k\rho^{1/n})\rho.\] This equation can be integrated into \[T=T_*\left[1\pm(\rho/\rho_*)^{1/n}\right](w+n+1)/(w+1)(\rho/\rho_*)^{w/(w+1)},\] where $T_*$ is constants of integration and $\rho_*\equiv[(1+w)/|k|]^n$ [see P-H. Chavanis, arXiv:1208.0797], the upper sign corresponds to $k>0$, and the lower sign corresponds to $k<0$. This relation can be viewed as a generalization of Stefan-Boltzmann law.

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.

solution

For the generalized polytropic equation of state the velocity of sound is given by \[c_s^2=\frac{d p}{d\rho}=w\pm(w+1)\frac{n+1}{n}\left(\frac\rho{\rho_*}\right)^{1/n},\] \[\rho_*\equiv\left[\frac{w+1}{|k|}\right]^n.\] The upper sign corresponds to $k>0$, and the lower sign corresponds to $k<0$. As we have seen in the previous problem \[T=T_*\left[1\pm(\rho/\rho_*)^{1/n}\right](w+n+1)/(w+1)(\rho/\rho_*)^{w/(w+1)}.\] The extremum of temperature (when it exists) is located at \[\rho_e=\rho_*\left[\mp\frac{wn}{(w+1)(n+1)}\right]^n.\] It is easy to see that the velocity of sound vanishes at the point where the temperature is extremum.