Difference between revisions of "Thermodynamics of Non-Relativistic Gas"

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

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

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

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 \[\frac{dp}{dT}=\frac1T(\rho+p).\]

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

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

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.

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.