Following the method of Appendix C, replacing equation (C.4) by the integral

$${\int }_0^{\infty }{e}^{-r}{r}^{2}dr=2$$

show that

$$V_{3N}=\int \underset{{\int }_{0\le 1}^N{r}_i\le R}{}\dots \prod _{i=1}^N\left(4\pi r_i^{2}d{r}_i\right)={\left(8\pi R^3\right)}^N/\left(3N\right)!$$

Using this result, compute the "volume" of the relevant region of the phase space of an extreme relativistic gas $(\epsilon =pc)$ of $N$ particles moving in three dimensions. Hence, derive expressions for the various thermodynamic properties of this system and compare your results with those of Problem 1.7.

Data From Problem 1.7

Study the statistical mechanics of an extreme relativisitic gas characterized by the single-particle energy states

$$\epsilon (n_x,n_y,n_z)=\frac{hc}{2L}{(n_x^2+n_y^2+n_z^2)}^{1/2}$$

instead of (1.4.5), along the lines followed in Section 1.4. Show that the ratio $C_P/C_V$ in this case is $4/3$, instead of $5/3$.