Discussion
The ratio of pressure exerted by photon gas to that of the ideal gas particles is ...... to ratio of their energy densities.
Answer: D
The pressure ratio is half the ratio of the energy densities.
Pressure and energy density are related differently for radiation and for a non-relativistic ideal gas:
\[ \begin{aligned} P_{\text{photon}} &= \frac{u_{\text{photon}}}{3} \\ P_{\text{gas}} &= \frac{2}{3}u_{\text{gas}} \end{aligned} \]
Dividing,
\[ \begin{aligned} &\frac{P_{\text{photon}}}{P_{\text{gas}}} \\ &= \frac{1}{2}\cdot\frac{u_{\text{photon}}}{u_{\text{gas}}} \end{aligned} \]
The 1/3 for photons comes from their energy being \(pc\), linear in momentum; the 2/3 for the gas from \(p^{2}/2m\). 'Twice' inverts the factor, 'equal' would need the same equation of state for both, and 'thrice' uses only the photon's 1/3.
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