Question #42138

A thermally insulated, solid radioactive sphere of initial radius $R_0$, mass $m$, and specific heat capacity $s$ is placed in a vacuum. At $t=0$, it contains $N_0$ active nuclei with a decay constant $\lambda$. Each decay releases energy $Q$, which is completely absorbed by the sphere. The coefficient of linear thermal expansion of the sphere's material is $\alpha$. Gamma photons emitted from the center of the sphere travel to the surface, undergoing attenuation according to $I = I_0 e^{-\tau}$, where the optical depth $\tau = \mu R$, and the linear attenuation coefficient $\mu$ is directly proportional to the instantaneous density $\rho$ of the sphere. If the initial optical depth is $\tau_0$, which of the following expressions represents the optical depth $\tau(t)$ at time $t$? (Assume $\alpha \Delta T \ll 1$).