Question #43820

A spherical black body of surface area $A$ contains a radioactive isotope that generates heat at a time-dependent rate $P(t) = P_0 e^{-\lambda t}$, where $P_0$ is the initial power and $\lambda$ is the decay constant. The sphere is located in deep space where the ambient temperature is $0\text{ K}$. An ideal Carnot engine operates between the sphere (acting as the hot reservoir) and a cold reservoir maintained at a constant temperature $T_c$.

The engine is designed to continuously extract a constant fraction $\alpha$ ($0 < \alpha < 1$) of the heat generated by the isotope, while the remaining $(1-\alpha)$ fraction of the generated heat is radiated away into space by the sphere. Assume the heat capacity of the sphere is negligible, so it is always in instantaneous thermal equilibrium. The engine operates until the temperature of the sphere drops to $T_c$.

Given that the initial temperature of the sphere is $T_0 = \left( \frac{(1-\alpha)P_0}{\sigma A} \right)^{1/4}$ (where $\sigma$ is the Stefan-Boltzmann constant), what is the total work $W$ done by the Carnot engine during its entire period of operation?