Question #43693

A sample contains $N_0$ identical hypothetical heavy nuclei, each initially having a mass number $A_0$. These nuclei undergo a continuous decay process by emitting neutrons. The binding energy per nucleon for these nuclei is given by the functional approximation $B(A) = \alpha - \beta A$, where $\alpha$ and $\beta$ are positive constants, and $A$ is the instantaneous mass number. The energy released per unit decrease in mass number is defined as the instantaneous Q-value, $Q(A)$. The decay process is statistical but can be modeled continuously such that the rate of decrease of mass number for each nucleus is directly proportional to its instantaneous Q-value, given by $\frac{dA}{dt} = -k Q(A)$ (where $k$ is a positive constant). Assuming $A_0 > \frac{\alpha}{2\beta}$, which of the following expressions represents the total power $P(t)$ released by the sample as a function of time $t$?