Question 2

Find the number of nuclei ($N$) at time $t$: The rate of change of $N$ is $\frac{dN}{dt} = \alpha - \lambda N$. Rearranging and integrating: $\int_0^N \frac{dN}{\alpha - \lambda N} = \int_0^t dt$. $-\frac{1}{\lambda} \ln\left(\frac{\alpha - \lambda N}{\alpha}\right) = t \implies \alpha - \lambda N = \alpha e^{-\lambda t} \implies \lambda N = \alpha(1 - e^{-\lambda t})$

Find the rate of decay ($R$): $R = \lambda N = \alpha(1 - e^{-\lambda t})$.

Find the rate of heat production ($P$): Each decay releases $E_0$, so $P = R E_0 = \alpha E_0(1 - e^{-\lambda t})$.

Find the rate of temperature increase: Power $P = ms \frac{dT}{dt} \implies \frac{dT}{dt} = \frac{P}{ms}$ $\frac{dT}{dt} = \frac{\alpha E_0}{ms}(1 - e^{-\lambda t})$.