Question

A radionuclide with half-life $1620s$ is produced in a reactor at a constant rate of $1000$ nuclei per second. During each decay energy, $200MeV$ is released. If the production of radionuclides started at $t=0$ , then the rate of release of energy at $t=3240s$ is

• A. $1.5\times 10^{5}MeV{s}^{-1}$
• B. $1.5\times 10^{2}MeV{s}^{-1}$
• C. $2.5\times 10^{2}MeV{s}^{-1}$
• D. $3.5\times 10^{5}MeV{s}^{-1}$

Answer 1

Answer: (A)

Let N be the number of nuclei at time t, then net rate of increase of nuclei at instant t is,z
$\frac{dN}{dt}=\alpha -\lambda N$ (where α = rate of production of nuclei)
$\begin{array}{l}{\int }_0^N\frac{dN}{\alpha -\lambda N}={\int }_0^{t}dt\\ N=\frac{\alpha }{\lambda }\left(1-e^{-\lambda t}\right)\end{array}$
Rate of decay at this instant $R=\lambda N=\alpha \left(1-e^{-\lambda t}\right)$
Hence, rate of release of energy at this time $=R$ (energy released in each decay)
$=\alpha \left(1-e^{-\lambda t}\right)(200)MeV/s$
Substituting the values, we have $\text{ rate of release of energy }=1000\left(1-e^{-\frac{0.693}{1620}\times 3240}\right)(200)$
$=1.5\times 10^{5}MeV/s$