Question

Consider the situation of the previous problem. Suppose the production of the radioactive isotope starts at t = 0. Find the number of active nuclei at time t.

Answer 1

Let the number of atoms present at t = 0 be N₀.
Let N be the number of radio-active isotopes present at time t.
Then,
N = N₀e^−λt
Here, $\lambda$ = Disintegration constant
∴ Number of radioactive isotopes decayed = N₀ − N = N₀ − N₀e^−λt
= N₀ (1−e^−λt) ...(1)
Rate of decay $\left(R\right)$ is given by
R = λN₀ ...(2)
Substituting the value of N₀ from equation (2) to equation (1), we get
$$\begin{gathered} N=N_0(1-e^{-\mathrm{\lambda }t}) \\ =\frac{R}{\lambda }(1-e^{-\mathrm{\lambda }t}) \end{gathered}$$