Question

If ${}^{\prime }{f}^{\prime }$ denotes the ratio of the number of nuclei decayed $\left(N_d\right)$ to the number of nuclei at $t=0\left(N_0\right)$ then for a collection of radioactive nuclei, the rate of change of ${}^{\prime }{f}^{\prime }$ with respect to time is given as :
[$\lambda$ is the radioactive decay constant]

• A. $-\lambda e^{-\lambda t}$
• B. $-\lambda (1-e^{-\lambda t})$
• C. $\lambda e^{-\lambda t}$
• D. $\lambda (1-e^{-\lambda t})$

Answer 1

The correct option is C $\lambda e^{-\lambda t}$

According to law of radioactive decay, the amount of substance left after time $t$ is given by,

$N=N_0{e}^{-\lambda t}$

Since the number of nuclei decayed is given by,

$N_d=N_0-N$

Plugging the value of $N$ in the above equation,

$N_d=N_0(1-e^{-\lambda t})$

$\frac{N_d}{N_0}=f=1-e^{-\lambda t}$

Differentiating wrt $t$ we get the required rate as,

$\frac{df}{dt}=\lambda e^{-\lambda t}$

Hence, option $\left(\text{C}\right)$ is correct.