Question

Obtain the exponential law of radioactive disintegration. Explain the decay curve.

Answer 1

According to the Law of radioactive disintegration-

Rate of nuclei disintegrating in small time $dt$ is directly proportional to the number of non-disintegrated nuclei present in the sample.

$-\frac{dN}{dT}\propto N$ negative sign since $N$ is decreasing

$\frac{dN}{dT}=-\lambda N$. where $\lambda =$ disintegration constant

$⟹\frac{dN}{N}=-\lambda dt$

$[\ln N{]}_{N_o}^N=-\lambda t$

$⟹N=N_o{e}^{-\lambda t}$

At $t=0,N=N_o$

Theoretically, it will take infinite time for all particles to disintegrate however small is $N_o$.

But in about $\frac{5}{\lambda }$ time, more than $99\%$ of the nuclei get disintegrated.