Question

State radioactive decay law. Prove that radioactive decay is exponential in nature.

Answer 1

Radioactive decay law states that the number of nuclei undergoing the decay per unit time is proportional to the total number of nuclei in the sample.

$Radioactive$ $decay$ $law:$ $N=N_0{e}^{-\lambda t}$

$PROVE:$

$\frac{dN}{dt}=-\lambda N$

$\int \frac{dN}{N}=-\lambda \int dt$

$lo{g}_{e}N=-\lambda t+C$

initial number of nuclei$=N_0$

$C=lo{g}_e{N}_0$

$lo{g}_e\left(\frac{N}{N_0}\right)=-\lambda t$

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