Question

Which sample, $A$ or $B$ shown in figure has shorter mean-life?

Answer 1

According to the Radioactive Decay Law or Rutherford Soddy Law the decay rate of a nucleus is given by the formula of,

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

$\frac{dN}{dt}$ is the rate of decay of the nucleus with respect to time,

$\lambda$ is the decay rate constant of the nucleus

$N$ is the number of nuclei present in the sample

Initially at $t=0$ from figure given,

${\left(\frac{d{N}_0}{dt}\right)}_A={\left(\frac{d{N}_0}{dt}\right)}_B$

So $(N_0{)}_A=(N_0{)}_B$

Slope of the graph is given by the decay constant $\lambda$.

From the graph,

${\lambda }_A<{\lambda }_B$

Since $\tau =\frac{1}{\lambda }$

${\tau }_A>{\tau }_B$

So mean lifetime of sample $A$ is greater than of $B$

Final Answer: $B$ has shorter mean life as $\lambda$ is greater for $B$.