a) According to radioactive decay law, the rate of decay is proportional to amount of nuclei ,N. Thus, −dNdt∝N (negative sign come because the amount of nuclei decreases with time)
So, dNdt=−λN or dNN=−λt
Integrating, lnN=−λt+c where C is the integrating constant.
At t=0, N=N0 so lnN0=c
or ln(N/N0)=−λt or N/N0=e−λt
Hence, N=N0e−λt, this is the mathematical expression of radioactive decay of nuclei.
b) The relation between mean life (tavg) and decay constant (λ) : tavg=1λ