(a) Derive the mathematical expression for law of radioactive decay for a sample of a radioactive nucleus. (b) How is the mean life of a given radioactive nucleus related to the decay constant?
Medium
Open in App
Updated on : 2022-09-05
Solution
Verified by Toppr
a) According to radioactive decay law, the rate of decay is proportional to amount of nuclei ,N. Thus, −dtdN∝N (negative sign come because the amount of nuclei decreases with time) So, dtdN=−λN or NdN=−λt Integrating, lnN=−λt+c where C is the integrating constant. At t=0, N=N0 so lnN0=c Now, lnN=−λt+lnN0 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=λ1