Question

The relation between half life ($T$) and decay constant ($\lambda$) is

• A. $\lambda T=1$
• B. $\lambda T=\frac{1}{2}$
• C. $\lambda T=lo{g}_{e}2$
• D. $\lambda =log2T$

Answer 1

Answer: (C)

$\lambda T=lo{g}_{e}2$

As per radiaoactive decay law, the radioactive decays per unit time are directly proportional to the total number of nuclei of radioactive compounds in the sample.
f the number of nuclei in a sample is N and the number of radioactive decays per unit time dt is dN then,
$$\frac{dN}{dt}=-\lambda N$$
Integrating both limits
$${\int }_{N_0}^N\frac{dN}{N}=–\lambda \underset{t_0}{\overset{t}{\int }}dt$$
Taking initial time $t_0$=0 we have
$$ln\frac{N}{N_0}=-\lambda t$$
By definition of half life ( time period in which the number of nuclei, N, decayed by half value) i.e. $N_t$=$\frac{N_0}{2}$
$\therefore$
$ln2=\lambda t={\log }_{e}2$