Decay Constant, Half Life and Mean Life

Decay constant: We know that the rate of decay is directly proportional to number of radioactive atom.

$$\frac{N}{N_o} = e^{-\lambda t}$$
$$\frac{dN}{dt} = -\lambda N$$

Hence, decay constant is the ratio of decay to the number of radioactive sample available initially.

Half life: The time taken by a radioactive substance to decay by half amount s called half life of the substance.

From the law of radioactive disintegration, we have:
$$\frac{dN}{dt} = -\lambda N$$
$$or, \frac{N}{N_o} = e^{-\lambda t}$$
$$\text{At time} \text{T} = t_{\frac{1}{2}} and N = \frac{N_o}{2}$$
$$or, \frac{N_o}{2} = N_o e^{\lambda t 1/2}$$
$$or, \frac{1}{2} = e^{\lambda t 1/2}$$
$$or, e^{\lambda t 1/2} = log_e2$$
$$or, t_{1/2} = \frac{loge2}{\lambda}$$
$$\therefore t_{1/2} = \frac{0.693}{\lambda}$$

This is the relation between half life and decay constant of a radioactive substance.

Mean life: It is the ratio of sum fo life oa all the atoms to total number of atoms. It is the reciprocal of decay constant.
$$i.e., \text{Mean Life} = \frac{\text{Sum of life of all the atoms}}{\text{Total number of atoms}}$$

From half life equation, we have:
$$t_{1/2} = \frac{0.693}{\lambda}$$
$$or, t_{1/2} = \frac{1}{\lambda} \times 0.693 $$
$$or, t_{1/2} = \text{Mean Life} \times 0.693$$

From this we conclude that half life of a radioactive substance is always less than its mean life and the number of active nuclei in radioactive sample decrease with time. The graphical representation of radioactive decay with time is given below:

Graphical representation of radioactive decay with time

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