Half-Life Period and Average Life Period:
It is defined as “the time required for the decay or disintegration of half of the radioactive atoms initially present in a given sample” or “the time during which the concentration of a radioactive substance reduces to one-half of its initial concentration”. It is denoted by t1/2 or t0.5. For Example- the half-life for the radioactive decay of uranium-238 is 4.51 x 109 years and that of radium-226 is 1.62 x 103 years.
Derivation of Expression For Half-Life Period:
We know that rate of radioactive disintegration is expressed as-
|λ = 2.303/t log N0/N|
or t = 2.303/λ log N0/N
At half-life period, t = t1/2 or t0.5 and N = N0/2
∴ t1/2 = 2.303/λ log N0/N0/2 = 2.303/λ log 2 = 2.303/λ X 0.3010 (∵ log 2 = 0.3010)
or t1/2 = 0.693/λ
Since λ is a constant, the time required for the disintegration of one-half of the original amount of radioactive substance is independent of the initial amount of radioactive substance.
Average Life Period:
Since the whole of a radioactive substance never disintegrates, although its actual quantity becomes too small to be measured i.e. the time of complete decay of radioactive element will be infinity. Thus, instead of total life period, the term Average Life Period is used which is defined as “the reciprocal of the decay constant (λ)”. It is denoted by 𝜏 i.e. 𝜏 = 1/λ.
|We know that, Half-life Period, t1/2 = 0.693/λ or λ = 0.693/t1/2|
∴ 𝜏 = 1/0.693/t1/2 = t1/2 X 1/0.693 = 1.44 X t1/2
i.e. 𝜏 = 1.44 X t1/2
∴ Average Life Period = 1.44 X Half Life Period