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प्रश्न
Obtain an expression for the half-lifetime of radioactive material. Hence state the relation between an average life and half life time of radioactive material.
Derive an expression for ‘Half Life Time’ of a radioactive material using the ‘Law of Radioactive Decay’.
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उत्तर
Expression for half-life period (T1/2):
From the law of radioactive decay,
N = N0 e⁻λT ...(i)
where,
N → is the number of nuclei present at any instant ‘t’
N0 → is the number of parent atoms at time ‘t’ = 0
λ → is Decay constant.
Substituting N = `N_0/2` and t = T1/2 ....(where T is half life period) in equation (i),
`"N"_0/2 = "N"_0 e^(-λT)`
`1/2 = e^(-λT)`
∴ 2 = eλT
Taking log on both sides,
∴ loge 2 = λT1/2
∴ λT1/2 = loge2 = 2.303 log102
∴ λT1/2 = 2.303 × 0.3010
∴ T1/2 = `0.693/λ`
This is the required relation for the half-life in terms of the decay constant of a radioactive element.
Relation between average life (τ) and a half lifetime of radioactive material:
T1/2 = τ ln 2 = 0.693 τ
Notes
Students should refer to the answer according to the question.
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