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Question
Derive the relationship between half-life and decay constant of a radioelement.
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Solution
Equation for the decay constant is given as,
`lambda = 2.303/"t" log_10 "N"_0/"N"` ....(i)
Where, λ = Decay constant
N = Number of nuclei (atoms) present at time t
At t = 0, N = N0.
Hence, at t = t1/2, N = N0/2
Substitution of these values of N and t in equation (i) gives,
`lambda = 2.303/"t"_(1//2) log_10 "N"_0/("N"_0/2)`
`= 2.303/"t"_(1//2) log_10 2 = 2.303/"t"_(1//2) xx 0.3010 = 0.693/"t"_(1//2)`
Hence, `lambda = 0.693/"t"_(1//2) or "t"_(1//2) = 0.693/lambda`
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