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Question
Obtain a relation between the half-life of a radioactive substance and decay constant (λ).
Solution
At t = T1/2 (half life ), N = `(N_0)/2`
Putting this, into
N = N0e-λt [where λ = decay constant]
We get `1/2 = e^(-lambdaT_(1/2)`
⇒ `1/2 = 1/(e^(lambdaT_(1/2))) ⇒ e^(lambdaT_(1/2)) = 2`
` e^(lambdaT_(1/2)) = 2`
Taking logarithm to both sides,
`lambdaT_(1/2) = "log "e^2`
∴ `T_(1/2) = ("log "e^2)/lambda`
i.e., `T_(1/2) = (2.303 xx "log"10^2)/lambda`
`T_(1/2) = (2.3 xx0.301)/lambda`
∴ `T_(1/2) =0.692/lambda`
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