Question

Obtain a relation between the half-life of a radioactive substance and decay constant (λ).

Answer 1

At t = T_1/2 (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`