#### Question

Define the term 'decay constant' of a radioactive sample. The rate of disintegration of a given radioactive nucleus is 10000 disintegrations/s and 5,000 disintegrations/s after 20 hr. and 30 hr. respectively from start. Calculate the half-life and the initial number of nuclei at t= 0.

#### Solution

The decay constant is the fraction of the number of atoms that decay in one second.

It is denoted by A.

Let N_{0} be the initial number of nuclei,

Let λ be the decay constant

Let t_{1/2} be the half-life

The instantaneous activity of radioactive material is given by `A = A_0e^(-lambdat)`

Where A_{0 }is activity at t = 0

Therefore, after 20 hours is 10,000 disintegrations per second

`10,000 = A_0e^(-lambda(20 xx 3600))` ...(1)

Activity after 30 hours is 5,000 disintegrations per second

`5000 = A_0e^(-lambda(30 xx 3600))` ..(2)

On dividing (1) by (2),

`2 = e^(lambda xx 3600)`

⇒ `lambda = ln 2/36000 = 1.92 xx 10^-5`

And half life is,

`ln 2/(1.92 xx 10^-5) = 36,000"s" = 10` hours

Since,

`(dN)/(dt) = lambdaN`

`1000 = (1.92 xx 10^-5) xx N_1`

⇒ `N_1 = (10,000)/(1.92 xx 10^-5) = 5.208 xx 10^8`

Therefore, the half life is 10 hours, thus the initial number of nuclei is `N_0 = 10.416 xx 10^8`