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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. - Physics

Question

Answer in Brief

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 N0 be the initial number of nuclei,

Let λ be the decay constant

Let t1/2 be the half-life

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

Where Ais 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`

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