Question

State law of radioactive disintegration.derive relation betweenhalf life and decay constant ?

Answer 1

`lambda = (ln 2)/tau` or `tau = (ln 2)/lambda`

Explanation:

Law Of Radioactive Disintegration states, broadly, that the rate of radioactive material disintegration, `(dN)/(dt)`, is proportional to the number of atoms, `N`, present at that instant, or:

`(dN)/(dt) = - lambda N qquad triangle`, where `lambda` is the disintegration or decay constant

If we were to look at half-lives instead, we would say that for a half-life `tau` and initial amount `N_o` of atoms:

`N = N_o(1/2)^( t/tau) qquad square`

So:

• at `t = tau`, `N = 1/2 N_o`

• at `t = 2 tau`, `N = N_o * (1/2)^2 = 1/4 N_o`,

• and so on

We can now differentiate `square` and compare it to `triangle`:

`N = N_o(1/2)^( t/tau) implies ln N = ln N_o - t/tau ln 2`

By implicit differentiation:

`1/N (dN)/(dt ) = 0 - 1/tau ln 2`

Or:

`(dN)/(dt ) = - (ln 2)/tau \ N`

So we see that:

• `lambda = (ln 2)/tau` or `tau = (ln 2)/lambda`