The rate of disintegration of a radioactive element at any time t is proportional to its mass at that time. Find the time during which the original mass of 1.5 g m will disintegrate into its mass of 0.5 gm.

Explanation

Let m be the mass of the radiocative element at time t.
Then the rate of disintegration is `(dn)/dt ` which is proportional to m.
` (dm)/(dt) prop m`
` (dm)/(dt) = - km. " where " k gt 0`
`(dm)/(m) - k dt `
On integrating we get ,
`int 1/m dm = k int dt +c `
`log m = Kt = c`
Initially i.e., when t =0 , m =1 .5
`log (1.5) - k xx 0 +c " " therefore c= log (3/2)`
`log m = kt + log (3/2) " " log m - log 3/2 = -kt `
when m = 0.5 `1/2 ` then
` log ((2 xx 1/2)/3) = - kt therefore log (1/3) = -kt `
` log (3)^(-1) = - kt " " therefore - log 3 =-kt " " therefore t = 1/k log 3`
The original mass will disintegrate to 0.5 gm when ` t = 1/k log 3`