Question

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 gm will disintegrate into its mass of 0.5 gm.

Answer 1

Let m be the mass of the radioactive element at time t.

Then the rate of disintegration is `"dm"/"dt"` which is proportional to m.

∴ `"dm"/"dt" prop "m"`

∴ `"dm"/"dt"` = - km, where k > 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 × 0 + c      ∴ c = log`(3/2)`

∴ log m = - kt + log`(3/2)`

∴ log m - log`3/2` = - kt

∴ `log("2m"/3)` = - kt

When  m = 0.5 = `1/2`, then

`log ((2 xx 1/2)/3) = - "kt"`

∴ `log (1/3)` = - kt

∴ log(3)^-1 = - kt

∴ - log 3 = - kt

∴ t = `1/"k" log 3`

∴ the original mass will disintegrate to 0.5 gm when t = `1/"k" log 3`