#### Question

The decay constant of ^{238}U is 4.9 × 10^{−18} S^{−1}. (a) What is the average-life of ^{238}U? (b) What is the half-life of ^{238}U? (c) By what factor does the activity of a ^{238}U sample decrease in 9 × 10^{9} years?

#### Solution

Given:

Decay constant, `lambda = 4.9 xx 10^-18 "s"^-1`

(a) Average life of uranium (`tau`) is given by

`tau = 1/lambda`

= `1/(4.9 xx 10^-18)`

= `1/4.9 xx 10^18 "s"`

= `10^16/(4.9 xx 365 xx 24 xx 36) "years"`

= `10^16/(4.9 xx 365 xx 24 xx 36) "years"`

= `6.47 xx 10^-7 xx 10^16 "years"`

= `6.47 xx 10^9 "years"`

(b) Half-life of uranium (`T_"1/2"`) is given by

`T_"1/2" = 0.693/lambda = 0.693/(4.9 xx 10^-18)`

= `0.693/4.9 xx 10^18 "s"`

= `0.1414 xx 10^18 "s"`

= `(0.1414 xx 10^18)/(365 xx 24 xx 3600)`

= `(1414 xx 10^12)/(365 xx 24 xx 36)`

= `4.48 xx 10^-3 xx 10^12`

= `4.5 xx 10^9` years

(c) Time, t = 9 × 10^{9} years

Activity (A) of the sample, at any time t, is given by

`A = A_0/2^(t/T_"1/2")`

Here , `A_0` = Activity of the sample at t = 0

`therefore A_0/A = 2^((9 xx 10^9)/(4.5 xx 10^9)) = 2^2 = 4`