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Question
The decay constant of 238U is 4.9 × 10−18 S−1. (a) What is the average-life of 238U? (b) What is the half-life of 238U? (c) By what factor does the activity of a 238U sample decrease in 9 × 109 years?
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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 × 109 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`
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