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Question
4 × 1023 tritium atoms are contained in a vessel. The half-life of decay tritium nuclei is 12.3 y. Find (a) the activity of the sample, (b) the number of decay in the next 10 hours (c) the number of decays in the next 6.15 y.
Solution
Given:
Number of tritium atoms, N0 = 4 × 1023
Half-life of tritium nuclei, `T_"1/2"`= 12.3 years
Disintegration constant, `lambda = 0.693/T_"1/2" = 0.693/12.3 "years"^-1`
Activity of the sample (A)is given by
`A_0 = "dN"/"dt" = lambdaN_0`
⇒ `A_0 = 0.693/t_"1/2"N_0`
= `0.693/12.3 xx 4 xx 10^23` disintegration/year
= `(0.693 xx 4 xx 10^23)/(12.3 xx 3600 xx 24 xx 365)` disintegration/sec
= `7.146 xx 10^14` disintegration/sec
(b) Activity of the sample, A = 7.146 `xx` 1014 disintegration/sec
Number of decays in the next 10 hours= `7.146 xx 10^14 xx 10 xx 3600`
= `257.256 xx 10^17`
= `2.57 xx 10^19`
(c) Number of atoms left undecayed, N = `N_0e^(-lambdat)`
Here, N0 = Initial number of atoms
`therefore N = 4 xx 10^23 xx e ^((-0.693)/12.3 xx 6.15) = 2.83 xx 10^23`
Number of atoms disintegrated = `(N_0 - N) = (4 - 2.83) xx 10^23 = 1.17 xx 10^23`
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