The radium and uranium atoms in a sample of uranium mineral are in the ratio of `12.8xx10^(6)`. If half life period of radium is 1620 years , the half life period of uranium will be

Options

A. `45.3 xx 10^9` years
B. `45.3 xx 10^(10)` years
C. `4.53 xx 10^9` years
D. `4.53 xx 10^6` years

Answer

C

Explanation

According to radioactive equilibrium `lambda_A N_A = lambda_A N_B`
or `(0.693 xx N_A)/(t_(1//2)(A)) = (0.693 xx N_B)/(t_(1//2) (B)) [lambda = (0.693)/(t_(1//2))]`
Where `t_(1//2) (A)` and `t_(1//2) (B)` are half periods of A and B respectively
`therefore (N_A)/(t_(1//2) (A)) =(N_B)/(t_(1//2) (B)) ` or `(N_A)/(N_B) = (t_(1/2) (A))/(t_(1//2)(B))`
`therefore` At equilibrium A and B are present in the ratio of their half lives `1/(2.8xx10^6) = (1620)/("Half-life of Uranium")`
`therefore` Half -life of uranium
`= 2.8xx10^6 xx 1620 = 4.53 xx 10^9` years