For a substance the average life for `alpha-`emission is `1620` years and for `beta-`emission is `405` years. After how much time the `1//4` of the material remains after `alpha` and `beta` emission ?

Options

A. 1500 years
B. 300 years
C. 449 years
D. 810 years

Answer

C

Explanation

( c) `lamda_alpha = (1)/(1620)` per year and `lamda_beta = (1)/(405)` per year and it is given that the fraction of the remained activity
`(A)/(A_0) = (1)/(4)`
Total decay constant
`lamda = lamda_alpha +lamda_beta = (1)/(1620) + (1)/(405) = (1)/(324)` per year
We know that `A = A_0 e^(- lamda t) rArr t = (1)/(lamda) log_e (A_0)/(A)`
`rArr t = (1)/(lamda) log_e 4 - (2)/(lamda) log_e 2 = 324 xx 2 xx 0.693`
=`449 years`.