Mass spectrometic analysis of potassium and argon atoms in a Moon rock sample shows that the ratio of the number of (stable)`.^(40)`At atoms present to the number of (radioactive)`.^(40)K` atoms is `10.3`.Assume that all the argon atoms were produced by the decay of potassium atoms, with a half-life of `1.25 xx 10^(9) yr`. How old is the rock ?

Options

A. `2.95 xx 10^(11) yr`
B. `2.95 xx 10^(9) yr`
C. `4.37 xx 10^(9) yr`
D. `4.37 xx 10^(11)yr`

Answer

C

Explanation

`N_(K)= N_(0)e^(-lamda t)` [`K rarr` potassium]
`N_(Ar)= N_(0)- N_(K)`
`therefore N_(K)= (N_(Ar) + N_(K)) e^(-lamda t)`
`rArr lamda t== ln (1+ (N_(Ar))/(N_(K)))`
`rArr t= (T_(1//2) ln (1+ (N_(Ar))/(N_(K))))/(ln 2) = ((1.25 xx 10^(9)) [ln (1+ 10.3)])/(ln 2)`
`=4.37 xx 10^(9)`yr