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

If `N_(0)` potanssium atoms were present at the time the rock was formed by solidificatio from a molten form, the number of potassium atoms remaining at the time of analysis is,
`N_(K) = N_(0)e^(-lambda t)` ..(i)
in which `t` is the age of the rock.
For every potassium atom that decays, an argon atom is produced. Thus, the number of argon atoms present at the line o fthe analysis is
`N_(Ar) = N_(0) - N_(K)` (ii)
We cannot measure `N_(0)`, so let's eliminate it from Eqs.(i) and (ii). we find, after some algebra, that
`lambdat = In (1+ (N_(Ar))/(N_(K)))`
in which `N_(Ar)//N_(K)` can be measured. Sloving fro `t`
`t = (T_(1//2)In(1+N_(Ar)//N_(K)))/(In2)`
`= ((1.25xx10^(9))[In(1+10.3)])/(In2)`
`= 4.37 xx 10^(9)` years