Question

Radioactive decay of 40K atoms in an igneous rock has resulted in a ratio of 25 percent 40K atoms to 75 percent 40AR and 40CA atoms how many years old is the this rock

Answer 1

The answer is 6×10⁹ years.

K-Ar dating is radioactive dating method used to measure a time elapsed since the rock is cool enough to trap Ar after a radioactive decay of 40K to 40Ar. It can be expressed as:
$t= \frac{ t_{ \frac{1}{2} } }{ln(2)} ln( \frac{ K_{f} + \frac{ Ar_{f} }{0.109} }{ K_{f} } )$
where:
$t$ - elapsed time
$t_{ \frac{1}{2} }$ - half-life of K40
$K_{f}$ - amount of K40
$Ar_{f}$ - amount of Ar40

We know that a half-life of K40 is 1.251×10⁹ years:
$t_{ \frac{1}{2} } =1.251* 10^{9}$

We do not know absolute value of amount of K40 and Ar40, but we know percentage and can express them as following:
$K_{f} =25%=0.25$
$Ar_{f} =75%=0.75$

So:
$t= \frac{1.251* 10^{9} }{ln(2)} ln( \frac{0.25+ \frac{0.75}{0.109} }{0.25} )$

⇒ t = 6×10⁹

Thus, the rock is old 6×10⁹ years.