Question

Dating using potassium isotopes is one of the most important techniques in geochemistry. Using mass spectrometry, it was found that a certain rock contains 0.45 mg of argon-40 for every milligram of potassium-40. The half life for the decay of potassium-40 to argon-40 is 1.2 x 10⁹ years. How old is the rock?

Ans: 6.4 x 10⁸ years.

Answer 1

Ans. Radioactive decay follows first order kinetics.

First order kinetics-

ln ([A]_t / [A]₀) = -kt                                -equation 1

where, [A]_t = Final amount

            [A]₀ = Initial amount

            k = rate constant

            t = time of reaction

Also, Half-life of first order reaction, t_½ = 0.693 / k

            Or, k = 0.693 / t_½                                        - equation 2

Combining equation 1 and 2 –

            ln ([A]_t / [A]₀) = -(0.693 / t_½) t               

            or, ln [A]_t – ln [A]₀ = -(0.693 / t_½) t                    - equation 3

Given, there is 0.45 mg Argon-40 every 1.00 mg K-40.

Total initial amount of K-40, [A]₀ = Amount of Ar-40 formed + Amount of K-40 remaining

= 0.45 mg Ar-40 + 1.00 mg K40

= 1.45 mg

Amount of K-40 remaining in the sample = 1.00 mg

Given, t_½ = 1.2 x 10⁹ yr

Putting the values in equation 3-

            ln 1 – ln (1.45) = -(0.693 / 1.2 x 10⁹ yr ) x t

            or, 0 - 0.3716 = (- 5.775 x 10^-10 yr^-1) t

            or, (– 0.3716) / (- 5.775 x 10^-10 yr^-1) = t

            or, t = 6.43 x 10⁸ yr

Hence, the rock is 6.43 x 10⁸ years old.