Question

Recently, James found an oddly-colored rock in his backyard and surmised that it came from space. James sent the rock off for elemental analysis to confirm his theory. Isotopic analysis via mass spectrometry found appreciable

amounts of (26)^Al and (26)^Mg in the sample (275 μg and 81.3 μg, respectively). These isotopes are formed only

when cosmic species are bombarded with solar rays to form 26Al, which then radioactively decays into 26Mg, with t1/2 = 7.17 × 105 years. Given the above amounts of each isotope, determine the earthly age of James’ rock. (Hint: assume that 26Al production ceases upon entering the atmosphere; you may use multiple online references to solve this issue)

Answer 1

Ans. Given, present content in sample: [²⁶Al] = 275.0 ug           , ²⁶Mg =81.3 ug

Since, all Mg is derived from Al, the total initial Al in the sample must be equal to the summed of current amount of ²⁶Al and ²⁶Mg in the sample.

So, total INITIAL [²⁶Al] = 275.0 ug + 81.3 ug = 356.3 ug

And, Final [²⁶Al] remaining in the sample = 275.0 ug

Radioactive decay follows first order kinetics.

First order kinetics-

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

where, [A]_t = Final amount of isotope

            [A]₀ = Initial amount of isotope

            k = rate constant

            t = time of decay

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                - equation 3

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

# Putting the values in equation 3-

            ln (275.0 ug / 356.0 ug)= - (0.693 / 7.17 x 10⁵ yr) x t

            Or, -0.2590 = -(9.6653 x 10^-7 yr^-1) t

            Or, t = 0.2590 / 9.6653 x 10^-7 yr^-1 = 2.68 x 10⁵ yr

Hence, age of James’ rock = 2.68 x 10⁵ yr