Question

a rock returned from the surface of the moon has a ratio of Rb⁸⁷ to stable Sr⁸⁷ atomos equal to 14.45 assuming that this was all Rb⁸⁷ upon the formation of the solar system, estimate the age of the solar system.

ans key:

t= 4.537*10^{9 yr}

Answer 1

Ans. Using the formula: ln (A₀ / A_t) = kt            - equation 1 (equation 1 rearranged)

            Where,             

A_o = Initial amount

                        A_t = Amount remaining at time t

                        k = decay constant

                        t = time duration of radioactivity

Given, Radioactive ⁸⁷Rb decays into stable ⁸⁷Sr

            [⁸⁷Rb] : [⁸⁷Sr] = 14.45 : 1

Since, ⁸⁷Rb decays into ⁸⁷Sr, all ⁸⁷Sr is derived from ⁸⁷Rb. One ⁸⁷Sr nucleus decays into one ⁸⁷Rb nucleus. So, Total initial amount of ⁸⁷Rb is equal to present [⁸⁷Rb] + [⁸⁷Sr]

Or, total initial [⁸⁷Rb] = [⁸⁷Rb] + [⁸⁷Sr] = 14.45 + 1 = 15.45 -at present    ; given, [⁸⁷Rb] : [⁸⁷Sr] = 1: 14.45

Or, total initial [⁸⁷Rb] = 15.45

Thus, [A₀] = 15.45

Final (at present) [⁸⁷Rb] = Initial amount – amount decayed into ⁷⁸Sr

                                    = Initial amount – present amount ⁷⁸Sr

                                    = 15.45 – 1.0 = 14.45

Thus, [A_t] = 14.45

Known: Decay constant, k for ⁸⁷Rb = 1.42 × 10^-11 yr^-1

Putting the values in equation 1-

            ln (15.45 / 14.45) = (1.42 × 10^-11 yr^-1) x t

or, 0.066914588785 = (1.42 × 10^-11 yr^-1) x t

or, t = (0.066914588785) / (1.42 × 10^-11 yr^-1) = 0.0471 x 10¹¹ y

Hence, age of the solar system= 4.71 x 10⁹ years.

Note: The variation might be due to decays constant value. Please cross-check the vale in your book.