A radio isotope X with a half life 1.4 x 10⁹ years decays to Y which is stable

A radio isotope X with a half life 1.4 x 10⁹ years decays to Y which is stable. A sample of the rock from a cave was found to contain X and Y in the ratio 1:7. The age of the rock is

Options

(a) 1.96 x 10⁹ years
(b) 3.92 x 10⁹ years
(c) 4.20 x 10⁹ years
(d) 8.40 x 10⁹ years

Correct Answer:

4.20 x 10⁹ years

Explanation:

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  • X/Y=1/7
    Y=7X
    Let the initial atoms be N.
    Then,
    X+Y=N
    X+7X=N
    8X=N
    X=N/8
    Hence, the final atoms be N/8.
    N/8/N=(1/2)^n
    (1/8)^3=(1/2)^n
    n=3
    Then, t=n×T1/2
    t=3×1.4×10^9

    t=4.2×10^9

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