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2 March, 03:34

The half life for the decay of carbon - is years. Suppose the activity due to the radioactive decay of the carbon - in a tiny sample of an artifact made of wood from an archeological dig is measured to be. The activity in a similar-sized sample of fresh wood is measured to be. Calculate the age of the artifact.

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  1. 2 March, 04:07
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    The artifact is 570 years old. That is, 5.7 * 10² years.

    Explanation:

    Radioactive decay follows first order reaction kinetics.

    Let the initial activity for fresh Carbon-14 be A₀

    And the activity at any other time be A

    The rate of radioactive decay is given by

    dA/dt = - KA

    dA/A = - kdt

    Integrating the left hand side from A₀ to A₀/2 and the right hand side from 0 to t (1/2) (where t (1/2) is the radioactive isotope's half life)

    In [ (A₀/2) / A₀] = - k t (1/2)

    In (1/2) = - k t (1/2)

    - In 2 = - k t (1/2)

    k = (In 2) / t₍₁,₂₎

    t (1/2) is given in the question to be 5.73 * 10³ years

    k = (In 2) / 5730 = 0.000121 / year

    dA/A = - kdt

    Integrating the left hand side from A₀ to A and the right hand side from 0 to t

    In (A/A₀) = - kt

    A/A₀ = e⁻ᵏᵗ

    A = A₀ e⁻ᵏᵗ

    A = 2.8 * 10³ Bq.

    A₀ = 3.0 * 10³ Bq.

    2.8 * 10³ = 3.0 * 10³ e⁻ᵏᵗ

    0.9333 = e⁻ᵏᵗ

    e⁻ᵏᵗ = 0.9333

    -kt = In 0.9333

    - kt = - 0.06899

    t = 0.06899/0.000121 = 570.2 years = 5.7 * 10² years
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