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.

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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