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

acid rain does not usually kill trees directly. instead, it is more likely to weaken the trees by damaging their leaves, limiting the nutrients available to them, or poisoning them with toxic substances slowly released from the soil. the main atmospheric pollutants that affect trees are nitrates and sulphates.