Question

Carbon dating of small bits of charcoal used in cave paintings has determined that some of the paintings are from 10000 to 30000 y old. Carbon-14 has a half-life of 5730 y. In a 1μg-sample of carbon from a live tree, the activity of carbon-14 is 6.4μCi. If researchers determine that 1μg of charcoal from a prehistoric cave painting in France has an activity of 0.80 μCi, what is the age of the painting?

Answer 1

The age of painting was determined from the decay kinetics of the radioactive Carbon -14 present in the painting sample.

Given that the half life of Carbon-14 is 5730 years.

Radioactive decay reactions follow first order rate kinetics.

Calculating the decay constant from half life:

λ$= \frac{0.693}{t_{1/2} }$

        = $\frac{0.693}{5730 yr}$ = $1.21*10^{-4}yr^{-1}$

Setting up the radioactive rate equation:

$ln\frac{A_{t} }{A_{0} } =-kt$

Where $A_{t} = Activity after time t = 0.80microCi$

$A_{t} = initial activity = 6.4microCi$

k = decay constant = $1.21*10^{-4}yr^{-1}$

$ln\frac{0.80uCi}{6.4uCi} =-(1.21*10^{-4}yr^{-1})t$

ln 0.125 = $-(1.21*10^{-4}yr^{-1})t$

-2.079=$-(1.21*10^{-4}yr^{-1})t$

t=$\frac{2.07944}{1.21*10^{-4} } yr$

 = 17185 years

t = 17185 years

Therefore age of the painting based in the radiocarbon -14 dating studies is 17185 years