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
`= (0.693)/(t_(1/2) )`

=
`(0.693)/(5730 yr)` =
`1.21*10^(-4)yr^(-1)`

Setting up the radioactive rate equation:

`ln(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(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=
`(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