Radiocarbon dating. A sample from timbers at an archeological site containing 500 g of carbon provides 3070 decays/min. What is the age of the sample?

Question

asked Oct 16, 2020 49.5k views

1 Answer

KKL Michael · Answer 1 · 2020-10-22T14:44:00+0000

Answer: 7546.76 years

Step-by-step explanation:

This can be solved by the following equation:

$N_(t)=N_(o)e^(-\lambda t)$ (1)

Where:

N_(t) is the number of atoms of carbon-14 left after time

N_(o)=255 Bq/kg is the defined atmospheric carbon-14 (the number of atoms of C-14 in the original sample)

$\lambda$ is the rate constant for carbon-14 radioactive decay

is the time elapsed

On the other hand,
$\lambda$ has a relation with the half life
of the C-14, which is
5730 years :

$\lambda=(ln(2))/(h)=(ln(2))/(5730 years)=1.21(10)^(-4) years^(-1)$ (2)

In addition, we can calculate the value of
N_(t) knowing the mass
of the sample and the decay rate
:

m=500 g (1 kg)/(1000 g)=0.5 kg

d=3070 (decays)/(min) (1 min)/(60 s)=51.16 (decays)/(s)=51.16 Bq

Then:

N_(t)=(d)/(m)=(51.16 Bq)/(0.5 kg)=102.32 Bq/kg (3)

Now, we have to find the age of the sample
from (1):

$t=ln((N_(t))/(N_(o)))(-(1)/(\lambda))$ (4)

Substituting (2) and (3) in (4):

t=ln((102.32 Bq/kg)/(255 Bq/kg))(-(1)/(1.21(10)^(-4) years^(-1))) (4)

Finally:

t=7546.76 years This is the age of the sample