Question

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?

Answer 1

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
`t`

`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

`t` is the time elapsed

On the other hand,
`\lambda` has a relation with the half life
`h` 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
`m` of the sample and the decay rate
`d`:

`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
`t` 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