Question 28.11: Age of bone A bone found by an archeologist contains a small......
Age of bone
A bone found by an archeologist contains a small amount of radioactive carbon-14. The radioactive emissions from the bone produce a measured decay rate of 3.3 decays/s. The same mass of fresh cow bone produces 30.8 decays/s. Estimate the age of the sample.
Sketch and translate The situation is sketched below.
Simplify and diagram We assume that the concentration of radioactive carbon-14 in the atmosphere is the same today as it was at the time of death of the animal being studied.
Represent mathematically The decay rate and number of radioactive nuclei are proportional to each other (ΔN/Δt = -λN). Therefore, we can determine the ratio of carbon-14 in the old bone and what it was when the animal died.
\frac{(\Delta N / \Delta t)_{\text {now }}}{(\Delta N / \Delta t)_{\text {at death }}}=\frac{-\lambda N_{\text {now }}}{-\lambda N_{\text {at death }}}=\frac{N_{\text {now }}}{N_{\text {at death }}}=\frac{N}{N_0}
From here we can use Eq. (28.11) to determine the age of the bone.
t=-\frac{\ln \left(N / N_0\right)}{0.693} T (28.11)
Solve and evaluate The fraction of carbon-14 atoms remaining in the old bone is
\frac{N}{N_0}=\frac{3.3}{30.8}=0.107
Substituting this value into Eq. (28.11), we find the age of the bone:
t=-\frac{\ln (0.107)}{0.693}(3 \text { years })=18,500 \text { years }
The unit is correct and the order of magnitude is reasonable.
Try it yourself: The ratio of carbon-14 in an old bone compared to the number in a fresh bone of the same mass is 0.050. Determine the age of the bone.
Answer: 25,000 years.