Question

Bones found in the Lascaux Caves in France have a carbon-14 disintegration rate of 2.25 disintegrations per minute per gram of carbon. If the rate of decay rate of carbon-14 at the time of death is 13.6 counts per minute per gram of carbon, calculate the age of the remnants. The half-life of carbon-14 is 5730 years.

Answer 1

Answer:

14873 years

Step-by-step explanation:

The exponential decay of the ¹⁴C can be expressed as follows:

$N_{t} = N_{0}e^{(-\lambda t)}$              (1)

where Nt: is the disintegration per minute per gram of carbon that has not yet decayed after a time t = 2.25, N₀: is the initial disintegration per minute per gram of carbon = 13.6 (at the time of death), λ: is the decay constant and t: is time.  

The half-life $t_{1/2}$ of carbon-14 is related to the decay constant by the following equation:

$t_{1/2} = \frac{ln(2)}{\lambda}$              

$\lambda = \frac{ln(2)}{t_{1/2}}$              (2)

Hence, by entering equation (2) into (1) we have:  

$N_{t} = N_{0}e^{(-\frac{ln(2)}{t_{1/2}} t)}$      (3)

Now, by solving equation (3) for t, we can find the age of the remnants:

$ln(\frac{N_{t}}{N_{0}}) = -\frac{ln(2)}{t_{1/2}} t$    

$t = -\frac{ln(\frac{N_{t}}{N_{0}})}{ln(2)} t_{1/2}$          

$t = -\frac{ln(2.25/13.6)}{ln(2)} (5730 y)$      

$t = 14873 y$                              

Therefore, the age of the remnants is 14873 years.

I hope it helps you!