Answer to Question #174539 in Atomic and Nuclear Physics for Charlotte

Question

An ancient beam of wood is discovered during an archiological dig and carbon dated. They find that the sample contains 12.5% of the count rate of a similar mass of living wood. The half-life of the carbon isotope is 5600 years. Calculate the age of the ancient wood.

Answer 1

Let’s use the formula for radioactive decay:

$$N=N_0e^{-\lambda t},$$ $$\dfrac{N}{N_0}=e^{-\lambda t},$$ $$ln(\dfrac{N}{N_0})=ln(e^{-\lambda t}),$$ $$ln(\dfrac{N}{N_0})=-\dfrac{0.693}{T_{1/2}}t,$$ $$t=\dfrac{ln(\dfrac{N}{N_0})}{(-0.693)}T_{1/2},$$ $$t=\dfrac{ln(\dfrac{12.5\%}{100\%})}{(-0.693)}\cdot5600\ y=16803\ y.$$