Charcoal from the dwelling level of the Lascaux Cave in France gives an average count of 0.97 disintegrations of ^14 C per minute per gram of sample. Living wood gives 6.68 disintegrations per minute per gram. Estimate the date of occupation and hence the probable date of the wall painting in the Lascaux Cave. Hint: Disintegrations per minute per gram" has the same units as the time-derivative of concentration for a radioactive decay model. (You may use the fact that the half-life of ^14C is 5568 years.)

Explanation:

count given by old sample = .97 disintegrations per minute per gram

count given by fresh sample = 6.68 disintegrations per minute per gram

Half life of radioactive carbon = 5568 years

rate of disintegration

dN / dt = λ N

In other words rate of disintegration is proportional to no of radioactive atoms present . As number reduces rate also reduces .

Let initial no of radioactive be N₀ and after time t , number reduces to N

N₀ / N = 6.68 / .97

Now

λ is disintegration constant

λ = .693 / half life

= .693 / 5568

= .00012446 year⁻¹

Putting the values in the equation above

1.929577 = .00012446 t

t = 15503.6 years .