When charcoal is prepared from a living tree, it shows a disintegration rate of 15.3 disintegrations of 14C per gram per minute. A sample from an ancient piece of charcoal shows 14C activity to be 12.3 disintegrations per gram per minute. How old is this sample? Half-life of 14C is 5730 y.
Initial activity of charcoal, A0 = 15.3 disintegrations per gram per minute
Half-life of charcoal, `T_"1/2"` = 5730 years
Final activity of charcoal after a few years, A = 12.3 disintegrations per gram per minute
Disintegration constant, `lambda = 0.693/T_"1/2" = 0.693/5370 "y"^-1`
Let the sample take a time of t years for the activity to reach 12.3 disintegrations per gram per minute.
Activity of the sample, `A = A_0e^(-lambdat)`
`A = A_0e^(-0.693/5730 xx t)`
⇒`"In" 12.3/15.3 = (-0.693)/5730 t`
⇒ `0.218253 = 0.693/5730 t`
⇒ t = 1804.3 years
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