The half-life of carbon-14 is 5600 years. If a piece of charcoal made from the wood of a tree shows only 69% of the carbon-14 expected in living matter, when did the tree die?

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Answer 1 · 2018-04-10T00:58:31

Recall that radioactive decay follows first order kinetics,

#ln[A]_"t" = -kt + ln[A]_0#,

and by extension,

#t_(1/2) = ln(2)/k#

Let's derive #k#,

#=> k = ln(2)/t_(1/2) approx 1.24*10^-4"yr"^-1#

Moreover, let's rearrange our kinetics equation, noting,

#69% = 0.69/1#

Hence, the tree died approximately,

#ln(([A]_"t")/([A]_0)) = -kt#

#=> t = (ln(([A]_"t")/([A]_0)))/(-k) approx 3.0*10^3"yr"# ago.