# A wooden artifact from a Chinese temple has a 14C

A wooden artifact from a Chinese temple has a 14C activity of 38.0 counts per minute as compared with an activity of 58.2 counts per minute for a standard of zero age. From the half-life for 14C decay, 5715 yr, determine the age of the artifact.
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Debbie Moore
Half-life of the reaction is defined as the time required by a substance to reach half its initial concentration. It is represented by ${t}_{\frac{1}{2}}$
The equation for the half-life for first order reaction follows:
${t}_{\frac{1}{2}}=\frac{0.693}{k}$
where,
k = rate constant of first order reaction
Given value:
${t}_{\frac{1}{2}}=5715$ years
Plugging value in equation 1, we get:
$k=\frac{0.693}{5715}$
$k=1.21×{10}^{-4}y{r}^{-1}$
The integrated rate law expression for first order reaction follows:
$t=\frac{2.303}{k}\mathrm{log}\frac{a}{\left(a-x\right)}$
where,
t = time period
a = initial concentration of the reactant = 58.2 counts per minute
(a-x) = Concentration of reactant left after time t = 38.0 counts per minute
k = rate constant of first order reaction $=1.21×{10}^{-4}y{r}^{-1}$
Plugging value in equation, we get:
$t=\frac{2.303}{1.21×{10}^{-4}y{r}^{-1}}\mathrm{log}\frac{58.2}{38.0}$
$t=3523.77yr$