Question

A fabric remnant from a burial site has a${}^{\mathbf{\text{14}}}{\mathbf{\text{C/}}}^{\mathbf{\text{12}}}\mathbf{\text{C}}$ ratio of $\mathbf{\text{0.735}}$of the original value. How old is the fabric?

Short answer

The fabric is$2546.6$years old

Step-by-step solution

Carbon dating

All the living organism contains${}^{14}C$ . As the organism dies its amount decreases slowly. Using ${}^{14}C$half-life the age of the organism can be determined.

Calculation

_{${}^{\text{14}}{\text{C/}}^{\text{12}}{\text{C=0.735(whichmeansthatN}}_{\text{t}}{\text{/N}}_{\text{0}}\text{=0.735)}$}

- The half-life of_{${}^{\text{14}}{\text{Cist}}_{\text{1/2}}\text{=5730yr}$}.

First, let us calculate the decay constant of_{${}^{14}\text{C}$}

_{$\begin{array}{c}\text{k=}\frac{\text{ln(2)}}{{\text{t}}_{\text{1/2}}}\\ \text{=}\frac{\text{0.693}}{\text{5730yr}}\\ {\text{=1.209×10}}^{\text{-4}}{\text{yr}}^{\text{-1}}\end{array}$}

Now we can calculate how old the fabric is

_{$\begin{array}{c}\text{ln}\left(\frac{{\text{N}}_{\text{t}}}{{\text{N}}_{\text{0}}}\right)\text{=-kt}\\ \text{t=}\frac{\text{-ln}\left(\frac{{\text{N}}_{\text{1}}}{{\text{N}}_{\text{0}}}\right)}{\text{k}}\\ \text{=}\frac{\text{-ln(0.735)}}{{\text{1.209×10}}^{\text{-4}}{\text{yr}}^{\text{-1}}}\\ \text{=2546.6yr}\end{array}$}

Hence, the fabric is old$2546.6$years