Q24.44 P - Chemistry: Molecular Nature Of Matter And Change

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?

Step-by-Step Solution

Answer

The fabric is $2546.6$ years old

Step 1: 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.

Step 2: Calculation

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

- The half-life of  _{${}^{\text{14}}{\text{C is t}}_{\text{1/2}}\text{=5730yr}$}.

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