Chapter 7.8, Problem 7.29SC

Interpretation Introduction

Interpretation:

The reasonability of using $\text{C-14}$ dating for the analysis of the sample with age of $100,000$ years has to be examined.

Concept Introduction:

Half-life of a radioactive material is the time taken to decay half of the amount of that material into a stable product.

The Half-life of $\text{C-14}$ is $\text{5715 years}\text{.}$

Radioactive decay being a first order reaction, the integrated rate equation can written as follows,

$\text{kt = 2}{\text{.303 log}}_{\text{10}}\left(\frac{{\text{A}}_{\text{0}}}{\text{A}}\right)$

Where,

    $\text{k}$   = rate constant or decay constant

    $\text{t}$    = time taken for the reaction

    ${\text{A}}_{\text{0}}$ = initial concentration of reactant

    $\text{A}$   = final concentration of the reactant

When ${\text{A}}_{\text{0}}$ the initial concentration becomes half $\left(\text{A}=\frac{{\text{A}}_{\text{0}}}{2}\right)$  , then time taken $\text{t}$ becomes ${\text{t}}_{\text{1/2}}$.

On substituting these values in the integrated rate equation, the equation for ${\text{t}}_{\text{1/2}}$ will be obtained as follows,

${\text{t}}_{\text{1/2}}\text{ = }\frac{\text{2}{\text{.303 log}}_{\text{10}}\left(\frac{{\text{A}}_{\text{0}}}{\frac{{\text{A}}_0}{2}}\right)}{\text{k}}=\frac{0.693}{k}$