Question

What is the half-life for the first-order decay of carbon-14?

\({{\bf{\;}}_{\bf{6}}}^{{\bf{14}}}{\bf{C}}\)⟶\({_{\bf{7}}^{{\bf{14}}}}{\bf{N + }}{{\bf{e}}^{\bf{ - }}}\)

The rate constant for the decay is\({\bf{1}}{\bf{.21 \times 1}}{{\bf{0}}^{{\bf{ - 4}}}}{\bf{yea}}{{\bf{r}}^{{\bf{ - 1}}}}\).

Short answer

Half-life periods may be defined as the time period at which half the concentration of reactant converts into the product. The half-life period of the first-order decay of carbon-14 is 5700 years which has a rate constant \({\bf{1}}{\bf{.21 \times 1}}{{\bf{0}}^{{\bf{ - 4}}}}{\bf{yea}}{{\bf{r}}^{{\bf{ - 1}}}}\).

\(\)

Step-by-step solution

Reaction Rate

The reaction involved the effective collision of two reactants to produce the desired products. Reactions can be natural, which occur in the surrounding environment, whereas it can be artificially done in the laboratory to form a desired required product.

The reaction rate can be defined as the reaction speed to produce the products. The reaction rate can be slow, fast or moderate. The reaction can take a lesser than millisecond to produce products, or it can take years to produce the desired product.

The half-life period can be defined as the time period at which half the concentration of the reactants gets converted into a product.

Explanation

The half-life period of the first-order decay of carbon-14 is:

\(Half - life{\bf{ }}period{\bf{ }} = {\bf{ }}\frac{{\ln {\bf{ }}\left( 2 \right)}}{k}\)

The rate constant for the decay of Carbon-14= \({\bf{1}}{\bf{.21 \times 1}}{{\bf{0}}^{{\bf{ - 4}}}}{\bf{yea}}{{\bf{r}}^{{\bf{ - 1}}}}\)

\(Half - life{\bf{ }}period{\bf{ }} = {\bf{ }}\frac{{0.693}}{{1.21{\bf{ }} \times {\bf{ }}{{10}^{ - 4}}yea{r^{ - 1}}}}\)

Half-life period = 5727 years or 5730 years.

Hence, the half-life period of the first-order decay of carbon-14 is 5700 years which has rate constant \({\bf{1}}{\bf{.21 \times 1}}{{\bf{0}}^{{\bf{ - 4}}}}{\bf{yea}}{{\bf{r}}^{{\bf{ - 1}}}}\).