We’re being asked to determine the approximate age (yr) of the artifact that showed a carbon-14 decay rate of 13.8 disintegrations/min per gram of carbon. Carbon-14 has a half-life of 5715 years, and currently living organisms decay at the rate of 15.3 disintegrations/min per gram of carbon

Recall that ** radioactive/nuclear decay of isotopes** follows first-order kinetics, and the integrated rate law for first-order reactions is:

$\overline{){\mathbf{ln}}{\mathbf{}}{\mathbf{\left[}\mathbf{N}\mathbf{\right]}}_{{\mathbf{t}}}{\mathbf{=}}{\mathbf{-}}{\mathbf{kt}}{\mathbf{+}}{\mathbf{ln}}{\mathbf{}}{\mathbf{\left[}\mathbf{N}\mathbf{\right]}}_{{\mathbf{0}}}}$

where:

**[N] _{t}** = concentration at time t

**k** = decay constant

**t** = time

**[N] _{0}** = initial concentration.

Also, recall that ** half-life** is the time needed for the amount of a reactant to decrease by 50% or one-half.

The half-life of a first-order reaction is given by:

$\overline{){{\mathbf{t}}}_{\raisebox{1ex}{$\mathbf{1}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{2}$}\right.}{\mathbf{=}}\frac{\mathbf{ln}\mathbf{}\mathbf{2}}{\mathbf{k}}}$

An artifact showed a carbon-14 decay rate of 13.8 disintegrations/min per gram of carbon. Carbon-14 has a half-life of 5715 years, and currently living organisms decay at the rate of 15.3 disintegrations/min per gram of carbon. What is the approximate age (yr) of the artifact?

a. 658

b. 581

c. 423

d. 851

e. 42328

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