Chapter 20, Problem 7SAQ

Interpretation Introduction

Interpretation: The age of artifact is to be determined.

Concept Introduction:

The rate constant (k) of the reaction is determined by:

$t_{1/2}=\frac{\text{0}\text{.693}}{k}$

The integrated rate law is as follows:

$\begin{array}{l}\text{ln}\frac{N_t}{N_{\text{0}}}=-kt\\ \frac{N_t}{N_{\text{0}}}=N_{\text{0}}{\text{e}}^{-kt}\end{array}$

Here, N is the amount of sample left after time T and $N_0$ is the initial amount of the sample.

As radioactivity is a first-order process, the rate of decay is directly proportional to the number of nuclei in the sample. Thus, the initial rate of decay $\left(rat{e}_0\right)$ and the rate of decay at time t $\left(rat{e}_t\right)$ can also be used in the integrated rate law:

$\begin{array}{l}{\text{Rate}}_t=k{N}_t;{\text{Rate}}_0=k{N}_0\\ \frac{N_t}{N_0}=\frac{{\text{Rate}}_t/\cancel{k}}{{\text{Rate}}_0/\cancel{k}}=\frac{{\text{Rate}}_t}{{\text{Rate}}_0}\end{array}$

Substitute the value of $\frac{N_t}{N_0}=\frac{{\text{Rate}}_t}{{\text{Rate}}_0}$ in the above integrated rate law equation to get,

$\ln \frac{{\text{Rate}}_t}{{\text{Rate}}_0}=-kt$