Chapter 24, Problem 24.47P

Interpretation Introduction

Interpretation:

Age of the rock has to be determined using the given information.

Concept Introduction:

Half–life period: It is the time required for the reactant (substrate) concentration to reduce to the one-half of its initial concentration.

The half–life period for first order reaction is as follows:

  $\begin{array}{l}{\text{t}}_{\text{1/2}}\text{=}\frac{\ln \text{ 2}}{\text{k}}\text{Where,}\\ \\ \text{k = Rate constant}\text{;}{\text{t}}_{\text{1/2}}\text{= half-life}\end{array}$

Integrated rate law for a first order reaction is given below.

  $\begin{array}{c}t\text{ = }\frac{1}{k}\ln \frac{N_0}{N_t}\\ \text{where,}\\ \\ \text{ln}\text{=}\text{natural}\text{logarithm}\\ \\ {\text{N}}_{\text{0}}\text{= Amount}\text{of}\text{subsatnce}\text{at}\text{time}\text{t}\text{=}\text{0}\\ \\ {\text{N}}_{\text{t}}\text{= Amount}\text{of}\text{subsatnce}\text{after}\text{time, t}\\ \\ \text{k}\text{=}\text{rate}\text{constant}\text{.}\end{array}$

Radioactive decay is a first order reaction.