Chapter 24, Problem 24.100P

Interpretation Introduction

Interpretation:

Fraction of ${}^{235}U$ that would remain on earth after $2.8\times {10}^9\text{ yr}$ has to be determined where half-life of ${}^{235}U$ is $7.0\times {10}^8\text{ yr}$.

Concept Introduction:

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

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

Integrated rate law for a first order reaction is given below

  $\begin{array}{c}\ln \left(\frac{N_t}{N_0}\right)=-kt\\ \text{where,}\\ \\ \text{ln}\text{=}\text{natural}\text{logarithm}\\ \\ {\text{N}}_{\text{0}}\text{= Amount}\text{of}\text{subsatnce}\text{at}\text{time}\text{t}\text{=}\text{0}\\ \\ {\text{N}}_{\text{t}}\text{= Amount}\text{of}\text{subsatnce}\text{after}\text{time, t}\\ \\ \text{k}\text{=}\text{Rate}\text{constant}\end{array}$

Radioactive decay is a first order reaction.