Chapter 20, Problem 45E

Interpretation Introduction

Interpretation:

The amount of time required for the decay of theU-235 (uranium isotope) to reach 10.0% of its initial amount with a half-life of $703\text{ million years }$ is to be determined.

Concept introduction:

The half-life of a substance is the numerical value in which the given radioactive substance is assumed to be reduced to half of its initial amount. The half-life for a given substance is represented by t_1/2.

In case, the decay of a radioactive substance is exponential, it will remain constant for the life time of the substance.

After each half-life period, the amount of the substance is reduced to half of the initial number.

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.