Chapter 14, Problem 14.103PAE

Interpretation Introduction

Interpretation:

The fraction of the original ²³⁷Np remains if there was some amount present when Earth was formed $4.54\times {10}^9\text{ years}$ ago should be calculated (The half-life of ²³⁷Np is $2.14\times {10}^6\text{ years}$ ).

Concept Introduction:

All radioactive reactions follow first-order kinetics.

For a first order reaction, rate constant and half-life time are related to each other as follows:

$k=\frac{0.693}{t_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}$

Here,

$t_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ is half-life time

k is the decay constant of the reaction

First order rate equation is as follows:

$k=\frac{2.303}{t}\log \frac{A_o}{A_t}$

Here,

t is the time-period

$A_t$ is concentration of the nuclei at time t

$A_o$ is the initial concentration of the nuclei