Chapter 6, Problem 6F.1E

Interpretation Introduction

Interpretation:

In the given reaction, the time at which $\text{Y}$ will be most abundant after $\text{X}$ is first formed has to be determined.

Concept Introduction:

Radioactive decay: The decay constant used to define the quantity of radioactive material that remains after time period has gone.  The fraction of material that leftovers after a time period gas gone is the ratio of the number of radioactive nuclei (${\text{N}}_{\text{g}}$) in the real sample when the measurement happening.  This segment is frequently expressed as a percent.

The equation relating fraction of nuclei after a period of time has elapsed to the decay constant $\text{(ln}\frac{{\text{N}}_{\text{t}}}{{\text{N}}_{\text{g}}}\text{= -kt)}$

Half-life period:

Half-life is defined as the time taken for amount of the sample to reduce to half of its starting value.  The symbol is ${\text{t}}_{\raisebox{1ex}{$\text{1}$}\!\left/ \!\raisebox{-1ex}{$\text{2}$}\right.}$.

  $t_{\frac{1}{2}}=\frac{\ln \left(2\right)}{k}$

Half-life for a first order reaction is, $t_{\frac{1}{2}}=\frac{0.693}{k}$

First order reaction $-kt=\ln \left(\frac{\left[A\right]}{\left[A_0\right]}\right)$

$\left[A\right]$ is the concentration of reactant A at time t ${\left[A\right]}_0$ is the  initial concentration of reactant, k is the rate constant.