Chapter 11, Problem 112QRT

(a)

Interpretation Introduction

Interpretation:

It has to be shown that at time $t$ the concentration ${\left[A\right]}_t$ of reactant is related to the initial concentration ${\left[A\right]}_0$ as given below.

  ${\left[A\right]}_t={\left[A\right]}_0{\left(\frac{1}{2}\right)}^x$

Concept Introduction:

The integrated rate law for first-order reaction is given below.

  $\ln {\left[A\right]}_t=-kt+\ln {\left[A\right]}_0$

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

The half-life period for first-order reaction is given below.

  $t_{1/2}=\frac{\ln 2}{k}$

(b)

Interpretation Introduction

Interpretation:

The following relation has to be proved.

  $\log \left(\frac{{\left[A\right]}_t}{{\left[A\right]}_0}\right)=\frac{t}{t_{1/2}}\left(-\log 2\right)=\frac{t}{t_{1/2}}\left(-0.301\right)$

(c)

Interpretation Introduction

Interpretation:

It has to be shown that for any first-order reaction a plot of $\log \left[A\right]$ versus $t$ will be a straight line and that half-life can be obtained from the slope of the line.