Chapter 10, Problem 47P

Interpretation Introduction

(a)

Interpretation:

The number of decays in the first two-minute interval needs to be determined.

Concept Introduction:

All the radioactive reactions follow first order kinetics. According to integrated rate law for the first order reaction, the relation between final concentration, initial concentration and half-life is represented as follows:

  $N\left(t\right)=N_o{\left(\frac{1}{2}\right)}^{t/t_{1/2}}$

Here, $N\left(t\right)$ is amount at time t, $N_o$ is initial amount and $t_{1/2}$ is half-life time.

The average number of decays per unit time or change in the radioactive nuclei present is known as activity. It is represented as follows:

  $A\left(t\right)=-\frac{dN}{dt}$

Also,

  $A\left(t\right)=\lambda N\left(t\right)$

Here, $\lambda$ is decay constant.

The relation between half-life time and decay constant is as follows:

  $t_{1/2}=\frac{0.6932}{\lambda }$

Here, $t_{1/2}$ is half-life time.

Interpretation Introduction

(b)

Interpretation:

The number of decays in the second two-minute interval needs to be determined.

Concept Introduction:

All the radioactive reactions follow first order kinetics. According to integrated rate law for the first order reaction, the relation between final concentration, initial concentration and half-life is represented as follows:

  $N\left(t\right)=N_o{\left(\frac{1}{2}\right)}^{t/t_{1/2}}$

Here, $N\left(t\right)$ is amount at time t, $N_o$ is initial amount and $t_{1/2}$ is half-life time.

The average number of decays per unit time or change in the radioactive nuclei present is known as activity. It is represented as follows:

  $A\left(t\right)=-\frac{dN}{dt}$

Also,

  $A\left(t\right)=\lambda N\left(t\right)$

Here, $\lambda$ is decay constant.

The relation between half-life time and decay constant is as follows:

  $t_{1/2}=\frac{0.6932}{\lambda }$

Here, $t_{1/2}$ is half-life time.