Chapter 19, Problem 19.56QP

(a)

Interpretation Introduction

Interpretation:

Balanced equation for manganese-50 decays through ${}_{\text{+1}}^{\text{0}}\text{β}$ particle beta emission has to be written.

Concept Introduction:

On accordance with law of conservation of mass, for any chemical reaction, total masses of reactants and products must be equal.

${\displaystyle \sum \text{reactant}\text{mass}\text{number=}{\displaystyle \sum \text{product}\text{mass}\text{number}}}$

Similarly for any chemical reaction, sum of the reactant atomic number is equal to the sum of the product atomic number.

${\displaystyle \sum \text{reactant}\text{atomic}\text{number=}{\displaystyle \sum \text{product}\text{atomic}\text{number}}}$

${}_Z^{A}X$, where X is the element, A is the mass number and Z is the atomic number.

(b)

Interpretation Introduction

Interpretation:

The half-life that has elapsed during manganese-50 decays through ${}_{\text{+1}}^{\text{0}}\text{β}$ particle beta emission has to be determined.

Concept Introduction:

• The rate of radioactive decay at any time $t$ is given by,

  $Rateofdecayattimet=\lambda N$, where $\lambda$ is the first-order rate constant and $N$ is the number of radioactive nuclei present at any time $t$.

• Number of radioactive nuclei at time zero $\left(N_0\right)$ and time $t$$\left(N_t\right)$ is given by $\ln \frac{N_t}{N_0}=-\lambda t$
• Half-life of the reaction is the time taken by the quantity to reduce to half of its initial amount. Half-life of the reaction is given by

$t_{1/2}=\frac{0.693}{\lambda }$