Chapter 18, Problem 35QRT

(a)

Interpretation Introduction

Interpretation:

Balanced nuclear equation has to be written for radon-222 emitting alpha particle.

Concept Introduction:

Nuclear reaction can be written in the form of nuclear equation.  Nuclear equation considers that atomic number and mass number of the elements that is involved in the reaction.  Balanced nuclear equation is the one that has mass number balance and atomic number balance equal on both sides of the equation.

Alpha particle emission results in formation of product nucleus that has mass number reduced by 4 and atomic number reduced by 2.  Alpha particle is represented as $\text{H}\text{e}$.

(b)

Interpretation Introduction

Interpretation:

Time required for the radon-222 radioisotope to decrease to $10.0\%$ of its original activity has to be calculated if the half-life is 3.82 days.

Concept Introduction:

For a radioactive isotope, the relative instability is expressed in terms of half-life.  Half-life is the time required for half of the given quantity of the radioisotope to undergo radioactive decay.  Half-life of the radioactive isotope is independent of the quantity of the radioactive isotope.  Half-life of a radioactive isotope can be expressed as,

    $t_{1/2}=\frac{\text{0}\text{.693}}{k}--------------(1)$

Number of disintegrations per unit time is known as the activity of the sample.  The rate of radioactive decay can be expressed as,

    $\text{Rate}\text{of}\text{radioactive}\text{decay}\text{=}\text{activity}\text{=}A=kN$

Where,

    $A$ is the activity of sample.

    $N$ is the number of radioactive atoms.

    $k$ is the first-order rate constant or decay constant.

If initial activity of the sample is $A_0$ at $t_0$, then after time $t$, this will have a smaller activity $A$.  If the number of radioactive atoms present initially is $N_0$, then after time $t$, the number of radioactive atoms present will be $N$.

    $\begin{array}{l}\frac{A}{A_0}=\frac{kN}{k{N}_0}\\ \frac{A}{A_0}=\frac{N}{N_0}----------------(2)\end{array}$

$N/N_0$ can be calculated using the integrated rate equation.

    $\ln N=-kt+\ln N_0--------------(3)$

Rearranging equation (3) as shown below,

    $\ln \frac{N}{N_0}=-kt------------------(4)$

Considering equation (2), the equation (3) can be rewritten in terms of fraction of radioactive atoms that is present after time $t$ as,

    $\ln \frac{A}{A_0}=-kt------------------(5)$