Chapter 31, Problem 58GP

(a)

To determine

The decay equation.

Answer to Problem 58GP

The decay equation is $\text{R}\text{d}\to \text{R}\text{n}+\text{H}\text{e}$

Explanation of Solution

Given:

Radon gas, $\text{R}\text{n}$

Calculation:

The decay equation is given by:

  $\text{R}\text{d}\to \text{R}\text{n}+\text{H}\text{e}$

Here, the atomic mass of parent nucleus is reduced by 4 units and the atomic number is reduced by 2 units.

Conclusion:

The decay equation is $\text{R}\text{d}\to \text{R}\text{n}+\text{H}\text{e}$

(b)

To determine

Kinetic energy of the alpha particle produced.

Answer to Problem 58GP

  $\mathrm{E}=4.8\text{ MeV}$

Explanation of Solution

Given:

Radon gas, $\text{R}\text{n}$

Calculation:

The kinetic energy can be calculated as follows:

  $\begin{array}{l}\mathrm{E}=\left({\mathrm{m}}_{\mathrm{R}\mathrm{d}}-{\mathrm{m}}_{\mathrm{R}\mathrm{n}}-{\mathrm{m}}_{\mathrm{H}\mathrm{e}}\right){\mathrm{c}}^2\\ \mathrm{E}=\left(226.025403\text{u}-222.01757\text{u}-4.002603\text{u}\right){\mathrm{c}}^2\\ \mathrm{E}=\left(226.025403-222.01757-4.002603\right)\left(931.5\right)\text{MeV}\\ \mathrm{E}=4.8\text{ MeV}\end{array}$

Conclusion:

Thus, the kinetic energy is $\mathrm{E}=4.8\text{ MeV}$

(c)

To determine

Momentum of alpha particle and the daughter nucleus.

Answer to Problem 58GP

The momentum of alpha particle and the daughter nucleus is $\mathrm{p}=191\text{ MeV/}\mathrm{c}$ and $\mathrm{p}=1409\text{ MeV/}\mathrm{c}$

Explanation of Solution

Given:

Radon gas, $\text{R}\text{n}$

Calculation:

The momentum of the alpha particle can be calculated as follows:

  $\mathrm{p}=\sqrt{2\mathrm{m}\mathrm{E}}$

Substituting all the given values, we get:

  $\begin{array}{l}\mathrm{p}=\sqrt{2\left(4.002603\text{u}\right)\left(931.5\text{MeV}\right)\left(4.8\text{ MeV}\right)}\\ \mathrm{p}=191\text{ MeV/}\mathrm{c}\end{array}$

The momentum of daughter nucleus is given by:

  $\begin{array}{l}\mathrm{p}=\sqrt{2\left(222.01757\right)\left(931.5\text{MeV}\right)\left(4.8\text{ MeV}\right)}\\ \mathrm{p}=1409\text{ MeV/}\mathrm{c}\end{array}$

Conclusion:

Thus, the momentum of alpha particle and the daughter nucleus is $\mathrm{p}=191\text{ MeV/}\mathrm{c}$ and $\mathrm{p}=1409\text{ MeV/}\mathrm{c}$

(d)

To determine

Kinetic energy of the daughter nucleus.

Answer to Problem 58GP

  $\mathrm{E}=0.0882\text{ MeV}$

Explanation of Solution

Given:

Radon gas, $\text{R}\text{n}$

Calculation:

The kinetic energy can be calculated as follows:

  $\begin{array}{l}\mathrm{E}=\frac{{\mathrm{p}}^2}{2\mathrm{m}}\\ \mathrm{E}=\frac{{\left(191\text{ MeV/}\mathrm{c}\right)}^2}{2\left(222.01757\right)\left(931.5\text{MeV}\right)}\\ \mathrm{E}=0.0882\text{ MeV}\end{array}$

Conclusion:

Thus, the kinetic energy is $\mathrm{E}=0.0882\text{ MeV}$