Chapter 3, Problem 3.7QP

(a)

Interpretation Introduction

Interpretation:

The velocity, the mass and identity of the given atoms should be calculated in the given statement by using the equation of kinetic energy

Concept Introduction:

Energy is the capacity to do work or transfer heat where work is the movement of a body using some force.  The SI unit of energy is joule ($\text{J}$).  Energy is in the form of kinetic energy or potential energy.  Kinetic energy is the energy associated with motion.  Kinetic energy (in joule) is calculated using the formula:

${\text{E}}_{\text{k}}=\frac{\text{1}}{\text{2}}{\text{mu}}^{\text{2}}$

where $\text{m}$ ‒ mass in kilograms; $\text{u}$ – velocity in meters per second.

Explanation of Solution

To find: Determine the velocity of $\text{Ne}$ atom that has ${\text{E}}_{\text{k}}\text{= 1}{\text{.86 × 10}}^{-\text{20}}\text{ J}$

Kinetic energy (in joule) is calculated using the formula: ${\text{E}}_{\text{k}}=\frac{\text{1}}{\text{2}}{\text{mu}}^{\text{2}}$

where $\text{m}$ ‒ mass in kilograms; $\text{u}$ – velocity in meters per second.  From this equation, velocity in meters per second is calculated using the formula:

$\text{u }=\sqrt{\frac{{\text{2E}}_{\text{k}}}{\text{m}}}$

By considering the given problem, the mass of $\text{Ne}$ atom $\text{m }=20.18\text{ amu}$; ${\text{E}}_{\text{k}}\text{= 1}{\text{.86 × 10}}^{-\text{20}}\text{ J}$.

The mass of $\text{Ne}$ atom in kilograms is

$\begin{array}{l}\text{m }=\text{ 20}\text{.18 amu × }\frac{\text{1}{\text{.661 × 10}}^{-\text{24}}\text{ g}}{\text{1 amu}}\times \frac{\text{1 kg}}{{\text{1 × 10}}^{\text{3}}\text{ g}}\\ \text{m }=\text{ 3}{\text{.352 × 10}}^{-26}\text{ kg}\end{array}$

${\text{E}}_{\text{k}}$ value in $\text{1}{\text{.86 × 10}}^{-\text{20}}\text{ J}$ is equal to ${\text{E}}_{\text{k}}$ value in $\text{1}{\text{.86 × 10}}^{-\text{20}}\text{ kg}\cdot {\text{m}}^{\text{2}}{\text{/s}}^{\text{2}}$ which is used for the purpose of making the unit cancellation. Substitute the given values in the formula,

$\begin{array}{l}\text{u }=\sqrt{\frac{\text{2 × (1}{\text{.86 × 10}}^{-\text{20}}\text{ kg}\cdot {\text{m}}^{\text{2}}{\text{/s}}^{\text{2}}\text{)}}{\text{3}{\text{.352 × 10}}^{-26}\text{ kg}}}\\ \text{u }=\text{ 1}{\text{.05 × 10}}^{\text{3}}\text{ m/s}\end{array}$

Therefore, the velocity of $\text{Ne}$ atom that has ${\text{E}}_{\text{k}}\text{= 1}{\text{.86 × 10}}^{-\text{20}}\text{ J}$ is $\text{1}{\text{.05 × 10}}^{\text{3}}\text{ m/s}$ (a).

(b)

Interpretation Introduction

Interpretation:

The velocity, the mass and identity of the given atoms should be calculated in the given statement by using the equation of kinetic energy

Concept Introduction:

Energy is the capacity to do work or transfer heat where work is the movement of a body using some force.  The SI unit of energy is joule ($\text{J}$).  Energy is in the form of kinetic energy or potential energy.  Kinetic energy is the energy associated with motion.  Kinetic energy (in joule) is calculated using the formula:

${\text{E}}_{\text{k}}=\frac{\text{1}}{\text{2}}{\text{mu}}^{\text{2}}$

where $\text{m}$ ‒ mass in kilograms; $\text{u}$ – velocity in meters per second.

Explanation of Solution

To find: Determine the velocity of $\text{Kr}$ atom that has ${\text{E}}_{\text{k}}=\text{ 7}{\text{.50 × 10}}^{-\text{21}}\text{ J}$

Kinetic energy (in joule) is calculated using the formula: ${\text{E}}_{\text{k}}=\frac{\text{1}}{\text{2}}{\text{mu}}^{\text{2}}$

where $\text{m}$ ‒ mass in kilograms; $\text{u}$ – velocity in meters per second.  From this equation, velocity in meters per second is calculated using the formula:

$\text{u }=\sqrt{\frac{{\text{2E}}_{\text{k}}}{\text{m}}}$

By considering the given problem, the mass of $\text{Kr}$ atom $\text{m }=8\text{3}\text{.80 amu}$; ${\text{E}}_{\text{k}}=\text{ 7}{\text{.50 × 10}}^{-\text{21}}\text{ J}$.

The mass of $\text{Kr}$ atom in kilograms is

$\begin{array}{l}\text{m }=\text{ 83}\text{.80 amu × }\frac{\text{1}{\text{.661 × 10}}^{-\text{24}}\text{ g}}{\text{1 amu}}\times \frac{\text{1 kg}}{{\text{1 × 10}}^{\text{3}}\text{ g}}\\ \text{m }=\text{ 1}{\text{.392 × 10}}^{-25}\text{ kg}\end{array}$

${\text{E}}_{\text{k}}$ value in $\text{7}{\text{.50 × 10}}^{-\text{21}}\text{ J}$ is equal to ${\text{E}}_{\text{k}}$ value in $\text{7}{\text{.50 × 10}}^{-\text{21}}\text{ kg}\cdot {\text{m}}^{\text{2}}{\text{/s}}^{\text{2}}$ which is used for the purpose of making the unit cancellation. Substitute the given values in the formula,

$\begin{array}{l}\text{u }=\sqrt{\frac{\text{2 × (7}{\text{.50 × 10}}^{-\text{21}}\text{ kg}\cdot {\text{m}}^{\text{2}}{\text{/s}}^{\text{2}}\text{)}}{\text{1}{\text{.392 × 10}}^{-\text{25}}\text{ kg}}}\\ \text{u }=\text{ 328 m/s}\end{array}$

Therefore, the velocity of $\text{Kr}$ atom that has ${\text{E}}_{\text{k}}=\text{ 7}{\text{.50 × 10}}^{-\text{21}}\text{ J}$ is $\text{328 m/s}$ (b).

(c)

Interpretation Introduction

Interpretation:

The velocity, the mass and identity of the given atoms should be calculated in the given statement by using the equation of kinetic energy

Concept Introduction:

Energy is the capacity to do work or transfer heat where work is the movement of a body using some force.  The SI unit of energy is joule ($\text{J}$).  Energy is in the form of kinetic energy or potential energy.  Kinetic energy is the energy associated with motion.  Kinetic energy (in joule) is calculated using the formula:

${\text{E}}_{\text{k}}=\frac{\text{1}}{\text{2}}{\text{mu}}^{\text{2}}$

where $\text{m}$ ‒ mass in kilograms; $\text{u}$ – velocity in meters per second.

Explanation of Solution

To find: Determine the mass and identity of an atom moving at $\text{385 m/s}$ that has ${\text{E}}_{\text{k}}\text{= 4}{\text{.812 × 10}}^{-\text{21}}\text{ J}$

Kinetic energy (in joule) is calculated using the formula: ${\text{E}}_{\text{k}}=\frac{\text{1}}{\text{2}}{\text{mu}}^{\text{2}}$

where $\text{m}$ ‒ mass in kilograms; $\text{u}$ – velocity in meters per second.  From this equation, mass in kilograms is calculated using the formula:

$\text{m }=\frac{{\text{2E}}_{\text{k}}}{{\text{u}}^{\text{2}}}$

By considering the given problem, the mass of a $\text{Kr}$ atom $\text{m }=\text{ 83}\text{.80 amu}$; ${\text{E}}_{\text{k}}\text{= 4}{\text{.812 × 10}}^{-\text{21}}\text{ J}$.  ${\text{E}}_{\text{k}}$ value in $\text{4}{\text{.812 × 10}}^{-\text{21}}\text{ J}$ is equal to ${\text{E}}_{\text{k}}$ value in $\text{4}{\text{.812 × 10}}^{-\text{21}}\text{ kg}\cdot {\text{m}}^{\text{2}}{\text{/s}}^{\text{2}}$ which is used for the purpose of making the unit cancellation.

The mass of an atom in kilograms is

$\begin{array}{l}\text{m }=\frac{\text{2 × (4}{\text{.812 × 10}}^{-\text{21}}\text{ kg}\cdot {\text{m}}^{\text{2}}{\text{/s}}^{\text{2}}\text{)}}{{\text{(385 m/s)}}^{\text{2}}}\\ \text{m }=\text{ 6}{\text{.493 × 10}}^{-26}\text{ kg}\end{array}$

If the mass in $\text{kg}$ is converted into the mass in $\text{amu}$, the identity of the atom will be determined.  The factor for conversion of $\text{kg }\to \text{ amu}$ is given as follows:

$\frac{{\text{1 × 10}}^{\text{3}}\text{ g}}{\text{1 kg}}\times \frac{\text{1 amu}}{\text{1}{\text{.661 × 10}}^{-\text{24}}\text{ g}}$

By substituting the mass value in the above expression, the identity of the atom will be determined as follows:

$\begin{array}{l}\text{6}{\text{.493 × 10}}^{-26}\text{ kg }\times \frac{{\text{1 × 10}}^{\text{3}}\text{ g}}{\text{1 kg}}\times \frac{\text{1 amu}}{\text{1}{\text{.661 × 10}}^{-\text{24}}\text{ g}}\\ \text{39}\text{.1 amu}\end{array}$

The atom with a mass of $\text{39}\text{.1 amu}$ is potassium.  Therefore, the mass and identity of an atom moving at $\text{385 m/s}$ that has ${\text{E}}_{\text{k}}\text{= 4}{\text{.812 × 10}}^{-\text{21}}\text{ J}$ are $\text{6}{\text{.493 × 10}}^{-26}\text{ kg}$ and potassium (c).