Chapter 7, Problem 41E

Interpretation Introduction

(a)

Interpretation:

The mass of $4.12\times {10}^{24}$ $\text{N}$ atoms is to be calculated.

Concept introduction:

The atomic mass of an atom is defined as the average of the total masses of the atoms in an element and is expressed in atomic mass unit $\left(\text{u}\right)$.

Molar mass is defined as the mass in grams of one mole of a substance and is expressed in ${\text{g mol}}^{-1}$.

Answer to Problem 41E

The mass of $4.12\times {10}^{24}$ $\text{N}$ atoms is $95.\text{9 g}$.

Explanation of Solution

There are $6.02\times {10}^{23}$ atoms of $\text{N}$ present in one mole. Thus, the moles of $\text{N}$ in $4.12\times {10}^{24}$ atoms of $\text{N}$ are calculated as follows,

$\begin{array}{rll}\text{Moles of N}& =& \frac{\text{1 mole of N }\times \text{4}\text{.12}\times {\text{10}}^{\text{24}}\text{ atoms of N}}{\text{6}\text{.02}\times {\text{10}}^{\text{23}}\text{ atoms of N}}\\ & =& 6.84\text{4 moles}\end{array}$

The molar mass of $\text{N}$ is $14.0{\text{1 g mol}}^{-1}$.

Therefore, the mass of one mole of $\text{N}$ is $14.0\text{1 g}$.

The mass of $6.84\text{4 moles}$ $\text{N}$ is calculated as follows,

$\begin{array}{rll}\text{Mass of N}& =& 14.0{\text{1 g mol}}^{-1}\times 6.84\text{4 moles}\\ & =& 95.\text{9 g}\end{array}$

Hence, the mass of $4.12\times {10}^{24}$ $\text{N}$ atoms is $95.\text{9 g}$.

Conclusion

The mass of $4.12\times {10}^{24}$ $\text{N}$ atoms is $95.\text{9 g}$.

Interpretation Introduction

(b)

Interpretation:

The mass of $4.12\times {10}^{24}$ ${\text{N}}_2$ molecules is to be calculated.

Concept introduction:

The atomic mass of an atom is defined as the average of the total masses of the atoms in an element and is expressed in atomic mass unit $\left(\text{u}\right)$.

Molar mass is defined as the mass in grams of one mole of a substance and is expressed in ${\text{g mol}}^{-1}$.

Answer to Problem 41E

The mass of $4.12\times {10}^{24}$ $\text{N}$ molecules is $\text{192 g}$.

Explanation of Solution

There are $6.02\times {10}^{23}$ molecules of ${\text{N}}_2$ present in one mole. Thus, the moles of ${\text{N}}_2$ in $4.12\times {10}^{24}$ molecules of ${\text{N}}_2$ are calculated as follows,

$\begin{array}{rll}\text{Moles of N}& =& \frac{\text{1 mole of N }\times \text{4}\text{.12}\times {\text{10}}^{\text{24}}\text{ molecules of N}}{\text{6}\text{.02}\times {\text{10}}^{\text{23}}\text{ molecules of N}}\\ & =& 6.84\text{4 moles}\end{array}$

The molar mass of $\text{N}$ is $14.0{\text{1 g mol}}^{-1}$. Thus, the molar mass of ${\text{N}}_2$ is calculated as follows,

$\begin{array}{rll}{\text{Molar mass of N}}_2& =& 2\times 14.0{\text{1 g mol}}^{-1}\\ & =& 28.02{\text{ g mol}}^{-1}\end{array}$

Therefore, the mass of one mole of ${\text{N}}_2$ is $28.02\text{ g}$.

The mass of $6.84\text{4 moles}$ ${\text{N}}_2$ is calculated as follows,

$\begin{array}{rll}{\text{Mass of N}}_2& =& 28.02{\text{ g mol}}^{-1}\times 6.84\text{4 moles}\\ & =& 192\text{ g}\end{array}$

Hence, the mass of $4.12\times {10}^{24}$ $\text{N}$ molecules is $\text{192 g}$.

Conclusion

The mass of $4.12\times {10}^{24}$ $\text{N}$ molecules is $\text{192 g}$.

Interpretation Introduction

(c)

Interpretation:

The number of atoms present in $4.1\text{2 g}$ $\text{N}$ is to be calculated.

Concept introduction:

The atomic mass of an atom is defined as the average of the total masses of the atoms in an element and is expressed in atomic mass unit $\left(\text{u}\right)$.

Molar mass is defined as the mass in grams of one mole of a substance and is expressed in ${\text{g mol}}^{-1}$.

Answer to Problem 41E

The number of atoms present in $4.1\text{2 g}$ $\text{N}$ is $\text{1}\text{.77}\times {\text{10}}^{\text{23}}\text{ atoms}$.

Explanation of Solution

The molar mass of $\text{N}$ is $14.0{\text{1 g mol}}^{-1}$.

Therefore, the mass of one mole of $\text{N}$ is $14.0\text{1 g}$.

The number of moles of $\text{N}$ is calculated by the formula,

$\text{Number of moles}=\frac{\text{Mass}}{\text{Molar mass}}$

The given mass is $4.1\text{2 g}$.

Substitute the values of mass and molar mass in the above formula to calculate the number of moles of $\text{N}$.

$\begin{array}{rll}\text{Number of moles}& =& \frac{4.1\text{2 g}}{14.0{\text{1 g mol}}^{-1}}\\ & =& 0.\text{294 moles}\end{array}$

Thus, $4.1\text{2 g}$ of $\text{N}$ has $0.\text{294 moles}$ of $\text{N}$. There are $6.02\times {10}^{23}$ atoms of $\text{N}$ present in one mole. Therefore, the number of atoms of $\text{N}$ present in $0.\text{294 moles}$ is calculated as follows,

$\begin{array}{rll}\text{Number of N atoms}& =& 6.02\times {10}^{23}\times 0.\text{294}\\ & =& \text{1}\text{.77}\times {\text{10}}^{\text{23}}\text{ atoms}\end{array}$

Hence, $\text{1}\text{.77}\times {\text{10}}^{\text{23}}\text{ atoms}$ are present in $4.1\text{2 g}$ of $\text{N}$.

Conclusion

The number of atoms present in $4.1\text{2 g}$ $\text{N}$ is $\text{1}\text{.77}\times {\text{10}}^{\text{23}}\text{ atoms}$.

Interpretation Introduction

(d)

Interpretation:

The number of molecules present in $4.1\text{2 g}$ ${\text{N}}_2$ is to be calculated.

Concept introduction:

The atomic mass of an atom is defined as the average of the total masses of the atoms in an element and is expressed in atomic mass unit $\left(\text{u}\right)$.

Molar mass is defined as the mass in grams of one mole of a substance and is expressed in ${\text{g mol}}^{-1}$.

Answer to Problem 41E

The number of molecules present in $4.1\text{2 g}$ ${\text{N}}_2$ is $\text{8}\text{.85}\times {\text{10}}^{\text{22}}$.

Explanation of Solution

The molar mass of ${\text{N}}_2$ is $28.02{\text{ g mol}}^{-1}$.

Therefore, the mass of one mole of ${\text{N}}_2$ is $28.02\text{ g}$.

The number of moles of ${\text{N}}_2$ is calculated by the formula,

$\text{Number of moles}=\frac{\text{Mass}}{\text{Molar mass}}$

The given mass is $4.1\text{2 g}$.

Substitute the values of mass and molar mass in the above formula to calculate the number of moles of ${\text{N}}_2$.

$\begin{array}{rll}\text{Number of moles}& =& \frac{4.1\text{2 g}}{28.02{\text{ g mol}}^{-1}}\\ & =& 0.147\text{ moles}\end{array}$

Thus, $4.1\text{2 g}$ of ${\text{N}}_2$ has $0.\text{147 moles}$ of ${\text{N}}_2$. There are $6.02\times {10}^{23}$ molecules of ${\text{N}}_2$ present in one mole. Therefore, the number of molecules of ${\text{N}}_2$ present in $0.\text{147 moles}$ is calculated as follows,

$\begin{array}{rll}\text{Number of N molecules}& =& 6.02\times {10}^{23}\times 0.\text{147}\\ & =& \text{8}\text{.85}\times {\text{10}}^{\text{22}}\text{ molecules}\end{array}$

Hence, $\text{8}\text{.85}\times {\text{10}}^{\text{22}}$ molecules are present in $4.1\text{2 g}$ of ${\text{N}}_2$.

Conclusion

The number of molecules present in $4.1\text{2 g}$ ${\text{N}}_2$ is $\text{8}\text{.85}\times {\text{10}}^{\text{22}}$.

Interpretation Introduction

(e)

Interpretation:

The number of atoms present in $4.1\text{2 g}$ ${\text{N}}_2$ is to be calculated.

Concept introduction:

The atomic mass of an atom is defined as the average of the total masses of the atoms in an element and is expressed in atomic mass unit $\left(\text{u}\right)$.

Molar mass is defined as the mass in grams of one mole of a substance and is expressed in ${\text{g mol}}^{-1}$.

Answer to Problem 41E

The number of atoms present in $4.1\text{2 g}$ ${\text{N}}_2$ is $1.77\times {\text{10}}^{\text{23}}$.

Explanation of Solution

The molar mass of ${\text{N}}_2$ is $28.02{\text{ g mol}}^{-1}$.

Therefore, the mass of one mole of ${\text{N}}_2$ is $28.02\text{ g}$.

The number of moles of ${\text{N}}_2$ is calculated by the formula,

$\text{Number of moles}=\frac{\text{Mass}}{\text{Molar mass}}$

The given mass is $4.1\text{2 g}$.

Substitute the values of mass and molar mass in the above formula to calculate the number of moles of ${\text{N}}_2$.

$\begin{array}{rll}\text{Number of moles}& =& \frac{4.1\text{2 g}}{28.02{\text{ g mol}}^{-1}}\\ & =& 0.147\text{ moles}\end{array}$

Thus, $4.1\text{2 g}$ of ${\text{N}}_2$ has $0.\text{147 moles}$ of ${\text{N}}_2$. There are $6.02\times {10}^{23}$ molecules of ${\text{N}}_2$ present in one mole. Therefore, the number of molecules of ${\text{N}}_2$ present in $0.\text{147 moles}$ is calculated as follows,

$\begin{array}{rll}\text{Number of N molecules}& =& 6.02\times {10}^{23}\times 0.\text{147}\\ & =& \text{8}\text{.85}\times {\text{10}}^{\text{22}}\text{ molecules}\end{array}$

Therefore, $\text{8}\text{.85}\times {\text{10}}^{\text{22}}$ molecules are present in $4.1\text{2 g}$ of ${\text{N}}_2$. One molecule of ${\text{N}}_2$ has two atoms of $\text{N}$. Thus, the number of atoms present in $4.1\text{2 g}$ ${\text{N}}_2$ molecule is calculated as follows,

$\begin{array}{rll}\text{Number of N atoms}& =& 2\times {\text{N}}_{\text{2}}\text{ molecules}\\ & =& 2\times \text{8}\text{.85}\times {\text{10}}^{\text{22}}\\ & =& 1.77\times {\text{10}}^{\text{23}}\text{ atoms}\end{array}$

Hence, $1.77\times {\text{10}}^{\text{23}}$ atoms are present in $4.1\text{2 g}$ of ${\text{N}}_2$.

Conclusion

The number of atoms present in $4.1\text{2 g}$ ${\text{N}}_2$ is $1.77\times {\text{10}}^{\text{23}}$.