Chapter 5, Problem 62E

Interpretation Introduction

(a)

Interpretation:

The average density of the carbon atom is to be stated.

Concept introduction:

The density of a substance is defined as the mass per unit volume of that substance. The average density is the average of densities of the substance in the multiple forms. For example, carbon has two isotopes having different masses. Then the average density of carbon is equal to the average mass of carbon divided by its volume.

Answer to Problem 62E

The average density of carbon atom is $\text{10}\text{.5 }\text{g}/{\text{cm}}^{\text{3}}$.

Explanation of Solution

The formula of the density of a substance is shown below.

$\text{Density}=\frac{\text{Mass}}{\text{Volume}}$

The average density of the substance is equal to the average of the densities of the substance.

${\text{Density}}_{\text{average}}=\frac{{\text{Density}}_{\text{1}}+{\text{Density}}_{\text{2}}+\mathrm{...}+{\text{Density}}_{\text{n}}}{\text{n}}$

In the case of isotopes, the mass of atom only changes not the volume of atom. Therefore, the average density in case of isotopes can be calculated by the formula shown below.

$\text{Density}=\frac{\text{Average Atomic Mass}}{\text{Volume of atom}}$…(1)

The average atomic mass of carbon atom is $12.01\text{ u}$.

The volume of carbon atom is given $1.9\times {10}^{-24}{\text{ cm}}^3$.

Substitute the values of average atomic mass and volume in equation (1).

$\text{Density}=\frac{12.01\text{ u}}{1.9\times {10}^{-24}{\text{ cm}}^3}$

The relation used to convert the atomic mass unit into grams is $\text{1g}=6.02\times {10}^{23}\text{ u}$.

$\begin{array}{rll}\text{Density}& =& \frac{12.01\text{ u}}{1.9\times {10}^{-24}{\text{ cm}}^3}\times \frac{\text{1 g}}{6.02\times {10}^{23}\text{ u}}\\ & =& \frac{12.01\text{ g}}{11.438\times {10}^{-1}{\text{ cm}}^3}\\ & =& \text{10}\text{.5 }\text{g}/{\text{cm}}^{\text{3}}\end{array}$

The average density of carbon atom is $\text{10}\text{.5 }\text{g}/{\text{cm}}^{\text{3}}$.

Conclusion

The average density of carbon atom is calculated as $\text{10}\text{.5 }\text{g}/{\text{cm}}^{\text{3}}$.

Interpretation Introduction

(b)

Interpretation:

The reason as to why the density of the carbon atom is so much larger than any of its form is to be stated.

Concept introduction:

The density of a substance is defined as the mass per unit volume of that substance. The average density is the average of densities of the substance in the multiple forms. For example, carbon has two isotopes having different masses. Then the average density of carbon is equal to the average mass of carbon divided by its volume.

Answer to Problem 62E

The density of carbon atom is more because in the carbon forms, the packing of atoms in the crystals is associated with an increase in the volume due to spaces in the crystal. The density is inversely proportional to the volume.

Explanation of Solution

The density of the carbon atom is larger than either of its form as calculated above in part (a). This is because while the formation of the carbon solid the packing of the atoms in the crystal is associated with some spaces. These spaces present in the crystal creates an extra volume due to which the density of the carbon forms falls down. The density is inversely proportional to the volume. More is the volume, lesser will be the density.

Conclusion

The reason why the carbon atom has more density than either of its forms is rightfully stated above.

Interpretation Introduction

(c)

Interpretation:

The volume of the nucleus of the carbon atom is to be stated.

Concept introduction:

The radius of the nucleus of an atom is very much smaller than the radius of the atom. All the mass of an atom lies inside the nucleus of an atom. The mass of an atom is due to its neutrons and protons and they are present in the nucleus.

Answer to Problem 62E

The volume of the nucleus of the carbon atom is $2\times {10}^{-39}{\text{ cm}}^3$.

Explanation of Solution

It is given that. the radius of the carbon atom is roughly $1\times {10}^5$ times larger than its volume. The atoms and nucleus are spherical in shape. The volume of the sphere is proportional to the cube of the radius. Therefore, the volume of carbon atoms is ${\left(1\times {10}^5\right)}^3$ times the volume of the nucleus.

Mathematically,

$\text{Volume of carbon atom}=\text{Volume of nucleus}\times {\left(1\times {10}^5\right)}^3$

The volume of carbon atom is given as $1.9\times {10}^{-24}{\text{ cm}}^3$.

Substitute the value of volume of carbon atom and rearrange to find out volume of nucleus as shown below.

$\begin{array}{rll}1.9\times {10}^{-24}{\text{ cm}}^3& =& \text{Volume of nucleus}\times {\left(1\times {10}^5\right)}^3\\ \text{Volume of nucleus}& =& \frac{1.9\times {10}^{-24}{\text{ cm}}^3}{{\left(1\times {10}^5\right)}^3}\\ & =& \frac{1.9\times {10}^{-24}{\text{ cm}}^3}{1\times {10}^{15}}\\ & =& 1.9\times {10}^{-39}{\text{ cm}}^3\approx 2\times {10}^{-39}{\text{ cm}}^3\end{array}$

The volume of nucleus of carbon atom is $2\times {10}^{-39}{\text{ cm}}^3$.

Conclusion

The volume of nucleus of carbon atom is calculated as $2\times {10}^{-39}{\text{ cm}}^3$.

Interpretation Introduction

(d)

Interpretation:

The average density of the carbon nucleus is to be stated.

Concept introduction:

The density of a substance is defined as the mass per unit volume of that substance. The average density is the average of densities of the substance in the multiple forms. For example, carbon has two isotopes having different masses. Then the average density of carbon is equal to the average mass of carbon divided by its volume.

Answer to Problem 62E

The average density of the carbon nucleus is $1\times {10}^{16}\text{g}/{\text{cm}}^{\text{3}}$.

Explanation of Solution

The formula of the density of a substance is shown below.

$\text{Density}=\frac{\text{Mass}}{\text{Volume}}$

The average density of the substance is equal to the average of the densities of the substance.

${\text{Density}}_{\text{average}}=\frac{{\text{Density}}_{\text{1}}+{\text{Density}}_{\text{2}}+\mathrm{...}+{\text{Density}}_{\text{n}}}{\text{n}}$

In the case of isotopes, the mass of nucleus only changes not the volume of nucleus therefore; the average density in case of isotopes can be calculated by the formula shown below.

$\text{Density}=\frac{\text{Average Mass of nucleus}}{\text{Volume of nucleus}}$…(1)

The average atomic mass of the carbon nucleus is $12.01\text{ u}$.

The volume of the carbon nucleus is calculated above is $2\times {10}^{-39}{\text{ cm}}^3$.

Substitute the values of average atomic mass and volume in equation (1).

$\text{Density}=\frac{12.01\text{ u}}{2\times {10}^{-39}{\text{ cm}}^3}$

The relation to convert the atomic mass unit into grams is $\text{1g}=6.02\times {10}^{23}\text{ u}$.

Therefore,

$\begin{array}{rll}\text{Density}& =& \frac{12.01\text{ u}}{2\times {10}^{-39}{\text{ cm}}^3}\times \frac{\text{1 g}}{6.02\times {10}^{23}\text{ u}}\\ & =& \frac{12.01\text{ g}}{12.04\times {10}^{-16}{\text{ cm}}^3}\\ & =& 0.9975\times {10}^{16}\text{g}/{\text{cm}}^{\text{3}}\approx 1\times {10}^{16}\text{g}/{\text{cm}}^{\text{3}}\end{array}$

The average density of carbon nucleus is $1\times {10}^{16}\text{g}/{\text{cm}}^{\text{3}}$.

Conclusion

The average density of carbon nucleus is calculated as $1\times {10}^{16}\text{g}/{\text{cm}}^{\text{3}}$.

Interpretation Introduction

(e)

Interpretation:

The mass of the sphere completely filled with carbon nuclei is to be stated.

Concept introduction:

The density of a substance is defined as the mass per unit volume of that substance. The average density is the average of densities of the substance in the multiple forms. For example, carbon has two isotopes having different masses. Then the average density of carbon is equal to the average mass of carbon divided by its volume.

Answer to Problem 62E

The mass of the sphere completely filled with carbon nuclei is $4\times {10}^5\text{ tons}$.

Explanation of Solution

The volume of the sphere is given $4\times {10}^5{\text{ cm}}^3$.

The formula of the density of a substance is shown below.

$\text{Density}=\frac{\text{Mass}}{\text{Volume}}$

The density of carbon nuclei is calculated above to be $1\times {10}^{16}\text{g}/{\text{cm}}^{\text{3}}$.

The mass of carbon nuclei in one cubic centimeter of the volume is $1\times {10}^{16}\text{ g}$.

Therefore, the mass of carbon nuclei in the $4\times {10}^5{\text{ cm}}^3$ will be calculated as shown.

$\begin{array}{rll}\text{Mass of carbon nuclei}& =& 4\times {10}^{-5}{\text{ cm}}^3\times \frac{1\times {10}^{16}\text{ g}}{1{\text{ cm}}^3}\\ & =& 4\times {10}^{11}\text{ g}\end{array}$

The following are relations to be used to convert the mass in grams to tons.

$\text{1 lb}=\text{454 g}$ and $\text{1 ton}=\text{2000 lb}$

Use these relations to convert the mass of carbon nuclei in grams to tons as shown below.

$\begin{array}{rll}\text{Mass of carbon nuclei}& =& 4\times {10}^{11}\text{ g}\\ & =& 4\times {10}^{11}\text{ g}\times \frac{1\text{ lb}}{454\text{ g}}\times \frac{1\text{ ton}}{2000\text{ lb}}\\ & =& 4.40\times {10}^5\text{ tons}\\ & \approx & 4\times {10}^5\text{ tons}\end{array}$

The mass of sphere completely filled with carbon nuclei is $4\times {10}^5\text{ tons}$.

Conclusion

The mass of sphere completely filled with carbon nuclei is calculated as $4\times {10}^5\text{ tons}$.