Chapter 3, Problem 56E

Calculate the number of atoms in the universe. The following steps will guide you through this calculation:

a. Planets constitute less than 1% of the total mass of the universe and can, therefore, be neglected. Stars make up most of the visible mass of the universe, so we need to determine how many atoms are in a star. Stars are primarily composed of hydrogen atoms and our Sun is an average-sized star. Calculate the number of hydrogen atoms in our Sun given that the radius of the Sun is $\text{7}\times {\text{10}}^{\text{8}}$ m and its density is $\text{1}{\text{.4 g/cm}}^{\text{3}}$. The volume of a sphere is given by

$V\text{=}\left(\frac{4}{3}\right)\times \pi r^3$

(Hint: Use the volume and the density to get the mass of the Sun.)

b. The average galaxy (like our own Milky Way galaxy) contains $\text{1}\times {\text{10}}^{\text{11}}$ stars, and the universe contains $\text{1}\times {\text{10}}^{\text{9}}$ galaxies. Calculate the number of atoms in an average galaxy and finally the number of atoms in the entire universe.

c. You can hold $\text{1}\times {\text{10}}^{\text{23}}$ atoms in your hand (five copper pennies constitute $\text{1}\text{.4}\times {\text{10}}^{\text{23}}$ copper atoms.) How does this number compare with the number of atoms in the universe?

Interpretation Introduction

Interpretation:

The number of hydrogen atoms in the sun and the number of atoms in a random galaxy are to be determined. Also, the number of hydrogen atoms in the universeis to be compared with the number of atoms that can be held in a hand.

Concept Introduction:

Density is the ratio of the mass of a substance to its volume.

$\text{Density of element }=\frac{\text{Mass of element in solution}}{\text{Volume of solutions}}$

The volume of a sphere is $\frac{4}{3}\pi r^3$.

Mole is the unit of the amount of a substance. It relates number of particles to the molar mass.

One mole of a substance contains Avogadro’s number ($6.022\times {10}^{23}$) of particles. The number remains constant regardless of the nature of the substance. One mole of an atom contains $6.022\times {10}^{23}$ atoms. One mole of a molecule contains $6.022\times {10}^{23}$ molecules.

It also relates to the molar mass of a substance in grams. The weight of one mole of a substance is equivalent to its molar mass in grams.

It is known that $\text{1m }=100\text{ cm}$, hence, conversion factor is as:

$\frac{\text{100 cm}}{\text{1 m}}$

Answer to Problem 56E

Solution: a) ${\text{6 × 10}}^{\text{56}}\text{ atoms}$ b) $6\times {\text{ 10}}^{76}\text{ atoms}$ c) The number of atoms that can be held in a hand is minuscule compared to the number of hydrogen atoms present in the universe.

Explanation of Solution

a) The number of hydrogen atoms present in the sun.

The radius of the sun is $7\times {10}^8\text{ m}$ and its density is $1.4{\text{ g/cm}}^3$.

The number of hydrogen atoms present in the sun is calculated as:

Conversion of the radius of the sun from m to cm is as:

$\begin{array}{c}\text{Radius of sun}=7\times {10}^8\text{ m }\\ \text{Radius of sun = 7}\times {10}^8\cancel{\text{m}}\text{ × }\frac{\text{100 cm}}{\text{1 }\cancel{\text{m}}}\\ \text{ = 7}\times {10}^{10}\text{ cm}\end{array}$

The volume of a sphere is $\frac{4}{3}\pi r^3$

Substitute the values in the above equation:

$\begin{array}{c}\text{Volume of sphere }=\frac{4}{3}\times 3.14\times {(}^7\\ =\text{ 1}\times {10}^{33}{\text{ cm}}^3\end{array}$

Thus, the volume of the sphere is $\text{ 1}\times {10}^{33}{\text{ cm}}^3$.

The mass of hydrogen in sun using the values of density and volume is calculated as:

$\begin{array}{c}\text{Density of hydrogen }=\frac{\text{Mass of hydrogen in sun}}{\text{Volume of sun}}\\ \text{Mass of hydrogen in sun }=\text{ Density of hydrogen}\times \text{ Volume of sun}\end{array}$

Substitute the values in the above equation:

$\begin{array}{c}\text{ Mass}=\text{1}\text{.4 }\frac{\text{g}}{\cancel{{\text{cm}}^{\text{3}}}}\times \text{ 1}\times {\text{ 10}}^{33}\cancel{{\text{cm}}^3}\\ \text{Mass}=1\times {10}^{33}\text{ g}\end{array}$

Therefore, the mass of hydrogen in sun is $1\times {10}^{33}\text{ g}$.

The number of atoms of hydrogen is calculated as follows:

$\begin{array}{c}\text{6}\text{.022 }\times {\text{10}}^{23}\text{ atoms of H }=\text{ 1 mole of H }=\text{ 1 g}\\ \text{1 g of H contains }=\text{ 6}\text{.022 }\times {\text{10}}^{23}\text{ atoms of H}\\ 1\times {\text{10}}^{33}\text{ g of H contains }=\frac{6.022\times {10}^{23}\text{ atoms}}{\text{1}\cancel{\text{g}}}\times 1\times {\text{10}}^{33}\cancel{\text{g}}\\ ={\text{ 6 × 10}}^{\text{56}}\text{ atoms (approx}\text{.)}\end{array}$

Hence, the number of atoms in hydrogen are ${\text{6 × 10}}^{\text{56}}\text{ atoms}$

b) The number of hydrogen atoms present in one galaxy and in universe.

The average galaxy consists of $1\times {10}^{11}\text{ stars}$ and a universe consists of $1\times {10}^9\text{ galaxies}$

The number of hydrogen atoms in one galaxy $\left({\text{N}}_{\text{g}}\right)$ is calculated as follows:

$\begin{array}{c}{\text{N}}_{\text{g}}=\text{ Number of hydrogen atoms in one star }\times \text{ Number of stars in a galaxy}\\ {\text{N}}_{\text{g}}=\text{ (6 }\times {\text{ 10}}^{56}\text{ atoms})\times \text{ (1}\times {\text{10}}^{11})\\ {\text{N}}_{\text{g}}=\text{6 }\times {\text{ 10}}^{67}\text{ atoms}\end{array}$

Therefore, hydrogen atoms present in one galaxy are $\text{6 }\times {\text{ 10}}^{67}\text{ atoms}$.

The number of hydrogen atoms in the universe $\left({\text{N}}_{\text{u}}\right)$ is calculated as follows: $\begin{array}{l}{\text{N}}_{\text{u}}=\text{ Number of hydrogen atoms in one galaxy }\times \text{ Number of galaxies in universe}\\ {\text{N}}_{\text{u}}=\text{ (6 }\times {\text{ 10}}^{67}\text{ atoms})\times \text{ (1}\times {\text{10}}^9)\\ {\text{N}}_{\text{u}}=6\times {\text{ 10}}^{76}\text{ atoms}\end{array}$

Thus, the number of hydrogen atoms in universe is $6\times {\text{ 10}}^{76}\text{ atoms}$.

c) Compare $1\times {10}^{23}$ number (can hold in hand) with number of atoms in the universe.

The atoms which can be held in a hand are $1\times {10}^{23}$.

The number of atoms that can be held in a hand are $1\times {10}^{23}$. It is minuscule in comparison to the number of hydrogen atoms present in the universe, which is $6\times {\text{ 10}}^{76}\text{ atoms}$.