Chapter 14, Problem 100A

(a)

Interpretation Introduction

Interpretation: Using the given volume of one hydrogen molecule and other information given in the question, the percentage of the volume of gas occupied by its molecules needs to be determined.

Concept Introduction: The number of moles, pressure, volume, and temperature are related to each other as follows:

  $PV=nRT$

Here, P is pressure, V is volume, n is the number of moles, R is the Universal gas constant, and T is temperature.

Explanation of Solution

The volume of the container is 0.10 L. The number of hydrogen gas molecules in the container at 100 kPa and $0{}^{\text{o}}\text{C}$ is $3.0\times {10}^{20}$ . If the volume of the hydrogen molecule is $6.7\times {10}^{-24}\text{ mL}$, the percentage of the volume of gas occupied by its molecules can be calculated as follows:

The volume of hydrogen gas in a 0.10 L container can be calculated using the ideal gas equation as follows:

  $PV=nRT$

Here, P is 101 kPa or 0.9968 atm and T is $0{}^{\text{o}}\text{C}$ or 273.15 K. Now, the number of ${\text{H}}_{\text{2}}$ molecules is given $3.0\times {10}^{20}\text{ molecules}$ , and the number of moles can be calculated as follows:

  $\begin{array}{c}n=\frac{1\text{ mol}}{6.022\times {10}^{23}}\left(3.0\times {10}^{20}\right)\\ =4.982\times {10}^{-4}\text{ mol}\end{array}$

Now, the volume of hydrogen gas can be calculated as follows:

  $V=\frac{nRT}{P}=\frac{\left(4.982\times {10}^{-4}\text{ mol}\right)\left(0.082\text{ L atm/K mol}\right)\left(273.15\text{ K}\right)}{\left(0.9968\text{ atm}\right)}$

Or,

  $V=0.0112\text{ L}$

The percentage of the volume of gas occupied by its molecule can be calculated by the unitary method. The total volume is taken as 0.0112 L. The volume of 1 hydrogen molecule is $6.7\times {10}^{-24}\text{ mL}$ thus, the volume of $3.0\times {10}^{20}$ hydrogen molecules can be calculated as follows:

  $\begin{array}{c}{V}_{{\text{H}}_2\text{ molecules }}=\frac{6.7\times {10}^{-24}\text{ mL}}{1{\text{ H}}_2\text{ molecule}}\times 3.0\times {10}^{20}{\text{ H}}_2\text{ molecules }\\ \text{=0}\text{.002 mL}\end{array}$

Now, the volume percentage of hydrogen gas will be:

  $\%V_{{\text{H}}_2\text{ molecules }}=\frac{V_{{\text{H}}_2\text{ molecules }}}{V}\times 100\%$

Put the values,

  $\begin{array}{c}\%V_{{\text{H}}_2\text{ molecules }}=\frac{0.002\text{ mL}}{0.0112\text{ L}\left(\frac{{10}^3\text{ mL}}{1\text{ L}}\right)}\times 100\%\\ =0.018\%\end{array}$

Therefore, the volume percentage of gas will be 0.018%.

(b)

Interpretation Introduction

Interpretation: The fraction of total volume occupied by hydrogen molecules needs to be calculated if pressure is increased to 100,000 kPa and the volume of the gas is $1\times {10}^{-4}\text{ L}$ .

Concept Introduction: The number of moles, pressure, volume, and temperature are related to each other as follows:

  $PV=nRT$

Here, P is pressure, V is volume, n is the number of moles, R is the Universal gas constant and T is temperature.

Explanation of Solution

The fraction of the total volume occupied by hydrogen molecules can be calculated as follows:

  $f_{V_{{\text{H}}_2\text{ molecules }}}=\frac{V_{{\text{H}}_2\text{ molecules }}}{V}$

Put the values,

  $\begin{array}{c}\%V{f}_{V_{{\text{H}}_2\text{ molecules }}}=\frac{0.002\text{ mL}}{\text{1}\times {\text{10}}^{-4}\text{ L}\left(\frac{{10}^3\text{ mL}}{1\text{ L}}\right)}\\ =0.02\end{array}$

Therefore, fraction of the total volume occupied by hydrogen molecules is 0.02.