Chapter 6, Problem 6.168AP

Interpretation Introduction

Interpretation: Uranus is given to have a total atmospheric pressure of $130\text{kPa}$ where the following gases are present: $83\%{\text{H}}_{\text{2}}$, $15\%\text{He}$, and $2\%{\text{CH}}_{\text{4}}$ by volume. The partial pressure of each gas in Uranus atmosphere is to be calculated.

Concept introduction: The mole fraction is defined as the ratio of the number of moles of each individual gas by the total numbers of moles which represent the entire component. The Dalton’s law of partial pressures state that “the total pressure exerted by a mixture of gases is equal to the sum of the partial pressures of the gases in the mixtures”.

To determine: The partial pressure of each gas in Uranus atmosphere.

Answer to Problem 6.168AP

Solution

The partial pressure of ${\text{H}}_{\text{2}}$ is $\underset{\_}{118.82kPa}$, the partial pressure of $\text{He}$ is $\underset{\_}{10.79kPa}$, the partial pressure of ${\text{CH}}_{\text{4}}$ is $\underset{\_}{0.26kPa}$.

Explanation of Solution

Explanation

The composition of gas mixtures are given in terms of their volumes, which is in mass percentage. First the mass percentage is converted into mole fraction of each component.

The mole fraction is defined as the ratio of the number of moles of each individual gas by the total numbers of moles which represent the entire component.

$\begin{array}{c}\text{Amount}\text{of}\text{sample}=83\text{g}\text{of}{\text{H}}_{\text{2}}+15\text{g}\text{of}\text{He}+2\text{g}\text{of}{\text{CH}}_4\\ =100\text{g}\end{array}$

The molar mass of each gas is calculated by the formula,

$\text{Mass}\text{of}\text{gass}=\text{Amount}\text{of}\text{gas}\times \frac{1\text{mol}\text{of}\text{gass}}{\text{Molecular}\text{mass}\text{of}\text{gas}}$

For hydrogen, the molecular mass of hydrogen molecule is $2.01588\text{g/mol}$.

Substitute the given values in above expression,

$\begin{array}{c}\text{Mass}\text{of}\text{gass}=\text{Amount}\text{of}\text{gas}\times \frac{1\text{mol}\text{of}\text{gass}}{\text{Molecular}\text{mass}\text{of}\text{gas}}\\ \text{Mass}\text{of}{\text{H}}_2=83\text{g}\text{of}{\text{H}}_{\text{2}}\times \frac{1\text{mol}\text{of}{\text{H}}_{\text{2}}}{2.0159\text{g}\text{of}{\text{H}}_{\text{2}}}\\ =41.17\text{moles}\text{of}{\text{H}}_{\text{2}}\end{array}$

For helium, the molecular mass of helium is $4.0026\text{g/mol}$.

Substitute the given values in above expression,

$\begin{array}{c}\text{Mass}\text{of}\text{gass}=\text{Amount}\text{of}\text{gas}\times \frac{1\text{mol}\text{of}\text{gass}}{\text{Molecular}\text{mass}\text{of}\text{gas}}\\ \text{Mass}\text{of}\text{He}=15\text{g}\text{of}\text{He}\times \frac{1\text{mol}\text{of}\text{He}}{4.0026\text{g}\text{of}\text{He}}\\ =3.748\text{moles}\text{of}\text{He}\end{array}$

For methane, the molecular mass of methane is $\text{16}\text{.0425}\text{g/mol}$.

Substitute the given values in above expression,

$\begin{array}{c}\text{Mass}\text{of}\text{gass}=\text{Amount}\text{of}\text{gas}\times \frac{1\text{mol}\text{of}\text{gass}}{\text{Molecular}\text{mass}\text{of}\text{gas}}\\ \text{Mass}\text{of}{\text{CH}}_4=2\text{g}\text{of}{\text{CH}}_4\times \frac{1\text{mol}\text{of}{\text{CH}}_4}{2.0159\text{g}\text{of}{\text{CH}}_4}\\ =0.1247\text{moles}\text{of}{\text{CH}}_4\end{array}$

$\begin{array}{c}\text{Total}\text{number}\text{of}\text{moles}=\text{Moles}\text{of}{\text{H}}_{\text{2}}+\text{Moles}\text{of}\text{He}+\text{Moles}\text{of}{\text{CH}}_{\text{4}}\\ =41.17\text{mol}+3.74\text{mol}+0.12\text{mol}\\ =45.03\text{mol}\end{array}$

The mole fraction of hydrogen gas is calculated by the formula,

${\chi }_{{\text{H}}_{\text{2}}}=\frac{\text{Moles}\text{of}{\text{H}}_{\text{2}}}{\text{Total}\text{numbers}\text{of}\text{moles}}$

Substitute the given values in the above expression,

$\begin{array}{c}{\chi }_{{\text{H}}_{\text{2}}}=\frac{\text{Moles}\text{of}{\text{H}}_{\text{2}}}{\text{Total}\text{numbers}\text{of}\text{moles}}\\ =\frac{41.17\text{mol}}{45.03\text{mol}}\\ =0.914\end{array}$

The mole fraction of helium gas is calculated by the formula,

${\chi }_{\text{He}}=\frac{\text{Moles}\text{of}\text{He}}{\text{Total}\text{numbers}\text{of}\text{moles}}$

Substitute the given values in the above expression,

$\begin{array}{c}{\chi }_{\text{He}}=\frac{\text{Moles}\text{of}\text{He}}{\text{Total}\text{numbers}\text{of}\text{moles}}\\ =\frac{3.74\text{mol}}{45.03\text{mol}}\\ =0.083\end{array}$

Similarly the mole fraction of ${\text{CH}}_4$ is calculated by the formula,

${\chi }_{{\text{CH}}_4}=\frac{\text{Moles}\text{of}{\text{CH}}_4}{\text{Total}\text{number}\text{of}\text{moles}}$

Substitute the given values in the above expression,

$\begin{array}{c}{\chi }_{{\text{CH}}_4}=\frac{\text{Moles}\text{of}{\text{CH}}_4}{\text{Total}\text{number}\text{of}\text{moles}}\\ =\frac{0.12\text{mol}}{45.03\text{mol}}\\ =0.002\end{array}$

According to Dalton’s law of pressure, partial pressure is the product of the mole fraction and the total atmospheric pressure.

The partial pressure of given gases is calculated by the formula,

$P_x={\chi }_x{P}_{\text{Total}}$

Where,

• $P_{\text{x}}$ is the partial pressure of $x=P_{{\text{H}}_{\text{2}}},P_{\text{He}},P_{{\text{CH}}_{\text{4}}}$.
• ${\chi }_x$ is the mole fraction of ${\chi }_{{\text{H}}_{\text{2}}}$, ${\chi }_{\text{He}}$, ${\chi }_{{\text{CH}}_{\text{4}}}$.
• $P_{\text{Total}}$ is the total atmospheric pressure $130\text{kPa}$.

Substitute the given values in the above expression,

$\begin{array}{c}{P}_{{\text{H}}_{\text{2}}}=0.914\times 130\text{kPa}\\ =\underset{\_}{118.82kPa}\end{array}$

$\begin{array}{c}{P}_{\text{He}}=0.083\times 130\text{kPa}\\ =\underset{\_}{10.79kPa}\end{array}$

$\begin{array}{c}{P}_{{\text{CH}}_{\text{4}}}=0.002\times 130\text{kPa}\\ =\underset{\_}{0.26kPa}\end{array}$

Therefore, the partial pressure of ${\text{H}}_{\text{2}}$ is $\underset{\_}{118.82kPa}$, the partial pressure of $\text{He}$ is $\underset{\_}{10.79kPa}$, the partial pressure of ${\text{CH}}_{\text{4}}$ is $\underset{\_}{0.26kPa}$.

Conclusion

The partial pressure of ${\text{H}}_{\text{2}}$ is $\underset{\_}{118.82kPa}$, the partial pressure of $\text{He}$ is $\underset{\_}{10.79kPa}$, the partial pressure of ${\text{CH}}_{\text{4}}$ is $\underset{\_}{0.26kPa}$