Chapter 6, Problem 6.99QP

Interpretation Introduction

Interpretation: A gas mixture is give to have ${\text{N}}_{\text{2}}$, ${\text{H}}_2$ and ${\text{CH}}_4$ gas. The mole fraction of ${\text{H}}_2$ gas, the pressure of the gas mixture and the partial pressure of each constituent gas are to be calculated.

Concept introduction: The mole fraction of ${\text{H}}_2$ gas is calculated by the formula,

$X_{{\text{H}}_{\text{2}}}=\frac{n_{{\text{H}}_{\text{2}}}}{n_{{\text{H}}_{\text{2}}}+n_{{\text{N}}_{\text{2}}}+n_{{\text{CH}}_{\text{4}}}}$

The partial pressure of the each constituent of gas is calculated by the ideal gas equation.

$PV=n\text{R}T$

The pressure of the gas mixture is sum of all partial pressure of each constituent of gas and it is calculated by the formula,

$P_{\text{mixture}}=P_{{\text{H}}_{\text{2}}}+P_{{\text{N}}_{\text{2}}}+P_{{\text{CH}}_{\text{4}}}$

To determine: The mole fraction of ${\text{H}}_2$ gas, the pressure of the gas mixture and the partial pressure of each constituent gas.

Answer to Problem 6.99QP

The mole fraction of ${\text{H}}_2$ gas is $\underset{\_}{0.20}$, the pressure of the gas mixture is $\underset{\_}{2.44atm}$ and the partial pressure of constituent ${\text{H}}_2$ gas is $\underset{\_}{0.49atm}$, the partial pressure of constituent ${\text{N}}_2$ gas is $\underset{\_}{1.7atm}$, the partial pressure of constituent ${\text{CH}}_4$ gas is $\underset{\_}{0.25atm}$.

Explanation of Solution

Given

The number of moles of ${\text{H}}_2$ gas is $0.20\text{mol}$.

The number of moles of ${\text{N}}_2$ gas is $0.70\text{mol}$.

The number of moles of ${\text{CH}}_4$ gas is $0.10\text{mol}$.

The temperature of the gas mixture is $\left(T\right)$ is $27°\text{C}$.

The volume of gas mixture $\left(V\right)$ is $10.0\text{L}$.

The value of universal gas constant $\left(\text{R}\right)$ is $0.082057{\text{LatmK}}^{-1}{\text{mol}}^{-1}$.

The relation between degree Celsius and Kelvin is,

$0°\text{C}=0+273.15\text{K}$

Apply this relation for the conversion of given temperature into Kelvin.

$\begin{array}{c}27°\text{C}=27+273.15\text{K}\\ =300.15\text{K}\end{array}$

The mole fraction of ${\text{H}}_2$ gas $\left(X_{{\text{H}}_{\text{2}}}\right)$ is calculated by the formula,

$X_{{\text{H}}_{\text{2}}}=\frac{n_{{\text{H}}_{\text{2}}}}{n_{{\text{H}}_{\text{2}}}+n_{{\text{N}}_{\text{2}}}+n_{{\text{CH}}_{\text{4}}}}$ (1)

Where,

• $n_{{\text{H}}_{\text{2}}}$ is number of moles of ${\text{H}}_2$ gas.
• $n_{{\text{N}}_2}$ is number of moles of ${\text{N}}_2$ gas.
• $n_{{\text{CH}}_4}$ is number of moles of ${\text{CH}}_4$ gas.

Substitute the values of $n_{{\text{H}}_{\text{2}}}$, $n_{{\text{N}}_2}$ and $n_{{\text{CH}}_4}$ in equation (1).

$\begin{array}{l}{X}_{{\text{H}}_{\text{2}}}=\frac{0.20\text{mol}}{0.70\text{mol}+0.70\text{mol}+0.10\text{mol}}\\ X_{{\text{H}}_{\text{2}}}=\frac{0.20\text{mol}}{1\text{mol}}\\ X_{{\text{H}}_{\text{2}}}=0.20\end{array}$

The partial pressure of ${\text{H}}_{\text{2}}$ gas $\left(P_{{\text{H}}_{\text{2}}}\right)$ is calculated by the ideal gas equation.

$P_{{\text{H}}_{\text{2}}}V=n_{{\text{H}}_{\text{2}}}\text{R}T$ (2)

Where,

• $V$ is volume of gas mixture.
• $T$ is temperature of gas mixture.
• $\text{R}$ is universal gas constant.
• $n_{{\text{H}}_{\text{2}}}$ is number of moles of ${\text{H}}_2$ gas.

Rearrange the equation (2).

$P_{{\text{H}}_{\text{2}}}=\frac{n_{{\text{H}}_{\text{2}}}\text{R}T}{V}$ (3)

Substitute the values of $V$, $T$, $\text{R}$ and $n_{{\text{H}}_{\text{2}}}$ in equation (3).

$\begin{array}{c}{P}_{{\text{H}}_{\text{2}}}=\frac{0.20\text{mol}\times 0.082057{\text{LatmK}}^{-1}{\text{mol}}^{-1}\times 300.15\text{K}}{10.0\text{L}}\\ =\frac{4.92\text{atmL}}{10.0\text{L}}\\ =0.49\text{atm}\end{array}$

Rewrite the equation (3) for the calculation of partial pressure of ${\text{N}}_2$ gas $\left(P_{{\text{N}}_2}\right)$.

$P_{{\text{N}}_2}=\frac{n_{{\text{N}}_2}\text{R}T}{V}$ (4)

Where,

• $n_{{\text{N}}_2}$ is number of moles of ${\text{N}}_2$ gas.

Substitute the values of $V$, $T$, $\text{R}$ and $n_{{\text{N}}_2}$ in equation (4).

$\begin{array}{c}{P}_{{\text{H}}_{\text{2}}}=\frac{0.70\text{mol}\times 0.082057{\text{LatmK}}^{-1}{\text{mol}}^{-1}\times 300.15\text{K}}{10.0\text{L}}\\ =\frac{17.24\text{atmL}}{10.0\text{L}}\\ =1.7\text{atm}\end{array}$

Rewrite the equation (3) for the calculation of partial pressure of ${\text{CH}}_4$ gas $\left(P_{{\text{CH}}_4}\right)$.

$P_{{\text{CH}}_4}=\frac{n_{{\text{CH}}_4}\text{R}T}{V}$ (5)

Where,

• $n_{{\text{CH}}_4}$ is number of moles of ${\text{CH}}_4$ gas.

Substitute the values of $V$, $T$, $\text{R}$ and $n_{{\text{CH}}_4}$ in equation (5).

$\begin{array}{c}{P}_{{\text{CH}}_4}=\frac{0.10\text{mol}\times 0.082057{\text{LatmK}}^{-1}{\text{mol}}^{-1}\times 300.15\text{K}}{10.0\text{L}}\\ =\frac{2.5\text{atmL}}{10.0\text{L}}\\ =0.25\text{atm}\end{array}$

The pressure of the gas mixture $\left(P_{\text{mixture}}\right)$ is sum of all partial pressure of each constituent of gas and it is calculated by the formula,

$P_{\text{mixture}}=P_{{\text{H}}_{\text{2}}}+P_{{\text{N}}_{\text{2}}}+P_{{\text{CH}}_{\text{4}}}$ (6)

Where,

• $P_{{\text{H}}_{\text{2}}}$ is partial pressure of ${\text{H}}_{\text{2}}$ gas.
• $P_{{\text{N}}_{\text{2}}}$ is partial pressure of ${\text{N}}_{\text{2}}$ gas.
• $P_{{\text{CH}}_4}$ is partial pressure of ${\text{CH}}_4$ gas.

Substitute the values of $P_{{\text{H}}_{\text{2}}}$, $P_{{\text{N}}_{\text{2}}}$ and $P_{{\text{CH}}_4}$ in equation (6).

$\begin{array}{c}{P}_{\text{mixture}}=0.49\text{atm}+1.7\text{atm}+0.25\text{atm}\\ =\text{2}\text{.44}\text{atm}\end{array}$

Thus, the mole fraction of ${\text{H}}_2$ gas is $\underset{\_}{0.20}$, the pressure of the gas mixture is $\underset{\_}{2.44atm}$ and the partial pressure of constituent ${\text{H}}_2$ gas is $\underset{\_}{0.49atm}$, the partial pressure of constituent ${\text{N}}_2$ gas is $\underset{\_}{1.7atm}$, the partial pressure of constituent ${\text{CH}}_4$ gas is $\underset{\_}{0.25atm}$.

Conclusion

The mole fraction of ${\text{H}}_2$ gas is $\underset{\_}{0.20}$, the pressure of the gas mixture is $\underset{\_}{2.44atm}$ and the partial pressure of constituent ${\text{H}}_2$ gas is $\underset{\_}{0.49atm}$, the partial pressure of constituent ${\text{N}}_2$ gas is $\underset{\_}{1.7atm}$, the partial pressure of constituent ${\text{CH}}_4$ gas is $\underset{\_}{0.25atm}$.