Chapter 10, Problem 74QP

Interpretation Introduction

Interpretation:

The mole fraction, partial pressure and total pressure are to be determined.

Concept introduction:

The mole fraction of an individual gas for the combination of gases is the ratio of the moles of the individual gas to the total number of moles of the gaseous mixture.

${\chi }_i=\frac{n_i}{n_{\text{total}}}$

Here, ${\chi }_i$ is the mole fraction, $n_i$ is the mole fraction of individual gas and $n_{\text{total}}$ is the total number of moles.

Also, the mole fraction of an individual gas for the combination of gases can be calculated from the ratio of the partial pressure of the individual gases with the total pressure of the combination.

${\chi }_i=\frac{P_i}{P_{\text{total}}}$

Here, ${\chi }_i$ is the mole fraction, $P_i$ is the partial pressure of individual gas and $P_{\text{total}}$ is the total pressure.

Ideal Gas Equation is given as

$\text{PV = nRT}$

where, $\text{P}$, pressure of the gas;  $\text{V}$, volume of the gas;  $\text{n}$, Number of Moles;  $\text{T}$, absolute temperature; $\text{R}$, Ideal Gas constant also known as Boltzmann Constant, $\text{0}{\text{.082057 L atm K}}^{\text{-1}}{\text{ mol}}^{\text{-1}}\text{.}$

Answer to Problem 74QP

Solution:

(a) All figures have the same mole fraction of gas A.

(b) Figure (iii)

(c) Figure (iii)

Explanation of Solution

a) Container that has the smallest mole fraction of gas A(red)

The red ball sphere represents the gas A.

The green ball sphere represents the gas B.

The blue ball sphere represents the gas C.

In the figure (i),

The number of moles of gas A is $3$.

The number of moles of gas B is $2$.

The number of moles of gas C is $4$.

Calculate the total number of moles as follows:

$n_{\text{total}}=n_{\text{A}}+n_{\text{B}}+n_{\text{C}}$

Substitute 3 for A, 2 for B and 4 for C in the above equation.

$\begin{array}{c}{n}_{\text{total}}=3+2+4\\ =9\end{array}$

Calculate the mole fraction of gas A as follows:

${\chi }_{{\text{A}}_1}=\frac{n_{\text{A}}}{n_{\text{total}}}$

Substitute $9$ for $n_{\text{total}}$ and $3$ for $n_{\text{A}}$ in the above equation.

$\begin{array}{c}{\chi }_{\text{A}}=\frac{3}{9}\\ =3\end{array}$

In the figure (ii),

The number of moles of gas A is $4$.

The number of moles of gas B is $3$.

The number of moles of gas C is $5$.

Calculate the total number of moles as follows:

$n_{\text{total}}=n_{\text{A}}+n_{\text{B}}+n_{\text{C}}$

Substitute 4 for A, 3 for B and 5 for C in the above equation.

$\begin{array}{c}{n}_{\text{total}}=4+3+5\\ =12\end{array}$

Calculate the mole fraction of gas A as follows:

${\chi }_{{\text{A}}_2}=\frac{n_{\text{A}}}{n_{\text{total}}}$

Substitute $12$ for $n_{\text{total}}$ and $4$ for $n_{\text{A}}$ in the above equation.

$\begin{array}{c}{\chi }_{{\text{A}}_2}=\frac{4}{12}\\ =3\end{array}$

In the figure (iii),

The number of moles of gas A is $5$.

The number of moles of gas B is $4$.

The number of moles of gas C is $6$.

Calculate the total number of moles as follows:

$n_{\text{total}}=n_{\text{A}}+n_{\text{B}}+n_{\text{C}}$

Substitute 5 for A, 4 for B and 6 for C in the above equation.

$\begin{array}{c}{n}_{\text{total}}=5+4+6\\ =15\end{array}$

Calculate the mole fraction of gas A as follows:

${\chi }_{{\text{A}}_3}=\frac{n_{\text{A}}}{n_{\text{total}}}$

Substitute $15$ for $n_{\text{total}}$ and $5$ for $n_{\text{A}}$ in the above equation.

${\chi }_{{\text{A}}_3}=\frac{5}{15}$

All the three diagrams have the same value of mole fraction of gas A, which is three.

Hence, all the three diagrams represent the same value of mole fraction of gas A.

b) Container that has the highest partial pressure of gas B (green)

The volume and temperature are constant for all the three figures, so, the total pressure will now depend directly on the number of moles.

For figure (i),

The total pressure of gas B in figure (i) is as follows:

$\begin{array}{c}{P}_{\text{total}}=n_{\text{total}}\\ =9\end{array}$

Calculate the mole fraction of gas B in figure (i) as follows:

${\chi }_{{\text{B}}_1}=\frac{n_{\text{B}}}{n_{\text{total}}}$

Substitute $9$ for $n_{\text{total}}$ and $2$ for $n_{\text{B}}$ in the above equation.

${\chi }_{{\text{B}}_1}=\frac{2}{9}$

Calculate the partial pressure of gas B in figure (i) as follows:

$P_{{\text{B}}_1}={\chi }_{\text{B}}\times P_{\text{total}}$

Substitute $\frac{2}{9}$ for ${\chi }_{\text{B}}$ and $9$ for $P_{\text{total}}$ in the above equation.

$\begin{array}{c}{P}_{{\text{B}}_1}=\frac{2}{9}\times 9\\ =2\end{array}$

For figure (ii),

The total pressure of gas B in figure (ii) is as follows:

$\begin{array}{c}{P}_{\text{total}}=n_{\text{total}}\\ =12\end{array}$

Calculate the mole fraction of gas B in figure (ii) as follows:

${\chi }_{{\text{B}}_2}=\frac{n_{\text{B}}}{n_{\text{total}}}$

Substitute $12$ for $n_{\text{total}}$ and $3$ for $n_{\text{B}}$ in the above equation.

$\begin{array}{c}{\chi }_{{\text{B}}_2}=\frac{3}{12}\\ =1/4\end{array}$

Calculate the partial pressure of gas B in figure (ii) as follows:

$P_{{\text{B}}_2}={\chi }_{\text{B}}\times P_{\text{total}}$

Substitute $\frac{1}{4}$ for ${\chi }_{\text{B}}$ and $12$ for $P_{\text{total}}$ in the above equation.

$\begin{array}{c}{P}_{{\text{B}}_{\text{2}}}=\frac{1}{4}\times 12\\ =3\end{array}$

For figure (iii),

The total pressure of gas B in figure (iii) is as follows:

$\begin{array}{c}{P}_{\text{total}}=n_{\text{total}}\\ =15\end{array}$

Calculate the mole fraction of gas B in figure (iii) as follows:

${\chi }_{{\text{B}}_2}=\frac{n_{\text{B}}}{n_{\text{total}}}$

Substitute $15$ for $n_{\text{total}}$ and $4$ for $n_{\text{B}}$ in the above equation.

${\chi }_{{\text{B}}_{\text{3}}}=\frac{4}{15}$

Calculate the partial pressure of gas B in figure (iii) as follows:

$P_{{\text{B}}_3}={\chi }_{\text{B}}\times P_{\text{total}}$

Substitute $\frac{4}{15}$ for ${\chi }_{\text{B}}$ and $15$ for $P_{\text{total}}$ in the above equation.

$\begin{array}{c}{P}_{{\text{B}}_3}=\frac{4}{15}\times 15\\ =4\end{array}$

Hence, the partial pressure of gas B is maximum in figure (iii).

c) Container that has the highest total pressure

The volume and temperature are constant for all the three figures, so, the total pressure will now depend directly on the number of moles.

Therefore, the total pressure for each diagram is calculated as follows:

The total pressure in figure (i) is:

$\begin{array}{c}{P}_{\text{total}}=n_{\text{total}}\\ =9\end{array}$

The total pressure in figure (ii) is:

$\begin{array}{c}{P}_{\text{total}}=n_{\text{total}}\\ =12\end{array}$

The total pressure in figure (iii) is:

$\begin{array}{c}{P}_{\text{total}}=n_{\text{total}}\\ =15\end{array}$

Hence, figure (iii) represents the maximum total pressure.