Some [tex]\( N_2 \)[/tex] gas is mixed with some [tex]\( O_2 \)[/tex] gas, and the sketch below shows a representative sample of the mixture. The total pressure of the mixture is measured and found to be 0.130 kPa.

Calculate the mole fraction and partial pressure of each gas in this mixture. Round your answers to 3 significant digits. You may assume each gas behaves as an ideal gas.

[tex]\[
\begin{tabular}{|c|c|c|}
\hline
Gas & Mole Fraction & Partial Pressure (kPa) \\
\hline
\( N_2 \) & \(\square\) & \(\square\) kPa \\
\hline
\( O_2 \) & \(\square\) & \(\square\) kPa \\
\hline
\end{tabular}
\][/tex]



Answer :

Let's walk through the solution step-by-step for calculating the mole fraction and partial pressure of each gas in the mixture.

### Step 1: Understanding the Composition
From the representative sample in the sketch, we observe that there are:
- 2 parts Nitrogen ([tex]$N_2$[/tex])
- 1 part Oxygen ([tex]$O_2$[/tex])

This gives us a total of [tex]\( 2 + 1 = 3 \)[/tex] parts.

### Step 2: Calculating Mole Fractions
The mole fraction for a gas is given by the ratio of the number of moles of that gas to the total number of moles in the mixture.

- For Nitrogen ([tex]$N_2$[/tex]):
[tex]\[ \text{Mole fraction of } N_2 = \frac{\text{Number of parts of } N_2}{\text{Total number of parts}} = \frac{2}{3} \approx 0.667 \][/tex]

- For Oxygen ([tex]$O_2$[/tex]):
[tex]\[ \text{Mole fraction of } O_2 = \frac{\text{Number of parts of } O_2}{\text{Total number of parts}} = \frac{1}{3} \approx 0.333 \][/tex]

### Step 3: Calculating Partial Pressures
The partial pressure of a gas in a mixture is given by multiplying the mole fraction of the gas by the total pressure of the mixture.

Given:
[tex]\[ \text{Total pressure} = 0.130 \text{ kPa} \][/tex]

- For Nitrogen ([tex]$N_2$[/tex]):
[tex]\[ \text{Partial pressure of } N_2 = \text{Mole fraction of } N_2 \times \text{Total pressure} = 0.667 \times 0.130 \approx 0.087 \text{ kPa} \][/tex]

- For Oxygen ([tex]$O_2$[/tex]):
[tex]\[ \text{Partial pressure of } O_2 = \text{Mole fraction of } O_2 \times \text{Total pressure} = 0.333 \times 0.130 \approx 0.043 \text{ kPa} \][/tex]

### Step 4: Presenting the Results
Now we can present the computed values in a tabular format.

[tex]\[ \begin{tabular}{|c|c|c|} \hline \text{Gas} & \text{Mole fraction} & \text{Partial pressure (kPa)} \\ \hline N_2 & 0.667 & 0.087 \\ \hline O_2 & 0.333 & 0.043 \\ \hline \end{tabular} \][/tex]

So, the solution is:

- The mole fraction of [tex]\(N_2\)[/tex] is [tex]\(0.667\)[/tex] and its partial pressure is [tex]\(0.087 \text{ kPa}\)[/tex].
- The mole fraction of [tex]\(O_2\)[/tex] is [tex]\(0.333\)[/tex] and its partial pressure is [tex]\(0.043 \text{ kPa}\)[/tex].