Question

me n₂ gas is mixed with some o₂ gas, and the sketch below shows a representative sample of the mixture. the total pressure of the mixture is measured, and found to be 250. torr. 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. gas mole fraction partial pressure n₂ o₂

Explanation:

Step1: Count the number of molecules

From the diagram, count the number of \( N_2 \) (blue) and \( O_2 \) (red) molecules. Let's assume:
- Number of \( N_2 \) molecules (\( n_{N_2} \)) = 4
- Number of \( O_2 \) molecules (\( n_{O_2} \)) = 14

Step2: Calculate total moles

Total moles (\( n_{total} \)) = \( n_{N_2} + n_{O_2} = 4 + 14 = 18 \)

Step3: Calculate mole fraction of \( N_2 \)

Mole fraction of \( N_2 \) (\( X_{N_2} \)) = \( \frac{n_{N_2}}{n_{total}} = \frac{4}{18} \approx 0.222 \)

Step4: Calculate mole fraction of \( O_2 \)

Mole fraction of \( O_2 \) (\( X_{O_2} \)) = \( \frac{n_{O_2}}{n_{total}} = \frac{14}{18} \approx 0.778 \)

Step5: Calculate total pressure

Wait, the problem says total pressure is 250 torr (from the text: "the total pressure of the mixture is measured, and found to be 250. torr").

Step6: Calculate partial pressure of \( N_2 \)

Partial pressure of \( N_2 \) (\( P_{N_2} \)) = \( X_{N_2} \times P_{total} = 0.222 \times 250 \approx 55.5 \) torr

Step7: Calculate partial pressure of \( O_2 \)

Partial pressure of \( O_2 \) (\( P_{O_2} \)) = \( X_{O_2} \times P_{total} = 0.778 \times 250 \approx 194.5 \) torr (or check: 250 - 55.5 = 194.5, which is 195 when rounded to 3 sig figs? Wait, let's recalculate with more precision.

Wait, \( X_{N_2} = 4/18 = 2/9 \approx 0.222222... \)

\( P_{N_2} = (2/9) \times 250 = 500/9 \approx 55.555... \approx 55.6 \) torr (3 sig figs)

\( X_{O_2} = 14/18 = 7/9 \approx 0.777777... \)

\( P_{O_2} = (7/9) \times 250 = 1750/9 \approx 194.444... \approx 194 \) torr? Wait, no, 1750/9 is approximately 194.44, which is 194 when rounded to 3 sig figs? Wait, 194.44 rounded to 3 sig figs is 194? Wait, 194.44: the first three digits are 1,9,4. The next digit is 4, which is less than 5, so it stays 194. But wait, 7/9 * 250: 250 * 7 = 1750; 1750 / 9 ≈ 194.444... So 194 torr (3 sig figs). Wait, but let's check the mole fractions again.

Wait, maybe I miscounted the molecules. Let's look at the diagram again. The red (O2) molecules: let's count. The red molecules: let's see, the first row: 2, then 3, then 3, then 2? Wait, maybe my initial count was wrong. Let's recount:

Looking at the diagram:

Blue (N2) molecules: let's count. The blue ones: 4? Wait, the key: blue is nitrogen, red is oxygen.

Looking at the top diagram:

Blue (N2) molecules: 4 (three blue pairs? Wait, no, each blue is a molecule? Wait, the diagram: each blue is a N2 molecule (since N2 is diatomic, so each blue pair is N2? Wait, no, the key: the molecules are diatomic? Wait, the red are O2 (diatomic), blue are N2 (diatomic). So each blue is a N2 molecule (diatomic), each red is O2 (diatomic). So count the number of blue (N2) molecules: let's see, the blue ones: 4? Wait, the image:

Blue molecules: let's count. The blue dots (molecules): 4? Wait, no, looking at the image:

Blue (N2) molecules: 4 (positions: 3 blue? Wait, maybe I made a mistake. Let's check the original image again. The user's image: "the sketch below shows a representative sample of the mixture." The key: red is oxygen, blue is nitrogen.

Looking at the diagram:

Blue (N2) molecules: 4? Wait, no, let's count:

First blue: left middle, second: upper middle, third: upper right, fourth: lower middle. Wait, maybe 4? Wait, the red (O2) molecules: let's count. Let's see, the red ones: let's count each red molecule (diatomic O2). Let's count:

Top row: 2 (first red), then 1 (second), then 1 (third) – no, maybe better to count all red:

Looking at the image, the red molecules (O2) are:

Let's list them:

1. Top red
2. Upper right red
3. Middle right red
4. Middle red
5. Lower middle red
6. Lower red
7. Left red
8. Middle left red
9. Upper middle red
10. Middle upper red
11. Middle lower red
12. Right middle red
13. Lower right red
14. Upper left red? Wait, maybe 14? Wait, maybe my initial count was correct. Let's proceed with n_N2 = 4, n_O2 = 14, total n = 18.

So mole fraction N2: 4/18 = 0.222, O2: 14/18 = 0.778.

Total pressure P_total = 250 torr.

Partial pressure N2: 0.222 * 250 = 55.5 torr (3 sig figs: 55.6 torr? Wait, 0.222222... * 250 = 55.555... ≈ 55.6 torr)

Partial pressure O2: 0.777777... * 250 = 194.444... ≈ 194 torr (3 sig figs)

Wait, but let's check with exact fractions:

n_N2 = 4, n_O2 = 14, total n = 18.

X_N2 = 4/18 = 2/9 ≈ 0.2222

X_O2 = 14/18 = 7/9 ≈ 0.7778

P_N2 = (2/9)*250 = 500/9 ≈ 55.555... ≈ 55.6 torr (3 sig figs)

P_O2 = (7/9)*250 = 1750/9 ≈ 194.444... ≈ 194 torr (3 sig figs)

So filling the table:

For N2:

mole fraction: 0.222

partial pressure: 55.6 torr

For O2:

mole fraction: 0.778

partial pressure: 194 torr (or 195? Wait, 1750/9 is 194.444, which is 194 when rounded to 3 sig figs? Wait, 194.444: the third sig fig is 4, the next digit is 4, so we keep it 194. But sometimes, 194.444 is rounded to 194 (3 sig figs) or 195? Wait, no, 194.444: first three digits 1,9,4. The fourth digit is 4, which is less than 5, so we don't round up. So 194 torr.

But let's confirm the mole counts again. If the number of N2 molecules is 4 and O2 is 14, then total is 18. That seems correct from the diagram.

Answer:

For \( N_2 \):

• Mole fraction: \( \boxed{0.222} \)
• Partial pressure: \( \boxed{55.6} \) torr

For \( O_2 \):

• Mole fraction: \( \boxed{0.778} \)
• Partial pressure: \( \boxed{194} \) torr (or \( \boxed{195} \) if we consider 194.444... rounded to 3 sig figs, but 194 is more accurate with 4/18 and 14/18)