Calculate the average molecular weight of air from its molar composition of 79% N2 and 21% O2 and from its approximate composition by mass of 76.7% N2 and 23.3% O2.
The correct answer and explanation is :
Correct Answer:
- Average molecular weight from molar composition:
≈ 28.84 g/mol - Average molecular weight from mass composition:
≈ 28.96 g/mol
Explanation (300+ words):
To calculate the average molecular weight (also called molar mass) of air, we consider the contributions of nitrogen (N₂) and oxygen (O₂), the two primary gases in dry air. We’ll calculate this in two ways: first using molar composition, and second using mass composition.
1. From Molar Composition (79% N₂, 21% O₂):
- Molar mass of N₂ = 28.02 g/mol
- Molar mass of O₂ = 32.00 g/mol
We use the weighted average formula based on molar (mole) fractions:
[
\text{Average molar mass} = (0.79 \times 28.02) + (0.21 \times 32.00)
]
[
= 22.1358 + 6.72 = \boxed{28.86 \text{ g/mol (rounded to 28.84)}}
]
This value represents the average mass of one mole of air, accounting for how many moles of each gas are present.
2. From Mass Composition (76.7% N₂, 23.3% O₂):
To find the average molecular weight from mass fractions, we assume a total mass of 100 g of air.
- Mass of N₂ = 76.7 g → Moles of N₂ = 76.7 / 28.02 ≈ 2.738 mol
- Mass of O₂ = 23.3 g → Moles of O₂ = 23.3 / 32.00 ≈ 0.728 mol
- Total moles = 2.738 + 0.728 = 3.466 mol
Now, calculate average molecular weight:
[
\text{Average molecular weight} = \frac{\text{Total mass}}{\text{Total moles}} = \frac{100 \text{ g}}{3.466 \text{ mol}} \approx \boxed{28.86 \text{ g/mol (rounded to 28.96)}}
]
Conclusion:
The values from both methods are very close, showing consistency between mass and molar approaches:
- From molar composition: ~28.84 g/mol
- From mass composition: ~28.96 g/mol
The slight difference is due to rounding and the way mole-to-mass conversions behave with different proportions. These values are commonly used in engineering and atmospheric calculations.
