Questions: Gas Mixture Thermodynamics and Dalton's Law

Question 1

A gas mixture is 21% O₂ and 79% N₂ by mole at sea level (total pressure 100 kPa). A climber ascends to high altitude where total pressure is 60 kPa. What is the mole fraction of O₂ at altitude?

• A. 12.6% — because reduced pressure compresses the mole fraction proportionally
• B. Still 21% — mole fraction is a ratio of amounts, unaffected by total pressure
• C. 35% — O₂ is heavier than N₂ and concentrates at lower pressures
• D. Cannot be determined without knowing the temperature at altitude

Answer: (B)

Mole fraction x_i = n_i / n_total depends only on the relative amounts of each component, not on the total pressure. The same number of O₂ molecules and N₂ molecules are present in a given parcel of air regardless of altitude — the composition hasn't changed. What does change is the partial pressure of O₂: P_O₂ = 0.21 × 60 kPa = 12.6 kPa, down from 21 kPa at sea level. It is this drop in partial pressure (not mole fraction) that reduces oxygen availability and causes altitude sickness.

Question 2

Dry air is 21% O₂ (M = 32 g/mol) and 79% N₂ (M = 28 g/mol) by mole. What is the mixture molecular weight of air?

• A. 30 g/mol — the simple average of 32 and 28
• B. 28.84 g/mol — the mole-fraction-weighted average: 0.21×32 + 0.79×28
• C. 32 g/mol — the heavier component dominates the mixture molecular weight
• D. 60 g/mol — molecular weights add for a two-component mixture

Answer: (B)

Mixture molecular weight is M_mix = Σ x_i M_i = 0.21×32 + 0.79×28 = 6.72 + 22.12 = 28.84 g/mol. This is a mole-fraction-weighted average, not a simple average (which would give 30). Because N₂ has a much larger mole fraction (0.79 vs 0.21), the mixture molecular weight is pulled closer to M_N₂ = 28. This value of ~28.84 g/mol gives the familiar specific gas constant for air: R_air = 8314 / 28.84 ≈ 287 J/(kg·K).

Question 3

For an ideal gas mixture, the partial pressure of a component equals the pressure that component would exert if it alone occupied the entire volume at the same temperature.

• T. True
• F. False

Answer: True

This is Dalton's law and follows directly from ideal gas behavior: each component acts independently, with no intermolecular interactions between unlike molecules. If component i alone were in the container at the same T and volume V, it would exert P_i = n_i R_u T / V. Summing over all components gives P_total = n_total R_u T / V, so P_i/P_total = n_i/n_total = x_i. Thus partial pressure, volume fraction, and mole fraction are all identical for ideal gas mixtures.

Question 4

As elevation increases, the percentage of oxygen in the air decreases, which is why breathing becomes harder at high altitude.

• T. True
• F. False

Answer: False

The mole fraction of oxygen in dry air is approximately 21% at all elevations — the composition of the atmosphere doesn't change with altitude. What decreases is the total atmospheric pressure, and with it, the partial pressure of oxygen: P_O₂ = 0.21 × P_total. At high altitude, P_total is much lower, so P_O₂ is lower, meaning less oxygen is available per breath. The difficulty breathing is caused by reduced oxygen partial pressure, not reduced oxygen percentage.

Question 5

Why does altitude affect a climber's ability to breathe, even though the mole fraction of oxygen in air stays constant at about 21%?

Model answer: Oxygen transport in the body depends on partial pressure, not mole fraction. Hemoglobin in the lungs picks up oxygen based on P_O₂ in the alveoli. At altitude, total atmospheric pressure falls, so P_O₂ = 0.21 × P_total also falls — there are fewer oxygen molecules per breath even though the proportion remains the same. Lower P_O₂ means hemoglobin is less saturated with oxygen after each breath, reducing oxygen delivery to tissues.

This is why the mole fraction vs. partial pressure distinction matters physiologically. A climber at 5,500 m (half sea-level pressure) breathes air that is still 21% O₂, but the partial pressure is only ~10.6 kPa instead of ~21.3 kPa — about half. The hemoglobin-oxygen dissociation curve tells us saturation drops substantially at lower partial pressures, which is the mechanism underlying altitude sickness and acclimatization.