P_total = P₁ + P₂ + P₃. Calculate partial pressure from mole fractions.

Dalton's Law of Partial Pressures — Gas Mixture Problem

Question

A container holds a mixture of nitrogen (N₂), oxygen (O₂), and carbon dioxide (CO₂). The mole fractions are:

N₂: 0.60
O₂: 0.30
CO₂: 0.10

If the total pressure of the mixture is 5 atm, find the partial pressure of each gas.

Solution — Step by Step

Dalton’s Law says the total pressure of a gas mixture equals the sum of the partial pressures of all component gases — as if each gas occupied the container alone.

P_{total} = P_1 + P_2 + P_3 + \cdots

Each gas exerts pressure independently. Molecules of different gases don’t “know” each other exists (ideal gas assumption).

Here’s the key relation that makes this problem solvable in seconds:

P_i = \chi_i \times P_{total}

where $\chi_i$ is the mole fraction of component $i$ . This comes directly from the ideal gas law — if N₂ makes up 60% of the moles, it contributes 60% of the total pressure.

P_{N_2} = \chi_{N_2} \times P_{total} = 0.60 \times 5 = \mathbf{3 \text{ atm}}

P_{O_2} = 0.30 \times 5 = \mathbf{1.5 \text{ atm}}

P_{CO_2} = 0.10 \times 5 = \mathbf{0.5 \text{ atm}}

Verification: $3 + 1.5 + 0.5 = 5$ atm ✓ Always verify this — JEE setters love putting a trap where mole fractions don’t sum to 1.

Why This Works

The ideal gas law for a single component in a mixture is $P_i V = n_i RT$ . For the total mixture: $P_{total} V = n_{total} RT$ . Dividing one by the other gives $P_i / P_{total} = n_i / n_{total}$ — which is exactly the mole fraction $\chi_i$ .

So the relation $P_i = \chi_i \times P_{total}$ isn’t a separate law — it’s a direct consequence of the ideal gas equation applied to each component. This is why Dalton’s Law holds only for ideal gases (or real gases at low pressure).

The physical picture: gas molecules in a mixture are far apart and don’t interact. Each gas fills the entire volume as if alone, contributing pressure proportional to its own molecular count.

Alternative Method

If you’re given moles directly (not mole fractions), calculate $\chi_i$ first.

Say you have 6 mol N₂, 3 mol O₂, 1 mol CO₂ with $P_{total} = 5$ atm:

n_{total} = 6 + 3 + 1 = 10 \text{ mol}

\chi_{N_2} = \frac{6}{10} = 0.60, \quad \chi_{O_2} = \frac{3}{10} = 0.30, \quad \chi_{CO_2} = \frac{1}{10} = 0.10

Then apply $P_i = \chi_i \times P_{total}$ as before. Same numbers — this is exactly the same problem restated with moles instead of fractions.

In JEE Main 2024, a variant asked for the partial pressure after removing one gas and re-sealing the container at the same temperature. The remaining gases still follow $P_i = \chi_i \times P_{total}$ — but now $P_{total}$ is recalculated using only the remaining moles. Mole fractions change; partial pressures change.

Common Mistake

Students confuse mole fraction with volume fraction or mass fraction. For ideal gases at the same T and P, mole fraction equals volume fraction — so that confusion usually doesn’t cause a wrong answer. But mass fraction is completely different. If a problem gives you “60% by mass is N₂”, you must first convert to moles before finding $\chi_i$ . Using mass percentage directly as mole fraction is one of the most common errors in this topic.

Also watch out: if the mole fractions given in a problem don’t sum to exactly 1.0, either there’s a rounding note or the problem has a typo. Always add them up before starting — takes 3 seconds and saves the calculation entirely if something’s off.

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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