6.2 · Apply Dalton’s law to ideal-gas mixtures

University of Alberta MEC E 340: Applied Thermodynamics

Gas Mixtures and Real Gases

Using mole fractions to find the pressure contributed by each gas

What you will learn

• State Dalton’s law and the assumptions behind applying it.
• Relate a gas component’s partial pressure to its mole fraction and the mixture’s total pressure.
• Use component amounts and the ideal-gas equation to solve mixture-pressure problems with SI units.
• Check that calculated partial pressures add to the total pressure and are physically plausible.

1. Review the pressure and composition ideas

• Define the mixture and its state before calculating.
• All components in the stated mixture share the same temperature and volume.
• Use mole fractions, not mass fractions, in the partial-pressure relation.

2. Apply Dalton’s law

• The partial pressures sum to the total pressure.
• A zero mole fraction gives a zero partial pressure.
• The sum of the mole fractions should be one, apart from small rounding differences.

3. A reliable problem-solving sequence

• Write down the state, system, components, and ideal-mixture assumption.
• Choose between the mole-fraction form and component ideal-gas equations based on the known data.
• Verify both the composition sum and the pressure sum.

4. Scope and interpretation

• Dalton’s law here is for ideal-gas mixtures.
• State clearly which components are included in the total.
• Do not confuse the mixture’s total pressure with any one component’s partial pressure.

Worked example

Find partial pressures from composition

1. Identify the mixture
   The closed tank contains two ideal-gas components at the same state, temperature, and volume. The total amount is the sum of their amounts.
   $$n=2.0+1.0=3.0\ \mathrm{mol}$$
2. Find mole fractions
   Divide each component amount by the total amount. The fractions should add to one.
   $$y_{\mathrm{N_2}}=\frac{2.0}{3.0},\qquad y_{\mathrm{O_2}}=\frac{1.0}{3.0}$$
3. Calculate component pressures
   Multiply each mole fraction by the given total pressure. These are the partial pressures under the mixture conditions.
   $$P_{\mathrm{N_2}}=200\ \mathrm{kPa},\qquad P_{\mathrm{O_2}}=100\ \mathrm{kPa}$$

Worked example

Find total pressure from component amounts

1. Set the shared conditions
   Both components occupy the same vessel volume and are at the same temperature. Use the ideal-gas equation for each component with the supplied gas constant.
   $$P_i=\frac{n_iRT}{V}$$
2. Calculate partial pressures
   Using pascals for pressure because the supplied gas constant is in joules, calculate each component’s pressure contribution.
   $$P_{\mathrm{He}}=49.9\ \mathrm{kPa},\qquad P_{\mathrm{Ne}}=74.8\ \mathrm{kPa}$$
3. Add the contributions
   Dalton’s law gives the total pressure as the sum of the component partial pressures.
   $$P=49.9+74.8=124.7\ \mathrm{kPa}$$

Worked example

Use partial pressure to recover component amount

1. Find the mole fraction
   Dalton’s law relates the component partial pressure to the total pressure through its mole fraction.
   $$y_{\mathrm{CO_2}}=\frac{30}{150}=0.20$$
2. Find the component amount
   Convert the partial pressure to pascals and use the ideal-gas equation for the carbon dioxide at the shared vessel temperature and volume.
   $$n_{\mathrm{CO_2}}=\frac{(30{,}000\ \mathrm{Pa})(0.050\ \mathrm{m^3})}{(8.314\ \mathrm{J/(mol\cdot K)})(320\ \mathrm{K})}=0.564\ \mathrm{mol}$$
3. Check against the total amount
   The total amount follows from the same ideal-gas equation using total pressure. The component amount should equal its mole fraction times this total.
   $$n=2.82\ \mathrm{mol},\qquad 0.20n=0.564\ \mathrm{mol}$$

Common mistakes and how to avoid them

Lesson summary

• For an ideal-gas mixture, components share the mixture temperature and volume.
• Mole fractions are component moles divided by total moles and sum to one.
• A component’s partial pressure is its mole fraction times total pressure.
• The sum of all component partial pressures equals total pressure.
• Use the ideal-gas equation when component amount, temperature, and volume are known, and check units and pressure sums.

Check your understanding

Key terms

Ideal-gas mixture

A mixture treated as ideal in which each gas component follows the ideal-gas model under the shared mixture conditions.

Mole fraction

The amount of one component divided by the total amount of the mixture.

Partial pressure

The pressure assigned to one component as if it alone occupied the mixture volume at the mixture temperature.

Dalton’s law

For an ideal-gas mixture, total pressure equals the sum of the component partial pressures.

Continue through MEC E 340

View the complete MEC E 340 University of Alberta MEC E 340: Applied Thermodynamics curriculum and lessons

• 6.1 · Convert between mole and mass fractions
• 6.3 · Estimate mixture gas constant and specific heat
• 6.4 · Use the compressibility factor for real gases
• 6.5 · Select ideal-gas or real-gas data without mixing models
• 1.1 · Define the system, working fluid, state, and process
• 1.2 · Read property tables and identify phase regions

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Before publication, the draft is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability, and then requires administrator approval. Errors can still occur, so corrections are welcomed.

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