Dalton’s Law of Partial Pressures - Chemistry

1. Fundamental Concepts

Dalton’s Law of Partial Pressures states that the total pressure of a mixture of non-reacting ideal gases is equal to the sum of the partial pressures of its individual components.

Core Formula: \( P_{\text{total}} = P_1 + P_2 + P_3 + ... + P_n \)

: Total pressure of the gas mixture
: Partial pressure of the nth component (the pressure the component would exert if it alone occupied the entire volume of the mixture at the same temperature)

2. Key Concepts

1) Independence of Partial Pressures: Each gas component contributes independently to the total pressure; their molecular interactions are negligible (ideal gas assumption).

2) Mole Fraction Relationship: The partial pressure of a component equals the total pressure multiplied by its mole fraction:

\( P_i = P_{\text{total}} \times X_i \), where \( X_i = \frac{n_i}{n_{\text{total}}} \)

\( X_i \): Mole fraction of the ith component

\( n_i \): Moles of the ith component

\( n_{\text{total}} \): Total moles of all components in the mixture

3) Applicability Conditions: Only valid for ideal gases and non-reacting gas mixtures.

3. Examples

Easy

A gas mixture contains O₂ with a partial pressure of 0.3 atm and N₂ with a partial pressure of 0.7 atm. No other gases are present. Calculate the total pressure of the mixture.

Solution:

\( P_{\text{total}} = P_{\text{O}_2} + P_{\text{N}_2} = 0.3\ \text{atm} + 0.7\ \text{atm} = 1.0\ \text{atm} \)

Medium

2 moles of O₂ and 3 moles of N₂ are mixed in a sealed container at constant temperature. The total pressure of the mixture is 5 atm. Calculate the partial pressure of O₂.

Solution:

1. Calculate the mole fraction of O₂:

\( X_{\text{O}_2} = \frac{n_{\text{O}_2}}{n_{\text{O}_2} + n_{\text{N}_2}} = \frac{2}{2+3} = 0.4 \)

2. Use the mole fraction relationship to find partial pressure:

\( P_{\text{O}_2} = P_{\text{total}} \times X_{\text{O}_2} = 5\ \text{atm} \times 0.4 = 2\ \text{atm} \)

Hard

0.5 moles of H₂ (initially at 1 atm, constant volume) and 0.5 moles of Cl₂ (initially at 2 atm, constant volume) are mixed in a rigid container at constant temperature. They react completely according to the equation:

\( \text{H}_2(g) + \text{Cl}_2(g) \rightarrow 2\text{HCl}(g) \)

Calculate the total pressure of the mixture after the reaction. (Assume all gases behave ideally, and volume/temperature remain constant.)

Revised & Correct Explanation:

1. Analyze the reaction stoichiometry:

1 mole of H₂reacts with 1 mole of Cl₂to produce 2 moles of HCl.

Initial moles: \( n_{\text{H}_2} = 0.5\ \text{mol} \), \( n_{\text{Cl}_2} = 0.5\ \text{mol} \) (1:1 ratio, so both reactants are completely consumed).

Moles of HCl produced: \( n_{\text{HCl}} = 2 \times 0.5 = 1.0\ \text{mol} \)

2. Relate pressure and moles (Avogadro’s Law for constant V, T):

For ideal gases at constant V and T: \( P \propto n \) (pressure is directly proportional to moles of gas).

First, find the total initial pressure of the unreacted mixture:

\( P_{\text{initial, total}} = P_{\text{H}_2} + P_{\text{Cl}_2} = 1\ \text{atm} + 2\ \text{atm} = 3\ \text{atm} \)

Total initial moles: \( n_{\text{initial, total}} = 0.5 + 0.5 = 1.0\ \text{mol} \)

Total final moles: \( n_{\text{final, total}} = 1.0\ \text{mol} \) (only HCl remains)

3. Calculate final total pressure:

Since \( n_{\text{final, total}} = n_{\text{initial, total}} \) and V, T are constant, the total pressure remains unchanged.

\( P_{\text{final, total}} = 3\ \text{atm} \)

Note: If the reaction produced a different total number of moles, we would use the ratio \( \frac{P_{\text{final}}}{P_{\text{initial}}} = \frac{n_{\text{final}}}{n_{\text{initial}}} \) to find the new total pressure.

4. Problem-Solving Techniques

Verify Applicability: First confirm the gas mixture meets the law’s conditions (ideal gas behavior, no chemical reactions).

Choose the Right Formula:

Use \( P_{\text{total}} = \sum P_i \) if partial pressures of all components are known.

Use \( P_i = P_{\text{total}} \times X_i \) if mole fractions (or moles) and total pressure are known.

Handle Reactions: For reacting gases, first use stoichiometry to calculate the moles of all components after the reaction, then apply Dalton’s Law to the final mixture.

Avoid Common Errors: Do not confuse partial pressure with volume fraction; partial pressure depends on mole fraction, not volume fraction (for ideal gases, mole fraction equals volume fraction at constant T and P, but this is a derived relationship, not the core of Dalton’s Law).