DAT General Chemistry: Dalton's Law of Partial Pressures

DAT High-Yield Concept

Dalton's Law Definition: Total pressure of a non-reacting gas mixture equals the sum of individual component partial pressures: $P_{\text{total}} = P_A + P_B + P_C + \dots$
Mole Fraction ( $X_i$ ): Represents the ratio of moles of gas $i$ to the total gas moles: $X_i = \frac{n_i}{n_{\text{total}}}$
Partial Pressure Formula: The partial pressure of any constituent gas is directly proportional to its mole fraction and the total pressure: $P_i = X_i P_{\text{total}}$
Sum of Mole Fractions: The sum of all mole fractions in a mixture always equals exactly 1 ( $\sum X_i = 1$ ).

1. Dalton's Law and Total Pressure

According to kinetic molecular theory, ideal gas particles act independently because intermolecular forces are negligible. When multiple non-reacting gases occupy a single container at a given volume and temperature, each gas exerts a pressure identical to what it would exert if it occupied the container alone.

Partial Pressure ( $P_i$ ): The individual pressure exerted by a single gas species in a mixture. Total pressure is independent of gas identity and depends only on total moles of gas particles.

Because each ideal gas particle contributes equally to collisions against the container walls regardless of its identity, the total pressure ( $P_{\text{total}}$ ) is simply the sum of all individual partial pressures:

P_{\text{total}} = \sum P_i = P_1 + P_2 + P_3 + \dots

Each gas is first shown alone in a 5.0 L container at one temperature, where it exerts its own partial pressure. Combining all three in the same volume at the same temperature gives a total pressure equal to the sum of those partial pressures, because in an ideal gas the pressure depends on how many particles are present and not on what they are. A gas that makes up half the particles therefore supplies half the total pressure, which is what the mole fraction expresses.

2. Gas-Phase Mole Fraction and Partial Pressure Calculations

The mole fraction ( $X_i$ ) is a dimensionless ratio expressing the proportion of a specific gas component relative to the total moles of gas present:

X_i = \frac{n_i}{n_{\text{total}}}

Illustration from Anatomy & Physiology, Connexions Web site. http://cnx.org/content/col11496/1.6/ , Jun 19, 2013. (Source: OpenStax College · CC BY 3.0 · Wikimedia Commons)

By substituting the ideal gas equation ( $P_i = \frac{n_i RT}{V}$ and $P_{\text{total}} = \frac{n_{\text{total}} RT}{V}$ ), we derive the foundational partial pressure relationship for DAT calculations:

P_i = X_i P_{\text{total}}

DAT Memory Trick: Think "Partial = X Marks the Total Pressure" (

P_i = X_i \times P_{\text{total}}

). Multiply the mole fraction by total pressure to find any gas's partial share.

Worked Example: Partial Pressure Calculation

Problem: A rigid container holds $\pu{2.00 mol}$ of $\ce{N2}$ , $\pu{1.00 mol}$ of $\ce{O2}$ , and $\pu{1.00 mol}$ of $\ce{Ar}$ at a total pressure of $\pu{800. mmHg}$ . Calculate the partial pressure of $\ce{O2}$ .

Solution:

Calculate total moles: $n_{\text{total}} = \pu{2.00 mol} + \pu{1.00 mol} + \pu{1.00 mol} = \pu{4.00 mol}$
Calculate mole fraction of $\ce{O2}$ : $X_{\ce{O2}} = \frac{\pu{1.00 mol}}{\pu{4.00 mol}} = 0.250$
Calculate partial pressure of $\ce{O2}$ : $P_{\ce{O2}} = X_{\ce{O2}} \times P_{\text{total}} = 0.250 \times \pu{800. mmHg} = \pu{200. mmHg}$

Common Exam Trap: DAT questions often give component quantities in grams instead of moles. Always convert mass to moles using molar mass before calculating mole fractions.

3. Gas Parameter Comparison

Parameter	Formula	Units / Range	Key Characteristic
Total Pressure ( $P_{\text{total}}$ )	$P_{\text{total}} = P_A + P_B + \dots$	$\pu{atm}$ , $\pu{mmHg}$ , $\pu{torr}$ , $\pu{kPa}$	Sum of collisions from all gas species against container walls
Partial Pressure ( $P_i$ )	$P_i = X_i P_{\text{total}}$	Same units as total pressure	Proportional to mole fraction; independent of other gases
Mole Fraction ( $X_i$ )	$X_i = \frac{n_i}{n_{\text{total}}}$	Unitless ( $0 \le X_i \le 1$ )	Sum of all component mole fractions must equal $1.00$

Critical Thinking Questions

If $\pu{1.00 mol}$ of inert $\ce{He}$ gas is added to a rigid container holding a mixture of $\ce{N2}$ and $\ce{O2}$ at constant temperature, what happens to the partial pressure of $\ce{N2}$ ? Explain why total pressure changes while $P_{\ce{N2}}$ remains constant.
In a gas mixture, species A has a molar mass double that of species B. If equal masses of gas A and gas B are introduced into a vessel, which gas exerts a higher partial pressure and what is the ratio $P_B : P_A$ ?
Why does Dalton's law apply strictly to ideal, non-reacting gases, and how would strong attractive intermolecular forces between two component gases affect measured total pressure?

References

Sources returned by Google Search grounding during lesson verification.

Stoichiometry of Gaseous Substances, Mixtures, and Reactions - OpenStax Chemistry 2e — openstax.org
Dalton's Law of Partial Pressures - Chemistry LibreTexts — chem.libretexts.org

Dalton's Law of Partial Pressures

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