Dalton’s Law of Partial Pressures and Gas Mixtures

Gases

Dalton’s Law of Partial Pressures

Dalton’s Law describes how the total pressure exerted by a mixture of non-reacting gases is equal to the sum of the partial pressures of each individual gas present. This principle is fundamental in understanding gas mixtures and their behavior in closed systems.

• Partial Pressure (Pgas): The pressure exerted by an individual gas in a mixture.

• Law of Partial Pressures: The total pressure is the sum of the partial pressures of all gases present.

Equation:

$P_{total} = P_{gas1} + P_{gas2} + P_{gas3} + \ldots$

• Example: A sample of neon gas exerts a pressure of 1.85 atm inside a cylinder. Some nitrogen gas is also present, at a pressure of 500 mm Hg. What is the total pressure inside the cylinder?

• Solution: Convert all pressures to the same unit and add:

  • Neon: 1.85 atm = 1.85 × 760 mm Hg = 1,406 mm Hg

  • Nitrogen: 500 mm Hg

  • Total Pressure: 1,406 mm Hg + 500 mm Hg = 1,906 mm Hg

Using Moles to Determine Partial Pressure (Ideal Gas Law)

The partial pressure of a gas can also be calculated using the Ideal Gas Law if the number of moles, volume, and temperature are known. This is especially useful when dealing with mixtures where the amount of each gas is specified.

• Ideal Gas Law: $P = \frac{nRT}{V}$

• Variables:

  • P: Pressure (atm)

  • n: Number of moles

  • R: Gas constant (0.0821 L·atm/mol·K)

  • T: Temperature (K)

  • V: Volume (L)

• Example: If 12.0 g helium and 20.0 g oxygen are placed inside a 5.00 L cylinder at 30°C, what is the partial pressure of the helium?

• Solution:

  • Convert mass to moles: He: $\frac{12.0\,g}{4.00\,g/mol} = 3.00\,mol$

  • Convert temperature to Kelvin: $30°C + 273 = 303\,K$

  • Apply Ideal Gas Law: $P = \frac{3.00 \times 0.0821 \times 303}{5.00} = 1.49\,atm$

Fractional Composition in Partial Pressure

When the percent composition of a gas in a mixture is given, its fractional composition can be used to determine its partial pressure. Fractional composition is the percent composition divided by 100.

• Fractional Composition: $\text{Fractional Composition} = \frac{\text{Gas %}}{100}$

• Dalton’s Law (using fractional composition): $P_{gas} = \text{Fractional Composition} \times P_{total}$

• Example: A cylinder of a gas mixture used for calibration of blood gas analyzers contains 5.0% O2, 12.0% CO2, and 83.0% N2. If the total pressure is 148 atm, what is the partial pressure of each component?

• Solution:

  • O2: $P_{O_2} = 0.05 \times 148 = 7.4\,atm$

  • CO2: $P_{CO_2} = 0.12 \times 148 = 17.8\,atm$

  • N2: $P_{N_2} = 0.83 \times 148 = 122.8\,atm$

Practice Problems

• A gas mixture contains 72.0% chlorine and 27.2% neon. What is the partial pressure of neon in the mixture if the total pressure is 880 mm Hg?

• Solution: $P_{Ne} = 0.272 \times 880 = 239\,mm\,Hg$

• The partial pressure of H2 in the air is 180 mm Hg at 1 atm. What is the partial pressure of H2 in a bubble of air outside the structure when the air is at 1.5 atm? What is the value when the pressure is 2.0 atm?

• Solution: $P_{H_2} = \frac{180}{760} \times 1.5 \times 760 = 270\,mm\,Hg$; $P_{H_2} = \frac{180}{760} \times 2.0 \times 760 = 360\,mm\,Hg$

Summary Table: Dalton’s Law and Partial Pressures

Concept	Equation	Application
Law of Partial Pressures	$P_{total} = P_{gas1} + P_{gas2} + \ldots$	Sum of individual gas pressures
Ideal Gas Law	$P = \frac{nRT}{V}$	Calculate pressure from moles, volume, temperature
Fractional Composition	$\text{Fractional Composition} = \frac{\text{Gas %}}{100}$	Percent of gas in mixture
Partial Pressure from Fractional Composition	$P_{gas} = \text{Fractional Composition} \times P_{total}$	Find partial pressure using percent composition