Dalton’s Law of Partial Pressures and Gas Mixtures

Gases

Gas Mixtures and Nonreacting Gas Behavior

In many chemical and physical processes, gases are found as mixtures rather than as pure substances. Understanding how these mixtures behave is essential for predicting pressures and other properties in real-world systems.

• Nonreacting Gases: When gases do not chemically react with each other, each gas in a mixture behaves as if it occupies the entire volume of the container alone.

• Atmospheric Example: The Earth's atmosphere is a mixture of several gases (e.g., N2, O2, Ar, CO2). If each component were placed in an identical empty cylinder, the pressure exerted by each would be the same as its contribution in the atmosphere.

Key Point: The total pressure of a gas mixture is the sum of the pressures each gas would exert if it were alone in the container.

Dalton’s Law of Partial Pressures

Dalton’s Law provides a quantitative way to determine the total pressure of a mixture of nonreacting gases.

• Definition: The total pressure exerted by a mixture of nonreacting gases is equal to the sum of the partial pressures of the individual gases.

Mathematical Form:

$P_T = P_A + P_B + P_C + \ldots$

• $P_T$ = total pressure

• $P_A, P_B, P_C, \ldots$ = partial pressures of gases A, B, C, etc.

Each partial pressure can be calculated using the ideal gas law for each component:

$P_A = \frac{n_A R T}{V}$ $P_B = \frac{n_B R T}{V}$ $P_C = \frac{n_C R T}{V}$

Therefore, the total pressure is:

$P_T = \frac{(n_A + n_B + n_C + \ldots) R T}{V}$

Mole Fraction and Partial Pressure

The mole fraction of a component in a gas mixture is the ratio of the number of moles of that component to the total number of moles in the mixture.

$X_A = \frac{n_A}{n_T}$

• $X_A$ = mole fraction of gas A

• $n_A$ = moles of gas A

• $n_T$ = total moles of all gases

The partial pressure of a gas can also be expressed in terms of its mole fraction:

$P_A = X_A P_T$

Example: If a mixture contains 0.01 mol of O2, 0.015 mol of N2, and 0.025 mol of CO2, the mole fraction of O2 is:

$X_{O_2} = \frac{0.01}{0.01 + 0.015 + 0.025} = 0.2$

So, O2 makes up 20% of the total moles, and its partial pressure is 20% of the total pressure.

Applications: Collecting Gases Over Water

Dalton’s Law is commonly applied when collecting gases over water. The total pressure in the collection vessel is the sum of the pressure of the collected gas and the vapor pressure of water.

• Equation: $P_{\text{total}} = P_{\text{gas}} + P_{\text{H}_2O}$

• To find the pressure of the collected gas, subtract the vapor pressure of water (at the collection temperature) from the total pressure.

Example: If the total pressure is 755 mmHg and the vapor pressure of water at the collection temperature is 23.8 mmHg, then:

$P_{\text{gas}} = 755\ \text{mmHg} - 23.8\ \text{mmHg} = 731.2\ \text{mmHg}$

Practice Problems

• Problem 1: Calculate the mole fraction and partial pressure of each gas in a mixture containing 0.01 mol O2, 0.015 mol N2, and 0.025 mol CO2 at a total pressure of 1.0 atm.

• Problem 2: If a mixture contains gases A and B with partial pressures 0.75 atm and 1.5 atm, respectively, what is the total pressure?

• Challenge Problem: A 6.19 g sample of PCl5 is placed in a 2.00 L flask and vaporized at 252°C. Calculate the pressure if no reaction occurs. If partial dissociation occurs, use Dalton’s Law and the observed pressure to find the partial pressures of PCl5 and its dissociation products.

Summary Table: Dalton’s Law of Partial Pressures

Concept	Equation	Description
Total Pressure	$P_T = P_A + P_B + P_C + \ldots$	Sum of partial pressures of all gases in the mixture
Partial Pressure (Ideal Gas Law)	$P_A = \frac{n_A R T}{V}$	Pressure exerted by gas A alone
Mole Fraction	$X_A = \frac{n_A}{n_T}$	Fraction of total moles contributed by gas A
Partial Pressure (Mole Fraction)	$P_A = X_A P_T$	Partial pressure as a fraction of total pressure

Additional info: The notes also include worked examples and practice problems to reinforce the application of Dalton’s Law in both theoretical and laboratory contexts, such as collecting gases over water and calculating partial pressures in chemical equilibrium scenarios.