Dalton’s Law of Partial Pressures and Gas Mixtures

Gas Mixtures and Partial Pressures

Introduction to Gas-Phase Systems

In real-world scenarios, gases are often found as mixtures rather than as pure substances. Understanding the behavior of gas mixtures is essential for predicting properties such as pressure, volume, and temperature in chemical systems.

• Gas mixtures behave predictably when the component gases do not chemically react with each other.

• Each gas in a mixture acts as if it occupies the entire volume of the container alone.

• Example: The Earth's atmosphere is a mixture of several gases, including N2, O2, Ar, CO2, and others.

Nonreacting Gas Behavior

When gases do not react, each gas in a mixture exerts its own pressure independently of the others. This is the foundation for Dalton's Law of Partial Pressures.

• Each gas in a mixture behaves as if it is the only gas present in the container.

• The total pressure is the sum of the pressures each gas would exert alone.

• Example: If you have separate cylinders of N2, O2, Ar, and CO2, and combine them into one container, the total pressure equals the sum of the individual pressures.

Dalton’s Law of Partial Pressures

Statement of Dalton’s Law

Dalton’s Law states that the total pressure exerted by a mixture of nonreacting gases is equal to the sum of the partial pressures of each individual gas.

• Partial pressure is the pressure that a gas in a mixture would exert if it occupied the entire volume alone.

Mathematical Expression:

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

Where $P_T$ is the total pressure, and $P_A$, $P_B$, $P_C$, etc., are the partial pressures of each gas.

Calculating Partial Pressures Using the Ideal Gas Law

The partial pressure of each gas can be calculated using the ideal gas law:

$P = \frac{nRT}{V}$

• For a mixture: $P_T = \frac{n_A RT}{V} + \frac{n_B RT}{V} + \frac{n_C RT}{V} + \ldots = \frac{(n_A + n_B + n_C + \ldots)RT}{V}$

• $n_T$ is the total number of moles: $n_T = n_A + n_B + n_C + \ldots$

Mole Fraction and Partial Pressure

The mole fraction ($X_A$) of a component gas is the ratio of the number of moles of that gas to the total number of moles in the mixture:

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

The partial pressure of a gas can be related to its mole fraction:

$P_A = X_A P_T$

• This relationship allows calculation of partial pressures when the composition of the mixture is known.

Applications of Dalton’s Law

Collecting Gases Over Water

Dalton’s Law is commonly applied when collecting gases over water, where the total pressure is the sum of the gas pressure and the vapor pressure of water.

• Atmospheric pressure ($P_{atm}$) is the sum of the pressure of the collected gas ($P_{gas}$) and the vapor pressure of water ($P_{H_2O}$):

$P_{atm} = P_{gas} + P_{H_2O}$

• To find the pressure of the collected gas, subtract the vapor pressure of water from the total pressure.

• Vapor pressure values depend on temperature and can be found in reference tables.

Example Calculation

• Suppose a mixture contains 0.01 mol $N_2$, 0.015 mol $O_2$, and 0.025 mol $Ar$ in a container. The total number of moles is $0.01 + 0.015 + 0.025 = 0.05$ mol.

• The mole fraction of $N_2$ is $\frac{0.01}{0.05} = 0.2$ (or 20%).

• If the total pressure is 1 atm, the partial pressure of $N_2$ is $0.2 \times 1 = 0.2$ atm.

Sample Problem

• Given: $P_T = 0.75$ atm ($N_2$) + $1.5$ atm ($O_2$) = $2.25$ atm (total pressure).

• Each partial pressure is calculated independently, then summed for the total pressure.

Challenge Problem Example

• 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 PCl5 partially dissociates, use Dalton’s Law and stoichiometry to find the partial pressures of each component.

Additional info: For dissociation problems, set up an ICE (Initial, Change, Equilibrium) table and use the ideal gas law for each component.

Summary Table: Dalton’s Law of Partial Pressures

Term	Definition	Formula
Total Pressure ($P_T$)	Sum of all partial pressures in a mixture	$P_T = P_A + P_B + P_C + \ldots$
Partial Pressure ($P_A$)	Pressure exerted by one gas in a mixture	$P_A = X_A P_T$
Mole Fraction ($X_A$)	Fraction of total moles contributed by one gas	$X_A = \frac{n_A}{n_T}$
Ideal Gas Law	Relates pressure, volume, temperature, and moles	$P = \frac{nRT}{V}$