Dalton’s Law of Partial Pressures and Gas Mixtures

Gas Mixtures and Dalton’s Law of Partial Pressures

Introduction to Gas Mixtures

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

• Gas-phase systems can consist of multiple gases that do not chemically react with each other.

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

• Example: The Earth's atmosphere is a mixture of gases such as nitrogen (N2), oxygen (O2), argon (Ar), and carbon dioxide (CO2).

Nonreacting Gas Behavior

When gases do not react with each other, their individual properties can be analyzed separately within the mixture.

• Each component gas exerts a pressure as if it were the only gas present in the container.

• This concept is foundational for understanding partial pressures in mixtures.

• Example: If N2, O2, Ar, and CO2 are placed in separate but identical containers, each exerts its own pressure. When combined in one container, the total pressure is 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 (Pi): The pressure exerted by a single component in a mixture.

• Total pressure (PT): The sum of all partial pressures in the mixture.

Mathematical Expression:

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

Where $P_A$, $P_B$, $P_C$, etc., are the partial pressures of gases A, B, C, etc.

Calculating Partial Pressures Using the Ideal Gas Law

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

$P_i = \frac{n_iRT}{V}$

• $n_i$ = moles of gas i

• $R$ = universal gas constant

• $T$ = temperature (in Kelvin)

• $V$ = volume of the container

For a mixture of gases:

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

Thus, the total pressure depends on the total number of moles of gas present.

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 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 and 0.04 mol total gas, the mole fraction of O2 is $0.01/0.05 = 0.2$ (or 20%).

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 is the sum of the pressure of the collected gas and the vapor pressure of water.

• Equation:

$P_{\text{atm}} = P_{\text{gas}} + P_{\text{H}_2O}$

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

Worked Example

• Problem: A sample of gas is collected over water at 25°C. The atmospheric pressure is 760 mmHg, and the vapor pressure of water at 25°C is 24 mmHg. What is the pressure of the dry gas?

• Solution: $P_{\text{gas}} = 760\ \text{mmHg} - 24\ \text{mmHg} = 736\ \text{mmHg}$

Practice Problems

Sample Questions

• Question 1: A 0.01 mol sample of O2 is mixed with 0.015 mol of N2 and 0.025 mol of Ar. What is the mole fraction of O2?

• Solution: $X_{O_2} = \frac{0.01}{0.01 + 0.015 + 0.025} = \frac{0.01}{0.05} = 0.2$ (20%)

• Question 2: If the partial pressures of two gases in a mixture are 0.75 atm and 1.5 atm, what is the total pressure?

• Solution: $P_T = 0.75\ \text{atm} + 1.5\ \text{atm} = 2.25\ \text{atm}$

Challenge Problem

• Problem: A 6.19 g sample of PCl5 is placed in a 2.00 L flask and completely vaporized at 252°C. Calculate the pressure in the flask if no chemical reaction occurs.

• Further: If PCl5 partially dissociates into PCl3 and Cl2, and the observed pressure is 1.00 atm, calculate the partial pressures of PCl5 and the products.

Additional info: These problems require using the ideal gas law and stoichiometry to relate mass, moles, and pressure, as well as equilibrium concepts for dissociation.