Gas Mixtures, Partial Pressures, and Kinetic Molecular Theory – General Chemistry I Study Notes

Chapter 5: Gases

Overview

This section covers the behavior of gas mixtures, the calculation of partial and total pressures, and the foundational concepts of the Kinetic Molecular Theory (KMT) as applied to ideal gases. These topics are essential for understanding how gases behave both individually and in mixtures, and for predicting their physical properties under various conditions.

Gas Mixtures and Partial Pressures

Introduction to Gas Mixtures

When dealing with mixtures of gases, it is important to understand how each component contributes to the overall behavior of the system. The total pressure of a gas mixture is the sum of the pressures that each gas would exert if it were present alone in the same volume at the same temperature.

• Partial Pressure: The pressure exerted by an individual gas in a mixture, assuming it alone occupied the entire volume.

• Total Pressure: The sum of all partial pressures in the mixture.

Dalton's Law of Partial Pressures

Dalton's Law states that the total pressure of a mixture of non-reacting gases is equal to the sum of the partial pressures of the individual gases.

Mathematical Expression:

$P_t = P_a + P_b + P_c + \ldots$

• $P_t$ = total pressure

• $P_a, P_b, P_c, \ldots$ = partial pressures of gases a, b, c, etc.

Calculating Partial Pressures Using Mole Fractions

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

$P_a = X_a P_t$

• $X_a$ = mole fraction of component a = $\dfrac{n_a}{n_t}$

• $n_a$ = moles of component a

• $n_t$ = total moles of all gases

Example Calculation:

• Given: 4.00 g CH4 ($n_{CH_4} = 0.249$ mol), 0.143 mol C2H6, 0.0688 mol C3H8 in a 1.5 L vessel at 273.15 K.

• Total moles: $n_t = 0.249 + 0.143 + 0.0688 = 0.461$ mol

• Total pressure: $P_t = \dfrac{n_tRT}{V} = \dfrac{0.461 \text{ mol} \times 0.08314 \text{ L bar mol}^{-1} \text{ K}^{-1} \times 273.15 \text{ K}}{1.5 \text{ L}} = 6.98 \text{ bar}$

• Partial pressure of CH4: $P_{CH_4} = \dfrac{0.249}{0.461} \times 6.98 \text{ bar} = 3.77 \text{ bar}$

• Repeat for other gases: C2H6 (2.17 bar), C3H8 (1.04 bar)

Example: Mixing Gases in Connected Flasks

When gases in separate containers are mixed by opening stopcocks, the final pressure can be calculated by considering the total moles and the final total volume.

Gas	Initial Pressure (atm)	Initial Volume (L)
CO2	2.13	1.50
H2	0.861	1.00
Ar	1.15	2.00

• Final total volume: $V_f = 1.50 + 1.00 + 2.00 = 4.50$ L

• For each gas: $P_{f} = \dfrac{V_i}{V_f} \times P_i$

• Example for Ar: $P_{f,Ar} = \dfrac{2.00}{4.50} \times 1.15 = 0.511$ atm

• Total pressure: $P_t = P_{f,Ar} + P_{f,H_2} + P_{f,CO_2} = 0.511 + 0.191 + 0.710 = 1.412$ atm

Kinetic Molecular Theory (KMT) of Gases

Introduction to KMT

The Kinetic Molecular Theory explains the physical behavior of gases at the molecular level. It provides a model for understanding gas laws and the properties of gases.

Postulates of Kinetic Molecular Theory

• Negligible Particle Size: The size of each gas particle is negligibly small compared to the distances between them.

• Average Kinetic Energy: The average kinetic energy of a gas particle is proportional to the temperature in Kelvin.

• Negligible Intermolecular Forces: Attractive and repulsive forces between gas molecules are negligible.

• Elastic Collisions: Collisions between gas particles and with the walls of the container are completely elastic (no energy is lost).

Implications of KMT

• If you increase the pressure on a gas sample by decreasing the volume, the gas particles will speed up.

• The pressure exerted by a gas depends on its molar mass.

• The speed and kinetic energy of a gas particle depend on its molar mass and temperature.

• Gas particles slow down when they collide with the walls of a container.

• The speed of gas particles is not affected by collisions with other gas particles.

• The speed of gas particles is affected by temperature.

Pressure and Molecular Collisions

Gas pressure results from collisions of molecules with the walls of the container. The magnitude of the pressure depends on both the frequency and the force of these collisions.

Effect of Volume and Temperature on Pressure

• Boyle's Law: At constant temperature, decreasing the volume increases the pressure because molecules collide with the walls more frequently.

• Effect of Temperature: At constant volume, increasing the temperature increases the pressure because molecules move faster, hit the walls more often, and with greater force.

Summary Table: Factors Affecting Gas Properties

Factor	Effect on Gas Behavior
Decrease Volume (constant n, T)	Increases pressure (Boyle's Law)
Increase Temperature (constant V, n)	Increases pressure (Gay-Lussac's Law)
Increase Molar Mass	Decreases average speed of particles
Increase Number of Moles	Increases pressure (if V and T constant)

Key Equations

• Dalton's Law: $P_t = P_a + P_b + P_c + \ldots$

• Partial Pressure via Mole Fraction: $P_a = X_a P_t$

• Mole Fraction: $X_a = \dfrac{n_a}{n_t}$

• Ideal Gas Law: $P V = n R T$

Examples and Applications

• Mixing Gases: When gases are mixed, the total pressure is the sum of the partial pressures, and each gas behaves independently if the gases are ideal.

• Calculating Partial Pressures: Use mole fractions and total pressure to find the partial pressure of each component in a mixture.

• Predicting Changes: Use KMT to explain how changes in temperature, volume, or number of moles affect pressure and molecular motion.

Additional info: For further practice, online gas law simulators are recommended (see: UTexas Gas Law Simulator and PhET Gas Properties).