Chapter 10: Gases

Gases: Kinetic Molecular Theory, Gas Laws, and Real Gases

Introduction

This chapter explores the behavior of gases through the kinetic molecular theory, the fundamental gas laws, and the deviations observed in real gases. It covers the relationships between pressure, volume, temperature, and amount of gas, and introduces the mathematical models used to describe and predict gas behavior under various conditions.

Kinetic Molecular Theory of Gases

Fundamental Postulates

• Negligible Particle Size: Gas particles are extremely small compared to the distances between them; their volume is negligible.

• Constant, Random Motion: Gas particles move in straight lines until they collide with other particles or the container walls.

• Elastic Collisions: Collisions between gas particles and with container walls are perfectly elastic; no energy is lost.

• Average Kinetic Energy: The average kinetic energy of gas particles is proportional to the absolute temperature (in Kelvin).

Example: At higher temperatures, gas particles move faster, increasing the pressure exerted on the container walls.

Pressure: The Result of Particle Collisions

Definition and Measurement

• Pressure (P): The force exerted by gas particles colliding with the walls of their container, divided by the area of the surface.

• Formula: $P = \frac{F}{A}$

• Units: Atmospheres (atm), pascals (Pa), torr, millimeters of mercury (mmHg).

Example: Atmospheric pressure at sea level is 1 atm = 101,325 Pa = 760 mmHg.

Pressure Units

Common Units and Conversions

• Atmosphere (atm): Standard unit for atmospheric pressure.

• Pascals (Pa): SI unit; 1 atm = 101,325 Pa.

• Millimeters of Mercury (mmHg) and Torr: 1 atm = 760 mmHg = 760 torr.

Unit	Equivalent at Sea Level
Atmosphere (atm)	1.00 atm
Millimeter of mercury (mmHg)	760 mmHg
Torr	760 torr
Pascals (Pa)	101,325 Pa

The Manometer: Measuring Pressure in the Laboratory

Principle and Use

• A manometer is a U-shaped tube containing a dense liquid (usually mercury) used to measure the pressure of a gas sample.

• The difference in liquid levels indicates the pressure exerted by the gas compared to atmospheric pressure.

Example: If the mercury level is higher on the side open to the atmosphere, the gas pressure is less than atmospheric pressure.

The Simple Gas Laws

Boyle's Law: Volume and Pressure

• Statement: The volume of a gas is inversely proportional to its pressure at constant temperature.

• Formula: $P_1V_1 = P_2V_2$

• Graph: Volume vs. pressure yields a hyperbolic curve.

Example: Compressing a gas sample to half its original volume doubles its pressure.

Charles's Law: Volume and Temperature

• Statement: The volume of a gas is directly proportional to its temperature (in Kelvin) at constant pressure.

• Formula: $\frac{V_1}{T_1} = \frac{V_2}{T_2}$

• Graph: Volume vs. temperature yields a straight line.

Example: Heating a balloon causes it to expand as the gas inside increases in temperature.

Avogadro's Law: Volume and Amount (Moles)

• Statement: The volume of a gas is directly proportional to the number of moles of gas at constant temperature and pressure.

• Formula: $\frac{V_1}{n_1} = \frac{V_2}{n_2}$

Example: Doubling the amount of gas in a container doubles its volume.

The Ideal Gas Law

Combining the Simple Gas Laws

• Statement: The ideal gas law combines Boyle's, Charles's, and Avogadro's laws.

• Formula: $PV = nRT$

• Variables: P = pressure, V = volume, n = moles, R = gas constant (0.08206 L·atm/mol·K), T = temperature in Kelvin.

Example: Calculate the volume occupied by 1 mole of an ideal gas at STP (0°C, 1 atm): $V = \frac{nRT}{P} = \frac{(1)(0.08206)(273)}{1} \approx 22.4$ L.

Applications of the Ideal Gas Law

Molar Volume, Density, and Molar Mass of a Gas

• Molar Volume at STP: 1 mole of an ideal gas occupies 22.4 L at standard temperature and pressure.

• Density of a Gas: $d = \frac{PM}{RT}$, where M is molar mass.

• Molar Mass Calculation: $M = \frac{dRT}{P}$

Gas	Density at STP (g/L)
Helium (He)	0.18
Nitrogen (N2)	1.25
Oxygen (O2)	1.43

Example: Calculate the density of O2 at STP: $d = \frac{(1)(32.00)}{0.08206 \times 273} = 1.43$ g/L.

Mixtures of Gases and Partial Pressures

Dalton's Law of Partial Pressures

• Statement: The total pressure of a mixture of gases is the sum of the partial pressures of each component.

• Formula: $P_{\text{total}} = P_1 + P_2 + P_3 + \ldots$

• Partial Pressure: $P_i = X_i P_{\text{total}}$, where $X_i$ is the mole fraction of component i.

Example: In a mixture of nitrogen and oxygen, if $X_{N_2} = 0.78$ and $P_{\text{total}} = 1.00$ atm, then $P_{N_2} = 0.78$ atm.

Temperature and Molecular Velocities

Kinetic Energy and Root Mean Square Velocity

• Average Kinetic Energy: $KE_{avg} = \frac{3}{2}RT$

• Root Mean Square Velocity (urms): $u_{rms} = \sqrt{\frac{3RT}{M}}$

• Relationship: Lighter gas molecules move faster than heavier ones at the same temperature.

Example: Calculate $u_{rms}$ for O2 at 298 K: $u_{rms} = \sqrt{\frac{3 \times 0.08206 \times 298}{32.00}}$

Mean Free Path, Diffusion, and Effusion of Gases

Definitions and Laws

• Mean Free Path: The average distance a gas particle travels between collisions.

• Diffusion: The mixing of gases due to random motion.

• Effusion: The escape of gas particles through a small hole.

• Graham's Law of Effusion: $\text{Rate}_1/\text{Rate}_2 = \sqrt{M_2/M_1}$

Example: Hydrogen effuses faster than oxygen because it has a lower molar mass.

Gases in Chemical Reactions: Stoichiometry Revisited

Using Gas Laws in Reactions

• Stoichiometric coefficients in chemical equations relate volumes of gases at the same temperature and pressure.

• Use the ideal gas law to calculate the amount of gas produced or consumed in a reaction.

Example: 2 H2 + O2 → 2 H2O: 2 volumes of H2 react with 1 volume of O2 to produce 2 volumes of H2O.

Real Gases: Effects of Size and Intermolecular Forces

Deviations from Ideal Behavior

• Finite Volume: Real gas particles occupy space; at high pressures, this volume becomes significant.

• Intermolecular Forces: Attractions between particles reduce pressure at low temperatures.

• Van der Waals Equation: Accounts for particle volume and intermolecular forces: $\left(P + a\frac{n^2}{V^2}\right)(V - nb) = nRT$

  • a = measure of intermolecular attraction

  • b = measure of particle volume

Example: CO2 deviates from ideal behavior at high pressure and low temperature due to significant intermolecular forces.

Key Terms and Concepts

• Pressure: Force per unit area exerted by gas particles.

• Kinetic Molecular Theory: Model explaining gas behavior based on particle motion.

• Boyle's Law, Charles's Law, Avogadro's Law: Fundamental relationships between gas properties.

• Ideal Gas Law: $PV = nRT$

• Dalton's Law of Partial Pressures: $P_{\text{total}} = P_1 + P_2 + \ldots$

• Root Mean Square Velocity: $u_{rms} = \sqrt{\frac{3RT}{M}}$

• Graham's Law of Effusion: $\text{Rate}_1/\text{Rate}_2 = \sqrt{M_2/M_1}$

• Van der Waals Equation: $\left(P + a\frac{n^2}{V^2}\right)(V - nb) = nRT$

Summary Table: Gas Laws and Equations

Law/Equation	Formula	Relationship
Boyle's Law	$P_1V_1 = P_2V_2$	V ∝ 1/P (T constant)
Charles's Law	$\frac{V_1}{T_1} = \frac{V_2}{T_2}$	V ∝ T (P constant)
Avogadro's Law	$\frac{V_1}{n_1} = \frac{V_2}{n_2}$	V ∝ n (P, T constant)
Ideal Gas Law	$PV = nRT$	Relates P, V, n, T
Dalton's Law	$P_{\text{total}} = \sum P_i$	Total pressure = sum of partial pressures
Density of Gas	$d = \frac{PM}{RT}$	Density from P, M, T
Root Mean Square Velocity	$u_{rms} = \sqrt{\frac{3RT}{M}}$	Velocity from T, M
Graham's Law	$\text{Rate}_1/\text{Rate}_2 = \sqrt{M_2/M_1}$	Effusion rates
Van der Waals Equation	$\left(P + a\frac{n^2}{V^2}\right)(V - nb) = nRT$	Real gas behavior

Conclusion

Understanding the behavior of gases is essential for predicting and explaining physical and chemical phenomena. The kinetic molecular theory and gas laws provide a foundation for describing ideal gases, while the van der Waals equation accounts for real gas deviations. Mastery of these concepts is crucial for success in general chemistry.