Chapter 6: Gases – Properties, Laws, and Molecular Theory

Gases: Properties and Behavior

Introduction to Gases

Gases are one of the fundamental states of matter, characterized by their ability to expand and fill any container. Their behavior is governed by several physical laws that relate pressure, volume, temperature, and the amount of gas present.

• Key Properties: Pressure (P), Volume (V), Temperature (T), Amount in moles (n)

• Applications: Atmospheric phenomena, respiration, industrial processes

Gas Pressure

Pressure is the force exerted per unit area by gas molecules as they collide with the surfaces around them.

• Definition: $\text{Pressure} = \frac{\text{Force}}{\text{Area}}$

• Gas molecules in constant motion exert pressure by colliding with container walls.

• Atmospheric pressure varies with altitude and weather conditions.

Units of Pressure

Pressure can be measured in several units, each commonly used in different contexts.

Unit	Symbol	Value at Sea Level
Pascals	Pa	101,325 Pa
Atmospheres	atm	1 atm
Millimeters of mercury	mmHg	760 mmHg
Torr	torr	760 torr
Pounds per square inch	psi	14.7 psi
Inches of mercury	inHg	29.92 inHg

Measuring Pressure: Barometers and Manometers

• Barometer: Measures atmospheric pressure using a column of mercury.

• Manometer: Measures the pressure of a gas sample in a container, comparing it to atmospheric pressure.

• Difference in liquid levels indicates pressure difference.

Simple Gas Laws

Boyle’s Law: Pressure and Volume

Boyle’s Law describes the inverse relationship between the pressure and volume of a gas at constant temperature and amount.

• Mathematical Form: $P_1 V_1 = P_2 V_2$

• As pressure increases, volume decreases proportionally.

• Graph of P vs V is a curve; P vs 1/V is a straight line.

• Example: Diving – pressure increases with depth, so the volume of air in lungs decreases.

Charles’s Law: Volume and Temperature

Charles’s Law states that the volume of a fixed amount of gas at constant pressure increases linearly with temperature (in kelvins).

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

• Temperature must be in kelvins: $T(K) = T(°C) + 273$

• At absolute zero (0 K), volume extrapolates to zero.

• Example: Balloon expands when moved from cold to hot water due to increased kinetic energy of gas particles.

Avogadro’s Law: Volume and Amount

Avogadro’s Law states that the volume of a gas is directly proportional to the number of moles of gas at constant temperature and pressure.

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

• Equal volumes of gases at the same temperature and pressure contain equal numbers of molecules.

Ideal Gas Law

The relationships described by Boyle’s, Charles’s, and Avogadro’s laws can be combined into the Ideal Gas Law, which relates all four properties.

• Equation: $PV = nRT$

• R is the gas constant: $R = 0.08206 \; \text{L} \cdot \text{atm} / \text{mol} \cdot \text{K}$

• Allows calculation of any one property if the others are known.

Standard Temperature and Pressure (STP) and Molar Volume

• STP: 0°C (273 K) and 1 atm

• Molar Volume at STP: 1 mole of any ideal gas occupies 22.4 L

• Identity of the gas does not affect molar volume at STP

Density and Molar Mass of Gases

Density of a gas is the ratio of its mass to volume, typically in g/L.

• Equation: $\text{Density} = \frac{\text{Molar Mass}}{\text{Molar Volume}}$

• Density is directly proportional to molar mass.

• Helium is less dense than air, so helium balloons float.

Mixtures of Gases and Partial Pressures

Dalton’s Law of Partial Pressures

In a mixture of gases, each component exerts a pressure independently of the others. The total pressure is the sum of the partial pressures.

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

• Partial pressure of a component: $P_i = \frac{n_i RT}{V}$

Mole Fraction

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

• Equation: $X_i = \frac{n_i}{n_{\text{total}}}$

• Partial pressure: $P_i = X_i P_{\text{total}}$

• Example: Nitrogen in air (78% by volume): $P_{N_2} = 0.78 \times 1.00 \; \text{atm} = 0.78 \; \text{atm}$

Collecting Gases Over Water

When gases are collected by displacement of water, the total pressure includes both the gas and water vapor.

• Partial pressure of water vapor depends only on temperature.

• Use tables to find vapor pressure at a given temperature.

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

Kinetic Molecular Theory

Postulates of Kinetic Molecular Theory

The kinetic molecular theory provides a molecular-level explanation for the behavior of gases.

• Gas particles are negligibly small compared to the distances between them.

• Particles are in constant, random motion.

• Collisions are perfectly elastic (no energy loss).

• Average kinetic energy is proportional to temperature in kelvins.

Temperature and Molecular Velocities

• Average kinetic energy: $KE_{\text{avg}} = \frac{1}{2} N_A m u^2$

• Root mean square velocity: $u_{\text{rms}} = \sqrt{\frac{3RT}{M}}$ (M = molar mass in kg/mol)

• Lighter molecules move faster at the same temperature.

Diffusion and Effusion

• Diffusion: Spread of molecules from high to low concentration.

• Effusion: Escape of gas molecules through a small hole into a vacuum.

• Rate of effusion is inversely proportional to the square root of molar mass.

• Graham’s Law: $\frac{\text{rate}_1}{\text{rate}_2} = \sqrt{\frac{M_2}{M_1}}$

Real Gases and Deviations from Ideal Behavior

Limitations of the Ideal Gas Law

At high pressures and low temperatures, real gases deviate from ideal behavior due to molecular volume and intermolecular attractions.

• Real gas molecules occupy space and experience attractions.

• At high pressure, volume is greater than predicted; at low temperature, pressure is less than predicted.

Van der Waals Equation

Johannes van der Waals modified the ideal gas law to account for real gas behavior.

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

• a = correction for intermolecular forces; b = correction for particle volume

• Each gas has unique a and b constants.

Summary Table: Ideal vs. Real Gas Behavior

Condition	Ideal Gas Prediction	Real Gas Observation
Low Pressure	PV/RT ≈ 1	PV/RT < 1 (intermolecular attractions)
High Pressure	PV/RT ≈ 1	PV/RT > 1 (molecular volume)
High Temperature	PV/RT ≈ 1	PV/RT ≈ 1 (ideal behavior)

Key Equations and Concepts

• Pressure: $P = \frac{F}{A}$

• Boyle’s Law: $P_1 V_1 = P_2 V_2$

• Charles’s Law: $\frac{V_1}{T_1} = \frac{V_2}{T_2}$

• Avogadro’s Law: $\frac{V_1}{n_1} = \frac{V_2}{n_2}$

• Ideal Gas Law: $PV = nRT$

• Density: $d = \frac{\text{molar mass}}{RT} P$

• Dalton’s Law: $P_{\text{total}} = P_1 + P_2 + \ldots$

• Graham’s Law: $\frac{\text{rate}_1}{\text{rate}_2} = \sqrt{\frac{M_2}{M_1}}$

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

Additional info: These notes expand on the original slides by providing full definitions, equations, and context for each law and concept, as well as examples and tables for clarity.