Chapter 10: Gases – Properties, Laws, and Calculations

Characteristics of Gases

Physical Properties of Gases

Gases differ significantly from liquids and solids in their physical properties due to the large distances between particles and weak intermolecular forces.

• Compressibility: Gases are highly compressible compared to liquids and solids.

• Expansion: Gases expand to fill the volume of their containers.

• Low Density: Gases have much lower densities than liquids and solids.

Condensed phases refer to liquids and solids, which are not as compressible and have higher densities.

Variables Affecting Gas Behavior

The behavior of gases is determined by four main variables:

• Amount (n): Number of moles of gas.

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

• Volume (V): Space occupied by the gas.

• Temperature (T): Measured in Kelvin, affects kinetic energy.

Pressure

Definition and Units

Pressure is the amount of force applied to an area. Atmospheric pressure is the weight of air per unit area at Earth's surface.

• Formula: $P = \frac{F}{A}$

• Standard atmospheric pressure: $1.00\ \text{atm} = 760\ \text{mm Hg (torr)} = 101.325\ \text{kPa}$

Common Units of Pressure

Unit	Abbreviation	Average Pressure at Sea Level
Pascal	Pa	101,325
Pounds per square inch	psi	14.7
Torr (mmHg)	torr	760
Inches of mercury	in Hg	29.92
Bar	bar	1.013
Atmosphere	atm	1.00

Kinetic-Molecular Theory

Main Tenets

The kinetic-molecular theory explains the behavior of gases at the molecular level:

• Continuous, random motion: Gas molecules move in constant, random motion.

• Elastic collisions: Collisions between molecules and container walls are perfectly elastic (no energy loss).

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

• Negligible volume: The volume of individual molecules is negligible compared to the total volume of the gas.

• Energy transfer: Energy can be transferred during collisions, but the average kinetic energy remains constant at constant temperature.

• Kinetic energy and temperature: The average kinetic energy of gas molecules is proportional to the absolute temperature ($KE \propto T$).

Formula for average kinetic energy: $KE = \frac{1}{2}mv^2$

Simple Gas Laws

Boyle's Law (Pressure-Volume Relationship)

At constant temperature and amount, the volume of a gas is inversely proportional to its pressure.

• Formula: $P_1V_1 = P_2V_2$

• Graph: As pressure increases, volume decreases.

• Example: If a gas at 1.0 atm has a volume of 10 L, increasing the pressure to 2.0 atm reduces the volume to 5 L.

Charles's Law (Temperature-Volume Relationship)

At constant pressure and amount, the volume of a gas is directly proportional to its absolute temperature.

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

• Graph: As temperature increases, volume increases.

• Example: Heating a gas from 200 K to 400 K doubles its volume.

Gay-Lussac's Law (Pressure-Temperature Relationship)

At constant volume and amount, the pressure of a gas is directly proportional to its absolute temperature.

• Formula: $\frac{P_1}{T_1} = \frac{P_2}{T_2}$

Avogadro's Law (Moles-Volume Relationship)

At constant pressure and temperature, the volume of a gas is directly proportional to the number of moles.

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

• Graph: As the amount of gas increases, volume increases linearly.

• Example: Doubling the number of moles doubles the volume.

Combined Gas Law

Combines Boyle's, Charles's, and Gay-Lussac's laws for a fixed amount of gas under two sets of conditions.

• Formula: $\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$

• Example: Used to calculate changes in volume, pressure, or temperature when conditions change.

Ideal Gas Law

Equation and Applications

The ideal gas law relates pressure, volume, temperature, and amount of gas in a single equation.

• Formula: $PV = nRT$

• R: Universal gas constant (0.0821 L·atm/mol·K or 8.314 J/mol·K)

• Useful for calculating any one variable when the others are known.

• Example: What volume does 0.118 mol of helium occupy at 0.97 atm and 305 K?

Molar Volume, Density, and Molar Mass of a Gas

Density and Molar Mass Calculations

Density and molar mass of gases can be determined using the ideal gas law.

• Density formula: $d = \frac{PM}{RT}$

• Molar mass formula: $M = \frac{mRT}{PV}$

• Example: Calculate the density of oxygen at 25°C and 25.0 atm.

Dalton's Law of Partial Pressures

Partial Pressure in Gas Mixtures

The total pressure of a mixture of gases equals the sum of the partial pressures of each component.

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

• Mole fraction formula: $\chi_A = \frac{n_A}{n_{\text{total}}}$

• Partial pressure formula: $P_A = \chi_A P_{\text{total}}$

• Example: Calculate the partial pressure of helium in a mixture.

Vapor Pressure of Water

Collecting Gases Over Water

When collecting gases over water, the vapor pressure of water must be subtracted from the total pressure to obtain the pressure of the dry gas.

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

• Example: Calculate the mass of oxygen gas collected over water at a given temperature and pressure.

Temp (°C)	Vapor Pressure (torr)
0	4.58
10	9.21
20	17.54
25	23.76
30	31.82

Additional info:

• These notes cover the core concepts of Chapter 10: Gases, including physical properties, gas laws, kinetic-molecular theory, and calculations involving the ideal gas law and partial pressures.

• Practice problems and examples are included throughout to reinforce understanding and application of the concepts.