Chapter 10: Gases – General Chemistry Study Notes

10.1 Physical Characteristics of Gases

General Properties of Gases

Gases exhibit unique physical properties that distinguish them from solids and liquids. They are composed mainly of nonmetallic elements with simple formulas and low molar masses.

• Expand to fill their containers: Gases do not have a fixed shape or volume.

• Highly compressible: Gases can be compressed much more than solids or liquids.

• Extremely low densities: The density of gases is much lower compared to other states of matter.

• Homogeneous mixtures: Two or more gases can mix completely to form a homogeneous mixture.

Pressure

Definition and Atmospheric Pressure

Pressure is a fundamental property of gases, defined as the amount of force applied to a given area. All gases exert pressure on any surface they contact, and atmospheric pressure is the weight of air per unit area at Earth's surface.

• Formula: $P = \frac{F}{A}$, where P is pressure, F is force, and A is area.

• Atmospheric pressure: Caused by the gravitational force acting on the air column above Earth's surface.

Units of Pressure

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

• Pascals (Pa): $1\,\text{Pa} = 1\,\text{N}/\text{m}^2$ (SI unit)

• Bar: $1\,\text{bar} = 10^5\,\text{Pa} = 100\,\text{kPa}$

• mm Hg or Torr: Based on the height difference in a mercury column (barometer).

• Atmosphere (atm): $1\,\text{atm} = 760\,\text{torr} = 760\,\text{mm Hg} = 101.325\,\text{kPa} = 1.01325\,\text{bar}$

Manometer and Standard Pressure

A manometer is used to measure the pressure of a gas in a vessel relative to atmospheric pressure. Standard atmospheric pressure at sea level is a reference point for many calculations.

• Manometer: Measures the difference in height of mercury columns to determine gas pressure.

• Standard pressure: $1\,\text{atm}$ is equal to $760\,\text{torr}$, $760\,\text{mm Hg}$, $101.325\,\text{kPa}$, $1.01325\,\text{bar}$.

10.2 The Gas Laws

Variables Defining Gas State

The physical state of a gas is defined by four variables: temperature, pressure, volume, and the amount of gas (in moles). The relationships between these variables are described by the gas laws.

• Temperature (K)

• Pressure

• Volume

• Amount of gas (moles)

Boyle's Law: Pressure-Volume Relationship

Boyle's Law states that the volume of a fixed quantity of gas at constant temperature is inversely proportional to the pressure.

• Mathematical relationship: $P_1V_1 = P_2V_2$

• Application: As pressure increases, volume decreases, and vice versa.

Charles's Law: Volume-Temperature Relationship

Charles's Law states that the volume of a fixed amount of gas at constant pressure is directly proportional to its absolute temperature.

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

• Application: As temperature increases, volume increases.

Avogadro's Law: Volume-Amount Relationship

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

• At STP (Standard Temperature and Pressure): One mole of any gas occupies 22.4 L.

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

10.3 The Ideal-Gas Equation

Derivation and Formula

The ideal-gas equation combines Boyle's, Charles's, and Avogadro's laws into a single relationship. It describes the behavior of ideal gases under various conditions.

• Combined relationship: $V \propto \frac{nT}{P}$

• Ideal-gas equation: $PV = nRT$

• R (Ideal Gas Constant): Value depends on units; common value is $0.08206\,\text{L} \cdot \text{atm}/\text{mol} \cdot \text{K}$

Using the Ideal-Gas Equation

The ideal-gas equation can be used to solve for unknown variables in gas problems, such as the number of moles, volume, pressure, or temperature.

• Example: Decomposition of CaCO3 to produce CO2 gas, collected in a flask.

• Calculation: $n = \frac{PV}{RT}$

Combined Gas Law

The combined gas law relates pressure, volume, and temperature for two sets of conditions, useful when the amount of gas is constant.

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

• Application: Used to solve problems involving changes in state variables.

Gas Densities and Molar Mass

The ideal-gas equation can be rearranged to calculate the density and molar mass of a gas.

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

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

Calculating the Molar Mass of a Gas

By measuring the mass, volume, pressure, and temperature of a gas, its molar mass can be determined using the ideal-gas equation.

• Example: Filling a flask with an unknown gas and using mass differences to calculate molar mass.

Volume of Gases in Chemical Reactions

Stoichiometry can be used to relate the volume of gases involved in chemical reactions, using balanced equations and the ideal-gas law.

• Steps: Use stoichiometric ratios, solve for moles using PV = nRT, and convert to desired units.

• Example: Air bag inflation by decomposition of sodium azide.

10.4 Gas Mixtures and Partial Pressures

Dalton's Law of Partial Pressures

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

• Formula: $P_t = P_1 + P_2 + P_3 + \ldots$

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

Mole Fraction and Partial Pressure

The mole fraction of a component in a gas mixture is the ratio of the number of moles of that component to the total number of moles. Partial pressure can be calculated using mole fraction.

• Mole fraction formula: $X_1 = \frac{n_1}{n_t}$

• Partial pressure formula: $P_1 = X_1 P_t$

Applications of Partial Pressures

Partial pressures are used in calculations involving gas mixtures, such as synthetic atmospheres for plant growth studies.

• Example: Calculating partial pressure and moles of O2 in a mixture.

10.5 Kinetic-Molecular Theory of Gases

Fundamental Concepts

The kinetic-molecular theory explains the behavior of gases based on the motion of their molecules.

• Gases consist of large numbers of molecules in continuous, random motion.

• The combined volume of all molecules is negligible relative to the total volume.

• Attractive and repulsive forces between molecules are negligible.

• Energy is transferred during collisions, but average kinetic energy remains constant if temperature is constant.

• Average kinetic energy is proportional to absolute temperature.

Application to Gas Laws

The kinetic-molecular theory provides explanations for the gas laws:

• Increase in volume at constant temperature: Pressure decreases due to fewer collisions with container walls.

• Increase in temperature at constant volume: Pressure increases due to more frequent and forceful collisions.

10.6 Molecular Speeds, Effusion, and Diffusion

Root-Mean-Square Speed

The root-mean-square (rms) speed of gas molecules is related to their kinetic energy and temperature.

• Formula: $u_{rms} = \sqrt{\frac{3RT}{M}}$

• Application: Lighter gases move faster than heavier gases at the same temperature.

Effusion and Diffusion

Effusion is the escape of gas molecules through a tiny hole, while diffusion is the spread of one substance throughout another.

• Effusion: Gas molecules pass through a small opening.

• Diffusion: Gas molecules spread out in a space or another substance.

Graham's Law of Diffusion and Effusion

Graham's Law relates the rates of effusion or diffusion of two gases to their molar masses. The lighter gas always effuses or diffuses faster.

• Formula: $\frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}}$

• Application: Used to identify unknown gases and compare rates of effusion.

Example: Identifying an Unknown Gas

By comparing the rate of effusion of an unknown gas to that of oxygen, its molar mass can be calculated and the gas identified.

• Calculation: Use Graham's Law to solve for molar mass.

Gas Law	Mathematical Relationship	Key Variable
Boyle's Law	$P_1V_1 = P_2V_2$	Pressure, Volume
Charles's Law	$\frac{V_1}{T_1} = \frac{V_2}{T_2}$	Volume, Temperature
Avogadro's Law	$\frac{V_1}{n_1} = \frac{V_2}{n_2}$	Volume, Moles
Ideal-Gas Law	$PV = nRT$	Pressure, Volume, Moles, Temperature
Combined Gas Law	$\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$	Pressure, Volume, Temperature
Dalton's Law	$P_t = P_1 + P_2 + P_3 + \ldots$	Partial Pressures
Graham's Law	$\frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}}$	Rates, Molar Mass