Chapter 10: Gases – Laws, Properties, and Kinetic Theory

Chapter 10: Gases

Gas Laws

The behavior of gases can be described by several fundamental laws, each relating different properties such as pressure, volume, and temperature. These laws are essential for understanding how gases respond to changes in their environment.

• Boyle’s Law: At constant temperature, the volume of a fixed amount of gas is inversely proportional to its pressure.

• Charles’s Law: At constant pressure, the volume of a fixed amount of gas is directly proportional to its absolute temperature.

• Gay-Lussac’s Law: At constant volume, the pressure of a fixed amount of gas is directly proportional to its absolute temperature.

• Combined Gas Law: Combines Boyle’s, Charles’s, and Gay-Lussac’s laws to relate pressure, volume, and temperature for a fixed amount of gas.

• Avogadro’s Law: At constant temperature and pressure, the volume of a gas is directly proportional to the number of moles of gas present.

Equations:

• Boyle’s Law: $P_1V_1 = P_2V_2$

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

• Gay-Lussac’s Law: $\frac{P_1}{T_1} = \frac{P_2}{T_2}$

• Combined Gas Law: $\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$

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

Example: If a 2.0 L sample of gas at 1.0 atm is compressed to 1.0 L at constant temperature, the new pressure is $P_2 = \frac{P_1V_1}{V_2} = \frac{1.0 \times 2.0}{1.0} = 2.0$ atm.

Kinetic Molecular Theory

The kinetic molecular theory explains the macroscopic properties of gases by considering their molecular composition and motion.

• Gases consist of a large number of particles in constant, random motion.

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

• Collisions between gas molecules and with the walls of the container are perfectly elastic.

• There are no intermolecular forces between gas particles.

• The average kinetic energy of gas particles is proportional to the absolute temperature.

Equation for average kinetic energy:

$KE_{avg} = \frac{3}{2}k_BT$

where $k_B$ is the Boltzmann constant and $T$ is the temperature in Kelvin.

The Ideal Gas Law

The ideal gas law combines the relationships between pressure, volume, temperature, and number of moles into a single equation.

• Equation: $PV = nRT$

• $P$ = pressure (atm), $V$ = volume (L), $n$ = moles, $R$ = ideal gas constant ($0.0821$ L·atm·mol−1·K−1), $T$ = temperature (K)

Example: Calculate the volume occupied by 2.0 mol of gas at 1.0 atm and 273 K: $V = \frac{nRT}{P} = \frac{2.0 \times 0.0821 \times 273}{1.0} = 44.8$ L.

Standard Temperature and Pressure (STP), Problem Solving, and Density

STP is a reference point for gas measurements, and density calculations are often required in gas law problems.

• STP: Standard temperature is 0°C (273.15 K), and standard pressure is 1 atm.

• At STP, 1 mole of an ideal gas occupies 22.4 L.

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

Example: Find the density of O2 at STP: $d = \frac{1.0 \times 32.0}{0.0821 \times 273} = 1.43$ g/L.

Stoichiometry of Gases

Gas stoichiometry involves using the relationships between moles, volume, and the ideal gas law to solve chemical reaction problems involving gases.

• At STP, use molar volume (22.4 L/mol) to convert between moles and volume.

• For non-STP conditions, use the ideal gas law to find moles or volume as needed.

Example: What volume of CO2 is produced at STP from the combustion of 4.0 g CH4? (Balanced equation: CH4 + 2O2 → CO2 + 2H2O)

• Moles CH4 = $\frac{4.0}{16.0} = 0.25$ mol

• Moles CO2 = 0.25 mol (1:1 ratio)

• Volume CO2 = $0.25 \times 22.4 = 5.6$ L

Mole Fraction of a Gas

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

• Equation: $X_A = \frac{n_A}{n_{total}}$

• The sum of all mole fractions in a mixture equals 1.

Example: In a mixture of 2 mol O2 and 3 mol N2, $X_{O_2} = \frac{2}{5} = 0.40$

Partial Pressure and Total Pressure of a Gas

Dalton’s Law of Partial Pressures states that the total pressure of a gas mixture is the sum of the partial pressures of each component.

• Partial pressure: $P_A = X_A \times P_{total}$

• Dalton’s Law: $P_{total} = P_A + P_B + P_C + ...$

Example: If $X_{O_2} = 0.40$ and $P_{total} = 1.0$ atm, then $P_{O_2} = 0.40$ atm.

Maxwell Distribution / Distribution of Molecular Speed

The Maxwell-Boltzmann distribution describes the range of speeds of molecules in a gas at a given temperature.

• Most molecules have speeds near the average, but some move much faster or slower.

• As temperature increases, the distribution broadens and the average speed increases.

• The root mean square (rms) speed is given by:

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

where $u_{rms}$ is the rms speed, $R$ is the gas constant, $T$ is temperature in Kelvin, and $M$ is the molar mass in kg/mol.

Example: Calculate the rms speed of O2 at 300 K: $u_{rms} = \sqrt{\frac{3 \times 8.314 \times 300}{0.032}} = 484$ m/s.

Summary Table: Gas Laws and Their Relationships

Law	Variables Held Constant	Relationship	Equation
Boyle’s Law	Temperature, moles	$P \propto \frac{1}{V}$	$P_1V_1 = P_2V_2$
Charles’s Law	Pressure, moles	$V \propto T$	$\frac{V_1}{T_1} = \frac{V_2}{T_2}$
Gay-Lussac’s Law	Volume, moles	$P \propto T$	$\frac{P_1}{T_1} = \frac{P_2}{T_2}$
Avogadro’s Law	Pressure, temperature	$V \propto n$	$\frac{V_1}{n_1} = \frac{V_2}{n_2}$
Combined Gas Law	moles	Relates $P$, $V$, $T$	$\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$
Ideal Gas Law	None	Relates $P$, $V$, $n$, $T$	$PV = nRT$

Additional info: This guide expands on the listed topics with definitions, equations, and examples to provide a comprehensive overview suitable for exam preparation in a general chemistry course.