Chapter 10: Gases – Laws, Properties, and Real Gas Behavior

Gases: Laws, Properties, and Real Gas Behavior

Boyle’s Law

Boyle’s Law describes the relationship between the pressure and volume of a gas at constant temperature and number of moles. It is a fundamental gas law used to predict how a change in pressure affects the volume of a gas sample.

• Statement: At constant temperature and moles, pressure is inversely proportional to volume.

• Mathematical Form: $P_1V_1 = P_2V_2$

• Conditions: Temperature (T) and amount of gas (n) are constant.

• Relationship: $P \propto \frac{1}{V}$

• Example: If a gas occupies 2.50 L at 1.20 atm, what is the new volume at 3.00 atm (constant T)? $1.20 \times 2.50 = 3.00 \times V_2$ $V_2 = 1.00$ L

Charles’s Law

Charles’s Law relates the volume and temperature of a gas at constant pressure and moles. It shows that volume increases with temperature (in Kelvin).

• Statement: At constant pressure and moles, volume is directly proportional to temperature (in Kelvin).

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

• Conditions: Pressure (P) and amount of gas (n) are constant.

• Temperature: Must be in Kelvin.

• Example: A balloon has a volume of 3.0 L at 300 K. What is its volume at 450 K? $V_2 = 3.0 \times \frac{450}{300} = 4.5$ L

Avogadro’s Law

Avogadro’s Law connects the volume of a gas to the number of moles at constant temperature and pressure. It is essential for understanding molar volume and stoichiometry in gases.

• Statement: At constant temperature and pressure, volume is directly proportional to moles of gas.

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

• Relationship: $V \propto n$

• At STP: 1 mole of an ideal gas occupies 22.4 L (STP: 1 atm, 273.15 K).

• Example: If 2.0 mol of gas occupies 44.8 L at STP, what volume would 3.0 mol occupy? $V_2 = 44.8 \times \frac{3.0}{2.0} = 67.2$ L

The Gas Constant (R)

The gas constant R is a proportionality constant in the ideal gas law. Its value depends on the units used for pressure and volume.

• Common Values:

  • 0.08206 L·atm/(mol·K)

  • 8.314 J/(mol·K)

  • 62.36 L·torr/(mol·K)

• Tip: Choose R to match the pressure units in your calculation.

Ideal Gas Law

The ideal gas law combines Boyle’s, Charles’s, and Avogadro’s laws into a single equation that relates pressure, volume, temperature, and moles of a gas.

• Equation: $PV = nRT$

• Variables: P = pressure, V = volume, n = moles, R = gas constant, T = temperature (K)

• Example: How many moles of gas are in a 10.0 L container at 2.00 atm and 300 K? $n = \frac{PV}{RT} = \frac{2.00 \times 10.0}{0.08206 \times 300} = 0.812$ mol

Dalton’s Law of Partial Pressures

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

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

• Partial Pressure Formula: $P_i = X_i P_{total}$

• Each gas behaves independently in the mixture.

Mole Fraction

The mole fraction expresses the ratio of the number of moles of a component to the total number of moles in a mixture. It is used to calculate partial pressures.

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

• Properties:

  • Dimensionless quantity

  • Sum of all mole fractions in a mixture equals 1

• Example: A mixture contains 2.0 mol N2 and 3.0 mol O2. Total pressure = 5.0 atm. Total moles = 5.0 mol $X_{N_2} = 0.40$ $P_{N_2} = 0.40 \times 5.0 = 2.0$ atm $P_{O_2} = 3.0$ atm

Kinetic Molecular Theory (KMT)

KMT explains the behavior of gases at the molecular level and provides the foundation for the gas laws.

• Main Assumptions:

  1. Gas particles are in constant random motion.

  2. Particle volume is negligible compared to the container volume.

  3. No intermolecular forces between particles.

  4. Collisions are perfectly elastic (no energy lost).

  5. Average kinetic energy depends only on temperature.

Molecular Speeds

The speed of gas molecules varies, but several characteristic speeds are defined in kinetic theory.

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

• Most Probable Speed: $u_{mp} = \sqrt{\frac{2RT}{M}}$

• Variables: M = molar mass (kg/mol), R = 8.314 J/(mol·K), T = temperature (K)

• Key Relationship: $u_{rms} > u_{avg} > u_{mp}$

• Trends:

  • Higher temperature → faster molecules

  • Lower molar mass → faster molecules

• Example: Which moves faster at the same temperature: He or O2? Lower molar mass (He) → faster speed.

Effusion and Diffusion

Effusion and diffusion describe the movement of gas molecules. Graham’s Law quantifies the rates based on molar mass.

• Diffusion: Spreading of gas molecules due to random motion.

• Effusion: Passage of gas through a tiny hole without collisions.

• Graham’s Law: $\frac{rate_1}{rate_2} = \sqrt{\frac{M_2}{M_1}}$

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

• Example: How much faster does H2 effuse than O2? $\frac{rate_{H_2}}{rate_{O_2}} = \sqrt{\frac{32}{2}} = \sqrt{16} = 4$ H2 effuses 4 times faster.

van der Waals Gases (Real Gases)

The ideal gas law does not accurately describe real gases under all conditions. The van der Waals equation introduces corrections for intermolecular attractions and molecular volume.

• When Ideal Gas Law Fails:

  • High pressure (particles are close together)

  • Low temperature (attractive forces become significant)

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

• Corrections:

  • a corrects for intermolecular attractions

  • b corrects for molecular volume

Concept Connections and Skills Checklist

Understanding the relationships among the gas laws and their molecular basis is essential for mastering gas behavior.

• Boyle’s, Charles’s, and Avogadro’s laws are special cases of the ideal gas law.

• Kinetic Molecular Theory explains gas laws at the microscopic level.

• Graham’s Law is derived from kinetic molecular theory.

• Deviations from ideality occur when KMT assumptions fail (high P, low T).

Skill	Description
Convert temperature to Kelvin	Add 273.15 to Celsius temperature
Choose correct R value	Match R to pressure units (atm, torr, Pa, etc.)
Rearrange PV = nRT algebraically	Solve for any variable (P, V, n, or T)
Calculate partial pressures	Use mole fraction and total pressure
Compare molecular speeds	Use rms speed formula and molar mass
Apply Graham’s law	Calculate relative rates of effusion/diffusion
Recognize when van der Waals equation is required	High pressure or low temperature conditions
Interpret qualitative trends	Predict effects of changing T, P, or M on gas behavior