Study Notes: Properties and Laws of Gases (Handout 5, Chapter 7)

Properties and Composition of Gases

General Characteristics of Gases

Gases are one of the fundamental states of matter, distinguished by their ability to fill any container and their high compressibility. The Earth's atmosphere is primarily composed of nitrogen and oxygen, with trace amounts of other gases.

• Major components of air: 78% N2, 21% O2, and 1% other gases.

• Common diatomic gas molecules: N2, O2, Cl2, F2, H2.

• Group 8 elements: Monoatomic noble gases (e.g., He, Ne, Ar).

Physical Properties of Gases

• Gases assume the volume and shape of their containers.

• They are the most compressible state of matter.

• Gases mix evenly and completely when confined together.

• They have much lower densities than liquids and solids.

• Gas molecules are in constant motion and exert pressure on surfaces they contact.

Gas Pressure and Measurement

Definition and Units of Pressure

Pressure is the force exerted per unit area by gas molecules colliding with surfaces. Atmospheric pressure is the pressure exerted by Earth's atmosphere.

• Pressure units: $1 ext{ Pa} = 1 ext{ N/m}^2$ $1.0 ext{ atm} = 760 ext{ mmHg} = 760 ext{ torr} = 101,325 ext{ Pa}$

• Barometer: Instrument for measuring atmospheric pressure.

• Standard atmospheric pressure is the pressure exerted by a column of mercury 760 mm high at 0°C.

Manometers

Manometers are devices used to measure the pressure of gases other than the atmosphere.

• Closed-tube manometer: Measures pressures below atmospheric pressure. Gas pressure is determined by the height difference (h) of mercury.

• Open-tube manometer: Can measure both above and below atmospheric pressure. The pressure is calculated by adding or subtracting the mercury column height from atmospheric pressure.

Gas Laws

Boyle’s Law: Pressure-Volume Relationship

Boyle’s law describes how the volume of a gas changes with pressure at constant temperature.

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

• Equation: $P imes V = k_1$ $P_1 V_1 = P_2 V_2$

• Example: If a sample of chlorine gas has a volume of 946 mL at 726 mmHg, and the volume is reduced to 154 mL at constant temperature, the new pressure is: $P_2 = \frac{P_1 V_1}{V_2} = \frac{726 \times 946}{154} = 4460 \text{ mmHg}$

Charles’s Law: Volume-Temperature Relationship

Charles’s law relates the volume of a gas to its absolute temperature at constant pressure.

• Statement: At constant pressure, the volume of a fixed amount of gas is proportional to its absolute temperature (in Kelvin).

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

• Example: A sample of CO gas occupies 3.2 L at 125°C (398 K). To find the temperature at which the volume is 1.54 L: $\frac{3.2}{398} = \frac{1.54}{T_2} \Rightarrow T_2 = 191.5 \text{ K}$

Avogadro’s Law: Volume-Mole Relationship

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

• Statement: $V \propto n$ (at constant P and T)

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

Combined Gas Law

The combined gas law incorporates Boyle’s, Charles’s, and Avogadro’s laws to relate pressure, volume, temperature, and moles.

• Equation: $\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}$

• Ideal Gas Law: $PV = nRT$ Where $R$ is the ideal gas constant ($0.08206 \text{ L atm/mol K}$ or $8.314 \text{ J/mol K}$).

Applications of the Ideal Gas Law

Calculating Volume, Pressure, and Moles

• Example: Calculate the volume occupied by 2.12 moles of NO at 6.54 atm and 76°C: $V = \frac{nRT}{P} = \frac{2.12 \times 0.082 \times (273+76)}{6.54} = 9.3 \text{ L}$

• Example: Find the volume occupied by 49.8 g of HCl at 1 atm and 0°C: $n = \frac{49.8}{36.5}$ $V = \frac{nRT}{P} = \frac{1.364 \times 0.082 \times 273}{1} = 30.5 \text{ L}$

Density and Molar Mass of Gases

• Density equation: $d = \frac{PM}{RT}$ Where $M$ is molar mass.

• Example: Density of uranium hexafluoride (UF6) at 779 mmHg and 62°C: $M = 352 \text{ g/mol}$ $P = 1.025 \text{ atm}$ $T = 335 \text{ K}$ $d = \frac{1.025 \times 352}{0.082 \times 335} = 13.1 \text{ g/L}$

Stoichiometry of Gaseous Reactions

Volume Relationships in Reactions

Under identical conditions, the volumes of gases involved in a chemical reaction are proportional to the coefficients in the balanced equation.

• Example: $2C_4H_{10} + 13O_2 \rightarrow 8CO_2 + 10H_2O$ 14.9 L of C4H10 requires: $14.9 \text{ L} \times \frac{13 \text{ L } O_2}{2 \text{ L } C_4H_{10}} = 96.85 \text{ L } O_2$

Dalton’s Law of Partial Pressures

Mixtures of Gases

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

• Equation: $P_{total} = P_A + P_B + P_C + ...$

• Mole fraction: $X_A = \frac{n_A}{n_{total}}$ $P_A = X_A P_{total}$

• Example: If a sample contains 8.24 mol CH4, 0.421 mol C2H6, and 0.116 mol C3H8 at 1.37 atm, calculate partial pressures using mole fractions.

Kinetic Molecular Theory of Gases

Postulates and Implications

The kinetic molecular theory explains the behavior of gases at the molecular level.

• Gases consist of small particles in constant, random motion.

• Collisions between particles and container walls are elastic (no loss of kinetic energy).

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

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

Root-Mean-Square (rms) Speed

• Equation: $u_{rms} = \sqrt{\frac{3RT}{M}}$ Where $M$ is molar mass in kg/mol.

• Example: Calculate $u_{rms}$ for N2 at 250 K: $M = 0.028 \text{ kg/mol}$ $u_{rms} = \sqrt{\frac{3 \times 8.314 \times 250}{0.028}} = 472 \text{ m/s}$

Summary Table: Gas Laws

Law	Relationship	Equation
Boyle's Law	Pressure & Volume (T constant)	$P_1 V_1 = P_2 V_2$
Charles's Law	Volume & Temperature (P constant)	$\frac{V_1}{T_1} = \frac{V_2}{T_2}$
Avogadro's Law	Volume & Moles (P, T constant)	$\frac{V_1}{n_1} = \frac{V_2}{n_2}$
Ideal Gas Law	P, V, n, T	$PV = nRT$
Dalton's Law	Partial Pressures	$P_{total} = \sum P_i$

Additional info: Some equations and examples have been expanded for clarity and completeness. The notes are based on the provided handout and standard general chemistry knowledge.