Chapter 10: Gases – Properties, Laws, and Theories

Chapter 10: Gases

10.1 Physical Characteristics of Gases

Gases are a fundamental state of matter with unique physical properties that distinguish them from solids and liquids. Most gases are composed of nonmetallic elements, often with simple molecular formulas and low molar masses.

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

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

• Low Density: Gases have much lower densities than other states of matter.

• Homogeneous Mixtures: Two or more gases mix completely to form homogeneous mixtures.

10.1 Pressure and Its Measurement

Pressure is defined as the force exerted per unit area. All gases exert pressure on any surface they contact, including the walls of their containers.

• Atmospheric Pressure: The pressure exerted by the weight of the atmosphere above a given point.

• Units of Pressure:

  • Pascals (Pa): SI unit of pressure.

  • Bar: 1 bar = 100 kPa.

  • Millimeters of Mercury (mmHg) or Torr: Based on the height difference in a mercury column.

  • Atmosphere (atm): 1 atm = 760 torr = 760 mmHg = 101.325 kPa = 1.01325 bar.

• Manometer: Device used to measure the pressure of a gas in a vessel relative to atmospheric pressure.

• Standard Pressure: Normal atmospheric pressure at sea level, defined as 1 atm.

10.2 The Gas Laws

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

Boyle’s Law: Pressure–Volume Relationship

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

• Mathematical Form: $P V = \text{constant}$

• For two sets of conditions: $P_1 V_1 = P_2 V_2$

• A plot of V versus P is not linear, but V versus 1/P is linear.

Charles’s Law: Volume–Temperature Relationship

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

• Mathematical Form: $\frac{V}{T} = \text{constant}$

• For two sets of conditions: $\frac{V_1}{T_1} = \frac{V_2}{T_2}$

• A plot of V versus T is linear.

Avogadro’s Law: Quantity–Volume Relationship

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

• Mathematical Form: $\frac{V}{n} = \text{constant}$

• At STP (Standard Temperature and Pressure: 0°C, 1 atm), 1 mole of any gas occupies 22.4 L.

10.3 The Ideal-Gas Equation

The ideal-gas equation combines Boyle’s, Charles’s, and Avogadro’s laws into a single relationship:

• Equation: $P V = n R T$

• R (Ideal Gas Constant): $R = 0.08206 \ \text{L} \cdot \text{atm} / (\text{mol} \cdot \text{K})$ (other values exist for different units)

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

Solving Ideal Gas Problems

• Tabulate known and unknown variables.

• Convert all quantities to consistent units.

• Choose the appropriate equation and solve for the unknown.

• Use dimensional analysis for unit conversions.

Gas Densities and Molar Mass

• Density ($d$) of a gas can be calculated using the ideal-gas equation:

• $d = \frac{P M}{R T}$, where M is molar mass.

• Molar mass can be determined if mass, volume, and temperature are known.

Volume of Gases in Chemical Reactions

• Stoichiometry can be used to relate volumes of gases in chemical reactions.

• If given volume, use stoichiometric ratios to find unknowns.

• If given P, V, and T, solve for n using $P V = n R T$, then use mole ratios to find other quantities.

10.4 Gas Mixtures and Partial Pressures

When two or more non-reacting gases are combined, each behaves independently as if it were alone in the container.

• Dalton’s Law of Partial Pressures: The total pressure of a mixture of gases equals the sum of the pressures each gas would exert if present alone.

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

• Mole Fraction (χ): $\chi_i = \frac{n_i}{n_{\text{total}}}$

• Partial pressure of a component: $P_i = \chi_i P_{\text{total}}$

10.5 Kinetic-Molecular Theory of Gases

The kinetic-molecular theory explains the observed behavior of gases and the gas laws.

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

• The combined volume of all molecules is negligible compared to the container volume.

• Attractive and repulsive forces between molecules are negligible.

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

• Average kinetic energy is proportional to absolute temperature.

• At the same temperature, all gases have the same average kinetic energy.

Distributions of Molecular Speed

• Temperature is related to average kinetic energy, but individual molecules have a range of speeds.

• Most probable speed, average speed, and root-mean-square speed are key statistical measures.

Application to Gas Laws

• Increasing volume at constant temperature decreases pressure (fewer collisions with container walls).

• Increasing temperature at constant volume increases pressure (more frequent and forceful collisions).

10.6 Molecular Speeds, Effusion, and Diffusion

• At a given temperature, all gases have the same average kinetic energy.

• Lighter gas molecules move faster than heavier ones at the same temperature.

• Effusion: Escape of gas molecules through a tiny hole into an evacuated space.

• Diffusion: Spread of one substance throughout a space or another substance.

• Graham’s Law: The rate of effusion or diffusion of a gas is inversely proportional to the square root of its molar mass.

• Mathematical Form: $\frac{\text{Rate}_1}{\text{Rate}_2} = \sqrt{\frac{M_2}{M_1}}$

10.7 Real Gases: Deviations from Ideal Behavior

Real gases deviate from ideal behavior at high pressures and low temperatures due to intermolecular forces and the finite volume of molecules.

• The assumptions of the kinetic-molecular theory break down under these conditions.

• Corrections can be made using the van der Waals equation, which adjusts for molecular volume and intermolecular attractions.

• van der Waals Equation: $\left( P + \frac{a n^2}{V^2} \right) (V - n b) = n R T$

• Constants a and b are specific to each gas and account for intermolecular forces and molecular size, respectively.