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

Chapter 10: Gases

10.1. Characteristics of Gases

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

• Homogeneous Mixtures: Two or more gases can form homogeneous mixtures, such as the atmosphere. In contrast, liquids or solids may or may not mix homogeneously depending on their chemical nature.

• Key Properties of Gases:

  • Expand to fill their containers

  • Highly compressible

  • Extremely low densities compared to liquids and solids

10.2. Pressure of Gases and Atmospheric Pressure

Pressure is a defining property of gases, resulting from the force exerted by gas molecules colliding with surfaces.

• Pressure (P): The amount of force applied to a unit area.

• Formula: $P = \frac{F}{A}$

• Atmospheric Pressure: The weight of air per unit area at Earth's surface.

Units of Pressure

• Pascal (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 mm of mercury columns (barometer).

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

Standard Pressure

• Standard atmospheric pressure at sea level is defined as:

  • 1 atm

  • 760 torr (760 mm Hg)

  • 101.325 kPa

Measuring Pressure: Manometer and Barometer

• Manometer: Measures the difference in pressure between atmospheric pressure and that of a gas in a vessel.

• Barometer: Measures atmospheric pressure.

Properties That Define the State of a Gas Sample

Four variables are needed to describe the state of a gas:

1. Pressure (P)

2. Temperature (T)

3. Volume (V)

4. Amount of gas (n, in moles)

The relationships among these variables are described by the gas laws.

10.3. Boyle's Law

Boyle's Law describes the inverse relationship between the pressure and volume of a gas at constant temperature and amount.

• Statement: For a fixed amount of gas at constant temperature, the volume increases as the pressure decreases.

• Equation: $PV = \text{constant}$

• Two-State Form: $P_1V_1 = P_2V_2$

• Graph: V vs. P is a curve; V vs. 1/P is linear.

• Example: Weather balloons expand as they rise due to decreasing atmospheric pressure.

Charles's Law

Charles's Law relates the volume of a gas to its absolute temperature at constant pressure and amount.

• Statement: The volume of a fixed amount of gas at constant pressure is directly proportional to its absolute temperature (Kelvin).

• Equation: $V = \text{constant} \times T$

• Two-State Form: $\frac{V_1}{T_1} = \frac{V_2}{T_2}$

• Graph: V vs. T is linear.

Gay-Lussac's Law

Gay-Lussac's Law describes the relationship between the pressure and temperature of a gas at constant volume and amount.

• Statement: As the temperature of an enclosed gas increases, the pressure increases (at constant volume).

• Equation: $P = \text{constant} \times T$

• Two-State Form: $\frac{P_1}{T_1} = \frac{P_2}{T_2}$

Avogadro's Law

Avogadro's Law relates the volume of a gas to the number of moles at constant temperature and pressure.

• Statement: The volume of a gas at constant temperature and pressure is directly proportional to the number of moles.

• Equation: $V = \text{constant} \times n$

• Two-State Form: $\frac{V_1}{n_1} = \frac{V_2}{n_2}$

• At STP (273 K, 1 atm): 1 mole of any gas occupies 22.4 L.

10.4. Ideal-Gas Equation

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

• Equation: $PV = nRT$

• R (Gas Constant): Value depends on units; commonly $R = 0.08206\,\text{L atm mol}^{-1}\text{K}^{-1}$ or $8.314\,\text{J mol}^{-1}\text{K}^{-1}$

• Temperature must be in Kelvin.

• STP Conditions: 0°C (273.15 K) and 1 atm (101.325 kPa)

• Example: 1 mol of ideal gas at STP occupies 22.41 L.

Fixed n (Combined Gas Law)

When the amount of gas is constant, the combined gas law relates pressure, volume, and temperature.

• Equation: $\frac{PV}{T} = \text{constant}$

• Two-State Form: $\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$

10.5. Density and Molar Mass of Gases

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

• Density (d): $d = \frac{m}{V} = \frac{MP}{RT}$

• Molar Mass (M): $M = \frac{mRT}{PV}$ or $M = \frac{dRT}{P}$

• Only molecular mass, pressure, and temperature are needed to calculate density.

Volume and Chemical Reactions

Gas laws can be used to relate the volume of gases involved in chemical reactions, using stoichiometry and the ideal-gas equation.

• Balanced equations provide mole ratios for reactants and products.

• Use $PV = nRT$ to relate gas volumes to moles in reactions.

• Example: Calculating the mass of sodium azide needed to inflate an airbag using stoichiometry and gas laws.

10.6. Dalton’s Law of Partial Pressures

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

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

• Each gas behaves as if it occupies the container alone.

Mole Fraction

• Mole Fraction ($X_i$): $X_i = \frac{n_i}{n_t}$, where $n_i$ is moles of component $i$ and $n_t$ is total moles.

• Partial Pressure: $P_i = X_i P_t$

Example: Applying Dalton's Law

• Calculate partial pressures and total pressure for a mixture of O2 and CH4 in a vessel using the ideal-gas equation and mole fractions.

Kinetic-Molecular Theory

The kinetic-molecular theory explains the observed properties and laws of gases by describing their molecular behavior.

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

• Average kinetic energy is proportional to absolute temperature.

10.9. Real Gases and 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.

• At high temperature and low pressure, gases behave more ideally.

• Deviations occur because the assumptions of the kinetic-molecular theory break down.

Corrections for Non-Ideal Behavior: van der Waals Equation

• The van der Waals equation introduces two constants to correct for intermolecular attractions (a) and molecular volume (b):

$\left(P + \frac{an^2}{V^2}\right)(V - nb) = nRT$

• a: Measures the strength of intermolecular attractions.

• b: Accounts for the finite volume of gas molecules.

Example: Nitrogen gas behaves more ideally as temperature increases.