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

Gases: Physical Characteristics

General Properties of Gases

Gases are a unique state of matter characterized by their ability to expand and fill containers, high compressibility, and extremely low densities compared to liquids and solids. Most gases are composed of nonmetallic elements with simple formulas and low molar masses. When mixed, gases form homogeneous mixtures regardless of their identities.

• Expansion: Gases fill any container they occupy.

• Compressibility: Gases can be compressed much more than liquids or solids.

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

• Homogeneous Mixtures: Multiple gases mix uniformly.

Common Gases at Room Temperature

Several compounds exist as gases at room temperature, each with distinct properties and hazards.

Formula	Name	Characteristics
HCN	Hydrogen cyanide	Very toxic, slight odor of bitter almonds
H2S	Hydrogen sulfide	Very toxic, odor of rotten eggs
CO	Carbon monoxide	Toxic, colorless, odorless
CO2	Carbon dioxide	Colorless, odorless
CH4	Methane	Colorless, odorless, flammable
C2H4	Ethene (Ethylene)	Colorless, ripens fruit
C3H8	Propane	Colorless, odorless, bottled gas
N2O	Nitrous oxide	Colorless, sweet odor, laughing gas
NO2	Nitrogen dioxide	Toxic, red-brown, irritating odor
NH3	Ammonia	Colorless, pungent odor
SO2	Sulfur dioxide	Colorless, irritating odor

Pressure and Its Measurement

Definition of Pressure

Pressure is defined as the amount of force applied to a given area. All gases exert pressure on surfaces they contact, and atmospheric pressure is the weight of air per unit area at Earth's surface.

• Formula: $\text{Pressure} = \frac{\text{Force}}{\text{Area}}$

• Atmospheric Pressure: Caused by the gravitational force acting on the air column above Earth's surface.

Units of Pressure

Pressure can be measured in several units, including pascals (Pa), bar, millimeters of mercury (mm Hg or torr), and atmospheres (atm). Conversion between these units is often necessary in calculations.

• Pascals (Pa): SI unit of pressure.

• Bar: 1 bar = 100 kPa.

• mm Hg or Torr: Based on the height difference in a mercury column.

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

Manometer

A manometer is used to measure the pressure difference between atmospheric pressure and the pressure of a gas in a vessel. The difference in height of the mercury column indicates the pressure.

Standard Pressure

Standard atmospheric pressure at sea level is used as a reference in many calculations. It is equivalent to 1 atm, 760 torr, 760 mm Hg, 101.325 kPa, or 1.01325 bar.

The Gas Laws

Variables Defining Gas State

The physical state of a gas is defined by four variables: temperature, pressure, volume, and amount (moles). The relationships between these variables are described by the gas laws.

• Temperature (T): Measured in Kelvin (K).

• Pressure (P): Measured in atm, Pa, etc.

• Volume (V): Measured in liters (L).

• Amount (n): Measured in moles.

Boyle’s Law: Pressure–Volume Relationship

Boyle’s Law states that the volume of a fixed quantity of gas at constant temperature is inversely proportional to the pressure.

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

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

• Graph: V vs. P is not linear; V vs. 1/P is linear.

Charles’s Law: Temperature–Volume Relationship

Charles’s Law states that the volume of a fixed amount of gas at constant pressure is directly proportional to its absolute temperature.

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

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

• Graph: V vs. T is linear.

Avogadro’s Law: Quantity–Volume Relationship

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

• Mathematical Relationship: $V = k n$

• At STP: One mole of any gas occupies 22.4 L.

The Ideal-Gas Equation

Derivation and Formula

The ideal-gas equation combines Boyle’s, Charles’s, and Avogadro’s laws into a single relationship. The proportionality constant R is known as the ideal gas constant.

• Formula: $P V = n R T$

• R Values: R has different numerical values depending on units.

Unit	Numerical Value
L·atm/(mol·K)	0.08206
J/(mol·K)	8.314
cal/(mol·K)	1.987
m3·Pa/(mol·K)	8.314
L·torr/(mol·K)	62.36

Solving Gas Law Problems

To solve gas law problems, tabulate information, convert units, select the correct equation, and use dimensional analysis.

Gas Densities and Molar Mass

Calculating Density

The density of a gas can be calculated using the ideal-gas equation and the molar mass.

• Formula: $d = \frac{M P}{R T}$

• Variables: M = molar mass, P = pressure, R = gas constant, T = temperature

Calculating Molar Mass

The molar mass of a gas can be determined if the mass, volume, and temperature are known.

• Formula: $M = \frac{d R T}{P}$

Volume of Gases in Chemical Reactions

Stoichiometry with Gases

Balanced chemical equations indicate the relative amounts of moles involved in reactions. Unknowns can be determined using stoichiometry and the ideal-gas equation.

• If given volume, use stoichiometric ratios.

• If given P, V, T, solve for n using $P V = n R T$, then use mole ratios and convert to desired units.

Gas Mixtures and Partial Pressures

Dalton’s Law of Partial Pressures

When two or more non-reacting gases are combined, each behaves as if it is alone in the container. The total pressure is the sum of the partial pressures of each gas.

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

Mole Fraction and Partial Pressure

The mole fraction of a gas in a mixture is used to calculate its partial pressure.

• Mole Fraction: $X_i = \frac{n_i}{n_{\text{total}}}$

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

Kinetic-Molecular Theory of Gases

Purpose and Statements

The kinetic-molecular theory explains why gases behave as described by the gas laws. It is based on several key assumptions:

• 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 if temperature is constant.

• Average kinetic energy is proportional to absolute temperature; at the same temperature, all gases have equal average kinetic energy.

Distributions of Molecular Speed

Temperature affects the distribution of molecular speeds. Three key speeds are defined:

• Most probable speed (vmp): The speed at which the largest number of molecules move.

• Average speed (vavg): The mean speed of all molecules.

• Root-mean-square speed (vrms): The speed associated with the average kinetic energy.

Application to Gas Laws

Kinetic-molecular theory explains the relationships observed in gas laws:

• Increasing volume at constant temperature decreases pressure due to fewer collisions.

• Increasing temperature at constant volume increases pressure due to more frequent and forceful collisions.

Molecular Speeds, Effusion, and Diffusion

Effusion and Diffusion

Effusion is the escape of gas molecules through a tiny hole into an evacuated space, while diffusion is the spread of one substance throughout another.

• Effusion: Escape through a pinhole.

• Diffusion: Mixing and spreading in a space.

Graham’s Law

Graham’s law relates the rate of effusion or diffusion to the molar mass of gases. Lighter gases effuse and diffuse faster than heavier gases.

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

Real Gases: Deviations from Ideal Behavior

Deviations from Ideal-Gas Behavior

Real gases deviate from ideal behavior at high pressures and low temperatures. The assumptions of the kinetic-molecular theory break down under these conditions.

• Gas molecules occupy a significant fraction of the total volume at high pressure.

• Attractive forces between molecules become significant at low temperature.

Van der Waals Equation

The van der Waals equation corrects the ideal-gas equation for real gases by accounting for molecular volume and intermolecular forces. Each gas has specific van der Waals constants (a and b).

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

• a: Corrects for intermolecular attractions.

• b: Corrects for molecular volume.

Substance	a (L2·atm/mol2)	b (L/mol)
He	0.0341	0.02370
Ne	0.211	0.0171
Ar	1.34	0.0322
Kr	2.32	0.0398
Xe	4.19	0.0510
H2	0.244	0.0266
N2	1.39	0.0391
O2	1.36	0.0318
F2	1.06	0.0290
Cl2	6.49	0.0562
H2O	5.46	0.0305
NH3	4.17	0.0371
CH4	2.25	0.0428
CO2	3.59	0.0427
CCl4	20.4	0.1383

Additional info: The van der Waals equation is essential for accurately describing gas behavior under non-ideal conditions, such as in industrial processes or extreme environments.