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

Gases: Properties and Characteristics

Physical Properties of Gases

Gases are a fundamental state of matter characterized by their ability to expand and fill any container, high compressibility, and low density. Most common gases are composed of nonmetallic elements with simple formulas and low molar masses. Gases form homogeneous mixtures regardless of their chemical identity.

• Expansion: Gases expand to fill their containers.

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

• Density: Gases have much lower densities than liquids or solids.

• Mixing: Two or more gases form homogeneous mixtures.

Brownian Motion

Gas molecules move chaotically, colliding with each other and the walls of their container. This random motion is known as Brownian motion and is responsible for the pressure exerted by gases.

Pressure and Its Measurement

Definition of Pressure

Pressure (P) is defined as the force (F) applied per unit area (A):

$P = \frac{F}{A}$

Atmospheric pressure is the weight of air per unit area at Earth's surface.

Units of Pressure

• Pascals (Pa): SI unit, 1 Pa = 1 N/m2

• Bar: 1 bar = 105 Pa = 100 kPa

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

• Atmosphere (atm): 1 atm = 760 torr = 101.325 kPa

Measuring Pressure: Barometers and Manometers

A barometer measures atmospheric pressure using a column of mercury. A manometer measures the pressure of a gas in a vessel relative to atmospheric pressure.

Standard Pressure and STP

Standard atmospheric pressure (STP) is defined as 1 atm (101.325 kPa, 760 mmHg) at 0°C.

Sample Pressure Calculations

Pressure due to a column of fluid is calculated as:

$P = d h g$

Manometer Calculations

The pressure of a gas in a flask can be determined using the difference in mercury heights in a manometer:

$P_{\text{gas}} = P_{\text{atm}} + h$

The Gas Laws

Boyle’s Law: Pressure-Volume Relationship

At constant temperature, the volume of a fixed quantity of gas is inversely proportional to its pressure:

$P_1 V_1 = P_2 V_2$

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

Charles’s Law: Temperature-Volume Relationship

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

$\frac{V_1}{T_1} = \frac{V_2}{T_2}$

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):

$\frac{V_1}{n_1} = \frac{V_2}{n_2}$

The Ideal Gas Equation

Derivation and Formulation

Combining Boyle’s, Charles’s, and Avogadro’s laws gives the ideal gas equation:

$PV = nRT$

• P: Pressure (atm, Pa, etc.)

• V: Volume (L, m3)

• n: Moles of gas

• R: Universal gas constant

• T: Temperature (K)

Gas Constant R

Units	Numerical Value
L atm/mol K	0.08206
J/mol K	8.314
m3 Pa/mol K	8.314
L torr/mol K	62.36
L kPa/mol K	8.314

Applications of the Ideal Gas Equation

Density and Molar Mass of Gases

The density (d) of a gas can be derived from the ideal gas equation:

$d = \frac{PM}{RT}$

Where M is the molar mass. If the mass, volume, and temperature are known, the molar mass can be found:

$M = \frac{mRT}{PV}$

Volumes of Gases in Chemical Reactions

Stoichiometry involving gases uses the ideal gas law to relate the volume of gases to the number of moles in a reaction. The coefficients in a balanced equation indicate the relative amounts of reactants and products.

Gas Mixtures and Partial Pressures

Dalton’s Law of Partial Pressures

The total pressure of a mixture of gases equals the sum of the pressures each would exert if present alone:

$P_t = P_1 + P_2 + P_3 + \ldots$

Partial Pressures and Mole Fractions

The partial pressure of a gas in a mixture is proportional to its mole fraction:

$P_1 = X_1 P_t$

Where $X_1 = \frac{n_1}{n_t}$ is the mole fraction.

Kinetic-Molecular Theory of Gases

Main Tenets

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

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

• Attractive and repulsive forces between molecules are negligible.

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

• The average kinetic energy is proportional to the absolute temperature.

Molecular Speeds and Temperature

At a given temperature, molecules have a distribution of speeds. The root-mean-square (rms) speed is related to the average kinetic energy:

$u_{\text{rms}} = \sqrt{\frac{3RT}{M}}$

Effusion and Diffusion

Effusion

Effusion is the process by which gas molecules escape through a tiny hole into an evacuated space. Graham’s Law relates the rates of effusion of two gases to their molar masses:

$\frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}}$

Real Gases and Deviations from Ideal Behavior

Deviations from Ideal Gas Law

Real gases deviate from ideal behavior at high pressures and low temperatures due to intermolecular forces and the finite volume of molecules. The ideal gas law assumes negligible volume and no intermolecular forces, which breaks down under these conditions.

Van der Waals Equation

The van der Waals equation corrects the ideal gas law for intermolecular attractions (a) and molecular volume (b):

$(P + \frac{n^2 a}{V^2})(V - nb) = nRT$

Substance	a (L2·kPa/mol2)	b (L/mol)
He	3.46	0.02370
Ne	21.4	0.0171
Ar	135	0.0322
N2	141	0.0391
O2	138	0.0318
CO2	364	0.0429
NH3	423	0.0371
CH4	228	0.0428
CCl4	2067	0.1383