Chemistry Chapter 10

Gases: Physical Properties and Characteristics

Overview of Gases

Gases are one of the fundamental states of matter, characterized by their ability to expand and fill any container. Unlike solids and liquids, gases have unique physical properties that are remarkably similar across different substances, despite their chemical differences.

• Composed of non-metallic elements

• Simple molecular formulas

• Low molar masses

• Highly compressible

• Form homogeneous mixtures with other gases

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

Pressure and Its Measurement

Definition and Units of Pressure

Pressure is defined as the force exerted per unit area by gas molecules as they collide with surfaces. Atmospheric pressure is the force exerted by the atmosphere on a given surface area.

• Common units of pressure:

  • 1 atm = 760 mm Hg = 760 torr = 1.01 × 105 Pa = 101.325 kPa = 1.01325 bar

Pressure conversions are essential in solving gas law problems.

Gas Laws

Variables Describing Gases

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

Boyle’s Law (Pressure-Volume Relationship)

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

$V = \frac{k_1}{P}$

or

$P_1V_1 = P_2V_2$

As pressure increases, volume decreases, and vice versa.

Charles’s Law (Temperature-Volume Relationship)

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

$V = kT$

or

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

Avogadro’s Law (Quantity-Volume Relationship)

At constant temperature and pressure, the volume of a gas is directly proportional to the number of moles:

$V = kn$

At standard temperature and pressure (STP: 0°C, 1 atm), 1 mol of any ideal gas occupies 22.4 L.

The Ideal Gas Law

Combining the Gas Laws

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

$PV = nRT$

Where R is the universal gas constant. Its value depends on the units used:

Units	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

Applications of the Ideal Gas Law

• Solving for any one variable when the others are known

• Relating changes in state variables (P, V, T, n) for a gas sample

Gas Density and Molar Mass

Calculating Density and Molar Mass

The density (d) of an ideal gas can be calculated using:

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

Where M is the molar mass of the gas.

Gas Mixtures and Partial Pressures

Dalton’s Law of Partial Pressures

In a mixture of gases, each gas exerts a pressure as if it were alone. The total pressure is the sum of the partial pressures:

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

The partial pressure of a component is related 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

Postulates of Kinetic-Molecular Theory (KMT)

The KMT explains the behavior of gases at the molecular level:

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

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

• Intermolecular forces are negligible.

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

Explaining Gas Laws with KMT

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

• Increasing temperature at constant volume increases pressure (molecules move faster and collide more energetically).

Effusion and Diffusion of Gases

Effusion

Effusion is the process by which gas molecules escape through a tiny hole into a vacuum. Lighter gases effuse faster than heavier gases.

Graham’s Law of Effusion

The rates of effusion of two gases are inversely proportional to the square roots of their molar masses:

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

Diffusion

Diffusion is the spread of one substance throughout a space or another substance. The path of a gas molecule is random due to collisions.

Molecular Speeds and Root-Mean-Square (rms) Speed

The average speed of gas molecules is described by the root-mean-square speed:

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

Lighter gases have higher rms speeds at the same temperature.

Gas Stoichiometry and Law of Combining Volumes

Volume Relationships in Reactions

At constant temperature and pressure, the volumes of reacting gases and their products are in simple whole-number ratios. For example:

• N2 (g) + 3 H2 (g) → 2 NH3 (g)

• 1 volume N2 : 3 volumes H2 : 2 volumes NH3

At STP, 1 mol of any gas occupies 22.4 L, which is useful for stoichiometric calculations involving gases.

Summary Table: Key Equations and Constants

Equation	Description
$PV = nRT$	Ideal Gas Law
$P_1V_1 = P_2V_2$	Boyle’s Law
$\frac{V_1}{T_1} = \frac{V_2}{T_2}$	Charles’s Law
$\frac{V_1}{n_1} = \frac{V_2}{n_2}$	Avogadro’s Law
$d = \frac{PM}{RT}$	Gas Density
$P_t = P_1 + P_2 + \ldots$	Dalton’s Law
$u_{rms} = \sqrt{\frac{3RT}{M}}$	Root-mean-square speed
$\frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}}$	Graham’s Law