Chapter 10: Gases – Properties, Laws, and Applications

Gases: Properties and Behavior

Characteristics of Gases

Gases are one of the fundamental states of matter, characterized by their ability to expand and fill any container. Despite differences in chemical properties, most gases share similar physical properties:

• Composed mainly of non-metallic elements

• Simple molecular formulas and low molar masses

• Highly compressible

• Form homogeneous mixtures with other gases

Examples of Common Gases at Room Temperature:

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 the force exerted by gas molecules as they collide with the surfaces of their container. Atmospheric pressure is the force exerted by the atmosphere on a given surface area.

• Common units: atm, mm Hg, torr, Pa, kPa, bar

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

Gas Laws

Variables Defining a Gas

The state of a gas is defined by four variables:

• Pressure (P)

• Volume (V)

• Temperature (T)

• Amount (n, in moles)

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: $P_1V_1 = P_2V_2$

• As pressure increases, volume decreases, and vice versa.

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).

• Mathematical form: $\frac{V_1}{T_1} = \frac{V_2}{T_2}$

• As temperature increases, volume increases.

Avogadro’s Law: Quantity-Volume Relationship

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

• Mathematical form: $\frac{V_1}{n_1} = \frac{V_2}{n_2}$

• At STP (0°C, 1 atm): 1 mol gas = 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:

• Equation: $PV = nRT$

• R is the 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 variable (P, V, n, T) when the others are known

• Relating changes in state: $\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$ (when n is constant)

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:

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

• Partial pressure relates to mole fraction: $P_1 = X_1 P_t$ where $X_1 = \frac{n_1}{n_t}$

Kinetic-Molecular Theory of Gases

Postulates of Kinetic-Molecular Theory (KMT)

KMT explains the macroscopic properties of gases by considering their molecular motion:

• Gases consist of many molecules in constant, random motion.

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

• Intermolecular forces are negligible.

• Average kinetic energy is proportional to 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, more collisions).

Effusion and Diffusion of Gases

Effusion and Graham’s Law

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

• Equation: $\frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}}$

• Lighter gases effuse faster than heavier gases.

Diffusion

Diffusion is the spread of one substance throughout another. Lighter gases diffuse more rapidly than heavier ones.

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

The speed of gas molecules is described by the root-mean-square speed ($u_{rms}$):

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

• Where R is the gas constant, T is temperature in Kelvin, and M is molar mass in kg/mol.

Gas Stoichiometry and Law of Combining Volumes

Law of Combining Volumes

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

• Example: $N_2 (g) + 3 H_2 (g) \rightarrow 2 NH_3 (g)$

• 1 volume N2 reacts with 3 volumes H2 to produce 2 volumes NH3

• At STP, 1 mol gas = 22.4 L

Sample Calculations

• Use the ideal gas law to relate volume, pressure, temperature, and moles in stoichiometric calculations.

• Convert between grams, moles, and volumes as needed for chemical reactions involving gases.

Summary Table: Key Equations

• Boyle’s Law: $P_1V_1 = P_2V_2$

• Charles’s Law: $\frac{V_1}{T_1} = \frac{V_2}{T_2}$

• Avogadro’s Law: $\frac{V_1}{n_1} = \frac{V_2}{n_2}$

• Ideal Gas Law: $PV = nRT$

• Dalton’s Law: $P_t = P_1 + P_2 + \ldots$

• Graham’s Law: $\frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}}$

• Root-mean-square speed: $u_{rms} = \sqrt{\frac{3RT}{M}}$