Chapter 10 Student Notes

Gases: Their Properties and Behavior

Overview of Gases

Gases are one of the fundamental states of matter in chemistry. Their observable properties are a result of the random motion of particles and the weak intermolecular forces between them. Understanding gases involves studying their pressure, volume, temperature, and amount (moles).

• Compressibility: Gases can be compressed much more than solids or liquids due to the large distances between particles.

• Expansion: Gases expand to fill their containers and do not have a fixed shape or volume.

• Low Density: Gases have much lower densities compared to solids and liquids.

Pressure

Pressure is a fundamental property of gases, defined as the force exerted per unit area.

• Definition: $P = \frac{F}{A}$, where $F$ is force and $A$ is area.

• SI Unit: Pascal (Pa), where $1\ \text{Pa} = 1\ \text{N}/\text{m}^2$.

• Other Units: Atmosphere (atm), torr, mmHg, bar.

Unit	Equivalent
1 atm	101,325 Pa = 760 mmHg = 760 torr
1 bar	100,000 Pa

The Gas Phase

Gases are typically found at high temperatures and low pressures. Their particles move rapidly and independently, resulting in frequent collisions with container walls, which is observed as pressure.

• Atmospheric Pressure: The pressure exerted by the atmosphere, measured with a barometer.

• STP (Standard Temperature and Pressure): $0^\circ$C (273.15 K) and 1 atm.

Gas Laws

The behavior of gases can be described by several empirical laws:

Boyle's Law

• At constant temperature, the volume of a gas is inversely proportional to its pressure.

$P_1V_1 = P_2V_2$

Charles's Law

• At constant pressure, the volume of a gas is directly proportional to its temperature (in Kelvin).

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

Avogadro's Law

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

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

Combined Gas Law

• Combines Boyle's, Charles's, and Avogadro's Laws for a fixed amount of gas.

$\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$

Ideal Gas Law

• Relates pressure, volume, temperature, and number of moles for an ideal gas.

$PV = nRT$

• $P$ = pressure (atm)

• $V$ = volume (L)

• $n$ = moles

• $R$ = ideal gas constant ($0.08206\ \text{L} \cdot \text{atm}/\text{mol} \cdot \text{K}$)

• $T$ = temperature (K)

Molar Volume and Gas Stoichiometry

• At STP, 1 mole of an ideal gas occupies 22.4 L.

• This allows for conversion between moles and volume for gases at STP.

Example: What volume of hydrogen gas at STP is produced from 1 mole of Zn reacting with excess HCl?

Dalton's Law of Partial Pressures

In a mixture of non-reacting gases, the total pressure is the sum of the partial pressures of each component gas.

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

• Partial Pressure: The pressure exerted by a single gas in a mixture.

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

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

Kinetic Molecular Theory of Gases

This theory explains the macroscopic properties of gases in terms of the motion of their particles.

• Gas particles are in constant, random motion.

• Collisions between particles are elastic (no energy lost).

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

$\text{Average kinetic energy} = \frac{3}{2}RT$

$\text{Root mean square velocity: } u_{rms} = \sqrt{\frac{3RT}{M}}$

• Lighter gases move faster than heavier gases at the same temperature.

Diffusion and Effusion

• Diffusion: The mixing of gases due to random motion.

• Effusion: The escape of gas particles through a small hole.

• Graham's Law: The rate of effusion is inversely proportional to the square root of the molar mass.

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

Deviations from Ideal Gas Behavior

Real gases deviate from ideal behavior at high pressures and low temperatures due to intermolecular forces and the finite volume of gas particles.

• van der Waals Equation: Corrects the ideal gas law for real gases.

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

• $a$ corrects for intermolecular attractions.

• $b$ corrects for the finite volume of gas molecules.

Summary Table: Gas Laws and Their Relationships

Law	Equation	Constant	Relationship
Boyle's Law	$P_1V_1 = P_2V_2$	$T, n$	$P \propto \frac{1}{V}$
Charles's Law	$\frac{V_1}{T_1} = \frac{V_2}{T_2}$	$P, n$	$V \propto T$
Avogadro's Law	$\frac{V_1}{n_1} = \frac{V_2}{n_2}$	$P, T$	$V \propto n$
Combined Gas Law	$\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$	$n$	All variables
Ideal Gas Law	$PV = nRT$	None	All variables

Key Terms

• Pressure (P): Force per unit area exerted by gas particles.

• Volume (V): Space occupied by the gas.

• Temperature (T): Measure of average kinetic energy of particles.

• Mole (n): Amount of substance.

• Partial Pressure: Pressure exerted by a single component in a mixture.

• Mole Fraction: Ratio of moles of one component to total moles.

• STP: Standard Temperature and Pressure (0°C, 1 atm).

Sample Problems and Applications

• Calculate the volume of a gas at non-STP conditions using the ideal gas law.

• Determine the partial pressure of a component in a gas mixture using Dalton's Law.

• Use Graham's Law to compare rates of effusion for different gases.

• Apply the van der Waals equation to correct for non-ideal behavior.

Example: Calculate the density of a gas at STP. $\text{Density} = \frac{\text{Molar mass}}{22.4\ \text{L}}$

Example: What is the partial pressure of oxygen in a mixture containing 0.30 moles of O2, 0.70 moles of N2, and 1.0 atm total pressure? $X_{O_2} = \frac{0.30}{0.30+0.70} = 0.30$ $P_{O_2} = X_{O_2} \times P_{\text{total}} = 0.30 \times 1.0\ \text{atm} = 0.30\ \text{atm}$

Example: If a gas sample effuses at a rate twice as fast as O2, what is its molar mass? $\frac{\text{Rate}_{\text{unknown}}}{\text{Rate}_{O_2}} = 2 = \sqrt{\frac{M_{O_2}}{M_{\text{unknown}}}}$ Solve for $M_{\text{unknown}}$.

Additional info: These notes include worked examples, practice problems, and summary tables to reinforce understanding of gas laws and their applications in General Chemistry.