Chapter 5: Gases – Properties, Laws, and Molecular Theory

Chapter 5: Gases – Properties, Laws, and Molecular Theory

Outline of Main Topics

• Result of Molecular Collisions

• Gas Laws: Boyle's Law, Charles's Law, and Avogadro's Law

• Ideal Gas Law: Molar Volume, Density, and Molar Mass

• Gases and Partial Pressures

• Chemical Reactions: Stoichiometry Revisited

• Kinetic Molecular Theory: A Model for Gases

• Mean Free Path, Diffusion, and Effusion of Gases

• Effects of Size and Intermolecular Forces

Breathing: Putting Pressure to Work

How Breathing Relates to Gas Laws

Breathing is a practical application of gas laws, involving the movement of air in and out of the lungs due to pressure differences. Muscles change the volume of the chest cavity, which alters the pressure inside the lungs and causes air to flow in or out.

• Diaphragm and chest muscles control lung volume.

• When lung volume increases, internal pressure decreases, and air flows in.

• When lung volume decreases (compression), internal pressure increases, and air flows out.

• This process is governed by the principles of pressure and volume relationships in gases.

Pressure: The Result of Molecular Collisions

Definition and Origin of Pressure

Pressure in gases arises from countless collisions of gas molecules with the walls of their container. The cumulative effect of these collisions produces a measurable force over a given area.

• Pressure (P) is defined as force (F) per unit area (A): $P = \frac{F}{A}$

• Gas molecules move rapidly and randomly, colliding with surfaces and each other.

• Examples: Air pressure can support jets or knock down buildings due to the sum of molecular collisions.

Units of Pressure

Pressure can be measured in several units, each with a standard value at atmospheric conditions.

Unit	Standard Pressure
bar	1.01 bar
Pa (Pascal)	101,325 Pa
kPa	101.3 kPa
Torr or mmHg	760 Torr
atm (atmosphere)	1.00 atm
psi	14.7 psi

Standard conditions for gases are typically 1 atm pressure and 0°C (273.15 K).

Gas Laws: Boyle's Law, Charles's Law, and Avogadro's Law

Boyle's Law

Boyle's Law describes the inverse relationship between pressure and volume for a fixed amount of gas at constant temperature.

• Mathematical form: $PV = \text{constant}$

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

• Example: Compressing a gas in a syringe decreases its volume as pressure increases.

Charles's Law

Charles's Law states that the volume of a fixed amount of gas is directly proportional to its temperature (in Kelvin) at constant pressure.

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

• As temperature increases, volume increases.

• Example: Heating a balloon causes it to expand.

Avogadro's Law

Avogadro's Law relates the volume of a gas to the number of moles present, at constant temperature and pressure.

• Mathematical form: $\frac{V}{n} = \text{constant}$

• Increasing the amount of gas increases its volume.

• Example: Adding more gas to a tire increases its volume.

Combined Gas Law

The combined gas law merges Boyle's, Charles's, and Avogadro's laws to relate pressure, volume, temperature, and amount of gas.

• Mathematical form: $\frac{P_1 V_1}{T_1 n_1} = \frac{P_2 V_2}{T_2 n_2}$

• Useful for calculations involving changes in multiple variables.

The Ideal Gas Law: Molar Volume, Density, and Molar Mass

Ideal Gas Law

The ideal gas law combines the relationships of pressure, volume, temperature, and amount of gas into a single equation.

• Equation: $PV = nRT$

• Where R is the gas constant ($R = 0.08206$ L·atm·mol−1·K−1 or $8.314$ J·mol−1·K−1$)

• Allows calculation of any one variable if the others are known.

Molar Volume

The molar volume is the volume occupied by one mole of an ideal gas at standard temperature and pressure (STP).

• At STP (0°C, 1 atm), molar volume is $22.4$ L/mol.

• At 1.00 bar and 273.15 K, molar volume is approximately $22.7$ L/mol.

Density and Molar Mass of a Gas

The ideal gas law can be rearranged to calculate the density and molar mass of a gas.

• Density: $d = \frac{PM}{RT}$

• Molar Mass: $M = \frac{dRT}{P}$

• Where d is density, M is molar mass, P is pressure, R is the gas constant, and T is temperature.

Mixtures of Gases and Partial Pressures

Dalton's Law of Partial Pressures

In a mixture of gases, each gas exerts a pressure independently of the others. The total pressure is the sum of the partial pressures.

• Equation: $P_{\text{total}} = P_A + P_B + P_C + \ldots$

• Partial pressure of a component: $P_A = x_A P_{\text{total}}$ where $x_A$ is the mole fraction of A.

• Mole fraction: $x_A = \frac{n_A}{n_{\text{total}}}$

Collecting Gases Over Water

When collecting gases over water, the total pressure includes both the gas and water vapor pressures.

• Equation: $P_{\text{total}} = P_{\text{gas}} + P_{\text{H}_2O}$

• To find the pressure of the dry gas, subtract the vapor pressure of water from the total pressure.

Temperature (°C)	Vapor Pressure of Water (Torr)
55	118
60	146
65	187
70	233
75	286
80	355
85	433
90	525
95	634
100	760

Stoichiometry of Gases in Chemical Reactions

Volume Relationships in Reactions

Gas volumes can be related to moles using the ideal gas law, allowing stoichiometric calculations for reactions involving gases.

• Use $PV = nRT$ to convert between volume and moles.

• Example: Calculating the mass of water produced from a given volume of hydrogen gas.

Kinetic Molecular Theory: A Model for Gases

Postulates of Kinetic Molecular Theory (KMT)

KMT explains the behavior of gases by modeling them as small particles in constant, random motion.

• Gas particles are negligibly small compared to the distances between them.

• Collisions between particles and with container walls are perfectly elastic.

• The average kinetic energy of particles is proportional to temperature in Kelvin.

Connection to Gas Laws

KMT provides a molecular explanation for the gas laws:

• Boyle's Law: Increased pressure results from more frequent collisions in a smaller volume.

• Charles's Law: Higher temperature increases kinetic energy, causing particles to move faster and occupy more volume.

• Avogadro's Law: More particles mean more collisions, increasing volume at constant pressure.

Molecular Velocities

At a given temperature, all gases have the same average kinetic energy, but lighter molecules move faster.

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

• Where M is molar mass in kg/mol.

Mean Free Path, Diffusion, and Effusion of Gases

Definitions

• Mean free path: Average distance a molecule travels between collisions.

• Diffusion: Process by which gas molecules spread out to evenly fill a container or mix with another gas.

• Effusion: Process by which gas escapes through a small hole into a vacuum.

Graham's Law of Effusion

The rate of effusion of a gas is inversely proportional to the square root of its molar mass.

• Equation: $\frac{\text{rate}_1}{\text{rate}_2} = \sqrt{\frac{M_2}{M_1}}$

• Lighter gases effuse faster than heavier gases.

Effects of Size and Intermolecular Forces

Deviations from Ideal Gas Behavior

Real gases deviate from ideal behavior at high pressures and low temperatures due to finite particle size and intermolecular forces.

• At high pressure, particle volume becomes significant.

• At low temperature, attractive forces between particles reduce pressure.

Van der Waals Equation

The Van der Waals equation introduces corrections for particle volume and intermolecular attractions.

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

• a = correction for intermolecular forces; b = correction for particle volume.

• Values of a and b vary for different gases.

Gas	a (L2·atm·mol−2)	b (L·mol−1)
CO2	3.610	0.0429
H2S	4.481	0.0542
NO	1.351	0.0371
N2O	3.799	0.0371
NO2	5.29	0.0305
O2	1.363	0.0319
HCl	3.648	0.0320

Conditions for Greatest Deviation

• Greatest deviation from ideal gas law occurs at low temperature and high pressure.

Additional info: Some equations and tables have been expanded for clarity and completeness. All key concepts from the provided notes are included and explained in academic context.