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

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

Outline of 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

• The Effects of Size and Intermolecular Forces

Breathing: Putting Pressure to Work

Mechanics of Breathing

Breathing is a physical process that involves the movement of air in and out of the lungs, driven by changes in pressure and volume within the chest cavity. Muscles such as the diaphragm and intercostal muscles play a crucial role in this process.

• Diaphragm and chest muscles expand the chest cavity, increasing lung volume.

• As lung volume increases, the pressure inside the lungs decreases below external atmospheric pressure.

• Air flows into the lungs to equalize the pressure.

• When the lungs are compressed, the process reverses and air is expelled.

Example: Approximately 10,000 L of air moves in and out of your lungs daily.

Pressure: The Result of Molecular Collisions

Definition and Origin of Pressure

Pressure in gases arises from the constant, random motion of atoms and molecules, which collide 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 per unit area:

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

• Where F is the force exerted and A is the area.

• Pressure can hold up jets or knock down buildings due to the sum of many molecular collisions.

Units of Pressure

Pressure is measured in several units, often depending on context or region. Standard conditions for gases are defined using these units.

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

Example: The height of a mercury column in a barometer is used to measure atmospheric pressure (mmHg or Torr).

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

Boyle's Law

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

• 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 gas increases with increasing temperature at constant pressure and amount.

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

• Temperature must be in Kelvin for calculations.

Example: Heating a balloon causes it to expand as the gas inside increases in volume.

Avogadro’s Law

Avogadro’s Law relates the volume of a gas to the amount (number of moles) 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 incorporates Boyle’s, Charles’s, and Avogadro’s laws to relate pressure, volume, temperature, and amount of gas.

$\frac{P_1 V_1}{T_1 n_1} = \frac{P_2 V_2}{T_2 n_2}$

Example: Calculating the final volume of a gas sample after changes in pressure and temperature.

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

Ideal Gas Law

The ideal gas law combines the relationships described by the previous laws into a single equation.

$PV = nRT$

• P = pressure

• V = volume

• n = number of moles

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

• T = temperature in Kelvin

Example: Calculating the volume occupied by 1 mole of an ideal gas at standard temperature and pressure (STP): $V = \frac{nRT}{P}$

Molar Volume

The molar volume is the volume occupied by one mole of an ideal gas at STP (0°C, 1 atm).

• Molar volume at STP: $22.4$ L/mol (at 1 atm, 0°C)

• At 1 bar and 0°C, molar volume is approximately $22.7$ L/mol.

Density and Molar Mass of a Gas

The density and molar mass of a gas can be determined using the ideal gas law.

• Density: $d = \frac{PM}{RT}$

• Molar mass: $M = \frac{dRT}{P}$

Example: Calculating the density of oxygen gas at a given temperature and pressure.

Mixtures of Gases and Partial Pressures

Dalton’s Law of Partial Pressures

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

• Partial pressure: $P_A = x_A P_{\text{total}}$

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

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

Example: Calculating the partial pressure of nitrogen in air at STP, given its mole fraction.

Collecting Gases Over Water

When gases are collected over water, the total pressure includes both the gas and water vapor. The partial pressure of the gas is found by subtracting the vapor pressure of water.

• Formula: $P_{\text{gas}} = P_{\text{total}} - P_{\text{H}_2O}$

Example: If the total pressure is 758.2 Torr and the vapor pressure of water at 25°C is 23.77 Torr, then $P_{\text{gas}} = 758.2 - 23.77 = 734.4$ Torr.

Gases in Chemical Reactions: Stoichiometry Revisited

Stoichiometry with Gases

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.

• Apply stoichiometric coefficients from balanced chemical equations.

Example: Calculating the mass of water produced from a given volume of hydrogen gas reacting with excess oxygen.

Kinetic Molecular Theory: A Model for Gases

Postulates of Kinetic Molecular Theory (KMT)

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

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

• Collisions between particles and container walls are perfectly elastic.

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

Example: KMT explains why pressure increases with temperature (more energetic collisions).

Molecular Velocities

All gases at the same temperature have the same average kinetic energy, but lighter particles move faster.

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

• M = molar mass in kg/mol

Example: Helium atoms move faster than oxygen molecules at the same temperature.

Mean Free Path, Diffusion, and Effusion of Gases

Definitions

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

• Diffusion: The process by which gas molecules spread out in response to a concentration gradient.

• Effusion: The process by which gas molecules escape through a small hole into a vacuum.

Graham’s Law of Effusion:

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

Example: Helium effuses 2.83 times faster than oxygen due to its lower molar mass.

The 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 reduce pressure.

Example: The ideal gas law underpredicts pressure when particle volume is significant.

Van der Waals Equation

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

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

• a = correction for intermolecular forces

• b = correction for particle volume

Example: Calculating the pressure of argon gas using Van der Waals constants.

Conditions for Maximum Deviation

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

Additional info: Van der Waals constants vary for different gases and are determined experimentally.