Thermodynamics and the Kinetic Theory of Gases

Ideal Gas Law and Kinetic Theory

The kinetic theory of gases explains the macroscopic properties of gases, such as pressure and temperature, in terms of the microscopic motions of gas molecules. The ideal gas law is a fundamental equation relating pressure, volume, temperature, and the number of particles in a gas.

• Pressure (p): The force exerted by gas particles colliding with the walls of a container per unit area.

• Volume (V): The space occupied by the gas.

• Temperature (T): A measure of the average kinetic energy of the gas particles, measured in Kelvins (K).

• Number of particles (N): The total number of gas molecules.

• Boltzmann constant (k):

• Ideal Gas Law:

• Alternate form (using moles n): where

• Avogadro's number:

Example: At room temperature (about 300 K), 1 mole of an ideal gas occupies approximately 22.4 L at standard pressure.

Assumptions of the Ideal Gas Model

An ideal gas is a theoretical gas composed of many randomly moving point particles that interact only through elastic collisions.

• No interactions among particles except for occasional elastic collisions.

• No phase changes (remains in the gas phase).

• The volume of individual particles is negligible compared to the volume of the container.

• Kinetic energy is much greater than potential energy between particles (valid for dilute, high-temperature gases).

Good approximation: The ideal gas law works well for gases at low pressure and high temperature, where intermolecular forces are negligible.

Kinetic Energy and Temperature

The average kinetic energy of a gas particle is directly proportional to the temperature of the gas.

• Translational kinetic energy per particle:

• Root-mean-square (rms) speed: where is the mass of a particle.

• For a mole of gas:

Example: For air at room temperature (T ≈ 300 K), for oxygen molecules is about 480 m/s.

Phases of Matter and Phase Changes

Matter exists in three main phases: solid, liquid, and gas. Phase changes occur when energy is added or removed from a substance.

• Solid to liquid: Fusion (melting), requires heat of fusion

• Liquid to gas: Vaporization, requires heat of vaporization

• Solid to gas: Sublimation, requires heat of sublimation

• During a phase change: Temperature remains constant; all heat is used to change the phase.

Formulas for heat during phase changes:

• Fusion:

• Vaporization:

• Sublimation:

Example: To melt 10 g of ice at 0°C, where for water is .

Humidity and Dew Point

Humidity refers to the amount of water vapor in the air. The dew point is the temperature at which air becomes saturated with water vapor and condensation begins.

• Problems involving humidity often require phase change calculations.

• Refer to textbook explanations for detailed problem-solving strategies.

First Law of Thermodynamics

The first law of thermodynamics is a statement of energy conservation for thermodynamic systems.

• Internal energy (U): The total energy contained within a system (thermal, not potential or kinetic of the whole system).

• Heat (Q): Energy transferred into the system due to temperature difference.

• Work (W): Energy transferred by the system doing work on its surroundings.

• First law equation:

• For an ideal gas, depends only on temperature, not on pressure or volume.

• Work done by the gas: (for constant pressure processes)

Example: If a gas absorbs 500 J of heat and does 200 J of work, .

Work Done by a Gas

Work is done by a gas when it expands or contracts against an external pressure.

• Work done by the gas:

• Sign convention: Work done by the gas is positive; work done on the gas is negative.

• In some textbooks, is used, so clarify the convention in use.

Internal Energy of Gases

The internal energy of an ideal gas is the sum of the kinetic energies of all its particles.

• Monatomic gas:

• Diatomic gas: (includes rotational energy; vibrational modes at higher T)

• For an ideal gas, internal energy depends only on temperature.

Summary Table: Phase Changes and Associated Heat

Additional info:

• Some details, such as the specific values for latent heats and the full explanation of humidity/dew point, were inferred based on standard physics curriculum.

• Sign conventions for work and heat may vary by textbook; always check your course's convention.