Kinetic Theory of Ideal Gases

Macroscopic and Microscopic Descriptions

The kinetic theory of gases provides a microscopic explanation for macroscopic properties such as pressure and temperature. It uses statistical averages over the motions of individual gas molecules to derive these properties, complementing the thermodynamic approach.

• Mole: A mole is the amount of substance containing Avogadro's number () of particles.

• Number of moles: , where is the number of molecules.

• Ideal Gas Law: or , where J/mol·K and J/K.

Work Done by an Ideal Gas (Isothermal Process)

During an isothermal expansion, the temperature remains constant and the work done by the gas is:

• Work is positive if .

• Example: The area under the curve in a p-V diagram represents the work done during an isothermal process.

Kinetic Theory Calculation of Pressure

Microscopic Basis of Pressure

Pressure arises from the collisions of gas molecules with the walls of their container. The change in momentum for a molecule colliding with a wall is .

• The rate of momentum transfer to the wall is , where is the length of the box.

• For molecules, the pressure is .

• Root-mean-square speed: .

• Pressure in terms of : .

Distribution of Molecular Speeds

Maxwell Distribution Function

The speeds of molecules in an ideal gas follow the Maxwell distribution:

• The fraction of molecules with speeds between and is .

• Average speed:

• Root-mean-square speed:

Thermal Energy and Equipartition Theorem

Average Kinetic Energy and Degrees of Freedom

The average kinetic energy of a molecule in an ideal gas is:

• Total thermal energy:

• For diatomic and polyatomic gases, rotational and vibrational degrees of freedom must be considered.

• Equipartition theorem: Each degree of freedom contributes per molecule.

Specific Heats of Ideal Gases

Molar Specific Heat Capacity

The molar specific heat at constant volume () is:

• , where is the number of degrees of freedom (3 for monatomic, 5 for diatomic, 6 for polyatomic).

• At constant volume:

• At constant pressure:

• Rotational and vibrational motions affect at different temperatures.

Adiabatic Expansion of an Ideal Gas

Adiabatic Processes

In an adiabatic process, no heat is transferred (). The relationship between pressure and volume is:

• , where

• For monatomic gases,

• Both pressure and temperature drop during adiabatic expansion.

Summary of Thermodynamic Processes for an Ideal Gas

Types of Processes

Several thermodynamic processes can be represented on a p-V diagram:

• Isobaric: Constant pressure

• Isothermal: Constant temperature

• Adiabatic: No heat transfer ()

• Isochoric: Constant volume

Adiabatic Free Expansion

Free Expansion Process

In an adiabatic free expansion, the gas expands into a vacuum without doing work and without heat transfer. The thermal energy remains unchanged, and the temperature does not change.

Examples and Applications

Calculations for Isothermal and Isobaric Processes

Example calculations show how to determine final pressure, temperature, and work done for isothermal and isobaric expansions using the ideal gas law and thermodynamic equations.

• Isothermal:

• Isobaric:

• Adiabatic: ,

Table: Degrees of Freedom and Specific Heat Capacities

Additional info: Vibrational degrees of freedom become important at high temperatures, increasing the specific heat capacity.