Kinetic Molecular Theory of Gases

Kinetic Molecular Theory of Gases

• The word kinetic means motion
• Maxwell and Boltzmann developed a mathematical theory to explain gas behavior and gas laws
• Gas is made up of many molecules in continuous motion

Assumptions of Kinetic Theory

• Gases consist of small particles called molecules dispersed in the container
• The actual volume of the molecules is negligible compared to the total volume of the gas
• Molecules of a given gas are identical and have the same mass
• Gas molecules are in constant random motion with high velocities
• Gas molecules move in straight lines with uniform velocity, changing direction upon collisions with other molecules or container walls
• The distance between molecules is large, so van der Waals attractive forces between them do not exist
• Gas collisions are elastic; no kinetic energy is lost during collisions
• The pressure of a gas is caused by the collisions of molecules on the container walls
• The average kinetic energy of gas molecules is directly proportional to the absolute temperature (Kelvin temperature); at a given temperature, average kinetic energy is the same

Difference Between Ideal and Real Gases

Derivation of Kinetic Gas Equation

• A certain mass of gas is enclosed in a cubic box at a fixed temperature
• The length of each side of the box is 1 cm
• Total number of gas molecules is n
• Mass of one molecule is m
• Velocity of a molecule is ν

Calculation of Molecular Velocities

• The RMS velocity (μ) of gas molecules at a given temperature can be calculated using the Kinetic Gas Equation (PV = 1/2 mNu²)
• For one mole of gas (n=1): μ= √(3RT/M)

Most Probable Velocity

• The most probable velocity (Vmp) is the velocity at which the largest number of gas molecules move. It is given by Vmp = √(2RT/M)

Distribution of Molecular Velocities

• Maxwell's Law describes the distribution of molecular velocities in a gas at a given temperature

Transport Properties

• The derivation of the Kinetic Gas equation does not account for collisions between molecules
• The molecules in a gas are constantly colliding with one another.
• Transport properties (diffusion, viscosity, mean free path) depend on molecular collisions

Mean Free Path (λ)

• At a given temperature, a molecule travels in a straight line before collision with another molecule
• The distance traveled by the molecule before collision is termed the free path
• The mean distance traveled by a molecule between two collisions is called the mean free path (λ)
• λ = 1/(√2πnσ²) where n is the number of molecules per unit volume and σ is the collision diameter

Collision Number

• The number of molecular collisions per second per unit volume Z= √2πnNσ²ν.

Specific Heat Ratio of Gases

• Specific heat (C): Amount of heat required to raise the temperature of one gram of a substance by 1°C
• Specific heat at constant volume (Cv): Amount of heat required to raise the temperature of one gram of a gas by 1 °C while the volume is constant
• Specific heat at constant pressure (Cp): Amount of heat required to raise the temperature of one gram of a gas by 1 °C while the pressure is constant
• The specific heat ratio (γ = Cp/Cv) is used to determine the atomicity of gas molecules

Deviations From Ideal Behaviour

• Ideal gas obeys gas laws at all pressures and temperatures.
• Real gases show deviations from ideal behavior, especially at low temperatures and high pressures

Compressibility Factor (Z)

• Z is a measure of the deviation of a real gas from ideal behavior.
• Z = PV/(RT)
• For ideal gas, Z = 1
• For real gases, Z >1 or Z< 1

Van der Waals Equation

• Van der Waals corrected the ideal gas equation (PV = nRT) to account for the volume of gas molecules and intermolecular forces
• Corrected equation: (P + a(n/V)²)(V-nb) = nRT, where 'a' and 'b' are van der Waals constants.

Critical Phenomena

• The critical temperature (Tc) is the temperature above which a gas cannot be liquefied, regardless of the pressure
• The critical pressure (Pc) is the minimum pressure required to liquefy a gas at its critical temperature
• The critical volume (Vc) is the volume occupied by one mole of the gas at its critical temperature and pressure
• Gases have characteristic critical constants (Tc, Pc, Vc).

Explore the Kinetic Molecular Theory of Gases, which describes the behavior of gas molecules in motion. Learn about the assumptions that govern gas behavior, including molecular volume, motion, and collisions. Perfect for students aiming to understand gas laws and their applications.