Lecture Week 10 Gas 5 Kinetic Theory

Kinetic Theory of Gases - Overview

This theory explores the relationship between bulk properties of gases (pressure, temperature, and volume) and molecular motion, providing critical insights into the behavior of gases in various conditions.

Learning Outcomes

• Understand the ideal gas equation in relation to kinetic theory.

• Calculate the kinetic energy of gases using various equations.

• Determine the mean square velocity of gases using kinetic energy equations.

Key Concepts in Kinetic Theory

Ideal Gas Equation:The equation that ties together pressure, volume, the number of moles, and temperature:[ PV = nRT ]Where:

• P = Pressure

• V = Volume

• n = Number of moles

• R = Ideal gas constant (8.314 J/(mol·K))

• T = Temperature (in Kelvin)

Assumptions of the Kinetic Theory

1. Negligible Molecular Volume:The volume occupied by individual gas molecules is negligible compared to the volume of the gas itself: ( V_{molecule} << V_{gas} ).

2. Elastic Collisions:Molecules behave like perfectly elastic spheres, meaning that energy is conserved during collisions.

3. No Intermolecular Forces:Molecules do not exert forces on each other except during collisions with the container walls.

4. Negligible Collision Duration:The duration of collisions with walls is negligible compared to the time between collisions.

Pressure and Momentum in Gases

Origin of Pressure:Pressure in a gas arises from collisions of gas molecules with the walls of its container.

Momentum Change Calculation:

• Before collision: ( Momentum = m u )

• After collision: ( Momentum = -m u )

• Change in momentum: ( \Delta Momentum = 2mu )

Calculating Pressure

For a cube of side length ( L ) containing ( N ) molecules:

1. Total Velocity Components:Combine individual velocity components as: ( c^2 = u^2 + v^2 + w^2 ).

2. Average Force on Wall:The average force exerted on the wall is given by: ( F_{average} = m u^2 / L ).

3. Pressure Expression:Pressure can be expressed as: ( P = \frac{m u^2}{L^3} ).

Mean Square Velocity

If ( N ) molecules each have unique velocities with the same mass:

• ( P = \frac{m}{L^3}(u_1^2 + u_2^2 + ... + u_N^2) )

• Mean square velocity is defined as: ( u^2 = \frac{1}{N} \Sigma (u_n^2) ).

Density and Pressure Relationship

Density Definition:Density (( \rho )) is calculated as:[ \rho = \frac{Total \ Mass}{Volume} = \frac{N m}{L^3} ]Pressure Relation:Pressure can be expressed in terms of density:[ P = \rho u^2; \text{ thus, } P = \frac{1}{3} \rho c^2 \text{ where } c^2 \text{ is the mean square speed.} ]

Kinetic Energy of the Gas

Total Kinetic Energy (KE):The total kinetic energy of gas molecules is: ( KE = \frac{1}{2} N m c^2 ).Temperature Relation to Kinetic Energy: Kinetic energy is related to temperature as follows: ( KE = \frac{3}{2} nRT ).

Calculating Velocity of Molecules

From kinetic energy equations, the speed of particles can be calculated as follows: ( c^2 = \frac{3nRT}{m} ).

Examples

Example 1:Given: 2 moles of nitrogen (N₂) in an 8.5 L container at 4.5 bar.Calculation:[ KE = \frac{3}{2}PV = \frac{3}{2} * (4.5 \times 10^5 Pa) * (8.5 \times 10^{-3} m^3) = 5737.5 J. ]Kinetic energy of one nitrogen molecule: ( KE_0 = \frac{KE}{2 * N_A} = 4.8 \times 10^{-21} J. )

Example 2:Given: Mass number of nitrogen = 28 g, temperature at 20°C = 293 K.Calculation:Mean square velocity: [ c^2 = \frac{3 * 8.31 \text{ J/(mol·K)} * 293 K}{28 \times 10^{-3} kg} = 260874 m^2/s^2. ]Root mean square velocity: [ c = \sqrt{260874 m^2/s^2} \approx 510 m/s. ]

Summary

Review the Kinetic Theory of Gases concepts and calculations, and complete Example Sheet 8 for practice to solidify understanding.