Kinetic-Molecular Theory: Explains the macroscopic properties of gases (pressure, volume, temperature) as a result of the microscopic properties (position and speed of gas molecules).
Objective: To relate microscopic properties to macroscopic properties, such as pressure and temperature.
Key Assumptions
Gas molecules do not interact except during collisions.
Collisions between molecules and with walls are elastic (conserve kinetic energy and momentum).
Core Concepts
Microscopic to Macroscopic Relationships
Speed of Molecules: If known, can predict pressure and vice versa.
Pressure and Force: Pressure is force per area. Force can be calculated using change in momentum (∆P/∆t).
Calculating Force
Force Equation:
Force (F) = m * 2 * v_x / ∆t
For elastic collisions, ∆v = 2v, as velocity is vectorial.
Time Between Collisions
Time Calculation:
∆t = 2L/v_x (for a cube with side L)
Used to substitute in force equation.
Pressure on a Wall
Average Force by Multiple Particles:
F = (m/L) * sum of v_x^2 across all particles
Average squared velocity (v_x^2) of particles is used.
Ideal Gas Law Derivation
Pressure Equation:
P = N * m * v_x^2 (average) / V
Relates pressure and volume to molecular velocity.
Total Velocity
Pythagorean Theorem in 3D:
V_total^2 = V_x^2 + V_y^2 + V_z^2
Used to find total average squared velocity.
Final Relationships
Pressure-Volume Relationship:
3 * P * V = N * m * V_total^2 (average)
Relates macroscopic variables to average kinetic energy of molecules.
Kinetic Energy Relationships
Average Kinetic Energy:
3/2 * P * V = N * average Kinetic Energy
Kinetic energy of single gas molecule = 3/2 * k * T
Internal Energy for Monatomic Ideal Gases:
U = 3/2 * N * k * T = 3/2 * n * R * T
Total internal energy = total kinetic energy.
Applications
Ideal Gas Law: Applications in predicting behavior of gases.
Thermal Systems: Total internal energy often referred to as total kinetic energy for monatomic gases, like helium or neon.
Conclusion
Kinetic-Molecular Theory provides a comprehensive way to relate microscopic molecular behavior to macroscopic gas properties, useful in various scientific applications.