Kinetic Theory
Matter is made of invisible atoms and molecules in constant motion. Kinetic theory connects the tiny (individual particles) to the observable (pressure, temperature, density).
Start with the simplest version: this lesson is about Kinetic Theory. If you can explain the core idea to a friend using everyday language, examples, and one clear reason why it matters, you have moved from memorising to understanding.
Matter is made of atoms and molecules in ceaseless motion. The kinetic theory derives the macroscopic gas laws from microscopic mechanics: pressure is the average momentum delivered per unit time by molecules colliding with the walls; temperature is a measure of the mean kinetic energy of those molecules.
The Ideal Gas Model
Assumptions: a large number of identical molecules of negligible volume; collisions are elastic and instantaneous; no intermolecular forces between collisions; molecules obey Newton's laws. Equation of state: PV = NkT = nRT, where k = 1.38 × 10⁻²³ J/K and R = N_A k.
Pressure of an Ideal Gas
From kinetic considerations: P = (1/3) (N/V) m v̄² = (1/3) ρ v̄², where v̄² is the mean-square speed. Combined with PV = NkT this gives ½ m v̄² = (3/2) k T.
Kinetic Interpretation of Temperature
Average translational kinetic energy per molecule: ⟨KE⟩ = (3/2) kT. Total internal energy of a monatomic ideal gas: U = (3/2) N k T = (3/2) n R T. Temperature, in this view, is just a label for the mean translational KE.
Molecular Speeds
For a Maxwell–Boltzmann distribution at temperature T:
v_rms = √(3kT/m) = √(3RT/M);
v_avg = √(8kT/(πm));
v_mp = √(2kT/m). Order: v_mp < v_avg < v_rms.
Law of Equipartition of Energy
In thermal equilibrium each quadratic degree of freedom (translation, rotation, vibration) carries average energy ½ k T. So a monatomic gas has U = (3/2) NkT (3 translational dof); a diatomic gas has U = (5/2) NkT (3 translational + 2 rotational at room temperature).
Specific Heats from Equipartition
For f degrees of freedom: C_v = (f/2) R per mole, C_p = ((f+2)/2) R, γ = C_p/C_v = (f+2)/f. Monatomic: γ = 5/3. Diatomic (f = 5): γ = 7/5.
Mean Free Path
λ = 1 / (√2 π d² n), where d is the molecular diameter and n = N/V is the number density. Average time between collisions τ = λ/v_avg. At STP, λ for air ≈ 70 nm.
Limitations of the Ideal Gas Model
Real gases deviate from PV = nRT at high pressure (molecular volume is no longer negligible) and low temperature (intermolecular attraction matters). The van der Waals equation (P + a/V²)(V − b) = RT corrects both effects.
Socratic Questions
- Why does pressure arise from molecular collisions with walls, not from molecules pushing each other?
- Two gases at the same temperature have the same average kinetic energy, but different rms speeds. Why?
- Why does γ for a diatomic gas decrease as temperature rises into the range where vibrational modes activate?
- How does the mean free path change if pressure is doubled at constant temperature?
- Why do real gases liquefy at low temperatures, while an ideal gas would not?