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Class XI · Chapter 12

Kinetic Theory

What pressure and temperature actually are, underneath the formulas — countless molecules colliding at random, adding up to laws precise enough to build an entire engine around.

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1Molecular Nature of Matter

All matter is made of atoms and molecules in constant, random motion — a fact confirmed experimentally by Brownian motion (the visible jittering of tiny particles suspended in a fluid, caused by countless unseen molecular collisions). Kinetic theory takes this microscopic picture and shows it can fully explain macroscopic gas behaviour: pressure, temperature, and the ideal gas laws all fall out as natural consequences of molecules simply bouncing around.

2Behaviour of Gases

Before kinetic theory explained why, experiment had already established how gases behave:

  • Boyle's Law: at constant temperature, PV = constant — compress a gas, and its pressure rises proportionally.
  • Charles' Law: at constant pressure, V/T = constant — heat a gas, and it expands proportionally (in kelvin).
Ideal Gas Equation (Combining Both)
PV = nRT = NkT
n = number of moles, N = number of molecules, k = Boltzmann constant (R/NA). Real gases follow this closely at ordinary pressures and temperatures, deviating mainly under extreme compression or near condensation.

3Assumptions of the Kinetic Theory

Kinetic theory derives gas behaviour from a small set of simplifying assumptions about molecules:

  • Molecules are in continuous, random motion, colliding with each other and the container walls.
  • Collisions are perfectly elastic — no kinetic energy is lost overall.
  • Molecules exert no force on each other except during a collision (no intermolecular attraction otherwise).
  • The total volume occupied by the molecules themselves is negligible compared to the container's volume.
  • Molecules obey Newton's laws of motion.

4Pressure of an Ideal Gas

Gas pressure isn't mysterious once you picture it correctly: it's simply the cumulative effect of countless molecules colliding with a container's walls every instant, each collision delivering a tiny impulse.

Kinetic Theory Pressure
P = (1/3) ρ vrms²
ρ = gas density, vrms = root-mean-square molecular speed. Faster molecules, or a denser gas, both mean more frequent and more forceful wall collisions — and therefore higher pressure.
RMS Speed
vrms = √(3RT/M) = √(3kT/m)
M = molar mass, m = mass of a single molecule. Lighter molecules move faster at the same temperature — exactly why helium leaks out of a balloon faster than air does.

5Kinetic Interpretation of Temperature

Average Translational Kinetic Energy per Molecule
KEavg = (3/2) k T
This is genuinely one of the most important results in the chapter: temperature isn't an abstract number on a scale — it's a direct, literal measure of a molecule's average kinetic energy. Absolute zero is exactly the temperature where that average kinetic energy would reach zero.

6Law of Equipartition of Energy

A molecule can store energy in more ways than just moving in a straight line (translation) — it can also rotate, and (at high enough temperature) vibrate. Each independent way energy can be stored is a degree of freedom.

Law of Equipartition of Energy In thermal equilibrium, energy is shared equally among all available degrees of freedom, with each contributing an average of ½kT per molecule.
  • Monatomic gas (like helium): only 3 translational degrees of freedom → total average energy = (3/2)kT per molecule.
  • Diatomic gas (like oxygen or nitrogen, at moderate temperature): 3 translational + 2 rotational degrees of freedom → total average energy = (5/2)kT per molecule.

7Specific Heat Capacities from Kinetic Theory

Equipartition directly predicts specific heat capacities — a genuinely satisfying payoff, connecting the microscopic molecular picture back to the macroscopic Chapter 11 quantities.

Monatomic Gas
Cv = (3/2)R, Cp = (5/2)R, γ = Cp/Cv = 5/3
Diatomic Gas
Cv = (5/2)R, Cp = (7/2)R, γ = 7/5
These predicted values match experimentally measured specific heats for real gases remarkably well — strong evidence that the kinetic theory's simple molecular picture genuinely captures what's physically happening.

8Mean Free Path

Between collisions, a molecule travels in a straight line for some distance before hitting another molecule. The mean free path is the average length of these straight-line stretches.

Mean Free Path
λ = 1 / (√2 π d² n)
d = molecular diameter, n = number density (molecules per unit volume). Denser gas or larger molecules both mean a shorter mean free path — more frequent collisions. This is directly connected to how gases diffuse and how quickly a smell spreads across a room.

Formula Summary

Ideal Gas Equation
PV = NkT
Pressure (Kinetic Theory)
P = (1/3)ρv_rms²
RMS Speed
v_rms = √(3RT/M)
Avg. KE per Molecule
(3/2)kT
Monatomic Cv
(3/2)R
Diatomic Cv
(5/2)R
Mean Free Path
λ = 1/(√2πd²n)

Solved Examples

Example 1 · RMS Speed

Find the rms speed of oxygen molecules at 300 K. (MO₂ = 32×10⁻³ kg/mol, R = 8.314 J/mol·K)

Solution: vrms = √(3RT/M) = √[(3 × 8.314 × 300) / 0.032] = √(233,831) ≈ 483.6 m/s.

Example 2 · Average Kinetic Energy

Find the average kinetic energy of a gas molecule at 27°C (= 300 K). (k = 1.38×10⁻²³ J/K)

Solution: KEavg = (3/2)kT = (3/2) × 1.38×10⁻²³ × 300 ≈ 6.21×10⁻²¹ J.

Notice this depends only on temperature — completely independent of which gas it is, whether helium or oxygen.

Quick Check

1. According to kinetic theory, temperature is fundamentally a measure of:
Gas pressure
✓ Average kinetic energy of molecules (correct)
Number of molecules
Molecular mass
2. For a diatomic gas at moderate temperature, the molar specific heat at constant volume (Cv) equals:
(3/2)R
✓ (5/2)R (correct)
(7/2)R
R
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Oscillations

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