Kinetic Theory

Kinetic Theory

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Kinetic theory links the macroscopic behaviour of an ideal gas — its pressure, temperature, volume, and internal energy — to the random motion of a huge number of identical point-like molecules that collide elastically with one another and with the container walls. The single relationship every NEET question pivots on is that the absolute temperature of a gas is a direct measure of the average translational kinetic energy per molecule: ⟨KE⟩ = (3/2) k_B T, with k_B = 1.38 × 10⁻²³ J K⁻¹.

• Pressure from impacts: P = (1/3) ρ v_rms², where ρ is mass density and v_rms is the root-mean-square speed.
• RMS speed: v_rms = √(3RT/M), with R = 8.314 J mol⁻¹ K⁻¹ and M = molar mass in kg mol⁻¹.
• Internal energy of N molecules of a monatomic ideal gas: U = (3/2) nRT.

Quantity	Formula	Depends on mass?
Average KE per molecule	(3/2) k_B T	No
v_rms	√(3RT/M)	Yes (∝ 1/√M)
Pressure	(1/3)(N/V)m v_rms²	Yes

🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Postulates and Pressure Derivation

The theory assumes: identical point molecules, negligible molecular volume compared with the container, no intermolecular force except during collisions, perfectly elastic collisions, and random motion averaged over a statistically large N. Pressure emerges from momentum transfer during wall collisions: each impact reverses the normal component of momentum, so force per unit area equals (1/3)(N/V)m⟨v²⟩, which is the kinetic pressure formula.

Temperature and Internal Energy

Because P V = (1/3) N m ⟨v²⟩ and also P V = N k_B T, equating gives ⟨(1/2) m v²⟩ = (3/2) k_B T. Temperature measures only translational kinetic energy per molecule; for one mole, U = (3/2) R T. Always use kelvin, never Celsius.

Equipartition of Energy

Each independent quadratic degree of freedom carries (1/2) k_B T of average energy. Degrees of freedom depend on molecular structure, which sets molar specific heats.

Molecule	f	C_v = (f/2)R	C_p = C_v + R	γ = C_p/C_v
Monatomic (He, Ar)	3	(3/2)R	(5/2)R	5/3
Diatomic, rigid (N₂, O₂ at room T)	5	(5/2)R	(7/2)R	7/5
Diatomic, vibrational ON	7	(7/2)R	(9/2)R	9/7

Mean Free Path

The average distance a molecule travels between collisions is λ = 1/(√2 · π d² n), where d is the effective molecular diameter and n = N/V. Mean free path controls viscosity, thermal conductivity, and diffusion in gases — a recurring cross-topic link in NEET.

• Convert all temperatures to kelvin before substituting into any kinetic-theory formula.
• v_rms > v_mean > v_mp in the ratio √3 : √(8/π) : √2.

🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Edge Cases and Real-Gas Behaviour

Kinetic theory works best at low pressure and high temperature where intermolecular separation is large. As P rises or T falls, the finite size of molecules and the attractive/repulsive forces between them distort the predictions. Van der Waals’ equation patches this by adding a volume correction (nb) and an attraction term (a n²/V²); NCERT and NEET frequently ask why a real gas deviates more near the liquefaction point.

Worked Example

Two moles of helium (monatomic, M = 4 × 10⁻³ kg mol⁻¹) at 300 K occupy 0.05 m³. Compute v_rms and internal energy.

1. v_rms = √(3RT/M) = √(3 × 8.314 × 300 / 0.004) ≈ √(1.871 × 10⁶) ≈ 1368 m s⁻¹.
2. U = (3/2) n R T = (3/2)(2)(8.314)(300) ≈ 7483 J.

Connections and Strategy

The same ⟨KE⟩ ∝ T result reappears in Calorimetry (linking heat capacity to molecular structure) and Thermodynamics (linking U, γ, and adiabatic exponents). One NEET MCQ every two years asks students to rank v_rms of H₂, He, N₂ at the same temperature — lighter molecules move faster. Assertion-reason questions often hinge on the trap that average KE is mass-independent while v_rms is mass-dependent.

Mistake	Correction
Using °C in v_rms = √(3RT/M)	Convert to K first
Writing C_p = (f/2)R	C_p = (f/2)R + R
Treating v_mean = v_rms	v_rms = 1.085 × v_mean
Ignoring vibrational modes above 1000 K	Switch diatomic f from 5 to 7

Practice Prompts

1. At what temperature will v_rms of O₂ equal 500 m s⁻¹? (Answer uses v_rms² = 3RT/M.)
2. Two vessels contain H₂ at 300 K and O₂ at 600 K. Compare their average molecular KE and their v_rms — does doubling T double v_rms?

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