Kinetic Theory - Class 11 Physics

Kinetic Theory

Overview: Microscopic study of gases. Relationship between Pressure, Volume, Temperature and molecular motion.

1. Ideal Gas

Gas Laws

Boyle's Law: PV = constant (at constant T).
Charles' Law: V/T = constant (at constant P).
Dalton's Law: Total P = Sum of partial pressures.

Equation of State

PV = nRT = (N/N_A) RT = N k_B T

Where k_B = Boltzmann constant = 1.38 × 10^-23 J/K.

2. Pressure & Kinetic Energy

Pressure Formula

Pressure due to molecular collisions.

P = 1/3 ρ v_rms²

Interpretation of Temperature

Average Kinetic Energy depends only on Temperature.

E_avg = 3/2 k_B T

v_rms = √(3RT/M) = √(3k_BT/m)

3. Degrees of Freedom (f)

Number of independent ways a molecule can possess energy.

Monoatomic (He, Ne): f = 3 (Translational).
Diatomic (H2, N2): f = 5 (3 Trans + 2 Rot) at normal T. f = 7 at high T (Vibrational).
Polyatomic: f ≥ 6.

Law of Equipartition of Energy

Energy is shared equally among all degrees of freedom. Energy per DOF = 1/2 k_B T.

Total Internal Energy U = f/2 nRT.

4. Mean Free Path (l)

Average distance between successive collisions.

l = 1 / (√2 n π d²)

Where n = number density (N/V), d = diameter of molecule.

Numericals - Kinetic Theory

Numericals

Ideal Gas Law

Q1. Calculate volume occupied by 2 moles of gas at 27°C and 1 atm.

PV = nRT

V = nRT / P

V = 2 × 8.314 × 300 / 101325

V = 4988.4 / 101325 = 0.049 m³

V = 49 Liters

RMS Speed

Q2. Calculate V_rms of Oxygen at 300K. (M = 32g/mol)

v_rms = √(3RT/M)

M = 32 × 10^-3 kg

v = √(3 × 8.314 × 300 / 0.032)

v = √(233857) ≈ 483 m/s

Energy per Molecule

Q3. Average kinetic energy of a molecule at 27°C?

E = 3/2 k_B T

E = 1.5 × 1.38 × 10^-23 × 300

E = 6.21 × 10^-21 J

Specific Heat Ratio

Q4. A gas has f=5. Calculate γ.

γ = 1 + 2/f

γ = 1 + 2/5

γ = 1.4

(Diatomic Gas)

Mixing of Gases

Q5. 1 mole of Ne mixed with 1 mole of O2. Find γ of mixture.

Cv_mix = (n1 Cv1 + n2 Cv2) / (n1 + n2)

Cv1 (Ne, mono) = 3/2 R. Cv2 (O2, dia) = 5/2 R.

Cv_mix = (1(3/2 R) + 1(5/2 R)) / 2 = (4R)/2 = 2R

Cp_mix = Cv_mix + R = 3R

γ_mix = 3R / 2R = 1.5

Mean Free Path

Q6. If pressure is doubled at constant Temp, how does mean free path change?

l ∝ 1/n. n = N/V.

PV = NkT → n = P/kT.

l ∝ T/P.

If P doubles, l becomes half.

RMS vs Temperature

Q7. At what temp will rms speed be double of that at 300K?

v ∝ √T

v2/v1 = √(T2/T1)

2 = √(T2/300)

4 = T2/300 → T2 = 1200 K

Pressure calculation

Q8. Find pressure if Vrms = 500 m/s and density = 1.2 kg/m³.

P = 1/3 ρ v²

P = 1/3 × 1.2 × (500)²

P = 0.4 × 250000 = 100000 Pa = 10⁵ Pa

Total Internal Energy

Q9. Find U for 2 moles of Helium at 300K.

U = n (f/2) RT

He is monoatomic, f=3.

U = 2 × 1.5 × 8.3 × 300

U = 3 × 2490 = 7470 J

Avogadro's Hypothesis

Q10. Two vessels have same V, T, P. One has H2, other has O2. Ratio of molecules?

PV = NkT.

If P, V, T same, N must be same.

Ratio = 1:1

Formulas & Facts - Kinetic Theory

Equations & Formulas

Concept	Formula
Ideal Gas Law	PV = nRT = NkT
Pressure	P = 1/3 ρ v_rms²
RMS Speed	v_rms = √(3RT/M)
Avg Speed	v_avg = √(8RT/πM)
Most Prob Speed	v_mp = √(2RT/M)
Avg KE	E = 3/2 kT
Internal energy	U = f/2 nRT
Mean Free Path	l = 1 / √2 n π d²
Mayer's Relation	C_p - C_v = R
Gamma	γ = 1 + 2/f

50 NEET Facts

Key points for Kinetic Theory.

1. Assumption 1 Gas molecules are point masses. Volume of molecules negligible vs Volume of gas.

2. Assumption 2 Collisions are perfectly elastic. No energy loss.

3. Assumption 3 No intermolecular forces. (Potential Energy = 0).

4. Pressure Cause Change in momentum during collisions with walls.

5. Pressure Factors Depends on Number density (n), Mass (m), and Velocity square (v^2).

6. Temperature Measure of mean kinetic energy of translational motion.

7. Absolute Zero Temp where RMS speed becomes zero. Molecular motion stops.

8. Light vs Heavy Gas At same T, lighter gas molecules move faster than heavier ones. (v ∝ 1/√M).

9. H2 vs O2 H2 moves 4 times faster than O2. (Mass ratio 16, sqrt 4).

10. Escape Velocity H2 escapes earth atmosphere because its v_rms is high.

11. Speed Distribution Maxwell-Boltzmann distribution. Curve flattens and shifts right as T increases.

12. Speed Order v_rms > v_avg > v_mp. (Ratio 1.73 : 1.6 : 1.41).

13. Degrees of Freedom Mono=3. Dia=5. Tri (Linear)=5. Tri (Non-linear)=6.

14. Vibrational Mode Active only at high temperatures. Adds +2 to f.

15. Equipartition Law Each quadratic term in energy expression contributes 1/2 kT per molecule.

16. Internal Energy U For ideal gas, U is purely kinetic.

17. Specific Heat of Solids C = 3R (Dulong Petit Law).

18. Specific Heat of Water C = 9R approx.

19. Mean Free Path (l) Distance between collisions. Depends inversely on density and diameter square.

20. Effect of Pressure on l At constant T, if P increases, l decreases.

21. Effect of Temp on l At constant P, if T increases, V increases, n decreases, l increases.

22. Brownian Motion Random motion of colloidal particles due to bombardment by fluid molecules. Proof of kinetic theory.

23. Avogadro Number N_A 6.02 × 10²³ molecules per mole.

24. STP Conditions P=1atm, T=273K, V=22.4L.

25. Ideal Gas Behavior Real gases behave ideally at Low Pressure and High Temperature.

26. Van der Waals Eq Corrects ideal gas eq. (P + a/V^2)(V - b) = RT.

27. 'a' constant Represents intermolecular attraction.

28. 'b' constant Represents finite volume of molecules.

29. Diffusion Rate of diffusion ∝ v_rms. Faster for lighter gases.

30. Dalton's Law Valid only for non-reacting gases.

31. Vapor vs Gas Vapor is below Critical Temp. Gas is above Critical Temp.

32. Critical Temperature Temp above which gas cannot be liquefied by pressure alone.

33. Boiling Point in Mountains Lower because P is lower.

34. Evaporation Cooling High energy molecules escape, lowering average KE of remaining liquid.

35. Pascal's Law Basis Pressure is isotropic because molecular motion is random.

36. Energy per mole E = 3/2 RT. Independent of nature of gas. Same for He and O2.

37. Energy per kg E = 3/2 (RT/M). Higher for lighter gases (Hydrogen has highest cal per kg).

38. Speed of Sound v_sound = √(γRT/M). Close to v_rms but v_sound < v_rms.

39. Newton's Sound Formula Assumed isothermal. Incorrect. Laplace corrected it to Adiabatic.

40. Humidity Effect Humid air is lighter (H2O < N2/O2). Sound travels faster in humid air.

41. Relaxation Time Time between collisions.

42. Transport Properties Viscosity, Conduction, Diffusion. All explained by Kinetic Theory.

43. Viscosity of Gas Increases with sqrt(T). Independent of Pressure (counter-intuitive).

44. Conductivity of Gas Also increases with sqrt(T).

45. Density of mixture Total Mass / Total Vol.

46. Atomic Size Approx 1 Angstrom (10^-10 m).

47. Interatomic Distance In gas ~ 10 Angstroms (10 times size).

48. Number of collisions Billions per second.

49. Isothermal Atmosphere P = P0 exp(-mgh/kT).

50. Vacuum Lowest possible pressure. Perfect vacuum impossible.

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