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Kinetic Theory

Kinetic Theory of Gases

Physics · Class 11

1. Thermodynamics vs KTG

Thermodynamics	Kinetic Theory of Gases
Study of gas at macroscopic level	Study of gas at microscopic level
Large scale quantities : temperature, pressure, volume of gas in vessel	Atomic / molecular scale : speed, momentum, kinetic energy of molecules

KTG relates the macroscopic properties (T, P, V) to the microscopic properties (speed, momentum, K.E.) of the molecules.

2. Ideal Gas Equation

PV = nRT

P = ρRTM (using n = mM, ρ = mV)

Symbol	Meaning
P	pressure of gas
V	volume of gas
n	amount of gas in moles
R	gas constant = 8.314 J/mol·K ≈ 253 J/mol·K
T	temperature in kelvin
ρ	density (kg/m³)
M	molar mass (in kg/mol)

e.g. N₂ gas : M = 28 g/mol = 28 × 10⁻³ kg/mol

3. Units

Pressure → SI unit pascal (Pa) = N/m² ; 1 atm ≈ 10⁵ Pa
Volume → SI unit m³ ; 1 litre = 10⁻³ m³ (1 m³ = 1000 litre)
1 litre = 1000 mL = 1000 cm³

4. Macroscopic Quantities

Pressure :

Force exerted by the gas due to collision of molecules with the walls (momentum transfer).

Collision is elastic : only the component ⊥ to wall (v_x) reverses ; v_y, v_z unchanged.
Momentum given to wall = m(2) − m(−2) = 2mv_x = 4m (along +x).

Volume :

Free space available for motion of gas molecules.

V_gas = V_vessel − (total volume of molecules) ≈ V_vessel

Density :

ρ = massvolume

Amount of gas :

Calculated in moles ; 1 mole = N_A molecules, N_A = 6.02 × 10²³

Temperature :

Property by virtue of motion of molecules : high temp. ⇒ high K.E. ; low temp. ⇒ low K.E.

5. Standard Assumptions (Ideal Gas)

Volume of molecules is negligible compared to volume of container ⇒ volume of gas = volume of container.
No intermolecular force ⇒ P.E. of ideal gas = 0 ⇒ internal energy is purely K.E.
No preferred direction ; motion is completely random.
Molecules move in straight lines (free motion) most of the time ; collision time is very small.
Collisions (molecule–molecule and molecule–wall) are perfectly elastic.
Motion is governed by Newton's laws.

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Kinetic Theory

6. Maxwell Distribution Curve

M = v_mp , A = v_avg , R = v_rms

Area of strip : dA = dNdv·dv = dN = no. of molecules having speed between v and v + dv.
Total area under curve = total no. of molecules N.
Peak of curve → most probable speed (largest no. of molecules near it).

7. Three Characteristic Speeds

v_mp = √2RTM

v_avg = √8RTπM

v_rms = √3RTM

R = 8.31 J/mol·K , M = molar mass in kg , T in kelvin

v_mp < v_avg < v_rms

	v_mp	v_avg	v_rms
factor	√2	√(8/π)	√3
value	1.414	1.596	1.732
ratio	1	1.13	1.22

All speeds ∝ √(T/M) (same factor √(RT/M) multiplied)

e.g. v_rms of N₂ at 300 K (M = 28 × 10⁻³ kg/mol) :
v_rms = √3 × 8.314 × 30028 × 10⁻³ = √(2.67 × 10⁵) ≈ 517 m/s

8. Effect of Temperature

On heating, peak shifts to higher speed (v_mp ∝ √T) and the curve becomes flatter & broader.
Area under both curves is the same (no. of molecules N unchanged).

9. Vapour Density

Mass of a certain volume of a gas divided by the mass of the same volume of hydrogen under identical conditions (same P, T).

Vapour density = mass of n molecules of gasmass of n molecules of H₂ = M2

(equal volumes at same P, T contain equal no. of molecules ; M_H₂ = 2 g/mol)

Molar mass = 2 × (Vapour density)

e.g. Vapour density of CO₂ = 22 ⇒ M = 2 × 22 = 44 g/mol

Key Points :

Area under Maxwell curve = total no. of molecules.
v_mp : v_avg : v_rms = √2 : √(8/π) : √3 ; all ∝ √(T/M).
Lighter gas (small M) → faster molecules at the same T.

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Kinetic Theory

10. Degrees of Freedom (f)

Number of independent parameters (independent ways of having energy) needed to define the state / configuration of a molecule.

Type	Remark
Translational	= 3 for every molecule (v_x, v_y, v_z)
Rotational	about x, y, z axes through COM ; counted only if moment of inertia ≠ 0
Vibrational	exhibited only at high temperatures

11. Monoatomic Gas (He, Ne, Ar …)

Translation

Rotation : I ≈ 0

Translational f = 3
Atom is tiny (point mass) ⇒ moment of inertia about any axis through it ≈ 0 ⇒ no rotational K.E. ⇒ rotational f = 0
Vibrational f = 0 (single atom) ⇒ f = 3

12. Diatomic Gas (H₂, N₂, O₂ …)

Translational f = 3 ; Rotational f = 2 (about y and z ; I about the bond axis = 0)
Vibrational f = 2 (atoms vibrate along the bond like a spring : K.E. + P.E. of vibration)

Normal temp. : f = 3 + 2 = 5 (vibration ignored)

High temp. : f = 3 + 2 + 2 = 7 (vibration significant)

13. Linear Polyatomic Gas (CO₂)

All atoms on one line ⇒ I about that line = 0
Translational f = 3 , Rotational f = 2 ⇒ f = 5

14. Non-linear Polyatomic Gas (SO₂, NO₂, H₂O)

Atoms not on one line ⇒ I ≠ 0 about all three axes
Translational f = 3 , Rotational f = 3 ⇒ f = 6

15. Summary of f (normal temperature)

	Mono	Di	Poly (L)	Poly (NL)
Translational	3	3	3	3
Rotational	0	2	2	3
Total f	3	5	5	6

Examples : Mono – He, Ne, Ar ; Di – H₂, N₂, O₂ ; Poly (L) – CO₂ ; Poly (NL) – SO₂, NO₂, H₂O

Key Points :

Translational f = 3 for every molecule.
Rotation about an axis counts only if I about it ≠ 0.
Diatomic : f = 5 normally, f = 7 at high temperature.
Vibrations are ignored unless high temperature is stated.

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Kinetic Theory

16. f, C_v, C_p and γ

C_v : molar specific heat capacity at constant volume
C_p : molar specific heat capacity at constant pressure
γ : ratio C_p / C_v

C_v = fR2

C_p = (f + 2)R2 = C_v + R

C_p − C_v = R (Mayer's relation)

γ = C_pC_v = f + 2f = 1 + 2f

f more (ज़्यादा) ⇒ γ less (कम). Always 1 < γ ≤ 5/3.

17. Table of C_v, C_p, γ

	Mono	Di	Poly (L)	Poly (NL)
f	3	5	5	6
C_v	3R/2	5R/2	5R/2	3R
C_p	5R/2	7R/2	7R/2	4R
γ	5/3	7/5	7/5	4/3

γ_mono > γ_dia > γ_{poly (NL)}

(γ_mono ≈ 1.67 , γ_dia = 1.4 , γ_{poly (NL)} ≈ 1.33)

18. Law of Equipartition of Energy

In thermal equilibrium the total energy of a molecule is equally divided among all its degrees of freedom.
Average energy associated with each degree of freedom :

E (per f) = 12 kT

k = Boltzmann constant = 1.38 × 10⁻²³ J/K , T = temperature (K)

19. Internal Energy of Ideal Gas (U)

Assumption : no interaction force between molecules ⇒ no interaction P.E.
⇒ Internal energy of gas is only due to motion of molecules (K.E.).

Energy of one molecule = f (½ kT)

n moles at temp. T have N = nN_A molecules :

U = Nf2kT = (nN_A)f2kT

U = nfRT2 = nC_vT

Symbol	Meaning
N	no. of molecules
n	no. of moles
N_A	Avogadro no. = 6.023 × 10²³
k	Boltzmann constant = 1.38 × 10⁻²³ J/K

R = N_A k = 8.314 J/mol·K

U depends only on T (for a given gas) — it is a state function.

20. Kinetic Energy Summary

Quantity	Formula
K.E. of a molecule	f2 kT
Translational K.E. of a molecule	32 kT
K.E. of gas (n moles)	nfRT2
Translational K.E. of gas	3nRT2

Translational K.E. per molecule = 32kT is the same for every gas at the same T (depends only on T).

Key Points :

C_v = fR/2 , C_p = (f + 2)R/2 , γ = 1 + 2/f.
Each degree of freedom gets ½ kT (per molecule) or ½ RT (per mole).
U = nfRT/2 = nC_vT ; R = N_Ak.

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Kinetic Theory

21. Mixture of Gases

Gases in the same vessel : volume and temperature of both are the same ; pressures may be different.

P₁V = n₁RT and P₂V = n₂RT

P₁, P₂ → partial pressures

P = P₁ + P₂ + … (Dalton's law)

Total gas : (P₁ + P₂)V = (n₁ + n₂)RT

Total internal energy :

U = n₁f₁RT2 + n₂f₂RT2 + … = (n₁ + n₂ + …) f_eq RT2

f_eq = n₁f₁ + n₂f₂ + …n₁ + n₂ + …

(C_v)_eq = n₁C_v1 + n₂C_v2 + …n₁ + n₂ + … = f_eqR2

(C_p)_eq = n₁C_p1 + n₂C_p2 + …n₁ + n₂ + … = (f_eq + 2)R2

γ_eq = (C_p)_eq(C_v)_eq = f_eq + 2f_eq

Note : γ_eq is not the simple average of γ₁ and γ₂ — always go through f_eq (or C_v).

e.g. 1 mol He (f = 3) + 1 mol O₂ (f = 5) :
f_eq = 1(3) + 1(5)2 = 4 ⇒ (C_v)_eq = 2R , (C_p)_eq = 3R
γ_eq = 4 + 24 = 1.5

22. Hypothetical Speed Distribution (Solved)

Q. Variation of dNdv with speed v of a hypothetical gas is as shown. Find total number of molecules, average speed, most probable speed and r.m.s. speed.

Line through origin : dNdv = λv₀ v (slope tanθ = λ/v₀) ⇒ dN = λvv₀dv

(1) Total molecules = area under curve :

N = ∫dN = ∫dNdvdv = area of triangle

N = ½ λ v₀

(2) Most probable speed :

dN/dv is maximum at v = v₀ (tallest strip ⇒ most molecules) ⇒ v_mp = v₀

(3) Average speed :

Average of y w.r.t. x = ∫y dx∫dx

v_avg = ∫v dN∫dN = ∫₀^v₀ v(λv/v₀) dv∫₀^v₀ (λv/v₀) dv = v₀³/3v₀²/2

v_avg = 2v₀3

(4) r.m.s. speed :

v_rms² = ∫v² dN∫dN = ∫₀^v₀ v³ dv∫₀^v₀ v dv = v₀⁴/4v₀²/2 = v₀²2

v_rms = v₀√2 ≈ 0.71 v₀

Here v_avg (0.67v₀) < v_rms (0.71v₀) < v_mp (v₀) — the order v_mp < v_avg < v_rms is for the Maxwell curve, not for every distribution (but v_avg ≤ v_rms always).

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Kinetic Theory

23. Gas in a Moving Vessel (Solved)

Q. A closed vessel has n moles of monoatomic gas (molar mass M) at temperature T. The vessel is kept in a car moving with velocity v. Find the total thermal internal energy and total kinetic energy of the gas.

Thermal internal energy is only due to random motion of molecules (w.r.t. COM) — it does not depend on v :

U = nfRT2 = 3nRT2

Total K.E. = K.E. of particles w.r.t. COM + K.E. of COM ; mass of gas = nM :

K.E._total = 3nRT2 + ½ (nM) v²

24. Mean Free Path (λ)

Average distance travelled by a molecule between two successive collisions.

λ = l₁ + l₂ + … + l_nn

λ = kT√2 π d² P

k = Boltzmann constant (1.38 × 10⁻²³ J/K), T = temp., d = diameter of molecule, P = pressure

λ ∝ TP and λ ∝ 1ρ

(P = ρRT/M ⇒ T/P = M/ρR , so for a given gas λ ∝ 1/ρ)

25. Relaxation Time & Collision Frequency

Mean relaxation time (τ) :

Average time between successive collisions.

τ = λv_avg

τ = kT / (√2 π d² P)√(8RT / πM) ⇒ τ ∝ √TP

Using P = nRTV : τ ∝ √T · VnRT ⇒ τ ∝ V√T (fixed n)

Average collision frequency (f) :

Mean rate of collisions (no. of collisions per second).

f = 1τ ⇒ f ∝ √TV

26. Quick Formula Revision

Quantity	Formula
Ideal gas	PV = nRT ; P = ρRT/M
Speeds	√(2RT/M) < √(8RT/πM) < √(3RT/M)
Molar mass	M = 2 × vapour density
Specific heats	C_v = fR/2 ; C_p = C_v + R
γ	1 + 2/f
Internal energy	U = nfRT/2 = nC_vT
Mixture	f_eq = Σn_if_i / Σn_i
Mean free path	λ = kT/(√2 π d² P)
Relaxation time	τ = λ/v_avg ; f = 1/τ

Key Points :

Motion of the container (COM) does not change temperature or internal energy.
λ ∝ T/P ; at constant pressure λ increases with T.
τ ∝ V/√T and f ∝ √T/V for a fixed amount of gas.
R = N_Ak links per-mole and per-molecule formulas.

Kinetic Theory of Gases

1. Thermodynamics vs KTG

2. Ideal Gas Equation

3. Units

4. Macroscopic Quantities

Pressure :

Volume :

Density :

Amount of gas :

Temperature :

5. Standard Assumptions (Ideal Gas)

6. Maxwell Distribution Curve

7. Three Characteristic Speeds

8. Effect of Temperature

9. Vapour Density

Key Points :

10. Degrees of Freedom (f)

11. Monoatomic Gas (He, Ne, Ar …)

12. Diatomic Gas (H₂, N₂, O₂ …)

13. Linear Polyatomic Gas (CO₂)

14. Non-linear Polyatomic Gas (SO₂, NO₂, H₂O)

15. Summary of f (normal temperature)

Key Points :

16. f, Cv, Cp and γ

17. Table of Cv, Cp, γ

18. Law of Equipartition of Energy

19. Internal Energy of Ideal Gas (U)

20. Kinetic Energy Summary

Key Points :

21. Mixture of Gases

Total internal energy :

22. Hypothetical Speed Distribution (Solved)

(1) Total molecules = area under curve :

(2) Most probable speed :

(3) Average speed :

(4) r.m.s. speed :

23. Gas in a Moving Vessel (Solved)

24. Mean Free Path (λ)

25. Relaxation Time & Collision Frequency

Mean relaxation time (τ) :

Average collision frequency (f) :

26. Quick Formula Revision

Key Points :

16. f, C_v, C_p and γ

17. Table of C_v, C_p, γ