Kinetic Theory of Gases
Pressure of an ideal gas, mean kinetic energy, equipartition, mean free path.
Pressure of an ideal gas
P = (1/3)ρv² derivation, ideal gas equation.
The kinetic theory of gases explains macroscopic properties (P, V, T) in terms of microscopic motion of molecules.
Assumptions of an ideal gas (KTG):
- Molecules are point particles (negligible volume).
- No intermolecular forces except during collision.
- Collisions are perfectly elastic.
- Molecular motion is random and obeys Newton's laws.
Pressure of an ideal gas:
P = (1/3) ρ v²_rms = (1/3) (nm/V) v²_rms
where v²_rms is the mean square speed and ρ is gas density.
From this and PV = nRT (n = moles, R = gas constant):
v_rms = √(3RT / M) (M = molar mass in kg/mol)
Or equivalently using Boltzmann's constant k_B:
v_rms = √(3k_B T / m) (m = molecular mass)
Three speed averages (for Maxwell-Boltzmann distribution):
- Average speed: v_avg = √(8RT / πM) ≈ 0.921 v_rms.
- Most probable speed: v_mp = √(2RT / M) ≈ 0.816 v_rms.
- RMS speed: v_rms = √(3RT / M).
Order: v_mp < v_avg < v_rms (ratio √2 : √(8/π) : √3 ≈ 1 : 1.13 : 1.22).
Kinetic energy per molecule:
KE_avg = (3/2) k_B T
Notice: depends only on temperature, not on the type of gas. All gases at the same T have the same average translational KE per molecule.
Total internal energy of n moles of ideal gas:
U = (3/2) nRT (for monoatomic).
U = (5/2) nRT (for diatomic at room temperature).
U = (6/2) nRT = 3nRT (for polyatomic).
Equipartition of energy theorem: in thermal equilibrium, each degree of freedom contributes (1/2) k_B T to the average energy per molecule.
| Gas | Degrees of freedom (room T) | C_v |
|---|---|---|
| Monoatomic (He, Ne, Ar) | 3 (translation) | (3/2) R |
| Diatomic (N₂, O₂, H₂) | 5 (3 trans + 2 rot) | (5/2) R |
| Polyatomic non-linear (H₂O, NH₃) | 6 (3 trans + 3 rot) | 3R |
| Polyatomic linear (CO₂) | 5 (3 trans + 2 rot) | (5/2) R |
At very high T, vibrational modes (2 per mode) also activate, adding to C_v.
Mayer's relation: C_p − C_v = R (for ideal gas).
Heat capacity ratio: γ = C_p / C_v.
- Monoatomic: γ = 5/3 ≈ 1.67.
- Diatomic: γ = 7/5 = 1.4.
- Polyatomic: γ ≈ 4/3.
Mean free path (average distance between collisions):
λ = k_B T / (√2 π d² P) = 1 / (√2 n π d²)
where d is the molecular diameter and n is number density. For air at room T and 1 atm: λ ≈ 70 nm.
Number of collisions per second per molecule: Z = √2 n π d² v_avg ≈ v_avg / λ.
Worked example. Find v_rms of nitrogen (M = 28 g/mol) at 300 K.
v_rms = √(3 × 8.314 × 300 / 0.028) = √(267,238) ≈ 517 m/s.
(For comparison, sound speed in air at 300 K is ~343 m/s — same order of magnitude, as expected since sound is essentially molecular collisions propagating.)
Kinetic energy and temperature
KE_avg = (3/2)kT, equipartition.
The kinetic theory of gases explains macroscopic properties (P, V, T) in terms of microscopic motion of molecules.
Assumptions of an ideal gas (KTG):
- Molecules are point particles (negligible volume).
- No intermolecular forces except during collision.
- Collisions are perfectly elastic.
- Molecular motion is random and obeys Newton's laws.
Pressure of an ideal gas:
P = (1/3) ρ v²_rms = (1/3) (nm/V) v²_rms
where v²_rms is the mean square speed and ρ is gas density.
From this and PV = nRT (n = moles, R = gas constant):
v_rms = √(3RT / M) (M = molar mass in kg/mol)
Or equivalently using Boltzmann's constant k_B:
v_rms = √(3k_B T / m) (m = molecular mass)
Three speed averages (for Maxwell-Boltzmann distribution):
- Average speed: v_avg = √(8RT / πM) ≈ 0.921 v_rms.
- Most probable speed: v_mp = √(2RT / M) ≈ 0.816 v_rms.
- RMS speed: v_rms = √(3RT / M).
Order: v_mp < v_avg < v_rms (ratio √2 : √(8/π) : √3 ≈ 1 : 1.13 : 1.22).
Kinetic energy per molecule:
KE_avg = (3/2) k_B T
Notice: depends only on temperature, not on the type of gas. All gases at the same T have the same average translational KE per molecule.
Total internal energy of n moles of ideal gas:
U = (3/2) nRT (for monoatomic).
U = (5/2) nRT (for diatomic at room temperature).
U = (6/2) nRT = 3nRT (for polyatomic).
Equipartition of energy theorem: in thermal equilibrium, each degree of freedom contributes (1/2) k_B T to the average energy per molecule.
| Gas | Degrees of freedom (room T) | C_v |
|---|---|---|
| Monoatomic (He, Ne, Ar) | 3 (translation) | (3/2) R |
| Diatomic (N₂, O₂, H₂) | 5 (3 trans + 2 rot) | (5/2) R |
| Polyatomic non-linear (H₂O, NH₃) | 6 (3 trans + 3 rot) | 3R |
| Polyatomic linear (CO₂) | 5 (3 trans + 2 rot) | (5/2) R |
At very high T, vibrational modes (2 per mode) also activate, adding to C_v.
Mayer's relation: C_p − C_v = R (for ideal gas).
Heat capacity ratio: γ = C_p / C_v.
- Monoatomic: γ = 5/3 ≈ 1.67.
- Diatomic: γ = 7/5 = 1.4.
- Polyatomic: γ ≈ 4/3.
Mean free path (average distance between collisions):
λ = k_B T / (√2 π d² P) = 1 / (√2 n π d²)
where d is the molecular diameter and n is number density. For air at room T and 1 atm: λ ≈ 70 nm.
Number of collisions per second per molecule: Z = √2 n π d² v_avg ≈ v_avg / λ.
Worked example. Find v_rms of nitrogen (M = 28 g/mol) at 300 K.
v_rms = √(3 × 8.314 × 300 / 0.028) = √(267,238) ≈ 517 m/s.
(For comparison, sound speed in air at 300 K is ~343 m/s — same order of magnitude, as expected since sound is essentially molecular collisions propagating.)
Mean free path and degrees of freedom
λ = 1/(√2 nπd²), monoatomic vs diatomic vs polyatomic.
The kinetic theory of gases explains macroscopic properties (P, V, T) in terms of microscopic motion of molecules.
Assumptions of an ideal gas (KTG):
- Molecules are point particles (negligible volume).
- No intermolecular forces except during collision.
- Collisions are perfectly elastic.
- Molecular motion is random and obeys Newton's laws.
Pressure of an ideal gas:
P = (1/3) ρ v²_rms = (1/3) (nm/V) v²_rms
where v²_rms is the mean square speed and ρ is gas density.
From this and PV = nRT (n = moles, R = gas constant):
v_rms = √(3RT / M) (M = molar mass in kg/mol)
Or equivalently using Boltzmann's constant k_B:
v_rms = √(3k_B T / m) (m = molecular mass)
Three speed averages (for Maxwell-Boltzmann distribution):
- Average speed: v_avg = √(8RT / πM) ≈ 0.921 v_rms.
- Most probable speed: v_mp = √(2RT / M) ≈ 0.816 v_rms.
- RMS speed: v_rms = √(3RT / M).
Order: v_mp < v_avg < v_rms (ratio √2 : √(8/π) : √3 ≈ 1 : 1.13 : 1.22).
Kinetic energy per molecule:
KE_avg = (3/2) k_B T
Notice: depends only on temperature, not on the type of gas. All gases at the same T have the same average translational KE per molecule.
Total internal energy of n moles of ideal gas:
U = (3/2) nRT (for monoatomic).
U = (5/2) nRT (for diatomic at room temperature).
U = (6/2) nRT = 3nRT (for polyatomic).
Equipartition of energy theorem: in thermal equilibrium, each degree of freedom contributes (1/2) k_B T to the average energy per molecule.
| Gas | Degrees of freedom (room T) | C_v |
|---|---|---|
| Monoatomic (He, Ne, Ar) | 3 (translation) | (3/2) R |
| Diatomic (N₂, O₂, H₂) | 5 (3 trans + 2 rot) | (5/2) R |
| Polyatomic non-linear (H₂O, NH₃) | 6 (3 trans + 3 rot) | 3R |
| Polyatomic linear (CO₂) | 5 (3 trans + 2 rot) | (5/2) R |
At very high T, vibrational modes (2 per mode) also activate, adding to C_v.
Mayer's relation: C_p − C_v = R (for ideal gas).
Heat capacity ratio: γ = C_p / C_v.
- Monoatomic: γ = 5/3 ≈ 1.67.
- Diatomic: γ = 7/5 = 1.4.
- Polyatomic: γ ≈ 4/3.
Mean free path (average distance between collisions):
λ = k_B T / (√2 π d² P) = 1 / (√2 n π d²)
where d is the molecular diameter and n is number density. For air at room T and 1 atm: λ ≈ 70 nm.
Number of collisions per second per molecule: Z = √2 n π d² v_avg ≈ v_avg / λ.
Worked example. Find v_rms of nitrogen (M = 28 g/mol) at 300 K.
v_rms = √(3 × 8.314 × 300 / 0.028) = √(267,238) ≈ 517 m/s.
(For comparison, sound speed in air at 300 K is ~343 m/s — same order of magnitude, as expected since sound is essentially molecular collisions propagating.)