Lecture covers Kinetic Theory of Gases: assumptions, degrees of freedom, pressure derivation, energy partition, kinetic interpretation, and key formulas.
Emphasis on applications to thermodynamics, specific heats, and exam-relevant numericals.
Teacher derives core relations connecting microscopic motion to macroscopic quantities.
Key Assumptions Of Kinetic Theory
Gas consists of extremely small particles (molecules/atoms), identical for a given gas.
Particles treated as rigid, perfectly elastic spheres.
Collisions between particles and with container walls are elastic.
Molecular size is negligible compared to interparticle distance.
No appreciable intermolecular attraction or repulsion (can be ignored).
Gravitational forces between molecules are negligible.
Particle speeds are high and vary between 0 and infinity (practically nonzero).
Degrees Of Freedom (f)
Degree of freedom: number of independent coordinates needed to specify a particle's position/configuration.
General formula for a molecule: f = 3n − r
n = number of atoms in the molecule
r = number of independent internal constraints (relations)
Typical cases:
Monatomic (n = 1): f = 3
Diatomic (n = 2): f = 5 (3n − r → 6 − 1)
Triatomic linear (e.g., CO2): f = 7? (3n − r → 9 − 2 = 7)
Triatomic non-linear (e.g., H2O): f = 6 (3n − r → 9 − 3 = 6)
Degree of freedom affects energy distribution and heat capacities.
Law Of Equipartition (Partition) Of Energy
For a system in thermal equilibrium, energy is equally partitioned among all quadratic degrees of freedom.
Each degree of freedom contributes (1/2)kB T per particle to average energy.
For f degrees of freedom, average energy per particle = (f/2) kB T.
For monatomic gas (f = 3): average kinetic energy per particle = (3/2) kB T.
Pressure Of An Ideal Gas — Kinetic Derivation
Consider N particles in a cubical/rectangular closed container with cross-sectional area A and length x (volume V = A x).
Change in momentum for one particle colliding elastically with a wall (x-direction): Δp = 2 m vx.
Number of particles in the volume: N_total = n V (where small n denotes number density, particles per unit volume).
Using collision frequency and momentum change, force on wall from all particles leads to:
Pressure p = (1/3) n m ⟨v^2⟩
Here n = particles per unit volume, m = mass of one particle, ⟨v^2⟩ = mean square speed.
Using equipartition and ideal gas law, relations follow (see next section).
Relation Between Kinetic Energy And Temperature
Combine kinetic pressure expression with ideal gas law PV = NkB T (or PV = n_moles R T).
Average translational kinetic energy per particle: (1/2) m ⟨v^2⟩ = (3/2) kB T for f = 3.
General form for f degrees of freedom: average energy per particle = (f/2) kB T.
Boltzmann constant kB = R / NA, where R = 8.314 J mol−1 K−1 and NA = Avogadro's number.
Summary Table: Degrees Of Freedom, Energy, Specific Heat Ratios
Molecule Type
Degrees Of Freedom f
Energy per Particle
Monatomic (e.g., He)
3
(3/2) kB T
Diatomic (approx.)
5
(5/2) kB T
Triatomic Linear (e.g., CO2)
7
(7/2) kB T
Triatomic Non-linear (e.g., H2O)
6
(6/2) kB T
For molar quantities, internal energy per mole = (f/2) R T.
Specific heats: Cv = (f/2) R, Cp = Cv + R = ((f+2)/2) R.
From (1/2) m ⟨v^2⟩ = (f/2) kB T /? (for translational part use f = 3):
For translational motion (monatomic ideal gas), rms speed:
vrms = sqrt(3 R T / M)
M = molar mass (kg mol−1), R = 8.314 J mol−1 K−1.
More generally, mean-square speed scale uses factor f as appropriate when deriving from energy per particle.
Mean Free Path (λ, average free path)
Mean free path x̄ (λ) — average distance a molecule travels between successive collisions.
Formula: λ = 1 / (√2 n π d^2) (notation adapted: often written λ = 1 / (√2 n π d^2))
n = number density (particles per unit volume)
d = molecular diameter
This relation arises from collision geometry and can be obtained by dimensional reasoning.
Important Formulas (Selected)
Ideal gas law (molecular form): pV = NkB T (N = number of molecules)
Pressure from microscopic motion: p = (1/3) n m ⟨v^2⟩
Average translational kinetic energy (monatomic): (1/2) m ⟨v^2⟩ = (3/2) kB T
General energy per particle: Eavg = (f/2) kB T
Molar internal energy: U = (f/2) R T (per mole)
Heat capacities: Cv = (f/2) R, Cp = Cv + R
Heat-capacity ratio: γ = Cp / Cv = (f+2) / f
RMS speed: vrms = sqrt(3 R T / M) for monatomic gases
Mean free path: λ = 1 / (√2 n π d^2)
Key Definitions
Number density (n): number of molecules per unit volume.
Mean square speed ⟨v^2⟩: average of the square of molecular speeds.
vrms: root-mean-square speed = sqrt(⟨v^2⟩).
Boltzmann constant kB = R / NA.
Avogadro's number NA ≈ 6.022 × 10^23 mol−1.
Molar mass M: mass per mole of gas (kg mol−1).
Action Items / Exam Tips
Memorize core formulas: p = (1/3) n m ⟨v^2⟩, Eavg = (f/2) kB T, vrms formula, λ expression.
Practice degree-of-freedom calculations using f = 3n − r; determine r by counting independent internal constraints.
For numericals, carefully distinguish between small n (number density) and N or n_moles (mole count).
Convert between molecular and molar forms using kB = R/NA and N = n_moles NA.
Use correct f for computing Cv, Cp, and γ; check whether molecule is linear or non-linear for triatomic cases.
Remember assumptions: elastic collisions and negligible intermolecular potentials — these justify using kinetic-only energy (no potential energy term).