Kinetic Theory of Gases and Radiation
This domain connects the microscopic motion of molecules with measurable gas behaviour and with thermal radiation. Maharashtra HSC questions typically ask for the derivation of P = (1/3)ρv(rms)², the three molecular speeds, degrees of freedom and specific heats, plus Stefan-Boltzmann and Wien law numericals.
Core rules
- PV = nRT and PV = (1/3)M v(rms)².
- v(rms) = √(3RT/M), v(avg) = √(8RT/πM), v(most probable) = √(2RT/M); their ratio is √3 : √(8/π) : √2.
- Law of equipartition: each quadratic degree of freedom carries ½k(B)T of energy. Monatomic f = 3, diatomic f = 5 at ordinary temperatures, polyatomic f = 6.
- C(V) = (f/2)R, C(P) = C(V) + R, γ = 1 + 2/f. Monatomic γ = 5/3, diatomic γ = 7/5.
- Mean free path λ = k(B)T/(√2 πd²P).
- Stefan-Boltzmann: P = σAeT⁴, with e = 1 for a perfect blackbody.
- Wien: λ(max)T = 2.898×10⁻³ m K. Kirchhoff: a good absorber at a wavelength is a good emitter there. Newton's law of cooling follows from Stefan's law for small temperature differences.
Worked example
For oxygen (M = 0.032 kg/mol) at 300 K: v(rms) = √(3 × 8.314 × 300/0.032) = √(2.34×10⁵) ≈ 484 m/s. For a blackbody at 6000 K, λ(max) = 2.898×10⁻³/6000 = 483 nm, in the visible green - consistent with the Sun's colour.
Traps
- Entering molar mass in grams per mole; SI requires kg per mole.
- Omitting the emissivity e, or using T in degrees Celsius in Stefan's and Wien's laws.
- Quoting γ = 5/3 for a diatomic gas; it is 7/5 unless vibrational modes are active.