← All printables Chapter 03 · Thermodynamics · 2 sheets · read the chapter →
Vedatom Physics
FORMULA CARD · PHYSICS.VEDATOM.COM
Chapter 03 · Part III · Thermodynamics
Revised on ____________________
Chapter 03 · Kinetic Theory of Gases

Temperature is the kinetic energy of a molecule

2 sheets
2 diagrams
24 results

Formulas and conditions only — no derivations, no solved numbers. The diagrams are the reader's own.

I

Pressure and temperature

FROM COLLISIONS ALONE
I.Pressure  P = 13 ρ vrms² = ⅓ (N/V) m vrms²
II.As energy density  P V = 23 N ⟨KE⟩
III.Mean kinetic energy  ⟨KE⟩ = 32 kB T per molecule, translational only
IV.Per mole  32 R T independent of the gas
V.Ideal gas law  P V = n R T = N kB T
VI.Dalton  P = Σ Pi each gas fills the volume alone
Temperature measures the translational kinetic energy only. Rotation and vibration store energy too — which is why CV differs between gases at the same temperature.
II

The three speeds

vp < v < vrms, ALWAYS
I.Root mean square  vrms = √( 3 R TM ) pressure and energy
II.Mean  v = √( 8 R Tπ M ) effusion, collisions, mean free path
III.Most probable  vp = √( 2 R TM ) the peak of the distribution
IV.Ratio  vp : v : vrms = 1 : 1.128 : 1.225
V.Temperature  v ∝ √T four times the T doubles the speed
VI.Graham's law  rate ∝ 1√M effusion, so v governs it
f(v)speed vT1T2 > T1v_pv̄v_rms
III

Degrees of freedom

COUNT THEM, THEN EVERYTHING FOLLOWS
I.Equipartition  12 kB T per quadratic term m g h is not quadratic — it does not count
II.Internal energy  U = f2 n R T
III.Molar heat capacity  CV = f2 R , CP = ( f2 + 1 ) R
IV.Ratio  γ = 1 + 2f
V.Monatomic  f = 3, γ = 5/3 diatomic at room T: f = 5, γ = 7/5
VI.Mixture  U = Σ fi2 ni R T never an averaged f
monatomicf = 3 (trans)rotdiatomicf = 5 (+2 rot)
IV

Mean free path and transport

WHERE THE √2 LIVES
I.Mean free path  λ = 1√2 n π d² n = molecules per unit volume
II.In terms of P and T  λ = kB T√2 π d² P
III.Collision frequency  z = √2 n π d² v the √2 is the relative-speed factor
IV.At constant T  η and D are independent of pressure
V.Viscosity  η ∝ √T
VI.Diffusion  D ∝ T3/2P
Boltzmann
kB = 1.38×10−23 J K−1
Gas constant
R = 8.314 J mol−1 K−1
Avogadro
6.022×1023
Molar volume, STP
22.4 L
N2 at 300 K
vrms ≈ 517 m s−1
Air, λ
≈ 68 nm
V

Where the marks go

FOUR ERRORS, EVERY YEAR
Where marks are lost in this chapter
1Using v in the pressure formula or vrms in Graham's law. Pressure and energy take vrms; effusion takes v.
2Dropping the √2 from the collision frequency. It comes from the mean relative speed and is worth 41 %.
3Reading γ = 7/5 as f = 7. A diatomic gas has f = 5; the 7 is f + 2. Use γ = 1 + 2/f.
4Believing viscosity rises with pressure. At fixed T, η is pressure-independent because n ∝ p and λ ∝ 1/p cancel — λ itself is not: λ = kT/(√2 σp).