Lesson 13 of 13 · 18 min
Chapter review
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Must-know facts
20 facts
- 01Atom ≈ 1 Å; spacing ≈ 2 Å in solids and liquids, tens of Å in gases; mean free path thousands of Å.
- 02Avogadro: equal volumes at the same T and P hold equal numbers of molecules; 22.4 L at STP holds 6.02 × 10²³.
- 03PV = μRT = k_BNT, R = N_Ak_B = 8.314 J mol⁻¹ K⁻¹, k_B = 1.38 × 10⁻²³ J K⁻¹.
- 04Real gases approach ideal behaviour at low pressure and high temperature.
- 05Dalton: total pressure is the sum of partial pressures.
- 06Water molecule: mass 3 × 10⁻²⁶ kg, radius ≈ 2 Å; spacing in vapour ≈ 40 Å.
- 07Each wall hit hands over momentum 2mv_x; P = ⅓nm⟨v²⟩.
- 08PV = ⅔E; average KE per molecule = (3/2)k_BT, independent of the gas.
- 09Internal energy of an ideal gas depends only on T.
- 10v_rms = √(3k_BT/m); nitrogen at 300 K: 516 m s⁻¹.
- 11Speed ratio at the same T: v₁/v₂ = √(M₂/M₁); Ar : Cl₂ = 1.33, UF₆ isotopes differ by 0.44%.
- 12Compressing a gas with a piston speeds up the molecules (2V + u), so it heats.
- 13Equipartition: ½k_BT per translational or rotational degree of freedom, k_BT per vibrational mode.
- 14Monatomic γ = 5/3, rigid diatomic 7/5, vibrating diatomic 9/7, polyatomic (4 + f)/(3 + f).
- 1544.8 L of helium at STP, heated by 15 °C at fixed volume: 374 J.
- 16Solids: C = 3R ≈ 24.9 J mol⁻¹ K⁻¹; carbon is the exception.
- 17Mean free path l = 1/(√2nπd²); air at STP: τ = 6.1 × 10⁻¹⁰ s, l = 2.9 × 10⁻⁷ m.
- 18Water vapour at 373 K: l ≈ 4 × 10⁻⁷ m, about 100 times the molecular spacing.
- 19⟨v²⟩ is not in general equal to ⟨v⟩².
- 20Air does not settle to the floor because mgh for ordinary heights is far below the molecules' kinetic energy.
Common traps
Where marks are lost
Formulas
14 to know
Ideal-gas equation
PV = μRT = k_B N T; P = n k_B T; P = ρRT/M₀
R = N_A k_B = 8.314 J mol⁻¹ K⁻¹; k_B = 1.38 × 10⁻²³ J K⁻¹.
Number of moles
μ = M/M₀ = N/N_A
N_A = 6.02 × 10²³; 1 mol fills 22.4 L at STP.
Dalton's law
P = P₁ + P₂ + …; P_i = μ_i RT/V
Non-reacting ideal gases.
Kinetic pressure
P = ⅓ n m ⟨v²⟩
n is number density, m the molecular mass.
Pressure and energy
PV = ⅔E; E = (3/2) N k_B T
E is the translational kinetic energy.
Mean kinetic energy
½ m⟨v²⟩ = (3/2) k_B T
Same for every ideal gas at a given T.
rms speed
v_rms = √(3k_BT/m) = √(3RT/M₀)
M₀ in kg mol⁻¹; N₂ at 300 K gives 516 m s⁻¹.
Equipartition
½ k_B T per squared term; k_B T per vibrational mode
Translation, rotation: ½k_BT each.
Monatomic gas
C_v = (3/2)R; C_p = (5/2)R; γ = 5/3
3 degrees of freedom.
Rigid diatomic gas
C_v = (5/2)R; C_p = (7/2)R; γ = 7/5
With a vibration: (7/2)R, (9/2)R, 9/7.
Polyatomic gas
C_v = (3 + f)R; C_p = (4 + f)R; γ = (4 + f)/(3 + f)
f vibrational modes.
Difference of molar specific heats
C_p − C_v = R
Any ideal gas.
Solids
U = 3RT; C = 3R
Carbon is an exception.
Mean free path
τ = 1/(nπ⟨v⟩d²); l = 1/(√2 nπd²)
d is the molecular diameter.
Key terms
15 terms
- Atomic hypothesis
- All matter is made of atoms in constant motion that attract when slightly apart and repel when pressed together.
- Avogadro's hypothesis
- Equal volumes of all gases at the same temperature and pressure contain the same number of molecules.
- Avogadro number
- 6.02 × 10²³, the number of molecules in one mole; one mole of gas fills 22.4 L at STP.
- Boltzmann constant
- k_B = 1.38 × 10⁻²³ J K⁻¹, the gas constant per molecule, R/N_A.
- Ideal gas
- A model gas that obeys PV = μRT exactly at all pressures and temperatures.
- Partial pressure
- The pressure one gas of a mixture would exert if it alone filled the vessel at the same temperature.
- Number density
- Number of molecules per unit volume, n = N/V.
- Elastic collision
- A collision in which total kinetic energy is conserved as well as momentum.
- rms speed
- The square root of the mean of the squared molecular speeds, √(3k_BT/m).
- Degree of freedom
- An independent way a molecule can move or store energy; each squared energy term counts once.
- Law of equipartition of energy
- In thermal equilibrium each squared energy term averages ½k_BT.
- Rigid rotator
- A molecule that rotates but does not vibrate, like a dumbbell.
- Ratio of specific heats (γ)
- C_p/C_v, 5/3 for a monatomic gas and 7/5 for a rigid diatomic gas.
- Mean free path
- Average distance a molecule travels between successive collisions.
- Dynamic equilibrium
- A steady state in which molecules keep moving and colliding while the averages stay fixed.