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Must-know facts
20 facts
- 1Atom ≈ 1 Å; spacing ≈ 2 Å in solids and liquids, tens of Å in gases; mean free path thousands of Å.
- 2Avogadro: equal volumes at the same T and P hold equal numbers of molecules; 22.4 L at STP holds 6.02 × 10²³.
- 3PV = μRT = k_BNT, R = N_Ak_B = 8.314 J mol⁻¹ K⁻¹, k_B = 1.38 × 10⁻²³ J K⁻¹.
- 4Real gases approach ideal behaviour at low pressure and high temperature.
- 5Dalton: total pressure is the sum of partial pressures.
- 6Water molecule: mass 3 × 10⁻²⁶ kg, radius ≈ 2 Å; spacing in vapour ≈ 40 Å.
- 7Each wall hit hands over momentum 2mv_x; P = ⅓nm⟨v²⟩.
- 8PV = ⅔E; average KE per molecule = (3/2)k_BT, independent of the gas.
- 9Internal 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
Saying heavier molecules have more kinetic energy at the same temperature.
Putting the molar mass in grams into v_rms = √(3RT/M₀).
Using °C in PV = μRT or in v_rms.
Giving a vibrational mode ½k_BT.
Counting three rotations for a diatomic molecule.
Using C_p for a gas heated in a sealed rigid cylinder.
Thinking the proportions of a mixture change each gas's rms speed.
Taking ⟨v²⟩ = ⟨v⟩².
Imagining gas molecules are thousands of diameters apart.
Forgetting the ½ in the number of molecules that hit a wall.
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.