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Theme B — The Particulate Nature of Matter

B.3 Gas Laws — IB Physics HL Revision Notes & Topical Questions

5 min read 10 parts Lesson 24 of 53 1 quiz

📚 IB Physics HL Hub / Topical Revision
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Curriculum Equation Key قوانين الرقم الهيدروجيني والاتزان الأيوني - pH & Acid-Base Equilibrium FormulasFast Exam Solution
pH = -log[H⁺] | pOH = -log[OH⁻] | pH + pOH = 14 | [H⁺][OH⁻] = 10⁻¹⁴
Word Equation & Conditions: الرقم الهيدروجيني = -لوغاريتم تركيز أيون الهيدروجين | pH + pOH = 14
Location & Specifications: الاتزان الأيوني، كيمياء 3 ثانوي، التحصيلي، و AP Chemistry.
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Lesson summary

B.3 Gas Laws — IB PHYSICS HL notes: Apply pressure, the mole, the empirical gas laws, the ideal gas equation and kinetic theory to ideal gases.Pressure = force per unit area, P = F/A, in pascals (Pa).Fluid pressure acts at 90° to a surface.Force over a large area gives small pres…

  • Force over a large area gives small pressure
  • Molar mass mr
  • For an ideal gas PV ∝ T, so PV/T
  • The constant in PV/T is nR or NkB
  • No intermolecular forces except during collisions
  • The number of molecules is very large

Part 1 of 8Pressure

Pressure
Pressure
Pressure force per unit area, P = F/A, in pascals (Pa).
Fluid pressure acts at 90° to a surface.
Force over a large area gives small pressure
  • Force over a large area gives small pressure over a small area gives large pressure.
    • (drawing pin, high heels)
Gas pressure comes from molecules bouncing off the container walls.
Use the cross-sectional area on which the force acts.

Part 2 of 8Amount of substance

The mole
The mole the amount of substance containing as many elementary entities as there are atoms in 12 g of carbon-12.
  • NA = 6.02 × 1023 mol-1 (Avogadro constant).
n
N / NA — the number of moles from the number of particles.
Molar mass mr
Molar mass mr m / n in g mol-1.
  • One mole of helium (relative atomic mass 4) has mass 4 g; water (2 × 1 + 16) has 18 g.
Multiplying moles by molar mass gives grams, not kilograms.

Part 3 of 8Empirical gas laws

  • For an ideal gas PV ∝ T, so PV/T constant — for a fixed mass of gas.
LawConstantRelation
BoyleTemperatureP ∝ 1/V, P1V1 = P2V2
CharlesPressureV ∝ T, V1/T1 = V2/T2
Gay-Lussac (pressure law)VolumeP ∝ T, P1/T1 = P2/T2
Empirical gas laws
T must be in kelvin in every gas law.
Graphs: P against 1/V is a straight line through the origin, V against T (K) and P against T (K) are straight lines through the origin.
P–V diagram paths.
  • Horizontal lineconstant pressure.
  • Vertical lineconstant volume.
  • Curve along an isothermconstant temperature.
  • Curve between isothermsP, V and T all change.

Part 4 of 8Ideal gas equation

Ideal gas equation
PV
PV nRT and PV = NkBT.
  • R = 8.31 J K-1 mol-1; kB = 1.38 × 10-23 J K-1.
kB
R / NA — R is macroscopic and kB describes microscopic particles.
Ideal gas
Ideal gas a gas that obeys the ideal gas equation at all pressures, volumes and temperatures.
The constant in PV/T is nR or NkB
  • The constant in PV/T is nR or NkB not to be mixed up.
The values of R, NA and kB are given in the data booklet.

Part 5 of 8Kinetic theory assumptions

1
Many identical molecules move randomly at high speed.
2
Molecule volume is negligible compared with the container (point particles).
3
Molecules obey Newton's laws and collide elastically with each other and the walls.
4No intermolecular forces except during collisions
  • No intermolecular forces except during collisions and collision time is negligible.
5
External forces such as gravity are ignored.
6The number of molecules is very large
  • The number of molecules is very large so average behaviour is used.
7
Each collision with a wall exerts a force; the average of many gives uniform pressure.
8
Real gases behave ideally at low pressure, low density and temperatures well above boiling point.
  • At high pressure molecules are close and volume and attraction matter; at low temperature gas condenses.
  • Air at room temperature and pressure behaves very nearly ideally.

Part 6 of 8Kinetic theory of gases equation

1
One molecule rebounding from a wall changes momentum by Δp = −2mv.
2
The force on the wall is 2mv/Δt with Δt = 2l/v, giving mv2/l.
3
Pressure from one molecule on a wall of area l2 is mv2/l3; N molecules give Nmv2/l3.
4
Molecules move equally in three dimensions — v2 = 3vx2, so P = Nmv2/3V.
5
With density ρ = Nm/V this becomes P = ⅓ρ⟨v2⟩.
1
⟨v2⟩ is the mean square speed.
2The derivation is not examined
  • The derivation is not examined but the result is used.

Part 7 of 8Average kinetic energy and internal energy

1No potential energy in an ideal gas
  • No potential energy in an ideal gas internal energy equals total kinetic energy.
2Average kinetic energy of a molecule Ek
½m⟨v2⟩ = (3/2)kBT.
  • Higher temperature means higher average kinetic energy.
3Internal energy of a monatomic ideal gas U
(3/2)NkBT = (3/2)nRT.
4U is proportional to T
  • U is proportional to T heating a fixed volume of gas raises U and T.
5Applies to monatomic gases such as helium, neon and argon
  • Applies to monatomic gases such as helium, neon and argon diatomic molecules also have rotational energy.
6Ek
(3/2)kBT is for one molecule — multiply by N for the whole gas.

Part 8 of 8More from the specification

More from the specification
Ideal Gas Equation of State
  • An ideal gas is described by pV = nRT = N * kB * T,
  • where kB is the Boltzmann constant relating thermodynamic temperature to individual molecular kinetic energy.
    • (1.38 x 10-23 J K-1
Combined Gas Law Calculations
  • For a fixed quantity of gas changing state, you substitute macroscopic quantities into (p1 * V1) / T1 = (p2 * V2) / T2 and rearrange to solve for unknown final pressure, final volume,
  • or final temperature given initial conditions.
Absolute Temperature and Kelvin Scale
  • Thermodynamic temperature is defined on the absolute Kelvin scale where 0 K represents absolute zero;
    • calculations must convert initial temperature and final temperature from degrees Celsius using T(K) = theta(°C) + 273.15.
Microscopic Kinetic Energy and Temperature
  • The average translational kinetic energy of a gas molecule is defined directly by Ek = 3/2 * kB * T, demonstrating that temperature is a direct measure of random thermal motion.
Boltzmann constant
  • K = R/NA, the gas constant per molecule
Thermodynamic temperature
  • The absolute temperature in kelvin.
N = nNA
What the symbols mean: N = number of particles · n = amount (mol) · NA = Avogadro constant (6.02 × 1023 mol−1)

Key terms

TermMeaning
MoleAmount of substance containing 6.02 × 1023 elementary entities.
Avogadro constantNumber of particles per mole, 6.02 × 1023 mol-1.
Ideal gasGas obeying PV = nRT at all pressures, volumes and temperatures.
Boltzmann constantkB = R/NA = 1.38 × 10-23 J K-1.
Mean square speedAverage of the squares of molecular speeds.
Boyle's lawP ∝ 1/V at constant temperature.
PressureForce per unit area.

Avoid these mistakes

1
Using temperature in Celsius in gas law equations.
2
Mixing the constant nR with Nk_B.
3
Treating a real gas as ideal at high pressure or low temperature.
4
Giving molar mass in kilograms when using g mol-1.
5
Using Ek = (3/2)kBT for the whole gas rather than one molecule.

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