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Theme B — The Particulate Nature of Matter

B.3.4 Ideal Gas Equation — IB Physics HL Revision Notes & Topical Questions

5 min read 10 parts Lesson 28 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.3.4 Ideal Gas Equation — IB PHYSICS SL notes: Use pV = nRT and pV = NkBT to solve gas problems, and explain when a real gas behaves ideally.Combining the laws — Boyle's, Charles's and the pressure law give one constant for a fixed amount of gas, which is proportional to the amo…

  • Equation of state
  • What the symbols mean
  • Definition
  • Gas constant R
  • Boltzmann constant kB
  • Constants in the data booklet

Section 1 of 10The ideal gas equation

Combining the laws
  • Boyle's, Charles's and the pressure law give one constant for a fixed amount of gas, which is proportional to the amount.
Equation of state
  • Equation of state pV = nRT.
    What the symbols mean: p = pressure (Pa) · V = volume (m3) · n = amount of gas (mol) · R = 8.31 J mol−1 K−1 · T = temperature (K = °C + 273)
  • p is the absolute pressure in Pa.
  • V is the volume in m3 (1 dm3 = 10−3 m3).
  • n is the amount of gas in mol.
  • R = 8.31 J K−1 mol−1 is the ideal gas constant.
  • T is the temperature in K.
Units
  • Units pV has the dimensions of energy, so the units of R follow, and all quantities must be in SI.
Definition
  • Definition an ideal gas obeys the ideal gas equation at all pressures, volumes and temperatures.

Section 2 of 10Constants

1Gas constant R
  • Gas constant R the macroscopic constant, linked to volume and temperature.
2Boltzmann constant kB
  • Boltzmann constant kB the microscopic constant, linked to the thermal energy of particles.
3Relation
  • Relation kB = R/NA = 1.38 × 10−23 J K−1.
4Constants in the data booklet
  • R, NA and kB are all given.

Section 3 of 10Molecular form

Equation
  • Equation pV = NkBT, using N = nNA.
When to use
  • When to use when the number of molecules, a number density N/V or a molecular mass is given.
Rearranged
  • Rearranged p = (N/V)kBT, so at constant temperature the pressure is proportional to the number of molecules per unit volume.
Link
  • Link the two forms are connected by nR = NkB.

Section 4 of 10Assumptions of the ideal gas

Small molecules
  • Small molecules their volume is negligible compared with their separation.
No intermolecular forces
  • No intermolecular forces except in collisions, so there is no potential energy.
Elastic collisions
  • Elastic collisions with each other and with the walls.
Random motion
  • Random motion the time of a collision is negligible compared with the time between collisions.
Internal energy
  • Internal energy all of it is random kinetic energy of the molecules.

Section 5 of 10Real gases

Close to ideal
  • Close to ideal at low pressure, low density and a temperature well above the boiling point.
  • The molecules are far apart, so their volume is negligible and the forces are small.
High pressure
  • High pressure the molecules are close and occupy a significant fraction of the volume, and attractive forces appear.
Low temperature
  • Low temperature the attractive forces become important compared with kinetic energy and the gas condenses.
Most ideal
  • Most ideal gases of weakly interacting molecules, such as helium.
Air
  • Air at normal room temperature and pressure it behaves very nearly like an ideal gas.
pV/nRT graph
  • pV/nRT graph it is a horizontal line at 1 for an ideal gas, dips below 1 at moderate pressures for real gases and rises above 1 at very high pressures.

Section 6 of 10Strategy for problems

1
List n or N, p, V and T, and convert to SI units.
2
If the same gas changes state, use pV/T = constant, which avoids R and n.
3
If the amount changes, as when gas leaks or is pumped in, use n = pV/RT at each stage.
4
For a gas under a piston or at depth, find the absolute pressure first.
5
To find the number of particles, use N = nNA.
  • Check
    • at 273 K and 1.01 × 105 Pa one mole occupies about 22.4 dm3.

Section 7 of 10Density form

Density
  • ρ = m/V = pM/RT, where M is the molar mass in kg mol−1.
Fixed pressure
  • Fixed pressure density is inversely proportional to absolute temperature, which is why hot air rises.
Fixed temperature
  • Fixed temperature density is proportional to molar mass, which is why a helium balloon floats.
Air
  • Air with M = 0.029 kg mol−1 at 300 K and 1.01 × 105 Pa, ρ ≈ 1.2 kg m−3.

Section 8 of 10More from the specification

1Kinetic collisions of ideal gas atoms
  • Kinetic collisions of ideal gas atoms ideal gas laws assume negligible intermolecular attractive forces between rapidly colliding gaseous molecules or atoms.
2Ideal Gas Equation of State
  • The equation of state for an ideal gas is p·V = n·R·T = N·kB·T, relating pressure p, volume V, temperature T in kelvin,
  • and quantity of substance in moles n confined within a rigid container.
    • (or number of molecules N)
3Kinetic Theory of Gases and Pressure
  • Kinetic theory models gas molecules in continuous random motion;
    • pressure results from elastic collisions with container walls where molecules rebound with equal speed, imparting momentum changes Δp = 2·m·v per collision.
4Molecular Speed Distribution and Temperature
  • The average translational kinetic energy of gas molecules is directly proportional to absolute temperature (½·m·(crms)² = 3/2·kB·T);
    • this means higher temperatures result in a higher root-mean-square speed of particles.
5Theory
  • A well-tested, wide-ranging explanation supported by evidence.
6Momentum
  • P = mv, a vector.
7Kinetic theory
  • Particles in constant random motion.
8Avogadro constant
  • The number of particles per mole, 6.02 × 10²³ mol⁻¹
9Average speed
  • Total distance divided by total time.
10Thermodynamic temperature
  • The absolute temperature in kelvin.

Section 9 of 10Key terms

TermMeaning
Ideal gasA gas that obeys the ideal gas equation at all pressures, volumes and temperatures
Ideal gas constant (R)R = 8.31 J K−1 mol−1
Boltzmann constant (kB)kB = R/NA = 1.38 × 10−23 J K−1
Equation of stateRelation pV = nRT linking pressure, volume, amount and temperature
Real gasA gas that deviates from ideal behaviour at high pressure and low temperature
Number densityNumber of molecules per unit volume, N/V

Section 10 of 10Exam technique

1
Using °C or litres directly in the equation — use kelvin and m3.
2
Mixing R with kB — use R with n and kB with N.
3
Using gauge pressure — use absolute pressure.
4
Using pV = nRT when the amount changes between states — apply it to each state.
5
Expecting ideal behaviour at high pressure and low temperature — real gases deviate there.

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