In simple terms
A friendly intro before the formal notes — no formulas yet.
Equation of state
Cambridge 9702 Paper 4 — Equation of state (15.2). Senpai Corner diagram-backed pilot with premium structure and live visuals.
- 1
The gas consists of a large number of identical molecules in continuous, random motion.
- 2
The volume of the molecules themselves is negligible compared to the volume of the container they occupy.
- 3
There are no intermolecular forces of attraction or repulsion between molecules; their potential energy is zero.
- 4
All collisions between molecules and with the walls of the container are perfectly elastic (no kinetic energy is lost).
What this topic covers
The official Cambridge syllabus points this lesson works through.
- 15.2.1
Understand that a gas obeying , where T is the thermodynamic temperature, is known as an ideal gas
- 15.2.2
Recall and use the equation of state for an ideal gas expressed as , where n = amount of substance (number of moles) and as , where N = number of molecules
- 15.2.3
Recall that the Boltzmann constant k is given by
Explore the concept
Use the live diagram, PhET or GeoGebra sim, and synced steps — play it, drag controls, or tap a step.
Step-synced diagram — highlights what to look for in the simulation above.
Step 1
The gas consists of a large number of identical molecules in continuous, random motion.
5 more simulations for this topic — run them in the Simulations section below
Simulations
Every simulation here runs the real model — try the steps on a card, then check what you see against the notes.
5 simulations · 4 to start with
Start herein this order — each one shows a different piece of the topic
- 3JCN PhysicsStart here · 19702 15.2 · 9702 15.1 · IB B.3
Ideal Gas
Vary P, V, n and T of an ideal gas and watch the equation of state balance
Why this one: Change any one of p, V, n, T and watch the others rebalance so that pV = nRT always holds.
Try this
- Halve V at fixed n and T and read P.
- Double T at fixed V and n and read P.
- Double n at fixed V and T and read P.
Look for pV = nRT: pressure doubles when volume halves, when temperature doubles, or when the amount of gas doubles.
3JCN Physics Simulation by Thomas Nguyen · CC BY 4.0
- The Physics ClassroomStart here · 29702 15.2 · 9702 15.3 · IB B.3
Ideal Gas Behavior
A gas cylinder with piston, thermometer, pressure gauge, volume and number of moles; set an initial state and change it
Why this one: Push the piston on a real-looking cylinder and read the gauge, thermometer and volume together.
Try this
- Hold the temperature and halve the volume.
- Hold the volume and raise the temperature.
- Add moles of gas and compare.
Look for pV divided by nT stays constant.
Physics Interactives by The Physics Classroom · Licensed to MarkScheme (site terms otherwise permit linking only)
- 3JCN PhysicsStart here · 39702 14.1 · 9702 14.2 · 9702 15.2
Absolute Zero Temp.
Cool a gas at constant volume and extrapolate the P-T line to absolute zero
Why this one: Plot p against T at constant volume; the line aims at −273 °C, so T must be in kelvin.
Try this
- Cool the gas at constant volume and record pressure against temperature.
- Extrapolate the P-T line back to zero pressure and read the temperature.
Look for Pressure falls linearly with temperature and the line meets zero pressure at -273 °C, absolute zero.
3JCN Physics Simulation by Thomas Nguyen · CC BY 4.0
- PhETStart here · 49702 15.2 · IB B.3
Gases Intro
A simpler gas box: change the number of particles, heat them, and read the pressure.
Why this one: Pump in more particles at fixed volume and watch pressure rise in proportion to N.
Try this
- Add 50 heavy particles and heat — the pressure gauge rises with T.
- On “Laws”, lock the volume and double the particles — p doubles.
Look for More particles or faster particles → more frequent, harder wall collisions → higher p.
Simulation by PhET Interactive Simulations, University of Colorado Boulder · Licensed to MarkScheme (public licence CC BY-NC 4.0 since 2026-03-30)
More simulations1 more on this topic — core ones first
- 3JCN PhysicsCore9702 14.1 · 9702 14.2 · 9702 15.2
Measure Abs Zero T
Measure absolute zero by plotting pressure against temperature for a gas
Try this
- Plot pressure against temperature for the gas at several temperatures.
- Extend the straight line to zero pressure and read the intercept.
Look for Every constant-volume line extrapolates to the same intercept, about -273 °C.
3JCN Physics Simulation by Thomas Nguyen · CC BY 4.0
Key formulas
Tap any symbol to reveal exactly what it means and its units.
Tap a symbol — great for exam definitions
Tap a symbol — great for exam definitions
Tap a symbol — great for exam definitions
Tap a symbol — great for exam definitions
Tap a symbol — great for exam definitions
Tap a symbol — great for exam definitions
Full topic notes
Formal explanation with the rigour you need for the exam.
The Ideal Gas Model
An ideal gas is a theoretical model that simplifies the complex behaviour of real gases by making several key assumptions. This model allows us to create a simple relationship between pressure, volume, and temperature. A gas that perfectly follows the relationship pV ∝ T is defined as an ideal gas.
Assumptions of the Ideal Gas Model
The gas consists of a large number of identical molecules in continuous, random motion.
The volume of the molecules themselves is negligible compared to the volume of the container they occupy.
There are no intermolecular forces of attraction or repulsion between molecules; their potential energy is zero.
All collisions between molecules and with the walls of the container are perfectly elastic (no kinetic energy is lost).
The duration of a collision is negligible compared to the time between collisions.
Always remember to convert temperatures into Kelvin (K) for all gas law calculations. Failing to do so is a very common mistake in exams!
The Building Blocks: Individual Gas Laws
Before reaching the universal ideal gas equation, physicists discovered three experimental laws describing how pressure, volume, and temperature interact when one variable is kept constant for a fixed mass of gas. These laws form the foundation for understanding ideal gas behaviour.
Boyle's Law: Pressure & Volume
If you squeeze a gas (decrease its volume) while keeping its temperature constant, the pressure it exerts will increase. This inverse relationship is Boyle's Law.
Charles' Law: Volume & Temperature
When a gas is heated (increasing its absolute temperature) at a constant pressure, its volume expands. This direct proportionality is Charles' Law.
Pressure Law (Gay-Lussac's Law): Pressure & Temperature
If you heat a gas in a sealed, rigid container (constant volume), the pressure inside will rise. This direct relationship between pressure and absolute temperature is the Pressure Law.
The Combined Gas Law
The three individual laws can be merged into a single, more versatile equation called the Combined Gas Law. It is extremely useful for problems where a fixed mass of gas undergoes changes in pressure, volume, and temperature simultaneously.
The Universal Equation: Ideal Gas Law (Macroscopic Form)
These gas laws culminate in the Ideal Gas Equation. It relates pressure, volume, and absolute temperature to the amount of substance (number of moles) of the gas, not just for changing states but for a single state.
P: Pressure in Pascals (Pa)
V: Volume in cubic metres (m³)
n: Number of moles (mol)
R: Molar gas constant (8.31 J mol⁻¹ K⁻¹)
T: Absolute temperature in Kelvin (K)
Connecting to Individual Molecules: Ideal Gas Law (Microscopic Form)
Sometimes you might deal with the total number of molecules (N) instead of moles (n). For this, we use the Boltzmann constant (k), which links the average kinetic energy of gas particles to temperature. The Ideal Gas Equation can then be expressed in terms of individual molecules.
N: Total number of molecules (N = n × N_A)
k: Boltzmann constant (k = R/N_A ≈ 1.38 × 10⁻²³ J K⁻¹)
N_A: Avogadro constant (6.02 × 10²³ mol⁻¹)
Worked examples
See the formulas applied — reveal one step at a time, like the exam.
A cylinder contains 0.50 mol of an ideal gas at a pressure of 2.0 × 10⁵ Pa and a temperature of 27 °C. Calculate the volume of the gas.
- 1
Identify knowns and unknowns:
A sealed container of volume 1.5 × 10⁻² m³ contains an ideal gas at a pressure of 3.0 × 10⁵ Pa and a temperature of 350 K. Calculate the number of gas molecules in the container. (Boltzmann constant, k = 1.38 × 10⁻²³ J K⁻¹)
- 1
Identify knowns and unknowns:
How it all connects
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Glossary
Key terms for this topic — skim now; the Check step will test them.
- Pressure Law
For a fixed mass of gas at constant volume, its pressure is directly proportional to its absolute temperature (P/T = constant).
- Avogadro's constant
It is the number of constituent particles (e.g., atoms or molecules) per mole of a substance. Its value is 6.02 × 10²³ mol⁻¹.
Quick check
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Teach it back
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Teach it back
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Revision flashcards
Guess first, then flip — retrieval beats re-reading.
Key takeaways
Review these before you close the topic — retrieval beats re-reading.
The gas consists of a large number of identical molecules in continuous, random motion.
The volume of the molecules themselves is negligible compared to the volume of the container they occupy.
There are no intermolecular forces of attraction or repulsion between molecules; their potential energy is zero.
All collisions between molecules and with the walls of the container are perfectly elastic (no kinetic energy is lost).
The duration of a collision is negligible compared to the time between collisions.
Practice — then mark it
The whole point: a real Cambridge question, marked mark-by-mark.
The volume V of the gas in (b) is now varied, keeping its pressure constant. On Fig. 3.1, sketch the variation with V of the internal energy U of the gas.
A sample of 0.26 m³ of an ideal gas is at pressure 2.0 × 10⁵ Pa and temperature 290 K. Determine: the number N of molecules of the gas
Extra simulations & links
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Frequently asked
Checkpoint
One marked question is worth ten re-reads — close the loop before you move on.
Reading it isn’t knowing it — prove it.
Before you move on: do 9702/41 · Q3(c) on paper, snap a photo, and get examiner-style feedback on exactly where you win and lose marks.
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