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
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Key formulas
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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
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Quick check
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Revision flashcards
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What are the two fundamental assumptions for an ideal gas?
Its molecules have no intermolecular forces and occupy negligible volume.
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.
9702/41 · Q3(c)
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.
9702/42 · Q4(c)
State three conclusions about the gases and their samples that may be drawn from Fig. 4.1 and Fig. 4.2. The conclusions may be qualitative or quantitative. Use the space below for any working that you need.
Extra simulations & links
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Frequently asked
Checkpoint
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