In simple terms
A friendly intro before the formal notes — no formulas yet.
Internal energy
Cambridge 9702 Paper 4 - Internal energy (16.1). Senpai Corner diagram-backed pilot with premium structure and live visuals.
- 1
Definition: Internal energy (U) is the sum of the random kinetic and potential energies of a system's molecules.
- 2
Kinetic Energy (KE): Related to the motion of particles. An increase in temperature leads to an increase in the average KE of the particles.
- 3
Potential Energy (PE): Related to the intermolecular forces between particles and their separation. PE changes significantly during a phase change (e.g., melting or boiling).
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Temperature Link: For any substance, increasing its temperature increases the random kinetic energy of its particles, thus increasing its internal energy.
What this topic covers
The official Cambridge syllabus points this lesson works through.
- 16.1.1
Understand that internal energy is determined by the state of the system and that it can be expressed as the sum of a random distribution of kinetic and potential energies associated with the molecules of a system
- 16.1.2
Relate a rise in temperature of an object to an increase in its internal energy
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
Definition: Internal energy (U) is the sum of the random kinetic and potential energies of a system's molecules.
9 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.
9 simulations · 4 to start with
Start herein this order — each one shows a different piece of the topic
- PhETStart here · 19702 15.3 · 16.1
Gas Properties
Look inside the box: particle speeds, collisions with the walls, and the speed distribution.
Why this one: Heat an ideal gas and only the speeds change: its internal energy is purely kinetic, proportional to T.
Try this
- On “Energy”, show the “Speed” histogram and heat the gas — the distribution shifts right.
- Mix light and heavy particles at the same T — which species moves faster?
- Turn on “Collisions” counter and heat — collisions per unit time rise.
Look for Pressure comes from momentum change at the walls; ½m⟨c²⟩ = 3/2 kT so lighter particles move faster at the same T.
Simulation by PhET Interactive Simulations, University of Colorado Boulder · Licensed to MarkScheme (public licence CC BY-NC 4.0 since 2026-03-30)
- PhETStart here · 29702 16.1 · IB B.1
Atomic Interactions
Drag two atoms apart and watch the Lennard-Jones force and potential-energy curves change with separation and atom pair.
Why this one: Pull two atoms apart and watch the potential energy curve; separating particles is where the PE part of U lives.
Try this
- Pull the atoms apart slowly — find the separation where the force reads zero.
- Switch to the potential-energy view — the force-zero point sits at the bottom of the well.
- Change the atom pair — compare how deep the well is.
Look for Internal energy includes this potential energy: atoms rest at the well minimum, and separating them costs energy.
Simulation by PhET Interactive Simulations, University of Colorado Boulder · Licensed to MarkScheme (public licence CC BY-NC 4.0 since 2026-03-30)
- 3JCN PhysicsStart here · 39702 16.2 · 9702 16.1 · IB B.4
Isovolumetric Pro
Heat a gas at constant volume and watch pressure rise with no work done
Why this one: Heat at constant volume: no work is done, so every joule of heating goes into internal energy.
Try this
- Heat the gas at constant volume and watch the pressure rise.
- Read the work done on the PV diagram.
Look for With no volume change no work is done, so all heat supplied raises internal energy.
3JCN Physics Simulation by Thomas Nguyen · CC BY 4.0
- 3JCN PhysicsStart here · 49702 16.2 · 9702 16.1 · IB B.4
Adiabatic Process
Compress a gas with no heat exchange; see temperature rise on the adiabat
Why this one: Compress with no heating and watch temperature rise — work alone has raised the internal energy.
Try this
- Compress the gas with no heat exchange and watch the temperature.
- Compare the adiabat's steepness with an isotherm through the same point.
Look for With q = 0 the work done on the gas raises its internal energy, so temperature rises and the adiabat is steeper than an isotherm.
3JCN Physics Simulation by Thomas Nguyen · CC BY 4.0
More simulations5 more on this topic — core ones first
- 3JCN PhysicsCore9702 16.2 · 9702 16.1 · IB B.4
A Cyclic Process
Run a gas around a closed PV cycle and read net work from the enclosed area
Try this
- Run the gas around the closed PV cycle.
- Read the net work from the enclosed area.
- Note that the internal energy change over the full cycle is zero.
Look for Over a complete cycle ΔU = 0, so net heat in equals net work out, the area enclosed.
3JCN Physics Simulation by Thomas Nguyen · CC BY 4.0
- 3JCN PhysicsCore9702 16.2 · 9702 16.1 · IB B.4
Isobaric Process
Expand a gas at constant pressure; compute work from the PV area
Try this
- Expand the gas at constant pressure and read the PV area.
- Compare the area with p × ΔV.
Look for Work done equals pΔV, the rectangle under the isobar.
3JCN Physics Simulation by Thomas Nguyen · CC BY 4.0
- 3JCN PhysicsCore9702 16.2 · 9702 16.1 · IB B.4
Isothermal Process
Compress a gas at constant temperature and trace the isotherm on a PV diagram
Try this
- Compress the gas at constant temperature and trace the isotherm.
- Halve the volume and read the pressure.
Look for At constant T, pV is constant, so the isotherm is a hyperbola and all heat supplied equals work done.
3JCN Physics Simulation by Thomas Nguyen · CC BY 4.0
- 3JCN PhysicsCore9702 16.2 · 9702 16.1 · IB B.4
T in Adiabatic Proc.
Track temperature change during an adiabatic compression or expansion
Try this
- Compress adiabatically and track the temperature.
- Expand adiabatically and track it again.
Look for Adiabatic compression heats the gas and adiabatic expansion cools it, with no heat flowing either way.
3JCN Physics Simulation by Thomas Nguyen · CC BY 4.0
- PhET9702 3.2 · 16.1 · IB A.2 · B.1
Friction
Rub two books together and watch the atoms at the surfaces jiggle faster as the thermometer rises.
Try this
- Drag the top book back and forth — watch the thermometer.
- Rub faster — the atoms vibrate more and the temperature climbs quicker.
- Stop rubbing — the surfaces slowly cool.
Look for Work done against friction becomes internal energy: random vibration of atoms, not useful motion.
Simulation by PhET Interactive Simulations, University of Colorado Boulder · Licensed to MarkScheme (public licence CC BY-NC 4.0 since 2026-03-30)
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
Full topic notes
Formal explanation with the rigour you need for the exam.
Understanding Internal Energy
At its core, a substance's internal energy (U) is the grand total of all the chaotic kinetic and potential energies possessed by its individual particles (atoms or molecules). Imagine countless tiny particles jiggling, vibrating, and colliding randomly. Their movement contributes to kinetic energy, while the forces between them contribute to potential energy. It’s the sum of all these microscopic energies throughout the entire system.
The balance between kinetic and potential energy differs by state. In solids, particles vibrate in a fixed lattice, possessing both vibrational KE and significant PE from strong bonds. In liquids, particles have translational and vibrational KE, with PE from forces that are weaker than in solids but still significant. In gases, particles move freely with high KE, and the PE from intermolecular forces is negligible, especially in an ideal gas.
Definition: Internal energy (U) is the sum of the random kinetic and potential energies of a system's molecules.
Kinetic Energy (KE): Related to the motion of particles. An increase in temperature leads to an increase in the average KE of the particles.
Potential Energy (PE): Related to the intermolecular forces between particles and their separation. PE changes significantly during a phase change (e.g., melting or boiling).
Temperature Link: For any substance, increasing its temperature increases the random kinetic energy of its particles, thus increasing its internal energy.
Internal Energy of an Ideal Gas
For the special case of an ideal gas, the model assumes there are no intermolecular forces between particles. This means the potential energy component of the internal energy is zero. Therefore, the internal energy of an ideal gas consists entirely of the sum of the random kinetic energies of its molecules. This leads to a crucial conclusion: the internal energy of a fixed mass of an ideal gas is directly proportional to its absolute temperature (in Kelvin).
For a monatomic ideal gas, this relationship can be expressed quantitatively as , where is the number of moles, is the ideal gas constant, and is the absolute temperature. This shows that if the temperature of an ideal gas is constant, its internal energy does not change.
The First Law of Thermodynamics
The First Law of Thermodynamics is a cornerstone of physics, essentially a statement of the conservation of energy applied to thermodynamic systems. It explains how energy can be transferred into or out of a system, changing its internal energy, through two primary mechanisms: heating or doing work. The law provides a mathematical relationship between the change in internal energy (), the heat added to the system (), and the work done on the system ().
: Change in internal energy. A positive value means an increase, a negative value means a decrease.
: Energy transferred as heat. Positive when heat enters the system. Negative when heat leaves the system.
: Work done. Positive when work is done on the system (e.g., compressing a gas). Negative when work is done by the system (e.g., an expanding gas pushes a piston).
Worked examples
See the formulas applied — reveal one step at a time, like the exam.
A gas in a cylinder absorbs 300 J of heat from its surroundings. At the same time, the gas expands, doing 120 J of work on the piston. Calculate the change in the internal energy of the gas.
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Identify given values with correct signs:
A piston compresses a gas in an insulated cylinder, doing 500 J of work on the gas. During the compression, the gas loses 200 J of heat to the surroundings. Calculate the change in the internal energy of the gas.
- 1
Identify given values with correct signs:
How it all connects
The big idea sits in the middle — tap a linked idea to explore the link.
Tap a linked idea to see how it connects back to the main topic — that connection is what examiners reward.
Glossary
Key terms for this topic — skim now; the Check step will test them.
- internal energy (U)
At its core, a substance's internal energy (U) is the grand total of all the chaotic kinetic and potential energies possessed by its individual particles (atoms or molecules).
- kinetic energy
Their movement contributes to kinetic energy, while the forces between them contribute to potential energy. It’s the sum of all these microscopic energies throughout the entire system.
- ideal gas
For the special case of an ideal gas, the model assumes there are no intermolecular forces between particles.
- First Law of Thermodynamics
The First Law of Thermodynamics is a cornerstone of physics, essentially a statement of the conservation of energy applied to thermodynamic systems.
Quick check
Write your answer first, then compare it with the model one — the gap is what you would have lost.
Teach it back
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Teach it back
Explain this topic as if teaching a friend. We name the gaps an examiner would still dock.
Revision flashcards
Guess first, then flip — retrieval beats re-reading.
Key takeaways
Review these before you close the topic — retrieval beats re-reading.
Definition: Internal energy (U) is the sum of the random kinetic and potential energies of a system's molecules.
Kinetic Energy (KE): Related to the motion of particles. An increase in temperature leads to an increase in the average KE of the particles.
Potential Energy (PE): Related to the intermolecular forces between particles and their separation. PE changes significantly during a phase change (e.g., melting or boiling).
Temperature Link: For any substance, increasing its temperature increases the random kinetic energy of its particles, thus increasing its internal energy.
Practice — then mark it
The whole point: a real Cambridge question, marked mark-by-mark.
State what is meant by the internal energy of a system.
the internal energy of the gas. Explain your reasoning.
Extra simulations & links
PhET, GeoGebra and other curated tools — open in a new tab.
Frequently asked
Checkpoint
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Reading it isn’t knowing it — prove it.
Before you move on: do 9702/42 · Q3(a) on paper, snap a photo, and get examiner-style feedback on exactly where you win and lose marks.
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