Thermal Energy and Internal Energy

This lesson covers the fundamental concepts of thermal energy and internal energy as required by AQA A-Level Physics specification section 3.6. Understanding internal energy is the foundation upon which all thermal physics is built. You must be able to define internal energy precisely, explain what happens to it during heating and during changes of state, and apply the first law of thermodynamics.

What Is Internal Energy?

Key Definition: The internal energy of a system is the sum of the randomly distributed kinetic energies and potential energies of all its molecules.

Internal energy has two components:

Kinetic energy — associated with the random translational, rotational, and vibrational motion of the molecules. The faster the molecules move, the greater the kinetic energy component.
Potential energy — associated with the intermolecular forces (bonds) between molecules. When molecules are closer together and interacting more strongly, the potential energy component is more significant.

Key Points About Internal Energy

Internal energy is a property of the system as a whole, not of individual molecules.
It depends on the temperature (which determines the average kinetic energy) and on the state of the substance (which affects the potential energy).
For an ideal gas, there are no intermolecular forces (except during collisions), so the internal energy is entirely kinetic. This is a crucial simplification used throughout the gas laws topic.
In solids and liquids, both the kinetic and potential energy components are significant because the molecules are close enough to interact via intermolecular forces.

Exam Tip: When defining internal energy, you must mention both kinetic and potential energy AND state that the energies are randomly distributed. Stating just "the total energy of all the molecules" is insufficient for full marks.

Temperature and Internal Energy

Temperature is a measure of the average kinetic energy of the molecules in a substance. When you heat a substance (without causing a change of state), you increase the average kinetic energy of the molecules, which means:

The temperature rises.
The kinetic energy component of internal energy increases.
The potential energy component may also change slightly as molecules vibrate more vigorously and move further apart on average, but the dominant change is in kinetic energy.

What Happens During a Change of State?

During a change of state (melting, boiling, freezing, condensing), the temperature remains constant even though energy is being supplied (or removed). This is one of the most important and frequently tested ideas in thermal physics.

When a solid melts:

Energy is supplied to the substance.
The temperature does not change.
The kinetic energy of the molecules does not change (since temperature is constant).
The energy goes into increasing the potential energy component of internal energy by breaking or weakening intermolecular bonds.
The internal energy increases, but only the potential energy part changes.

When a liquid boils:

The same principle applies — energy increases the potential energy component.
More energy is required for boiling than for melting because all remaining intermolecular bonds must be broken to separate molecules completely.

Exam Tip: A very common exam question asks what happens to internal energy during a change of state. The model answer must state three things: (1) temperature remains constant, (2) kinetic energy does not change, and (3) only the potential energy component of internal energy increases. Many candidates lose marks by failing to mention that kinetic energy stays the same.

The First Law of Thermodynamics

The first law of thermodynamics is a statement of conservation of energy applied to thermal processes:

ΔU = Q − W

where:

ΔU is the change in internal energy of the system (J)
Q is the energy transferred to the system by heating (J)
W is the work done by the system on its surroundings (J)

Sign Convention

Quantity	Positive means...	Negative means...
Q	Energy transferred INTO the system (heating)	Energy transferred OUT of the system (cooling)
W	Work done BY the system (expansion)	Work done ON the system (compression)
ΔU	Internal energy increases	Internal energy decreases

Applying the First Law

Case 1: Heating at constant volume If a gas is heated in a rigid container, it cannot expand, so W = 0. Therefore ΔU = Q — all the energy supplied goes into increasing the internal energy.

Case 2: Heating at constant pressure If a gas is heated at constant pressure, it expands and does work on its surroundings. The work done is W = pΔV. Therefore ΔU = Q − pΔV — some of the energy supplied goes into work, and the rest increases the internal energy.

Case 3: Adiabatic expansion If a gas expands without any heat transfer (Q = 0), then ΔU = −W. The gas does work at the expense of its own internal energy, so it cools down. This is why gas escaping from a pressurised container feels cold.

Case 4: Free expansion into a vacuum An ideal gas expanding into a vacuum does no work (there is nothing to push against), so W = 0. If the container is insulated, Q = 0 as well. Therefore ΔU = 0 — the temperature of an ideal gas does not change during free expansion. For a real gas, the temperature drops slightly because work is done against intermolecular forces.

Worked Example — First Law of Thermodynamics

Question: A gas is heated, receiving 500 J of energy. During this process, the gas expands and does 200 J of work on a piston. Calculate the change in internal energy of the gas.

Solution:

Using ΔU = Q − W:

ΔU = 500 − 200 = 300 J

The internal energy of the gas increases by 300 J. The remaining 200 J of the 500 J supplied was used to do work pushing the piston outwards.

Worked Example — Compression of a Gas

Question: A bicycle pump is used to compress air. The pump does 150 J of work on the gas, and 40 J of heat escapes to the surroundings during the process. Calculate the change in internal energy of the gas.

Solution:

Work is done ON the gas, so W = −150 J (the system is compressed, not expanding). Heat is lost from the system, so Q = −40 J.

ΔU = Q − W = (−40) − (−150) = −40 + 150 = 110 J

The internal energy increases by 110 J. The gas heats up because the work done on it exceeds the heat lost to the surroundings.

Exam Tip: Be very careful with the sign convention in first law problems. If the question says work is done ON the gas, W is negative in the equation ΔU = Q − W. If heat is lost, Q is negative. Read the question carefully and assign signs before substituting.

Heating, Cooling, and Energy Transfer

When two objects at different temperatures are placed in thermal contact, energy flows spontaneously from the hotter object to the cooler object until they reach thermal equilibrium (the same temperature). This is a consequence of the second law of thermodynamics.

The direction of energy transfer depends on temperature, not on internal energy. A large cold object may have more total internal energy than a small hot object, but energy still flows from hot to cold.

Methods of Energy Transfer

Energy can be transferred by three mechanisms:

Conduction — energy is transferred through a material by the vibration of particles and (in metals) by the movement of free electrons.
Convection — energy is transferred by the bulk movement of a heated fluid (liquid or gas).
Radiation — energy is transferred by electromagnetic waves (infrared radiation), which can travel through a vacuum.

Summary Table

Concept	Key Formula	Notes
Internal energy	U = total KE + total PE	Sum of randomly distributed molecular energies
First law	ΔU = Q − W	Conservation of energy for thermal processes
Change of state	Temperature constant	Only PE component of U changes
Heating (no state change)	Temperature rises	Mainly KE component increases
Ideal gas internal energy	U = total KE only	No intermolecular forces, so PE = 0

Common Misconceptions

"Internal energy is the same as temperature." This is wrong. Temperature measures the average kinetic energy per molecule, while internal energy is the total energy of all molecules. A bath of warm water has far more internal energy than a red-hot needle, even though the needle is at a much higher temperature.
"During a change of state, no energy is being supplied." Energy is being supplied (or removed), but it does not cause a temperature change — it changes the potential energy component.
"All the heat supplied to a gas goes into raising its temperature." This is only true at constant volume. At constant pressure, some energy goes into doing work (expanding against the surroundings).
"Work is always positive." Work is positive when done BY the system; it is negative when done ON the system. The sign matters in the first law.