Internal Energy

In Lesson 1 we introduced temperature as a measure of the average kinetic energy of the particles in a substance. We now move from average to total. A body contains a huge number of atoms or molecules, each in ceaseless motion and each interacting with its neighbours. The total amount of energy stored in those microscopic motions and interactions is called the internal energy of the body. This is the key quantity of thermal physics — it is what changes when a body is heated, cooled, compressed or allowed to expand. Module 5.1.2 of the OCR A-Level Physics A specification (H556) requires you to understand what internal energy is, what contributes to it, and how it responds to changes in temperature and state.

---

Definition of Internal Energy

Internal energy U is the sum of the randomly distributed kinetic and potential energies of the particles within a system.

Two points matter in this definition:

1. "Randomly distributed" — ordinary macroscopic kinetic energy (e.g. a football flying through the air) does not count. The whole football has a lot of kinetic energy, but all the atoms of the football are moving together in the same direction. That is bulk or ordered kinetic energy, not random, and it is not part of the internal energy. If you warm the football with your hands while holding it still, on the other hand, the random motion of its atoms increases, and you raise its internal energy.
2. "Kinetic and potential energies" — both contribute. The kinetic part comes from the motion of the particles (translation, rotation and vibration). The potential part comes from the forces the particles exert on one another through their separations.

Internal energy is symbolised by U and measured in joules (J), like any other energy.

---

The Kinetic Component

Every particle in the body is in motion. In a gas, the motion is free translation through empty space, punctuated by occasional collisions. In a liquid, the motion is hindered by near neighbours, so particles jostle about in a confined space rather than flying freely. In a solid, the particles are held at fixed lattice sites and can only vibrate about them.

In every case, the kinetic energy of each particle is (1/2)mv² where v is that particle's instantaneous speed. Since every particle has a different v at any given moment, the total translational kinetic energy of the body is

KE_total = Σ (1/2) m_i v_i²

summed over all N particles.

At higher temperatures, the average kinetic energy per particle is larger. Lesson 10 will show precisely that for an ideal monatomic gas, each particle has an average translational kinetic energy of (3/2)kT, where k is the Boltzmann constant. Increasing the temperature therefore increases the kinetic component of the internal energy directly.

---

The Potential Component

The potential component arises because the particles are not independent: they attract or repel each other. The forces between molecules — the intermolecular forces — give the particles potential energy that depends on their separations.

In a solid

In a solid, particles are bound tightly to their equilibrium positions in a lattice. If you pull two neighbouring atoms apart, you do work against the attractive force between them, raising their potential energy. This stored energy is part of the internal energy of the solid.

In a liquid

In a liquid, particles are still in close contact with their neighbours, but they are not fixed in a lattice. They slide past one another. There is still significant intermolecular potential energy (which is why the density of a liquid is similar to that of a solid), but the arrangement of particles is constantly changing.

In a gas

In a gas (especially at low density), particles are widely separated. Except during brief collisions, they barely interact, and the potential component of the internal energy is therefore very small. In an ideal gas — which will be our main model from Lesson 5 onward — we assume there are no intermolecular forces, so the internal energy consists of kinetic energy alone.

graph LR
    U[Internal energy U] --> KE[Kinetic component]
    U --> PE[Potential component]
    KE --> K1[Translation]
    KE --> K2[Rotation]
    KE --> K3[Vibration]
    PE --> P1[Intermolecular forces]
    PE --> P2[Intramolecular bonds]

---

Internal Energy and Temperature

Since increasing the temperature increases the average kinetic energy of the particles, it follows that increasing the temperature increases the internal energy (provided the number of particles does not change). For an ideal gas at constant amount:

U = (3/2) N k T

for a monatomic gas, where N is the number of molecules. The internal energy is directly proportional to the absolute temperature. Doubling T doubles U.

But be careful: for a real substance — a solid, a liquid, or a non-ideal gas — the relationship is more complicated because the potential part of the energy also changes. And most importantly, during a change of state the temperature does not change at all, yet the internal energy does. We will return to this in Lesson 4 when we discuss latent heat.

---

Changes in Internal Energy

You can change the internal energy of a body in two ways:

1. By heating it — supplying thermal energy from a hotter body. The added energy increases the random motion of the particles.
2. By doing work on it — for example, compressing a gas with a piston. The work done on the gas increases its internal energy.

This is the content of the first law of thermodynamics, which you will meet formally only if you study physics beyond A-Level. For A-Level you simply need to know that any input of energy to the system — whether as heat or as work — increases its internal energy by the same amount.

---

Worked Example 1: Which motion counts as internal?

A 0.50 kg lump of metal is moving through the air at 10 m s⁻¹. Its temperature is 25 °C. Which of the following contributes to its internal energy?

1. Its kinetic energy of (1/2)(0.50)(10²) = 25 J as it flies through the air.
2. The vibrational energy of its atoms about their equilibrium positions.
3. The potential energy of the interatomic bonds in the lattice.

The answer: only 2 and 3. The 25 J of bulk kinetic energy is ordered motion of the whole lump, not random motion of the particles. It does not count as internal energy, even though it is certainly energy of the lump. Internal energy is strictly the random, microscopic part.

If you were to catch the lump and bring it to rest, that 25 J of ordered kinetic energy would be converted (by friction, air resistance or impact) into additional random motion — that would increase the internal energy of the lump and whatever it collides with.

---

Worked Example 2: Heating a solid