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Every object around you — a cup of tea, a block of iron, even the air in the room — is made up of an enormous number of particles (atoms or molecules) that are constantly in motion. These particles have kinetic energy because they are moving, and they have potential energy because of the forces between them. The internal energy of a system is the sum of the randomly distributed kinetic and potential energies of all the particles within it.
This is one of the most fundamental ideas in thermodynamics, and it underpins everything from heating a saucepan of water to understanding how engines work.
The particles in any substance are in constant, random motion. In a gas, this motion is primarily translational — the molecules fly around in straight lines between collisions. In solids, the particles vibrate about fixed positions. In liquids, the behaviour is somewhere in between.
Kinetic energy of particles depends on their speed. At higher temperatures, the particles move faster on average, so their mean kinetic energy increases. This is a crucial link: temperature is a measure of the average kinetic energy of the particles in a substance.
Potential energy of particles arises from the intermolecular forces (bonds) between them. In a solid, particles are held close together in a regular arrangement by strong forces, and they sit in potential energy "wells." In a liquid, the particles have enough energy to partially overcome these forces but remain loosely bound. In a gas, the particles have largely overcome the intermolecular forces and the potential energy contribution is very different from that in a solid or liquid.
Therefore:
Internal energy = total kinetic energy of all particles + total potential energy due to intermolecular forces
graph TD
A["Internal Energy U"] --> B["Total Kinetic Energy"]
A --> C["Total Potential Energy"]
B --> D["Translational KE\n(gas molecules)"]
B --> E["Vibrational KE\n(solid/liquid particles)"]
B --> F["Rotational KE\n(polyatomic molecules)"]
C --> G["Intermolecular bonds\n(attractive forces)"]
C --> H["Repulsive core\n(electron overlap)"]
style A fill:#f9f,stroke:#333,stroke-width:2px
When you heat a substance and its temperature rises, you are increasing the average kinetic energy of its particles. The particles move faster (or vibrate more vigorously), and the internal energy of the system increases.
However — and this is a critical distinction — internal energy is not the same as temperature. Temperature only reflects the average kinetic energy component. Internal energy also includes the potential energy between particles. Two objects at the same temperature can have very different internal energies if they contain different amounts of substance or are in different phases.
For example, 1 kg of water at 100 °C and 1 kg of steam at 100 °C are at the same temperature, but the steam has significantly more internal energy because additional energy was needed to break the intermolecular bonds during the change of state.
To appreciate how large the potential energy contribution can be, consider converting 1 kg of water at 100 °C to steam at 100 °C. The specific latent heat of vaporisation of water is 2 260 000 J kg⁻¹. This means 2.26 MJ of energy must be supplied — and every joule goes into potential energy (breaking hydrogen bonds), since the temperature does not change. Compare this to heating that same 1 kg of water from 0 °C to 100 °C, which requires only:
Q = mcΔθ = 1.0 × 4200 × 100 = 420 000 J = 0.42 MJ
The latent heat is more than five times the energy needed to heat the water through 100 °C. This vividly illustrates that potential energy is often the dominant contribution to internal energy changes during phase transitions.
When a substance changes state — for example, ice melting to water, or water boiling to steam — something remarkable happens: the temperature stays constant even though energy is being supplied.
Where does the energy go? It goes into increasing the potential energy of the particles. During a change of state, the energy supplied is used to overcome the intermolecular forces that hold the particles in their current arrangement, rather than increasing their kinetic energy.
Consider ice at 0 °C being heated:
The same principle applies at 100 °C when water boils to steam: the temperature remains constant while the energy input increases the potential energy of the molecules as they break free from the liquid.
This is worth emphasising because it is frequently examined. During a change of state:
A heating curve illustrates this clearly:
| Phase | What happens | KE | PE | Temperature |
|---|---|---|---|---|
| Solid being heated | Vibrations increase | Increases | Roughly constant | Rises |
| Solid → Liquid (melting) | Bonds breaking | Constant | Increases | Constant |
| Liquid being heated | Particles speed up | Increases | Roughly constant | Rises |
| Liquid → Gas (boiling) | Bonds breaking | Constant | Increases | Constant |
| Gas being heated | Molecules speed up | Increases | Roughly constant | Rises |
The flat sections of a heating curve correspond to changes of state, where all the supplied energy increases potential energy rather than kinetic energy.
The total internal energy differs between phases even at the same temperature:
This explains why steam burns are more severe than boiling water burns — the steam carries additional internal energy (the latent heat) that is released when it condenses on the skin.
For an ideal gas, the situation simplifies considerably. By definition, an ideal gas has no intermolecular forces between molecules (except during instantaneous elastic collisions). This means:
This is a critical result for the first law of thermodynamics. It means that during an isothermal (constant temperature) process involving an ideal gas, the internal energy does not change (ΔU = 0), regardless of what happens to the pressure and volume.
For a monatomic ideal gas (e.g., helium, neon, argon), each atom has three translational degrees of freedom, and the total internal energy is:
U = 3/2 NkT = 3/2 nRT
where N is the total number of atoms, n is the number of moles, k is the Boltzmann constant, and R is the molar gas constant.
A 2 kg block of ice at 0 °C is heated until it has completely melted into water at 0 °C. The specific latent heat of fusion of ice is 334 000 J kg⁻¹. What happens to the internal energy?
The energy supplied is:
Q = mL = 2 × 334 000 = 668 000 J
This 668 kJ of energy has increased the internal energy of the water compared to the ice. Since the temperature has not changed, the kinetic energy of the particles is the same. All 668 kJ has gone into increasing the potential energy of the particles — breaking the hydrogen bonds in the ice lattice so the water molecules can move more freely.
A sealed container holds 3.0 mol of an ideal monatomic gas at 400 K. Calculate the total internal energy of the gas. (R = 8.31 J mol⁻¹ K⁻¹)
For a monatomic ideal gas:
U = 3/2 nRT = 3/2 × 3.0 × 8.31 × 400
U = 1.5 × 3.0 × 8.31 × 400 = 14 958 J ≈ 15.0 kJ
If the gas is heated to 800 K (temperature doubled), the internal energy doubles to 30.0 kJ, since U is directly proportional to T for an ideal gas.
A copper calorimeter of mass 0.15 kg contains 0.30 kg of water, both initially at 20 °C. A 0.10 kg piece of aluminium at 250 °C is dropped in. Find the final temperature, assuming no heat loss. (c_Cu = 390 J kg⁻¹ K⁻¹, c_water = 4200 J kg⁻¹ K⁻¹, c_Al = 900 J kg⁻¹ K⁻¹)
Energy lost by aluminium = Energy gained by water + Energy gained by calorimeter
m_Al × c_Al × (250 − T) = m_water × c_water × (T − 20) + m_Cu × c_Cu × (T − 20)
0.10 × 900 × (250 − T) = (0.30 × 4200 + 0.15 × 390) × (T − 20)
90(250 − T) = (1260 + 58.5)(T − 20)
22 500 − 90T = 1318.5T − 26 370
22 500 + 26 370 = 1318.5T + 90T
48 870 = 1408.5T
T = 48 870 / 1408.5 = 34.7 °C
This example shows why it is important to include the calorimeter's heat capacity — ignoring it would give a higher final temperature (about 36.7 °C).
| Mistake | Why it is wrong |
|---|---|
| Saying "heat" and "internal energy" are the same | Heat is energy transferred due to a temperature difference; internal energy is the total KE + PE of particles |
| Claiming temperature rises during a change of state | Temperature is constant during phase changes — energy goes to PE, not KE |
| Saying an ideal gas has internal energy = 0 | An ideal gas has zero PE but non-zero KE; its internal energy depends on temperature |
| Confusing internal energy with thermal energy | Internal energy includes all microscopic KE and PE; "thermal energy" is an informal term |
| Forgetting that steam at 100 °C has more internal energy than water at 100 °C | The latent heat of vaporisation adds substantial PE to the steam |
Thermal storage systems exploit high internal energy changes during phase transitions. Phase-change materials (PCMs) such as paraffin wax or sodium acetate trihydrate are used in hand warmers, building insulation, and solar thermal storage. They absorb large amounts of energy when melting (storing latent heat) and release it when solidifying, all at a nearly constant temperature.
Freeze-thaw weathering is a geological process driven by internal energy changes. Water seeps into cracks in rocks. When it freezes, the expansion exerts enormous pressure on the rock, gradually breaking it apart over many cycles.