Internal energy

In brief

Lift a brick off the floor and it gains gravitational potential energy as a whole object; throw it and it gains kinetic energy as a whole object. Both are properties of the brick's motion and position relative to its surroundings. But heat the same brick with a torch, and it also stores more energy, even though it never moves and never rises. Where does that energy go? It goes into the brick's own particles: the atoms vibrate more vigorously about their lattice positions, and if you heated it enough to melt or vaporize it, the bonds holding those atoms in a rigid arrangement would be pulled apart, changing the potential energy stored in those bonds as well. Internal energy is the name for this total microscopic store, kinetic and potential energy at the particle scale, kept separate from whatever kinetic or potential energy the object has as a whole. It matters because heating, compressing, and melting a substance clearly do something to it, and internal energy is the single quantity that lets you say precisely what.

The full treatment

First look: a sealed box of bouncing balls

Picture a sealed, rigid box containing many small balls, bouncing around and colliding elastically with each other and the walls, with springs connecting some of them to represent chemical bonds. From outside, the box as a whole might sit perfectly still on a shelf, showing no kinetic energy and no change in height, hence no change in gravitational potential energy. Yet inside, there is plainly energy: the kinetic energy of every bouncing ball, added up over all of them, plus the potential energy stored in every stretched or compressed spring linking them. If you could total up every ball's kinetic energy and every spring's potential energy at one instant, that sum is a good mechanical picture of internal energy. Real matter is this box, with atoms as the balls and interatomic and intermolecular forces as the springs.

Building the idea: two buckets, kinetic and potential, at the particle scale

Internal energy divides naturally into two buckets, mirroring the mechanical energy accounting already familiar from single objects, but applied particle by particle rather than to the object as a whole.

The first bucket is microscopic kinetic energy: the sum, over every particle in the substance, of one-half times that particle's mass times its speed squared, counting translational motion (flying through space), rotational motion (tumbling), and vibrational motion (atoms jiggling within a molecule or lattice) wherever each applies. This bucket is precisely what temperature measures the average of, as established in the account of temperature as a microscopic energy distribution; a rise in this bucket, averaged per particle, is a rise in temperature.

The second bucket is microscopic potential energy: the energy stored in the configuration of the particles relative to each other, chiefly the energy of the bonds and intermolecular forces holding them in whatever arrangement they occupy. Pulling two atoms apart against an attractive bond, as happens when a solid melts or a liquid vaporizes, increases this bucket without necessarily changing the kinetic bucket at all, which is exactly why a substance can absorb a large amount of energy during melting or boiling while its temperature does not rise.

Internal energy, given the symbol U, is the sum of both buckets across every particle in the system: U = (sum of microscopic kinetic energies) + (sum of microscopic potential energies). Unlike heat, discussed elsewhere as energy in transit, U is a state quantity: at any given instant, a system in a definite condition, meaning a definite temperature, volume, and phase (solid, liquid, or gas), has one definite value of U, regardless of how it got there. This is what makes U usable for bookkeeping, you can ask how much U changed between two states without knowing the history of the process that connected them.

The formal model: what changes internal energy

Since U is the total microscopic kinetic and potential energy, only processes that alter one of those buckets change U. Three cases illustrate the model concretely.

Heating a substance while holding its volume and phase fixed increases the kinetic bucket, raising both U and, correspondingly, temperature; for many simple substances over modest ranges, the internal energy change is proportional to the temperature change, delta-U = (heat capacity) times (delta-T), where heat capacity measures how many joules it takes to raise the substance's temperature by one degree.

Melting or vaporizing a substance at constant temperature increases the potential bucket, since bonds are being stretched apart or broken, while the kinetic bucket, and hence temperature, stays fixed during the transition; the energy required is called latent heat, entirely a change in U with no accompanying change in temperature, which is why ice and water at the same zero degrees Celsius still have different internal energies.

Compressing a gas by pushing a piston can raise U by doing work on the gas, increasing the kinetic bucket through the piston striking and speeding up gas molecules at the boundary, without any heat crossing the boundary at all, which is why a bicycle pump warms up as you compress the air inside it.

Why the separation from heat and work matters

The essential discipline is this: U describes what a system has, at an instant, independent of history, while heat and work describe what happens to a system during a process, and depend on the particular path taken. A given change in U, say raising a gas's internal energy by five hundred joules, could be produced entirely by heat, entirely by work, or by any combination of the two, and the split between them depends on exactly how the process was carried out, even though the final value of U depends only on the final state. Keeping U as the state quantity and heat and work as the process quantities is what allows a coherent energy account of any thermal or mechanical process whatsoever, from a boiling kettle to a car engine.

Lineage

The recognition that matter stores energy invisibly, beyond its gross motion and position, developed alongside the resolution of the caloric theory of heat in the nineteenth century. Once James Prescott Joule's experiments established that mechanical work and heat are interconvertible, it became necessary to name the quantity that both heat and work change: the energy stored within the substance itself, distinct from the mechanical energy of the substance as a bulk object. Rudolf Clausius and William Thomson (Lord Kelvin) formalized this stored quantity as internal energy in the 1850s, treating it as a state function whose changes obey the conservation law now known as the first law. Fermi's and Callen's textbooks present the modern axiomatic version, and Feynman's lectures connect it back to the kinetic-theory picture of colliding, vibrating particles.

The strongest case for it

Splitting energy into a bulk mechanical part, kinetic and potential energy of the whole object, and an internal part, kinetic and potential energy of its particles, correctly predicts a wide range of otherwise puzzling behavior. It predicts that a substance can absorb substantial energy during melting or boiling with no temperature rise, matching measured latent heats for every known material. It predicts that compressing a gas without any heat exchange still raises its temperature, matching both laboratory adiabatic compression experiments and the everyday warming of a bicycle pump. It also treats internal energy as depending only on the current state, not on history, which is what allows engineers to compute the energy change across a process, such as a piston stroke in an engine, using only the initial and final conditions.

The strongest case against it

The clean split into microscopic kinetic and potential buckets is exact in principle but often impractical to compute directly, since it would require tracking energy across an astronomical number of particles; in practice, internal energy is inferred from macroscopic measurements, such as heat capacities and latent heats, rather than calculated particle by particle. For real substances with strong intermolecular forces, the two buckets are not always cleanly separable, since compressing a real gas changes both the average spacing between molecules, and hence potential energy, and their speeds, and hence kinetic energy, simultaneously, unlike the idealized case of a dilute gas where potential energy between particles is negligible. A common misconception is treating internal energy as identical to heat, saying an object "contains" a certain amount of heat; internal energy is a stored state quantity, while heat, as established elsewhere, is strictly energy in transit across a boundary, and the two must not be interchanged. Another misconception is assuming internal energy depends only on temperature; it also depends on volume and phase, which is exactly why melting changes U without changing temperature at all.

Where it stands now

Internal energy as the total microscopic kinetic and potential energy of a system's particles, and as a state function whose value depends only on the current state, is settled physics with broad consensus, forming one of the two central quantities, alongside heat and work, in classical thermodynamics. It is used without qualification across engineering thermodynamics, chemistry, and physics to account for heating, compression, and phase change in any material system.

Test yourself

A sealed, rigid, insulated container holds a fixed mass of ice at exactly zero degrees Celsius. You slowly add exactly enough energy, entirely as heat, to melt all the ice into water still at zero degrees Celsius, and no more. Explain what happened to the internal energy of the contents, why the temperature did not change even though energy was added, which of the two microscopic buckets, kinetic or potential, changed, and how your account would differ if instead the same amount of energy had been added to liquid water already well above zero degrees Celsius, with no phase change occurring at all.