The first law of thermodynamics

In brief

An engineer designing a steam engine faces an old temptation: could the engine be built to put out more energy than is fed into it, through some clever arrangement of pistons and valves? The first law of thermodynamics is the flat, tested answer: no. Whatever energy a system stores, its internal energy, can only change because energy crossed its boundary, either as heat, driven by a temperature difference, or as work, driven by an organized mechanical force acting through a distance. There is no third door, and nothing appears from nowhere. This is simply the law of conservation of energy, already familiar from a ball rolling down a hill, extended to include heat as a legitimate way energy moves. Stated as an equation it is compact: the change in internal energy equals the heat added to the system minus the work done by the system, delta-U = Q - W. That equation is the master ledger for every thermal and mechanical process there is.

The full treatment

First look: a sealed cylinder with a movable piston

Picture a cylinder of gas sealed by a piston that can slide in or out, sitting on a stove. Two things can happen to this gas. You can turn on the stove, letting energy flow in because the stove is hotter than the gas, which is heat. Or you can push the piston in by hand, doing mechanical work on the gas by force acting through a distance, or the gas can push the piston out, doing work on whatever is outside. Whatever combination of these two things happens, the gas's own stored energy, its internal energy, changes by exactly the net amount that came in through these two channels combined, no more and no less. If you heat the gas and simultaneously let it expand and push the piston outward, doing work on the surroundings, then part of the heat you added goes toward raising the gas's internal energy and the rest leaves again as the work the gas performs pushing the piston. The first law is simply the demand that this bookkeeping close exactly, with nothing left over and nothing missing.

Building the idea: naming every channel and every sign

To turn the cylinder story into a usable law, three quantities must be defined precisely, each already established elsewhere and now assembled together.

Internal energy, U, is the total microscopic kinetic and potential energy stored in the gas's own particles, a state quantity with a definite value at each instant, independent of history.

Heat, Q, is energy crossing the system's boundary because of a temperature difference between the system and its surroundings, taken positive when it flows into the system.

Work, W, is energy crossing the boundary through an organized mechanical action, a force acting through a distance at the moving boundary, taken here as positive when the system does work on its surroundings, for instance a gas pushing a piston outward, and negative when the surroundings do work on the system, such as a piston compressing the gas.

With these three defined and signed consistently, the first law states: delta-U = Q - W, where delta-U is the change in internal energy between an initial and final state, Q is the total heat added to the system during the process, and W is the total work done by the system during the process. Every term is measured in the same units, joules, since heat, work, and internal energy are all forms of energy.

The derivation: why this is just conservation of energy

The first law is not an independent new principle invented specially for thermodynamics; it is conservation of energy, already established for mechanical systems, extended to include the microscopic energy stored inside matter. Consider all the energy that could possibly exist for the gas and its surroundings combined: the gas's internal energy, plus whatever energy sits in the surroundings, including any device doing work on or receiving work from the piston, and any heat reservoir supplying or absorbing heat. Conservation of energy for this combined system demands that the total energy of gas plus surroundings never changes on its own, only moves between the two parts. Any energy that leaves the surroundings and enters the gas as heat, or as work done on the gas, must show up as an increase in the gas's own internal energy; any energy the gas gives up, as work done on the surroundings, must show up as a decrease in the gas's internal energy, unless offset by heat coming in. Rearranging that bookkeeping statement directly produces delta-U = Q - W: internal energy increases by whatever heat enters, and decreases by whatever work the system does on the outside world. There is no derivation deeper than this, because the first law simply asserts that heat and work are now counted as legitimate members of the energy ledger, alongside kinetic and potential energy already tracked in ordinary mechanics.

Worked mechanism: two ways to change the same gas

Applying the law to a fixed sample of gas in two contrasting processes shows how the same equation handles very different situations.

Heating at constant volume: if the piston is locked in place so the gas cannot expand, no work is done by the gas at all, so W = 0, and delta-U = Q exactly; every joule of heat added shows up as increased internal energy, and correspondingly as a temperature rise, if no phase change occurs.

Isothermal expansion against a piston with heat supplied: if the gas expands slowly while in contact with a heat source that keeps its temperature, and hence internal energy, constant, then delta-U = 0, forcing Q = W exactly; every joule of heat supplied is entirely converted to work done pushing the piston outward, with nothing accumulating as stored energy in the gas. Contrast this with free expansion into an evacuated, insulated chamber, where there is nothing on the other side of the boundary to push against, so W = 0 as well as Q = 0, and delta-U = 0 even though the volume changed, a case that surprises many learners until they see that work requires an opposing force, not merely a change in volume.

Lineage

The first law crystallized in the mid-nineteenth century from two convergent lines of work: James Prescott Joule's careful mechanical experiments in the 1840s, churning water with paddle wheels to measure how much mechanical work produced a given rise in temperature, established a fixed conversion rate between work and heat, showing they are the same kind of quantity, energy, rather than separate substances. Julius Robert von Mayer had independently proposed the equivalence of heat and mechanical work around the same period from physiological and gas-expansion arguments. Rudolf Clausius and William Thomson then assembled these results into the formal statement now called the first law, within the broader mathematical framework of classical thermodynamics they were simultaneously developing, which also produced the second law. Fermi's and Callen's textbooks present the modern, rigorous statement of the law as a foundational axiom of thermodynamics.

The strongest case for it

The first law's authority comes from its complete absence of exceptions across an enormous range of tested processes: chemical reactions, phase changes, gas compressions and expansions, biological metabolism, and every engine ever built or measured obey it exactly, once every channel of energy transfer is properly accounted for. It correctly forbids perpetual motion machines of the first kind, devices that would produce net work with no energy input, a claim tested repeatedly by inventors across centuries and never once achieved. It also predicts quantitative outcomes precisely: knowing the heat added and the work done in an engine cycle allows exact calculation of the internal energy change, matching calorimetric and mechanical measurements across every scale from microscopic chemical reactions to planetary energy budgets.

The strongest case against it

The first law by itself is silent on an important further fact: it does not forbid processes that never actually happen, such as heat spontaneously flowing from a cold object to a hot one, which would still balance the energy ledger perfectly but never occurs in practice, a gap filled only by the second law and its treatment of irreversibility, so the first law alone does not fully describe which conserving processes are physically allowed. The law also depends on drawing the system boundary correctly and identifying every channel of energy crossing it; a common error is forgetting a work term, for instance friction at a piston wall doing additional work not accounted for in an idealized frictionless model, which then makes the books appear not to balance when in fact a channel was simply missed. A common misconception is believing the first law says energy is never lost, full stop, when what it actually guarantees is that total energy including heat is conserved, even though usable, organized energy, such as the capacity to do further work, can still be degraded into less useful forms, a distinct concern addressed by the second law rather than the first. Another misconception is applying delta-U = Q - W to a system whose boundary is ambiguous or moving in an ill-defined way, which produces a nonsensical accounting; the law requires a clearly specified boundary before Q and W can even be measured.

Where it stands now

The first law of thermodynamics stands as one of the most thoroughly tested principles in all of physics, a direct extension of energy conservation with zero confirmed exceptions across every domain it has been applied to, from engine design to cellular metabolism to stellar structure. It is taught and used without qualification throughout physics and engineering, and no serious rival or revision has been proposed since Clausius and Thomson formalized it in the 1850s; refinements since then, chiefly the incorporation of mass-energy equivalence in relativistic contexts, extend rather than contradict the underlying conservation principle.

Test yourself

A gas-filled cylinder with a movable piston starts and ends at the same internal energy, having undergone a full cycle: it was heated and allowed to expand slowly, pushing the piston out and doing 300 joules of work on a load, and then compressed back to its starting volume by an external motor doing work on the gas while releasing heat to a cold reservoir. Using delta-U = Q - W applied separately to the expansion half and the compression half, and using the fact that the internal energy returns to its starting value over the full cycle, determine what must be true about the total heat added during expansion compared to the total heat released during compression, and explain why a cycle like this, which does net work over one full loop, does not violate the first law even though it looks, at a glance, like it produced work from nothing.