Internal energy and the first law

The first law of thermodynamics is nothing more exotic than energy conservation, written for a system's internal energy: ΔU = Q − W, where Q is heat added to the system and W is work done by the system on its surroundings. Add heat and internal energy rises; have the system do work on the outside world and internal energy falls, since that work had to come from somewhere. Neither term alone determines the outcome — a gas compressed (work done on it, negative W) while also losing heat can have its internal energy rise, fall, or stay put depending on which effect wins, which is exactly why the equation keeps both terms rather than assuming one dominates.

The crucial property that makes this bookkeeping work is that internal energy is a state function: its value depends only on the system's current state — its temperature, pressure, composition — and not at all on the history of how it got there. Heat and work, by contrast, are not state functions; they depend entirely on the path taken. Two different processes connecting the same starting and ending states can involve wildly different amounts of heat and work individually, yet ΔU = Q − W must come out identical for both, because ΔU only cares about the endpoints. This is why measuring the change in a state function is often far easier than tracking the process itself in detail — you only need the beginning and the end.

Heat and work are not the same kind of quantity either, despite both being forms of energy transfer. Heat is energy transferred because of a temperature difference, moving spontaneously from hot to cold; work is energy transferred by an organised, directed force acting through a displacement. The first law does not require distinguishing their microscopic mechanisms — both simply add or remove energy from the system's internal energy — but conflating them conceptually obscures why the second law of thermodynamics treats them so differently.