Thermodynamic processes

A gas has several variables — pressure, volume, temperature — and a thermodynamic process is what happens when it moves between states. Naming a process by which variable it holds fixed is not a formality: that single constraint, combined with the first law, completely determines how heat, work and internal energy trade off for that process.

An isothermal process holds temperature fixed. Since an ideal gas's internal energy depends on temperature alone, ΔU = 0 throughout, so by the first law every joule of heat added must leave again as work done, and vice versa — heat and work are in lockstep. An isochoric (constant-volume) process holds volume fixed, and since work in this context is force through a changing volume, a process with no volume change does zero work regardless of how much the pressure or temperature shifts; every joule of heat added goes directly into internal energy. An isobaric process holds pressure fixed, letting volume and temperature move together, with heat splitting between raising the internal energy and doing work.

An adiabatic process holds no thermodynamic variable fixed — what it fixes is the exchange of heat, Q = 0, not temperature. This is the misconception to watch for: "adiabatic" and "isothermal" are opposite constraints, not synonyms. With Q = 0, the first law reads ΔU = −W directly: if the gas does work by expanding, that work can only be paid for out of its own internal energy, and since internal energy is tied to temperature, an adiabatically expanding gas necessarily cools — no heat leaves, but the gas still gets colder, because energy has to leave through work even when it cannot leave as heat. This is exactly why a bicycle pump body warms during rapid compression (adiabatic, work done on the gas raises its internal energy) and why gas released rapidly from a canister feels cold (adiabatic expansion, the gas does work at its own internal energy's expense).