fundamental thermodynamic relation
The fundamental thermodynamic relation is the master equation that ties together the headline quantities of thermodynamics in a single line. If the laws of thermodynamics are the constitution, this is the sentence that says how energy, heat, temperature, pressure, and volume must move together whenever a system changes a little. Almost every other formula in the subject can be unfolded from it.
It is usually written dU = TdS − pdV: a tiny change in a system's internal energy U equals the heat it absorbs (temperature T times the change in entropy S) minus the work it does pushing out its boundary (pressure p times the change in volume V). It marries the first law (energy is conserved) with the second (the heat term is tied to entropy), and it holds for any reversible change.
Why it matters: from this one relation flow the definitions of the free energies, the Maxwell relations, and the rules for how every thermodynamic quantity depends on the others. The honest subtlety: written this way it assumes a closed system of fixed composition. When matter can be added or a reaction runs, you must tack on a chemical-potential term, μdn, to keep the books straight.
Hold the volume fixed (dV = 0) and the relation collapses to dU = TdS — any heat added simply raises internal energy. Insulate the system instead (dS = 0 for a reversible adiabat) and it reads dU = −pdV: the energy to do work comes straight out of U.
Pin one variable and the master equation gives you a simple special case.
Although derived for a reversible path, the relation connects only state functions (U, S, V), so the final result holds between any two equilibrium states however the change actually occurred. That is what makes it so widely useful.