What is the name of the equation dU = T dS - P dV?

The equation dU = T dS − P dV is called the fundamental thermodynamic relation. It expresses how a system's internal energy changes with entropy and volume when temperature and pressure describe the equilibrium state.

Here, U is internal energy, S is entropy, T is absolute temperature, P is pressure, and V is volume. The equation follows from combining the first law, dU = δQ − δW, with the reversible relations δQ = T dS and δW = P dV. Although that derivation uses a reversible path, the resulting relation connects state functions and therefore applies between equilibrium states generally.

The same thermodynamic information can be written using other potentials. For example, dH = T dS + V dP applies to enthalpy, while dF = −S dT − P dV applies to Helmholtz free energy. With changing chemical composition, an additional sum of chemical-potential terms, μᵢ dnᵢ, must be included.

The equation is not the Gibbs–Duhem relation or a Maxwell relation; those are related consequences of thermodynamic state-function mathematics.