Which thermodynamic identity relates internal energy changes to entropy, volume, and particle number?

The fundamental thermodynamic relation relates changes in internal energy to entropy, volume, and particle number.

For a simple multicomponent system, it is written dU = T dS − P dV + Σμᵢ dNᵢ. Here U is internal energy, S entropy, V volume, Nᵢ the amount of each particle species, T temperature, P pressure, and μᵢ chemical potential. The equation combines the first and second laws with the energy associated with changing composition.

A common mix-up is the shorter form dU = T dS − P dV. That version assumes fixed particle numbers and only pressure–volume work. The generalized form is needed for chemical reactions, diffusion, phase changes, or particle exchange. Related equations for enthalpy, Helmholtz energy, and Gibbs energy are obtained by changing thermodynamic potentials, but they are alternative forms of the same underlying framework—not different answers to this question.

The relation is especially important because its coefficients identify measurable intensive variables: temperature, pressure, and chemical potential are conjugate to entropy, volume, and particle number.