Fundamental thermodynamic relation

The fundamental equation of thermodynamics (also fundamental relation or Gibbs fundamental equation after Josiah Willard Gibbs ) is the starting point of formal thermodynamics. It is the most important characteristic feature and describes the set of all the equilibrium points of a thermodynamic system as a function of state quantity of the internal energy U of all sizes extensive

In non-magnetic component systems, the natural variables to entropy S, volume V, and amount of substance n simplify:

This also applies analogously for non-magnetic multi-component systems with k different materials:

Equivalent function can also be specified in the form

Both functions each contain the entire thermodynamic information of the system under consideration. The mathematical structure of thermodynamics is thus established. Further, especially physical, content is found by connecting to the main theorems.

Often a differential notation is used:

The definitions for the temperature T, the pressure p and the chemical potential of the following:

Under the assumption of a constant amount of substance () to this further simplifies to:

It follows from this that the equations of state are the first derivatives of the fundamental equation of the principle.

For mathematical theorems on differentiable functions of several variables can relations of the second derivatives are found: the Maxwell relations. For these second derivatives of the experimentally important response coefficients can be derived, such as specific heat capacity, compressibility and thermal expansion coefficient.

The Legendre transform of the fundamental relation leads to the thermodynamic potentials: free energy, enthalpy and Gibbs free energy.