State variable

A macroscopic quantity with a definite value in each equilibrium thermodynamic state.

State variable

A state variable is a macroscopic quantity that is well-defined for each thermodynamic state in the class of states under consideration (typically equilibrium states). Equivalently, it is a coordinate-like function on state space: it assigns to each state a value that can, in principle, be measured without reference to the history of how the state was prepared.

A suitable set of independent state variables uniquely specifies the equilibrium state (the “state postulate”), and all other equilibrium properties are then state functions of that chosen set.

Physical interpretation. Common state variables include:

extensive variables such as volume , internal energy , entropy , and particle number ;
intensive variables such as temperature , pressure , and chemical potential .

The intensive/extensive distinction is formalized by intensive variables and extensive variables ; one often also uses normalized specific quantities (per mass, per mole) or densities.

Conjugate pairs from a fundamental relation. If equilibrium energy is described by a fundamental relation $U=U(S,V,N)$ for a single-component simple compressible system, then conjugate intensive variables are obtained as partial derivatives:

T = \left(\frac{\partial U}{\partial S}\right)_{V,N},\qquad P = -\left(\frac{\partial U}{\partial V}\right)_{S,N},\qquad \mu = \left(\frac{\partial U}{\partial N}\right)_{S,V}.

Here the derivatives are understood in the sense of the partial derivative .

Constraints and equations of state. State variables are not all independent: constraints and an equation of state link them. Differentiating these relations produces measurable response functions such as compressibilities and thermal expansion coefficients.