# State variables

We will call state variable (or function) any quantity which at equilibrium depends only on the thermodynamic variables of a system, rather than on its history. In other words if ${\displaystyle X_{1},\dots ,X_{N}}$ are the thermodynamic variables of a given system, any state variable ${\displaystyle A}$ will be a function ${\displaystyle A(X_{1},\dots ,X_{N})}$ of them. State variables can be either extensive or intensive.

By extensive we mean that it is proportional to the size of the system. In other words, if a system is composed of ${\displaystyle N}$ subsystems and ${\displaystyle X_{\alpha }}$ is the value of the thermodynamic variable ${\displaystyle X}$ relative to the ${\displaystyle \alpha }$-th subsystem, then the value ${\displaystyle {\overline {X}}}$ of the variable relative to the whole system is:

${\displaystyle {\overline {X}}=\sum _{\alpha =1}^{N}X_{\alpha }}$
Considering them as functions, a state variable ${\displaystyle A}$ will be extensive if it is a homogeneous function of degree one:
${\displaystyle A(\lambda X_{1},\dots ,\lambda X_{N})=\lambda A(X_{1},\dots ,X_{N})}$
where ${\displaystyle \lambda }$ is a positive real number. Examples of extensive state variables are the internal energy[1], the mass or the volume of the system.

By intensive we mean that it is independent of the size of the system, namely that the value of the variable relative to a subsystem is equal to that of the whole system.

Intensive state variables are homogeneous functions of degree zero:

${\displaystyle A(\lambda X_{1},\dots ,\lambda X_{N})=A(X_{1},\dots ,X_{N})}$
Examples of intensive variables are the pressure and the temperature of a system.

The fundamental hypothesis of thermodynamics is that any thermodynamic system can be characterized at equilibrium with a finite set of thermodynamic variables.

Finally, we also know that thermodynamic theory is based upon four laws (or postulates). For example, it is well known that the first law of thermodynamics is a restatement of the conservation of energy, namely a change of the internal energy ${\displaystyle dU}$ of a system can only occur if there is an exchange of heat ${\displaystyle \delta Q}$ and/or work ${\displaystyle \delta W}$, such that:

${\displaystyle dU=\delta Q-\delta W}$
with the convention that ${\displaystyle \delta Q>0}$ if the heat is absorbed by the system and ${\displaystyle \delta W>0}$ if the work is done by the system; the difference in notation is due to the known fact that ${\displaystyle dU}$ is an exact differential, while ${\displaystyle \delta Q}$ and ${\displaystyle \delta W}$ aren't.

1. Note that, strictly speaking, internal energy is not extensive, but it is so for large enough systems. In fact if we put two systems 1 and 2 in contact, the internal energy of the total system ${\displaystyle U_{\text{TOT}}}$ will be the sum of the energies ${\displaystyle U_{1}}$ and ${\displaystyle U_{2}}$ of the two initial systems and the energy ${\displaystyle U_{\text{int.}}}$ due to the surface interaction between the two, assuming that the components of the two systems interact via a short-range potential. However, the internal energies of the two systems are proportional to their volumes, while the interaction energy is proportional to their interaction surface; since the surface grows much slower than the volume with respect to the size (unless the systems have very exotic and strange shapes, a case which of course we ignore), for large systems this interaction energy can be serenely ignored.