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 are the thermodynamic variables of a given system, any state variable will be a function 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 subsystems and is the value of the thermodynamic variable relative to the -th subsystem, then the value of the variable relative to the whole system is:

Considering them as functions, a state variable will be extensive if it is a homogeneous function of degree one:
where 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:

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 of a system can only occur if there is an exchange of heat and/or work , such that:

with the convention that if the heat is absorbed by the system and if the work is done by the system; the difference in notation is due to the known fact that is an exact differential, while and 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 will be the sum of the energies and of the two initial systems and the energy 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.
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