order equations Just as with the internal energy version of the fundamental equation, the chain rule can be used on the above equations to find k+2 equations of state with respect to the particular potential. If Φ is a thermodynamic potential, then the fundamental equation may be expressed as: where the are the natural variables of the potential. If is conjugate to then we have the equations of state for that potential, one for each set of conjugate variables. Only one equation of state will not be sufficient to reconstitute the fundamental equation. All equations of state will be needed to fully characterize the thermodynamic system. Note that what is commonly called "the equation of state" is just the "mechanical" equation of state involving the Helmholtz potential and the volume: For an ideal gas, this becomes the familiar PV=NkBT. Euler integrals Because all of the natural variables of the internal energy U are extensive quantities, it follows from Euler's homogeneous function theorem that Substituting into the expressions for the other main potentials we have the following expressions for the thermodynamic potentials: Note that the Euler integrals are sometimes also referred to as fundamental equations. Gibbs–Duhem relationship Differentiating the Euler equation for the internal energy and combining with the fundamental equation for internal energy, it follows that: which is known as the Gibbs-Duhem relationship. The Gibbs-Duhem is a relationship among the intensive parameters of the system. It follows that for a simple system with r components, there will be r+1 independent parameters, or degrees of freedom. For example, a simple system with a single component will have two degrees of freedom, and may be specified by only two parameters, such as pressure and volume for example. The law is named after Willard Gibbs and Pierre Duhem. Second order equations There are many relationships that follow mathematically from the above basic equations. See Exact differential for a list of mathematical relationships. Many equations are expressed as second derivatives of the thermodynamic potentials (see Bridgman equations). Maxwell relations Maxwell relations are equalities involving the second derivatives of thermodynamic potentials with respect to their natural variables. They follow directly from the fact that the order of differentiation does not matter when taking the second derivative. The four most common Maxwell relations are: {| |- | |width="80"| | |- | |width="80"| | |} The thermodynamic square can be used as a tool to recall and