Legendre transforms are used to introduce intensive variables as natural variables in the fundamental equations of thermodynamics. Natural variables are important because when a thermodynamic potential can be determined as a function of its natural variables, all the other thermodynamic properties of the system can be obtained by taking partial derivatives. It is usually more convenient to use thermodynamic potentials that have intensive natural variables because they are often more easily controlled than the conjugate extensive variables. This is illustrated for chemical reaction systems in which it is of interest to specify the chemical potential of a species (for example, the pH in a biochemical system). Since the electric potentials of phases in a multi-phase system are not natural variables of the Gibbs free energy, it is useful to define a transformed Gibbs free energy, for which electric potentials of phases are natural variables. The use of a Legendre transform brings in a new set of thermodynamic properties, new Maxwell equations, Gibbs-Duhem equations, and Gibbs-Helmholtz equations.