General condition for equilibrium

A system is heterogeneous if some properties have different values in different portions of the system when the system is at equilibrium. Two scenarios may exist, where variations of the properties can be either continuous or discontinuous. In the scenario of continuous variations, the gradients of the variations must be coupled so that the system remains at equilibrium. The number of independent variables is thus reduced. These gradients must also be constrained along the boundaries between the system and the surroundings. This type of constrained equilibrium is not discussed in the book as it involves heterogeneous boundary conditions between the system and the surroundings and depends on the morphology of the system.

In the second scenario, with discontinuous variations, the properties have different values in different portions of the system, but remain homogenous within each portion. The system is in equilibrium as each portion is in equilibrium with all other portions of the system. The homogeneous portions represent different phases in the system, with the properties in each phase being homogeneous at equilibrium. In the previous chapter, it was been shown that all potentials are homogeneous in a homogeneous system.

For a heterogeneous system, the same conclusion can be obtained. If the temperature is inhomogeneous, heat can be conducted from high temperature locations to low temperature locations, and this process is irreversible based on the second law of thermodynamics because it increases the internal entropy of the system. If the pressure is inhomogeneous, the amounts of lower molar volume phases will increase to reduce the internal energy of the system. If the chemical potential of a component is inhomogeneous, the chemical potential difference of the component will drive that component to locations with a lower chemical potential in order to decrease the internal energy of the system. Therefore, it can be concluded that all potentials are homogeneous in a heterogeneous system at equilibrium, and the variables that are not homogeneous are thus their conjugate molar quantities. Under certain special circumstances, to be discussed later in this book, some molar quantities may also have the same values in difference phases.