All naturally occurring processes proceed spontaneously until the state of equilibrium is reached where no further net change occurs in the system. The implication of the equilibrium conditions is that the system is not interacting with the surroundings [4]. Understanding this implication is crucial to distinguish between a system at equilibrium and an open system at steady state. The open system, at steady state, is also characterized by time-invariant properties. However, it is engaged in the interexchange of mass and energy with the surroundings and is not at equilibrium [4]. As stated in section 8.1.1, the state of a system refers to time-invariant conditions present in the system. An implicit assumption in that definition was that the system is at equilibrium; that is, no net interaction with surroundings is occurring.
The second central concern of thermodynamics involves identifying the conditions that represent equilibrium in the system. In a chemical thermodynamic system, the equilibrium state is characterized by the minimum in thermodynamic potential, much the same way equilibrium in mechanical systems is characterized by a minimum in potential (or height), as shown in Figure 8.3.
Figure 8.3 Minimization principle in mechanical and thermodynamic systems.
Source: Matsoukas, T., Fundamentals of Chemical Engineering Thermodynamics with Applications to Chemical Processes, Prentice Hall, Upper Saddle River, New Jersey, 2013.
For chemical engineers, equilibrium problems invariably involve determining accurately the distribution of chemical species in different phases that are in contact with each other. This phase equilibrium problem is shown in Figure 8.4 [6].
Figure 8.4 Essence of a phase equilibrium problem.
Source: Prausnitz, J. M., R. M. Lichtenthaler, and E. G. de Azevedo, Molecular Thermodynamics of Fluid-Phase Equilibria, Third Edition, Prentice Hall, Upper Saddle River, New Jersey, 1999.
Phases α and β, each consisting of the same N components, coexist in contact with each other at a pressure P and temperature T. The composition in each phase is represented by the mole fractions of the components, the subscript representing the component and the superscript representing the phase. The phase equilibrium problem essentially consists of complete characterization of the intensive properties of both phases. In this particular problem, the composition of phase α is known along with either the temperature (or the pressure). The chemical engineer is required to find the composition of the other phase β and the pressure (or the temperature).
The concept of chemical potential provides the foundation for determining these equilibrium conditions. It can be readily understood that when a species is present in two phases that are in contact with each other and is able to distribute itself in both the phases—that is, it is able to cross the phase boundary that separates the two phases—then it will continue to do so until no driving force exists for its movement across the phase boundary. This driving force for the movement of the species across the phases is provided by the difference in the chemical potentials of the species in the two phases, much the same way the temperature difference between two bodies in contact provides the driving force for heat transfer. It follows that this two-phase system will reach equilibrium and there will be no net movement of the species between the phases when its chemical potentials in the two phases are equal, again similar to absence of heat transfer between two bodies that are present at identical temperatures. Mathematically, the equilibrium condition is represented as follows [6]:
Here, μ represents the chemical potential,5 subscript i refers to the species i, and α and β are the two phases.
5. Chemical potential of a species is also equal to its partial molar Gibbs energy, the discussion of equivalence left to the courses in thermodynamics.
Equation 8.8 provides the basis for determining the equilibrium state for the system: It requires computing the chemical potentials of the species in the two phases. However, determining chemical potential is rather complicated, and the equilibrium condition is often expressed in terms of quantity called fugacity, which is a measure of the escaping tendency of the species. The mathematical development of the fugacity and its relationship with the chemical potential are two key topics of chemical engineering thermodynamics courses. Although this mathematical treatment is not covered here, the equality of fugacities as a necessary and sufficient condition for equilibrium can be easily understood from its significance as the escaping tendency of the species. In the following equation, fi represents the fugacity of the species i:
where 
is the escaping tendency of species i from the phase α. Since the phase α is in contact with the phase β, the species i will escape into the phase β. However, the species also has a fugacity 
in phase β, which indicates its tendency to escape from phase β to phase α. So long as the two fugacities are not equal, the species will move from one phase to another depending on in which phase it has a higher fugacity. However, once the two fugacities are equal, equilibrium is reached with no net transfer of the component, as there is no higher preference for escaping from either phase.
The determination of equilibrium of a system is thus essentially a matter of computing the fugacities of the species that are present in the phases in contact with each other. Fugacities, as with other thermodynamic properties, can be readily calculated from the volumetric properties of substances. The accuracy of fugacities is critically dependent on having an accurate mathematical expression that can explain the relationship among the pressure, temperature, and volume of a substance.
Equilibrium in reacting systems is based on similar principles. The equilibrium constant for a reaction can be related to the fugacities and, in turn, to the concentrations/pressures of the species involved in the reaction [8]. The equilibrium constant can be determined from the thermodynamic properties, and concentrations of species or conversion of a reaction can be determined from the equilibrium constant. Analyses of multiphase reacting systems involve simultaneous application of phase and reaction equilibria.
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