constant T and p equilibrium.

state of the system state of the system Fig.1.18 Equilibrium conditions for certain restraints

In the left-hand sketch, the "state of the system" might be temperature nonuniformity inside the impervious boundary. Or, it could be an unequal distribution of the material within the boundary, as illustrated in the example of Fig. 1.17. Off-equilibrium states in the right-hand diagram most often represent a chemically reactive mixture that has not attained chemical equilibrium.

In a multiphase system, additional requirements for equilibrium are the equality of temperature and pressure in each phase. Multicomponent systems entail additional equilibrium requirements related to the concentrations of the various species they contain: a property called the chemical potential of each species must be the same in all phases. The chemical potential of a species is related to (but is not equal to) its concentration. Detailed discussion of this topic is deferred to Chap. 7.

1.11.2 External Equilibrium

If the boundaries can transmit heat and matter and can expand contract, the system and its surroundings must satisfy the following:

• the temperatures of the system and its surroundings must be the same

• the pressure of the system and the surroundings must be equal

• the chemical potential of all species to which the boundary is permeable must be the same in the system and in the surroundings.

1.12 Components, Phases, and the Gibbs Phase Rule

7 Henceforth, the Gibbs free energy is termed "free energy"

Until now, the number of components in the system has not been addressed, and the examples illustrating concepts have been based on a single phase. Components are distinct chemical species whose quantities can be independently varied. Phases are regions of a system in which all properties are uniform and are distinct from other regions in the same system.

Single-phase systems are gas, liquid, or solid. Two-phase systems are combinations of these phases, and correspond to the following forward and reverse processes:

• gas-liquid: vaporization/condensation

• gas-solid: sublimation/condensation

• liquid-solid: melting (fusion)/freezing (crystallization)

Somewhat less common than the above combinations of phases are two-liquid systems and two solid systems. When two liquids do not mix, such as oil and water, they are said to be immiscible. Systems consisting of two coexisting solids are also quite common; metal and its oxide, such as rust on iron, are important in metallurgy and in the natural process of corrosion.

1.12.1 One-component systems

In a one-component system, coexistence of solid, liquid, and vapor occurs at a unique set of conditions called the triple point. The triple point of water is at 0.01oC and 0.61 kPa, and so is not observed except in the laboratory where the pressure can be reduced below atmospheric pressure (100 kPa). Every pure substance (element or compound) has a vapor-liquid-solid triple point with its associated unique p - T combination. The phase field of a pure (i.e., one component) substance in the pressure-temperature space is illustrated schematically in Fig. 1.19.

This diagram contains three curves that separate areas of stability of solid, liquid and vapor phases. Single-phase states exist over wide regions of the p - T space. States along the curves represent the three common two-phase systems mentioned above. The sublimation curve extends in principle to the absolute zero temperature, and the melting line continues, in theory, to unlimited pressures. However, the vaporization curve ends at a distinct p - T combination called the critical point. Beyond this point, there is no difference in the properties of liquid and vapor phases, and the interface between the two no longer exists.

1.12.2 Two-component systems

In much of materials science, there is a need to represent the state of multicomponent, multiphase systems in which the gas phase is excluded for convenience and chemical reactions are not involved. Such representations are called phase diagrams. They show the regions of existence of various phases on a plot with temperature as the ordinate and composition as the abscissa. A simple phase diagram of the binary A-B system is shown in Fig. 1.20. The ends of

0 i xB = mole fraction of component B

Fig. 1.20 The phase diagram for an ideal A-B binary system the lens-shaped portion of the diagram intersect at the melting points of pure A, TMA, on the left, and the corresponding temperature TMB for component B on the right. Below the lower curve, the system is a single-phase solid solution, in which A and B are homogeneously mixed (although in a specific crystal lattice structure called a). Above the upper curve, the system is a single liquid without, of course, a regular structure. In between these two curves is the two-phase region designated a+L, where the solid and the liquid coexist.

1.12.3 Counting components

Specification of the number of components in a system is less clear-cut than fixing the number of phases present. Each chemical species of fixed molecular makeup that can be mixed in arbitrary amounts in the system is considered to be a component. Composition is dictated by various measures of concentration. The most common is the mole fraction, or occasionally the mass fraction. If the system contains C components, only C-1 mole fractions need be specified in order to fix unambiguously the composition. For example, a mixture of O2 and N2 is a two-component (or binary) system, but the mole fraction of only one component is needed to specify the system's composition. In addition, if the composition of the mixture does not change during the process under consideration, the binary system can be treated as a pseudo single-component system. In flowing through a valve, for example, air can be considered as a single species with thermodynamic properties that are the concentration-weighted average of the pure-species values.

If a chemical reaction occurs between the species in a system, the number of components is reduced by one. The system M + MO2 + O2 contains three molecular (or elemental) species but only two components, M and O. Another example is a gas containing H2, O2 and H2O. At low temperature and in the absence of an ignition source, the hydrogen does not burn and the mixture is a true three-component (or ternary) system. At high temperatures, on the other hand, the chemical reaction: 2H2(g) + O2(g) = 2H2O(g) provides a relation between the concentrations of the three molecular species. This restraint effectively reduces the number of components from three to two.

1.12.4 Proof of the Phase Rule

Figures 1.19 and 1.20 suggest the existence of a fundamental relationship between the number of components in a system, the number of coexisting phases at equilibrium, and the number of variables (temperature, pressure, composition) that can be independently fixed without altering the number of phases present. The number of independently-controllable variables is called the degrees of freedom of the system, and is designated as F.

To be perfectly general, let the system contain C components, labeled A, B, C,... that are distributed among P phases designated a, P, y,... Because the sum of the mole fractions is unity in each phase, there are C -1 independent composition variables in each phase. Since the system contains P phases, the total number of independently adjustable compositions is P(C-1).

Other independent variables are the temperature and pressure in each phase. For the moment, we allow these properties to be different in each phase. These quantities add P variable temperatures and the same number of variable pressures. The total number of potentially independent variables describing the system is thus:

However, the system is required to be in a state of internal equilibrium, which provides restraints on the above property variations. The most obvious are the requirements of temperature and pressure equality in all phases. Each of these conditions provides P-1 restraints (i.e., if the system is two-phase a+P, the single temperature restraint is

Ta = T p). Together, temperature and pressure equality in all phases provides 2(P-1) restraints.

The final internal equilibrium requirement that the chemical potentials of each component must be the same in all phases provides P-1 restraints for each component. For a system containing C components, this particular equilibrium condition contributes C (P-1) restraints on the compositions in the phases of the system. Summing all of the restraints yields:

number of restraints = 2(P-1) + C(P-1) = (P-1)(C +2)

The difference between the number of potentially variable properties and the number of restraints is equal to the number of degrees of freedom actually allowed to the system. Subtracting the above two equations gives:

This equation is the famous Gibbs phase rule. Applying it to a pure substance (C = 1) gives F = 3-P. If only one phase is present (P=1), then F = 2. This result corresponds to the areas labeled solid, liquid, and vapor in Fig. 1.19. In these regions, both p and T can be independently set. However if two phases of a pure substance are to coexist (P=2), the phase rule permits only one degree of freedom (F=1). Consequently, if T is specified, p follows (or vice versa). This restriction applies along the sublimation, vaporization, and melting curves in Fig. 1.19. Finally, at the triple point, P=3, so F=0, which means that the conditions for simultaneous equilibrium of the three states of matter are unique.

Figure 1.20 represents a two-component system, so Eq (1.21) reduces to F = 4 - P. However, in phase diagrams involving only condensed phases, the total pressure is fixed (and has little influence in any case), thereby removing one degree of freedom. The phase rule then reduces to F = 3 - P. In the single-phase solid and liquid zones of Fig. 1.20, P=1 and F=2. This means that both temperature and composition can be independently varied without altering the single-phase character of the system. In the a+L two-phase region of Fig. 1.20, there is only one degree of freedom; if the temperature is specified and if the overall composition (including both phases) places the point inside the lens-shaped a+L region, the compositions of the coexisting solid and liquid phases are fixed. The simple binary phase diagram of Fig 1.20 has no invariant point analogous to the triple point in Fig. 1.19. However, more complex binary phase diagrams exhibit points where three phases coexist at a unique temperature.

The phase rule is modified if species in the system engage in an equilibrium chemical reaction, which in effect removes one degree of freedom. However, the number of components is equal to the number of molecular species, not the number of elements. For N multiple simultaneous equilibria, the phase rule reads:

Example: The gas-phase reaction CO + ^ O2 = CO2 contains a single phase and two components: either the elements C and O or the species CO, O2 and CO2 less one reaction connecting them. In either interpretation, F = 2 - i + 2 = 3. The degrees of freedom are temperature, total pressure and one composition.

The phase rule can be applied more generally than it has been to the examples discussed above. It is a powerful tool for making order from complex multicomponent, multiphase thermodynamic systems.


1. M. Abbott & H. C. Van Ness, Theory and Problems in Engineering Thermodynamics, McGraw-Hill (1989)

2. M. Potter and C. Somerton, Theory and Problems in Engineering Thermodynamics, McGraw-Hill (1993)

Problems for Chap. 1

1.1 Determine the degradation of high-quality input work, W, or energy, E, into heat Qirr (as the ratio Qirr/W or Qhr/E) for the following irreversible processes:

(a) Pulling at constant speed a solid block on a horizontal surface

(b) Expanding a gas in a cylinder with a piston. Because of friction between cylinder and piston, movement of the piston requires that the pressure of the gas (p) be greater than the pressure of the surroundings (psurr ).

(c) Suppling a current i from a battery to a circuit containing a resistance R.

(d) Loading a mass m horizontally on the top of a vertical spring, which has a force constant k. After the initial oscillations are damped out, the spring reaches an equilibrium compression distance x.

1.2 The process in problem 1.1(d) can be made reversible in the following manner. A number N of smaller masses Am that are arranged on vertically-stacked shelves next to the spring. The small weights are loaded on to the spring sequentially, each one compressing the spring a small amount. Loading of each small mass is done horizontally as the top of the spring reaches the elevation of the shelf holding the small mass.

Prove that the energy stored in the spring is equal to the loss of potential energy of the distributed masses, which means that the process is reversible.

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