Pure species phase equilibrium

An alternative criteria for chemical equilibrium between two phases of pure species i can be written ... 

This equation may be applied separately to the liquid phase and to the vapor phase to yield the pure-species values ( ) and ( ) For vapor/ liquid equilibrium (Eq. [4-280]), these two quantities are equal. Given parameters Oj and bj, the pressure P in Eq. (4-230) that makes these two values equal is the equihbrium vapor pressure of pure species i as predicted by the equation of state. 

When the standard states for the solid and liquid species correspond to the pure species at 1 atm pressure or at a low equilibrium vapor pressure of the condensed phase, the activities of the pure species at equilibrium are taken as unity at all moderate pressures. Consequently, the gas phase composition at equilibrium will not be... 

In a system in which there are P pure condensed phases and one chemical reaction at equilibrium, there are (P —1) components. The system is thus univariant and hence indifferent. The state of the system is defined by assigning a value to at least one extensive variable in addition to the mole numbers of the species. The extent of the reaction taking place within the system is dependent upon the value of the additional extensive variable. A simple example is a phase transition of a pure compound when the change of phase is considered as a reaction. We consider the two phases as two species in the one-component system. In order to define the state of the system, we assign values to the volume of the system in addition to the temperature and mole number of the component. For the given temperature and mole number, the number of moles of the component in each phase is determined by the assigned volume. 

The statement is incorrect. P° is the pressure of species of i at the standard state chosen. If i is a component in the gaseous phase, P° = 1 atm. If i is a component in the liquid or solid solution, P° is the vapour pressure which is in equilibrium with species / which is at the standard state. Therefore only if the pure state is chosen at the standard state for the species i, the statement given is true. 

Since coexisting phases of saturated liquid and saturated vapor are in < librium, the equality of fugacities as expressed by Eqs. (11.22) and (11.24) is criterion of vapor/liquid equilibrium for pure species. 

The third plane identified in Fig. 12.1 is the vertical one perpendicular to the composition axis and indicated by MNQRSLM. When projected on a parallel plane, the lines from several such planes present a diagram such as that shown by Fig. 12.4. This is the PT diagram lines t/C, and KC2 are vapor-pressure curves for the pure species, identified by the same letters as in Fig. 12.1. Each interior loop represents the PT behavior of saturated liquid and of saturated vapor for a mixture of fixed composition the different loops are for different compositions. Clearly, the PT relation for saturated liquid is different from that for saturated vapor of the same composition. This is in contrast with the behavior of a pure species, for which the bubble line and the dew line coincide. At points A and B in Fig. 12.4 saturated-liquid and saturated-vapor lines intersect. At such points a saturated liquid of one composition and a saturated vapor of another composition have the same T and P, and the two phases are therefore in equilibrium. The tie lines connecting the coinciding points at A and at B are perpendicular to the PT plane, as illustrated by the tie line VX in Fig. 12.1. 

To avoid some possible difficulties in determining chemical potentials, Lewis proposed a new property called the fugacity /. At low pressure and concentration, the fugacity is a well-behaved function. The fugacity function can define phase equilibrium and chemical equilibrium. For an ideal gas, the fugacity of a species in an ideal gas mixture is equal to its partial pressure. As the pressure decreases to zero, pure substances or mixtures of species approach an ideal state, and we have... 

The phase diagram of a pure species is a plot of pressure versus temperature that shows regions where the species exists as a solid, a liquid, or a gas curves that bound these regions where pairs of phases can coexist in equilibrium and a point (the triple point) where all three phases can coexist,... 

For a pure species coexisting liquid and vapor phases are in equilibrium when they have the same temperature, pressure, and fugacityJ... 

The Clapeyron equation, derived in Sec. 6.4 for the latent heat of phase transition of pure chemical species, is also applicable to pnre-gas adsorption eqnilibrinm. Here, however, the two-phase equilibrium pressure depends not only on temperature, but on surface coverage or the amount adsorbed. Thus the analogous equation for adsorption is written... 

For mixtures the presumption is that the equation of state has exactly the same form as when written for pure species. Equations (4-104) are therefore applicable, with parameters p and q given by Eqs. (4-105) and (4-106). Here, these parameters, and therefore b and a T), are functions of composition. Liquid and vapor mixtures in equilibrium in general have different compositions. The PV isotherms generated by an equation of state for these different compositions are represented in Fig. 4-9 by two similar lines the solid line for the liquid-phase composition and the dashed line for the vapor-phase composition. They are displaced from each other because the equation-of-state parameters are different for the two compositions. 

Note that the fugacity of the pure liquid, P), in Eq. 9.3-11 can be found from the methods of Sec. 7.4b.] As will be seen in Chapters 10 to 12, the calculation of the activity coefficient for each species in a mixture is an important step in many phase equilibrium calculations. Therefore, much of this chapter deals with models (equations) for G and activity coefficients. 

The fugacity function is central to the calculation of phase equilibrium. This should be apparent from the earlier discussion of this chapter and from the calculations of Sec. 7.5, which established that once we had the pure fluid fugacity, phase behavior in a pure fluid could be predicted. Consequently, for the remainder of this chapter we will be concerned with estimating the fugacity of species in gaseous, liquid, and solid mixtures. 

The determination of the activity coefficients of species that exist dominantly as neutral molecules, such as Si02(ag), H2S(ag) and C02(ag), is much simpler. In these cases it is usually possible to establish a two-phase equilibrium between the substance in its pure state (solid or gaseous) and the substance in its aqueous or dissolved state. This leads to a simple and rigorous determination of the activity coefficient in solutions of varying composition. 

Figure 5.1 presents the behaviour of a pure species that can exist as solid, liquid or vapour in a pressure-temperature diagram. We may have three types of two-phase equilibrium solid/liquid, vapour/liquid and solid/vapour. There is a point where all three phases coexist, designated by the triple point. Here the phase rule gives F=C+2-P= +2-3=Q degrees of freedom. Neither pressure nor temperature can be used to modify the equilibrium. If only two phases can be found at equilibrium F=l+2-2=l, and either pressure or temperature can vary. The most important equilibrium in process engineering is vapour-liquid equilibrium, abbreviate as VLE. It may be observed that the two phases will coexist up to a point where it is difficult to make a distinction between vapour and liquid. This is the critical point, a fundamental physical property characterised by critical parameters and. Above the critical point the state... 

The equilibrium condition for a pure species distributed between two phases a and p at the same and uniform P and T can be expressed as the equality of the Gibbs free energy in each phase ... 

A case of practical interest is a chemical reactor coupled with a separation section, from which the unconverted reactants are recovered and recycled. Let s consider the simplest situation, an irreversible reaction A—>B taking place in a CSTR coupled to a distillation column (Fig. 13.5). Here we present results obtained by steady state and dynamic simulation with ASPEN Plus and ASPEN Dynamics. The reader is encouraged to reproduce this example with his/her favourite simulator. The species A and B may be defined as standard components with adapted properties. In this case, we may take as basis the properties of n-propanol and iso-propanol, and assume ideal phase equilibrium. The relative volatility B/A increases at lower pressures, being approximately 1.8 at 0.5 atm. We consider the following data nominal throughput of 100 kmol/hr of pure A, reactor volume 2620 1, and reaction constant =10 s". For stand-alone operation the reaction time and conversion are r= 0.106 hr and = 0.36. 

For multicomponent mixtures, graphical representations of properties, as presented in Chapter 3, cannot be used to determine equilibrium-stage requirements. Analytical computational procedures must be applied with thermodynamic properties represented preferably by algebraic equations. Because mixture properties depend on temperature, pressure, and phase composition(s), these equations tend to be complex. Nevertheless the equations presented in this chapter are widely used for computing phase equilibrium ratios (K-values and distribution coefficients), enthalpies, and densities of mixtures over wide ranges of conditions. These equations require various pure species constants. These are tabulated for 176 compounds in Appendix I. By necessity, the thermodynamic treatment presented here is condensed. The reader can refer to Perry and Chilton as well as to other indicated sources for fundamental classical thermodynamic background not included here. 

Use the Soave-Redlich-Kwong correlation to estimate the compressibility factor, enthalpy (relative to zero-pressure vapor at 0°F) and K-values for the equilibrium phases. Necessary constants of pure species are in Appendix I. All values of fc, are 0.0. Compare estimated -values to exjjerimental /(-values. 

---