Solution-phase equilibria

When only the total system composition, pressure, and temperature (or enthalpy) are specified, the problem becomes a flash calculation. This type of problem requires simultaneous solution of the material balance as well as the phase-equilibrium relations. 

It follows that, because phase equilibrium requires that the chemical potential p. be the same in the solution as in the gas phase, one may write for the chemical potential in the solution ... 

In this chapter we shall consider some thermodynamic properties of solutions in which a polymer is the solute and some low molecular weight species is the solvent. Our special interest is in the application of solution thermodynamics to problems of phase equilibrium. 

These are the criteria of phase equilibrium apphed in the solution of practical problems. 

Vapor/liquid equilibrium (XT E) relationships (as well as other interphase equihbrium relationships) are needed in the solution of many engineering problems. The required data can be found by experiment, but such measurements are seldom easy, even for binaiy systems, and they become rapidly more difficult as the number of constituent species increases. This is the incentive for application of thermodynamics to the calculation of phase-equilibrium relationships. 

The N equations represented by Eq. (4-282) in conjunction with Eq. (4-284) may be used to solve for N unspecified phase-equilibrium variables. For a multicomponent system the calculation is formidable, but well suited to computer solution. The types of problems encountered for nonelectrolyte systems at low to moderate pressures (well below the critical pressure) are discussed by Smith, Van Ness, and Abbott (Introduction to Chemical Engineering Thermodynamics, 5th ed., McGraw-Hill, New York, 1996). 

Availability of large digital computers has made possible rigorous solutions of equilibrium-stage models for multicomponent, multistage distillation-type columns to an exactness limited only by the accuracy of the phase equilibrium and enthalpy data utilized. Time and cost requirements for obtaining such solutions are very low compared with the cost of manual solutions. Methods are available that can accurately solve almost any type of distillation-type problem quickly and efficiently. The material presented here covers, in some... 

Ternary-phase equilibrium data can be tabulated as in Table 15-1 and then worked into an electronic spreadsheet as in Table 15-2 to be presented as a right-triangular diagram as shown in Fig. 15-7. The weight-fraction solute is on the horizontal axis and the weight-fraciion extraciion-solvent is on the veriical axis. The tie-lines connect the points that are in equilibrium. For low-solute concentrations the horizontal scale can be expanded. The water-acetic acid-methylisobutylketone ternary is a Type I system where only one of the binary pairs, water-MIBK, is immiscible. In a Type II system two of the binary pairs are immiscible, i.e. the solute is not totally miscible in one of the liquids. 

The working capacity of a sorbent depends on fluid concentrations and temperatures. Graphical depiction of soration equilibrium for single component adsorption or binary ion exchange (monovariance) is usually in the form of isotherms [n = /i,(cd or at constant T] or isosteres = pi(T) at constant /ij. Representative forms are shown in Fig. I6-I. An important dimensionless group dependent on adsorption equihbrium is the partition ratio (see Eq. 16-125), which is a measure of the relative affinities of the sorbea and fluid phases for solute. 

Raoult s Law. The molar composition of a liquid phase (ideal solution) in equilibrium with its vapor at any temperature T is given by... 

There is one other three-phase equilibrium involving clathrates which is of considerable practical importance, namely that between a solution of Q, the clathrate, and gaseous A. For this equilibrium the previous formulas and many of the following conclusions also hold when replacing fiQa by fiQL, the chemical potential of Q in the liquid phase. But a complication then arises since yqL and the difference are not only... 

Let us first consider the three-phase equilibrium ( -clathrate-gas, for which the values of P and x = 3/( +3) were determined at 25°C. When the temperature is raised the argon content in the clathrate diminishes according to Eq. 27, while the pressure can be calculated from Eq. 38 by taking yA values following from Eq. 27 and the same force constants as used in the calculation of Table III. It is seen that the experimental results at 60°C and 120°C fall on the line so calculated. At a certain temperature and pressure, solid Qa will also be able to coexist with a solution of argon in liquid hydroquinone at this point (R) the three-phase line -clathrate-gas is intersected by the three-phase line -liquid-gas. At the quadruple point R solid a-hydroquinone (Qa), a hydroquinone-rich liquid (L), the clathrate (C), and a gas phase are in equilibrium the composition of the latter lies outside the part of the F-x projection drawn in Fig. 3. The slope of the three-phase line AR must be very steep, because of the low solubility of argon in liquid hydroquinone. 

The equilibria between clathrate and gas, and Qa, clathrate, and gas could be determined by using w-propanol as the auxiliary solvent.53 In the latter equilibrium, the composition of the clathrate is found from the amount of gas required for the conversion of a given amount of solid a-hydroquinone suspended in the propanol solution into clathrate at constant temperature and pressure. The dissociation pressure of the clathrate is given by the total pressure of the four-phase equilibrium -clathrate-solution-gas, corrected for the vapor pressure of w-propanol saturated with a-hydroquinone. Using this technique it was found that the equilibrium clathrates of hydroquinone and argon have yA = 0.34 at 25°C63 and 0.28 at 60°C.28... 

In most cases the critical temperature of the solute is above room temperature. As can be seen in the binary system H2S-H20 drawn in Fig. 6, the three-phase line HL2G is then intersected by the three-phase line HL G. The point of intersection represents the four-phase equilibrium HLXL2G and indicates the temperature... 

In our discussion of (vapor + liquid) phase equilibria to date, we have limited our description to near-ideal mixtures. As we saw in Chapter 6, positive and negative deviations from ideal solution behavior are common. Extreme deviations result in azeotropy, and sometimes to (liquid -I- liquid) phase equilibrium. A variety of critical loci can occur involving a combination of (vapor + liquid) and (liquid -I- liquid) phase equilibria, but we will limit further discussion in this chapter to an introduction to (liquid + liquid) phase equilibria and reserve more detailed discussion of what we designate as (fluid + fluid) equilibria to advanced texts. 

Other measurements of AfG involve measuring AG for equilibrium processes, such as the measurement of equilibrium constants, reversible voltages of electrochemical cells, and phase equilibrium measurements. These methods especially come into play in the measurement of Afand AfG for ions in solution, which are processes that we will now consider. 

The penetration theory has been used to calculate the rate of mass transfer across an interface for conditions where the concentration CAi of solute A in the interfacial layers (y = 0) remained constant throughout the process. When there is no resistance to mass transfer in the other phase, for instance when this consists of pure solute A, there will be no concentration gradient in that phase and the composition at the interface will therefore at all Limes lie the same as the bulk composition. Since the composition of the interfacial layers of the penetration phase is determined by the phase equilibrium relationship, it, too. will remain constant anil the conditions necessary for the penetration theory to apply will hold. If, however, the other phase offers a significant resistance to transfer this condition will not, in general, be fulfilled. 

In order to begin this presentation in a logical manner, we review in the next few paragraphs some of the general features of polymer solution phase equilibrium thermodynamics. Figure 1 shows perhaps the simplest liquid/liquid phase equilibrium situation which can occur in a solvent(l)/polymer(2) phase equilibrium. In Figure 1, we have assumed for simplicity that the polymer involved is monodisperse. We will discuss later the consequences of polymer polydispersity. 

The computational problem of polymer phase equilibrium is to provide an adequate representation of the chemical potentials of each component in solution as a function of temperature, pressure, and composition. 

From the outset, Flory (6) and Huggins (4,5 ) recognized that their expressions for polymer solution thermodynamics had certain shortcomings (2). Among these were the fact that the Flory-Huggins expressions do not predict the existence of the LCST (see Figure 2) and that in practice the x parameter must be composition dependent in order to fit phase equilibrium data for many polymer solutions 3,8). 

Values of p can be determined, in principle, from any phase equilibrium data. A small table of p 2 values is available in reference (2). However, one of the most straightforward ways of determining pf values is to fit phase equilibrium data for solvent sorption in concentrated polymer solutions. To do this, equations (2) and (13) are combined to solve for p utilizing experimental partial pressure data. 

There are two types of multicomponent mixtures which occur In polymer phase equilibrium calculations solutions with multiple solvents or pol ers and solutions containing poly-disperse polymers. We will address these situations In turn. 

There are relatively few phase equilibrium data relating to concentrated polymer solutions containing several solvents. Nevertheless, In polymer devolatilization, such cases are often of prime Interest. One of the complicating features of such cases Is that. In many Instances, one of the solvents preferentially solvates the polymer molecules, partially excluding the other solvents from Interaction directly with the polymer molecules. This phenomenon Is known as "gathering". 

We have reviewed here, in the brief space available, some recent developments in phase equilibrium representations for polymer solutions. With these recent developments, reliable tools have become available for the polymer process designer to use in considering effects of phase equilibrium properly. 

When a solution is saturated, there is dynamic equilibrium. Solute molecules move back and forth between the solute phase and the solution phase at equal rates. 

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