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#### Developments and Applications in Solubility

**By TM Letcher**

**The Royal Society of Chemistry**

**Copyright © 2007 The Royal Society of Chemistry**

All rights reserved.

ISBN: 978-0-85404-372-9

All rights reserved.

ISBN: 978-0-85404-372-9

CHAPTER 1

*Thermodynamics of Nonelectrolyte Solubility*

EMMERICH WILHELM *Institute of Physical Chemistry, University of Wien, Währinger Straße 42, Wien (Vienna) A-1090, Austria*

*Magic means rather different things to different people*. Brigadier Donald Ffellowes in *"The Kings of the Sea",* by S.E. Lanier, *The Magazine of Fantasy & Science Fiction*, November 1968.

**1.1 Introduction**

The liquid state is one of the three principal states of matter. The majority of chemical synthesis reactions are carried out in the liquid state, and separation processes usually involve liquid/fluid states, *i.e.* solutions. Thus, not surprisingly, for a century and a half experimental investigations of physical properties of solutions and of phase equilibria involving solutions (vapour–liquid equilibrium: VLE; liquid–liquid equilibrium: LLE; solid–liquid equilibrium: SLE; solid–vapour equilibrium: SVE) have held a prominent position in physical chemistry. The scientific insights gained in these studies can hardly be overrated, and have been of immense value for the development of the highly formalized, general discipline of mixture thermodynamics, for instance by providing idealized solution models, such as the one based on the Lewis–Randall (LR) rule, or the one based on Henry's law (HL). In addition to its profound theoretical interest, this topic includes many important practical, industrial applications in chemical process design, in the environmental sciences, in geochemistry, in biomedical technology and so forth. Water is the most abundant liquid on the earth, and because it sustains life as we know it, it is also the most important liquid solvent. The preponderance of scientific papers dealing with aqueous solutions is thus not surprising. We note that the study of the solubility in water of the rare gases and of simple hydrocarbons have provided fundamental information on hydrophobic effects that are thought to be of pivotal importance for the formation and stability of higher order structures of biological substances, such as proteins, nucleic acids, and cell membranes.

Evidently, this short review cannot possibly be comprehensive, and I shall focus on just a few topics which reflect my current research interests and idiosyncrasies. For instance, VLE with supercritical solutes, that is the solubility of gases in liquids, will be discussed in some detail, and so will the van't Hoff type analysis of high-precision solubility data. SLE and SVE will not be considered at all. Almost inevitably, pride of place will be given to the Henry fugacity, or Henry's law constant, which is one of most misunderstood thermodynamic quantities. The goal is to clarify some points often overlooked, and to dispel misconceptions frequently encountered in the literature.

**1.2 Thermodynamics**

In this section I will present a brief overview of classical thermodynamics applicable to nonelectrolyte solutions in general, and to solutions of gases in liquids in particular.' When discussing solutions, one is either interested in *single-phase properties*, such as partial molar volumes, or in quantities which characterize the equilibrium *solubility* itself, for instance the amount of substance *i*, the solute, dissolved in a given amount of solvent *j* in the presence of *both coexisting phases*. The equations governing VLE and LLE will be considered first. For details see refs. 1 and 2.

A general criterion for phase equilibrium at temperature *T* and pressure *P* is the equality of the chemical potential μπ[*i*] of each constituent component *i* in all coexisting phases π, or equivalently, the equality of the fugacity *f*π*i* of each component in all coexisting phases. Thus, for the specific case of VLE (π = V or L),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where *N* is the number of components present, each with mole fraction *X*V*i* in the vapour phase and *X*L*i* in the liquid phase. Similarly, for LLE (π = L' or L")

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

From now on, however, I shall confine attention to *binary* systems, where *i* = 1 or 2.

Two entirely *equivalent* formal procedures are commonly used to establish the link with experimental reality:

(I) When using the fugacity coefficient of component *i* in solution in phase π, which quantity is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

and adopting the convenient notation *X*V*i* = yi, *X*L*i* = *X i*, and dropping the superscript L where unambiguously permissible, the condition for thermodynamic equilibrium (VLE) may be expressed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

and for LLE as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

This approach is called, for obvious reasons, the (Φ, Φ) method.

(II) In the second procedure, the component fugacities in the vapour phase are again expressed in terms of fugacity coefficients, whereas the liquid-phase fugacities of the components are expressed in terms of appropriately normalized liquid-phase activity coefficients.

When based on the LR rule the convention is called *symmetric*, and the corresponding activity coefficient is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

where the superscript asterisk denotes, as always, a pure substance property: fL**i*(T,P) = PΦL**i* (*T, P*) is the *fugacity of pure component i* in either a real or a hypothetical liquid state at (*T,P*) of the liquid solution, and ΦL**i* (*T,P*) is its fugacity coefficient.

When based on HL the convention is called *unsymmetric*, and leads to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

where *hij (T,P*) is the *Henry fugacity of i dissolved in liquid j* at (*T,P*) of the liquid solution. This quantity is also known as Henry's law constant. It is defined for any phase π (L or V) by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

where all the operations are at *constant T and P*.

The VLE conditions may thus be recast into

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

or, equivalently, into

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

where the superscript π = L of the Henry fugacity has been dropped for convenience. This approach is called the (Φ, γ) method.

For LLE we may write either

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

or equivalently

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

For VLE, there exists in principle a third procedure in which the component fugacities in the liquid phase *as well as* in the vapour phase are expressed in terms of activity coefficients (γL,LR*i*,γV,LR*i*, γL,HL*i*,γV,HL*i*). However, to the best of my knowledge, it has never been utilized.

By definition, for component *i* in solution in *any* phase π, Equation (3) applies, hence according to Equation (8) the important, *generally valid* relation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

is obtained, where Φπ∞*i* (*T,P*) is the fugacity coefficient of *i* at infinite dilution in the phase π.

As this juncture, several points should be emphasized. While

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

is a property, at (*T,P*), of pure liquid *i* (GR,L*i denotes the residual molar Gibbs energy), the Henry fugacity defined by Equation (8) for the liquid solution phase (π = L) is a *liquid-phase* property which depends on (*T,P*) and the chemical identity *of both* solute *i* and solvent *j* (hence the double subscript!):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

Here, ΦL∞*i* is the fugacity coefficient of component *i* at infinite dilution in the liquid solvent *j*, and μR,L∞*i* is the corresponding residual chemical potential.

The various quantities corresponding to the two conventions introduced above are, of course, related. For instance,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

where γLR∞*i* denotes the activity coefficient at infinite dilution. For details see ref. 1 and the literature cited therein.

Equations (4), (9), and (10) may each serve as a rigorous thermodynamic basis for the treatment of VLE. The decision as to which approach should be adopted for solving actual problems is by and large a matter of taste and/or convenience, yet is subject to important practical constraints.

VLE involving fairly simple fluids may conveniently be treated in terms of the (Φ, Φ) approach, Equation (4), because the use of a single equation of state (EOS) valid for *both* phases V and L has some computational advantage and a certain aesthetic appeal. However, since no generally satisfactory EOS for dense fluids of practical, that is technical, importance has as yet been developed, this approach is rather limited. The situation is further aggravated by the sensitivity of results on the so-called mixing rules and combining rules, which have always an empirical flavour.

At low to moderate pressures, data reduction and VLE calculations are preferably based upon the classical (Φ, γ) formalism expressed by Equations (9) and (10). Here, an EOS is required only for the low-density vapour phase for which satisfactory models based on virial coefficients are available, while for the liquid phase a suitable activity coefficient model is introduced.

For LLE, similar comments apply: in the majority of cases the (γ, γ) method is used.

Gas solubilities are usually measured at isothermal conditions. Since the equilibrium composition varies with total pressure, for each composition the quantities ΦVi, ΦL*i*, γLR*i*, γHL*i*, fL**i* and *h*2,1 refer to a *different* pressure. For the reduction and correlation of solubility data it is customary and advantageous to select for each temperature the vapour pressure Ps,1(*T*) of the solvent as reference pressure (the subscript s always indicates saturation condition). For temperatures well below the critical temperature of the solvent, the respective correction terms, known as *Poynting integrals*, are usually quite small. If so desired, conversion to any other reference pressure is, in principle, straightforward.

According to Equation (8), the Henry fugacity of solute 2 dissolved in liquid solvent 1 is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

For VLE, because of the phase equilibrium criterion (*f*), fL2 may be set equal to the fugacity of the solute in the coexisting vapour phase, that is

fL2 = fV2 = ΦV2y2P (20)

At the vapour pressure Ps,1, the Henry fugacity pertaining to the *liquid phase* is thus rigorously accessible from *isothermal* VLE measurements at decreasing total pressure P [right arrow] Ps,1 according to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

Entirely equivalent expressions relating the Henry fugacity to limiting slopes, (see Equation (8)), may be derived. We note that from the VLE measurements at *P* >*P*s,1 the liquid-phase activity coefficient γHL2 may be extracted, though frequently experimental imprecision precludes obtaining reliable results.

Another versatile and widely used measure of the solubility of a gas in a liquid is the *Ostwald coefficient*. It is defined by

L2,1(*T,P*)= (ρL2/ρV2) (22)

where ρ2 = n2/v = x2/V = x2ρ, with the appropriate superscript L or V, is the amount-of-substance concentration of solute 2 in either the liquid-phase solution or in the *coexisting* vapour-phase solution at *T* and equilibrium pressure P.

The amounts of solvent 1 and solute 2 are denoted by *n*1 and *n*2, respectively, *v* = (*n*1 + *n*2) *V, V* = ρ-1 is the molar volume of the solution (L or V), and ρ is the (total) molar density of the solution. Thus in contradistinction to the Henry fugacity, the Ostwald coefficient is a *distribution coefficient* pertaining to the solute dissolved in the coexisting phases L and V. It therefore *always* refers to *T* and P of the *actual* VLE. After some algebraic manipulation one can show that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)

where *Z*V*s,1 = *P*s,1*V*V*s,1/*RT* is the compression factor of pure saturated solvent vapour, *V*L*s,1 is the molar volume of pure saturated solvent vapour, VL*s,1 is the molar volume of pure saturated liquid solvent, and ΦV∞2 is the fugacity coefficient of the solute in the vapour phase at infinite dilution. When correlating solubility data over wide temperature ranges up to the critical point, it might be advantageous to use *L*∞2,1 instead of *h*2,1.

The most important application of VLE relations is in the design of zseparation processes. A frequently used measure of the tendency of a given component to distribute itself in one or the other equilibrium phase is the *vapour-liquid distribution* coefficient or *K-value* of solute 2 in solvent 1, *K*2,1(*T,P*) = (*y*2/*x*2)equil. Using Equation (4) the *general* expression

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)

is obtained, which establishes the link with EOS calculations. The infinite-dilution limit of this quantity may thus also be expressed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)

or

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)

Infinite-dilution quantities are usually used for selecting selective solvents for extractive distillation or extraction (γLR∞*i*is needed) or gas absorption (*h*ij is needed) (see, for instance, ref. 15).

1.3 Subtleties of Approximation

Taking into account the pressure dependence of *hij (T,P,*) and γHL2(*T, P, x*2),the equilibrium criterion Equation (10) may be recast into the *key equation* for *isothermal* VLE data treatment (data reduction and correlation) within the *unsymmetric* convention:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28)

This equation provides the rigorous basis for the determination of the activity coefficients γHL2 from *isothermal* solubility data measured at various total pressures *P*. The argument of the logarithmic term on the right-hand side of Equation (28) is a dimensionless group containing the experimental data, the Henry fugacity already extracted therefrom *via* Equation (21), and the vapour-phase fugacity coefficient of the solute which must be either known from independent experiments or calculated from a suitable EOS, say, the virial equation. In order to evaluate the second term on the right-hand side, *i.e.* the Poynting integral, information is needed on the pressure dependence as well as the composition dependence of the partial molar volume *V*L2 of the solute in the liquid phase. Each data point thus yields a constant-temperature, constant-pressure activity coefficient γHL2(*T*, *P*s,1, x2), which may be represented as a function of composition by any appropriate correlating equation compatible with the number and the precision of the experimental results. This is, then, the reward for exacting and tedious experimental work on the solubility of a gas in a liquid: the *Henry fugacity**h*2,1(*T,P*s,1) and a *correlating equation* for γHL2 (*T, P*s,1,*x*2). This classical sequential approach is almost universally adopted in this field and simply reflects the focusing of interest on the solute in a composition range close to pure solvent.

*(Continues...)*

Excerpted fromDevelopments and Applications in SolubilitybyTM Letcher. Copyright © 2007 The Royal Society of Chemistry. Excerpted by permission of The Royal Society of Chemistry.

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