Solutions: thermodynamics

Chemical Thermodynamics of Materials by Svein St0len and Tor Grande 2004 John Wiley Sons, Ltd ISBN 0 471 492320 2 

In practice, many of the extensive functions (IJ, S, H, G), and their corresponding partial molar quantities, can only be determined up to an additive constant—absolute values of U, H, and can be obtained from simulation, but these then depend on the model of choice. Hence, their values are typically expressed with respect to a set of defined reference or standard states. These are usually taken as the pure solutions of each component at the same T and p. One can then define a series of mixing quantities such that 

This concept is also often applied to the volume of the solution even though absolute molar volumes of mixing can be determined quite easily. Other standard states are possible. The most common alternative involves an infinitely dilute solute, also known as the Henry s law standard state. 

A series of excess quantities may then be defined using the properties of ideal solutions such that 

Obtaining the derivatives on the right-hand side requires a fitting equation for the excess mixing quantities. The Wilson, Redlich-Kister, and nonrandom two liquid (NRTL) model equations are some of the most commonly used (Poling, Praunitz, and O Connell 2000). Some additional practical considerations are also provided in Section 1.3.9 and Section 4.2 in Chapter 4. 

New York, 1999. Smith, J. M., H. C. Van Ness, and M. M. Abbott, Introduction to Chemical Engineering Thermodynamics, 7th ed., McGraw-Hill, New York, 2005. Tester, J. W., and M. Modell, Thermodynamics and Its Applications, 3d ed., Prentice-Hall PTR, Upper Saddle River, N.J., 1997. Van Ness, H. C., and M. M. Abbott, Classical Thermodynamics of Nonelectrolyte Solutions With Applications to Phase Equilibria, McGraw-Hill, New York, 1982. 

Thermodynamics is the branch of science that lends substance to the principles of energy transformation in macroscopic systems, The general restrictions shown by experience to apply to all such transformations are known as the laws of thermodynamics. These laws are primitive they cannot be derived from anything more basic. 

The first law of thermodynamics states that energy is conserved, that although it can be altered in form and transferred from one place to another, the total quantity remains constant. Thus the first law of thermodynamics depends on the concept of energy, but conversely energy is an essential thermodynamic function because it allows the first law to be formulated. This coupling is characteristic of the primitive concepts of thermodynamics. 

The words system and surroundings are similarly coupled. A system can be an object, a quantity of matter, or a region of space, selected for study and set apart (mentally) from everything else, which is called the surroundings. An envelope, imagined to enclose the system and to separate it from its surroundings, is called the boundary of the system. 

When a system is isolated, it cannot be affected by its surroundings. Nevertheless, changes may occur within the system that are detectable with measuring instruments such as thermometers and pressure gauges. However, such changes cannot continue indefinitely, and the system must eventually reach a final static condition of internal equilibrium. 

The total number of gram-atoms of the component A is Nlt that of component C is N2, Then if nl,n2, and n3 are the number of moles of a, c, and ac species, respectively, conservation of the number of atoms of each type gives 

The Gibbs energy change upon mixing N2 gram-at. of A and N2 of C to 

Since AGM is a state function that is extensive in nu n2, and n3, i.e., a homogeneous function of the first degree in nu n2, and n3, Euler s theorem gives 

Thus the relative chemical potential of each component equals that of the corresponding uncombined species and we shall no longer distinguish between them. We note then that we can rewrite Eq. (26) as 

These are the first important relations of solution thermodynamics that we require. 

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We shall discuss three types of phenomena for polymer solutions thermodynamic properties in Chap. 8, frictional properties in Chap. 9, and lightscattering properties in Chap. 10. A common feature of virtually all phenomena in these areas is that they all depend on the molecular weight of the solute. Thus observations of these properties can be interpreted to yield values for M we shall use this capability as a unifying theme throughout these chapters. 

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. 

Idea.1 Liquid Solutions. Two limiting laws of solution thermodynamics that are widely employed are Henry s law and Raoult s law, which represent vapor—Hquid partitioning behavior in the concentration extremes. These laws are used frequently in equiUbrium problems and apply to a variety of real systems (10). 

P rtl IMol r Properties. The properties of individual components in a mixture or solution play an important role in solution thermodynamics. These properties, which represent molar derivatives of such extensive quantities as Gibbs free energy and entropy, are called partial molar properties. For example, in a Hquid mixture of ethanol and water, the partial molar volume of ethanol and the partial molar volume of water have values that are, in general, quite different from the volumes of pure ethanol and pure water at the same temperature and pressure (21). If the mixture is an ideal solution, the partial molar volume of a component in solution is the same as the molar volume of the pure material at the same temperature and pressure. 

The solvophobic model of Hquid-phase nonideaHty takes into account solute—solvent interactions on the molecular level. In this view, all dissolved molecules expose microsurface area to the surrounding solvent and are acted on by the so-called solvophobic forces (41). These forces, which involve both enthalpy and entropy effects, are described generally by a branch of solution thermodynamics known as solvophobic theory. This general solution interaction approach takes into account the effect of the solvent on partitioning by considering two hypothetical steps. Eirst, cavities in the solvent must be created to contain the partitioned species. Second, the partitioned species is placed in the cavities, where interactions can occur with the surrounding solvent. The idea of solvophobic forces has been used to estimate such diverse physical properties as absorbabiHty, Henry s constant, and aqueous solubiHty (41—44). A principal drawback is calculational complexity and difficulty of finding values for the model input parameters. 

Cullinan presented an extension of Cussler s cluster diffusion the-oiy. His method accurately accounts for composition and temperature dependence of diffusivity. It is novel in that it contains no adjustable constants, and it relates transport properties and solution thermodynamics. This equation has been tested for six very different mixtures by Rollins and Knaebel, and it was found to agree remarkably well with data for most conditions, considering the absence of adjustable parameters. In the dilute region (of either A or B), there are systematic errors probably caused by the breakdown of certain implicit assumptions (that nevertheless appear to be generally vahd at higher concentrations). 

Whatever the specific system or situation, the key issue in diffusion interphase adhesion is physical compatibility. This is once again, a thermodynamic issue and may be quantified in terms of mutual solubility. Most of the strategies for predicting diffusion interphase adhesion are based on thermodynamic compatibility criteria. Thus it is appropriate to review briefly the relevant issues of solution thermodynamics and to seek quantitative measures of compatibility between the phases to be bonded. 

A theory of regular solutions leading to predictions of solution thermodynamic behavior entirely in terms of pure component properties was developed first by van Laar and later greatly improved by Scatchard [109] and Hildebrand [110,1 11 ]. It is Scatchard-Hildebrand theory that will be briefly outlined here. Its point of departure is the statement that It is next assumed that the volume... 

Vapour Pressures of Dilute Solutions Thermodynamic Theory. 

The difficulties engendered by a hypothetical liquid standard state can be eliminated by the use of unsymmetrically normalized activity coefficients. These have been used for many years in other areas of solution thermodynamics (e.g., for solutions of electrolytes or polymers in liquid solvents) but they have only recently been employed in high-pressure vapor-liquid equilibria (P7). 

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). 

Pn gogine s work led to a complete representation of polymer solution thermodynamics. Because of the form of Prigogine s expressions, they are often referred to as free-volume expressions. 

Unfortunately, relatively little work has been done on the solution thermodynamics of concentrated polymer solutions with "gathering". The definitive work on the subject Is the article of Yamamoto and White (17). The corresponding-states theory of Flory (11) does not account for gathering. We therefore restrict our consideration here to multicomponent solutions where the solvents and polymer are nonpolar. For such solutions, gathering Is unlikely to occur. 

The corresponding-states theory of polymer solution thermodynamics, developed principally by Prigogine and Flory, has provided a reliable predictive tool requiring only minimal information. We have seen here several examples of the use of the corresponding-states theory. We have also seen that the corresponding-states theory is a considerable improvement over the older Flory-Huggins theory. 

The ultimate goal is to bring together knowledge of solution thermodynamics, rheology and polymerization kinetics so that the latter, which is well described at low conversion (12,13), may be better described in a continuous manner to complete conversion. 

This stipulation of the interaction parameter to be equal to 0.5 at the theta temperature is found to hold with values of Xh and Xs equal to 0.5 - x < 2.7 x lO-s, and this value tends to decrease with increasing temperature. The values of = 308.6 K were found from the temperature dependence of the interaction parameter for gelatin B. Naturally, determination of the correct theta temperature of a chosen polymer/solvent system has a great physic-chemical importance for polymer solutions thermodynamically. It is quite well known that the second viiial coefficient can also be evaluated from osmometry and light scattering measurements which consequently exhibits temperature dependence, finally yielding the theta temperature for the system under study. However, the evaluation of second virial... 

The values of % and 8 are much less widely available for aqueous systems than for nonaqueous systems, however. This reflects the relative lack of success of the solution thermodynamic theory for aqueous systems. The concept of the solubility parameter has been modified to improve predictive capabilities by splitting the solubility parameter into several parameters which account for different contributions, e.g., nonpolar, polar, and hydrogen bonding interactions [89,90],... 

We present a molecular theory of hydration that now makes possible a unification of these diverse views of the role of water in protein stabilization. The central element in our development is the potential distribution theorem. We discuss both its physical basis and statistical thermodynamic framework with applications to protein solution thermodynamics and protein folding in mind. To this end, we also derive an extension of the potential distribution theorem, the quasi-chemical theory, and propose its implementation to the hydration of folded and unfolded proteins. Our perspective and current optimism are justified by the understanding we have gained from successful applications of the potential distribution theorem to the hydration of simple solutes. A few examples are given to illustrate this point. 

Equations (2) and (3) relate intermolecular interactions to measurable solution thermodynamic properties. Several features of these two relations are worth noting. The first is the test-particle method, an implementation of the potential distribution theorem now widely used in molecular simulations (Frenkel and Smit, 1996). In the test-particle method, the excess chemical potential of a solute is evaluated by generating an ensemble of microscopic configurations for the solvent molecules alone. The solute is then superposed onto each configuration and the solute-solvent interaction potential energy calculated to give the probability distribution, Po(AU/kT), illustrated in Figure 3. The excess... 

With applications to protein solution thermodynamics in mind, we now present an alternative derivation of the potential distribution theorem. Consider a macroscopic solution consisting of the solute of interest and the solvent. We describe a macroscopic subsystem of this solution based on the grand canonical ensemble of statistical thermodynamics, accordingly specified by a temperature, a volume, and chemical potentials for all solution species including the solute of interest, which is identified with a subscript index 1. The average number of solute molecules in this subsystem is... 

Folded and unfolded proteins in solution are dense materials characterized in large part by different degrees of conformational flexibility and solvent exposure. Thus, packing is a foundational issue for their solution thermodynamic properties. Although the developments above... 

A practical basis for the calculation of solution thermodynamic properties... 

A useful tool for the development of physically motivated approximate models of solution thermodynamics, particularly in view of quasichemical extensions... 

To prevent or at least minimize such problems, we must better understand the environment at all levels, including the fundamental chemical processes that affect it. We have learned the lesson that when assessing the fate of new products in the environment, we should not underestimate the potential of these to appear in unexpected places. The recognition, avoidance, or solution of complex environmental problems requires the expertise of a variety of science and engineering disciplines. Only then will it be possible to produce realistic evaluations of how new compounds will be distributed and will act in the ecosystem. In addition to chemistry and biochemistry, fields such as solution thermodynamics and transport phenomena in which many chemical engineers work, as well as earth sciences and environmental engineering, have crucial contributions to make. 

Schantz, M.M., Martire, D.E. (1987) Determination of hydrocarbon-water partition coefficients from chromatographic data and based on solution thermodynamics and theory. J. Chromatogr. 391, 35-51. 

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