The thermodynamics of dilute solutions

the solute S is in dilute solution, and tire equation can be used across the enthe composition range of tire A-B binary solvent, when Aa + Ab is close to one. When the concentration of the dilute solute is increased, the more concentrated solution can be calculated from Toop s equation (1965) in the form 

This model is appropriate for random mixtures of elements in which tire pairwise bonding energies remain constant. In most solutions it is found that these are dependent on composition, leading to departures from regular solution behaviour, and therefore the above equations must be conhned in use to solute concentrations up to about 10 mole per cent. 

When there is a large difference between ys(A) and ys(B) in the equation above, there must be signihcant deparmres from dre assumption of random mixing of the solvent atoms around tire solute. In this case tire quasi-chemical approach may be used as a next level of approximation. This assumes that the co-ordination shell of the solute atoms is hlled following a weighting factor for each of tire solute species, such that 

The equation conesponding to die Darken equation quoted above is then 

In liquid metal solutions Z is normally of the order of 10, and so this equation gives values of Ks(a+B) which are close to that predicted by the random solution equation. But if it is assumed that the solute atom, for example oxygen, has a significantly lower co-ordination number of metallic atoms than is found in the bulk of die alloy, dieii Z in the ratio of the activity coefficients of die solutes in the quasi-chemical equation above must be correspondingly decreased to the appropriate value. For example, Jacobs and Alcock (1972) showed that much of the experimental data for oxygen solutions in biiiaty liquid metal alloys could be accounted for by the assumption that die oxygen atom is four co-ordinated in diese solutions. 

Many reactions encountered in extractive metallurgy involve dilute solutions of one or a number of impurities in the metal, and sometimes the slag phase. Dilute solutions of less than a few atomic per cent content of the impurity usually conform to Henry s law, according to which the activity coefficient of the solute can be taken as constant. However in the complex solutions which usually occur in these reactions, the interactions of the solutes with one another and with the solvent metal change the values of the solute activity coefficients. There are some approximate procedures to make the interaction coefficients in multicomponent liquids calculable using data drawn from binary data. The simplest form of this procedure is the use of the equation deduced by Darken (1950), as a solution of the ternary Gibbs-Duhem equation for a regular ternary solution, A-B-S, where A-B is the binary solvent 

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The steel will be considered to be an ideal ternary solution, and therefore at all temperatures a, = 0-18, Ani = 0-08 and flpc = 0-74. Owing to the Y-phase stabilisation of iron by the nickel addition it will be assumed that the steel, at equilibrium, is austenitic at all temperatures, and the thermodynamics of dilute solutions of carbon in y iron only are considered. 

There are some who question the usefulness of the Flory-Huggins solubility parameter for problems related to the solubilization of polymers, although it is agreed that it is useful for study of the thermodynamics of dilute solutions. Barton (1975) has referred to literature that cites its shortcomings as a practical criterion of solubility. Some of these are ... 

In some treatments of the thermodynamics of dilute solutions BaovM s law is taken as the fundamental datum, and the other relations concerning change in equilibrium temperatures are derived... 

As we have learned in Section 1.8, there are a few concentration regimes in the polymer solution. Chapter 2 will primarily focus on the thermodynamics of dilute solutions, that is, below the overlap concentration, although we will also look at how the thermodynamics of the solution deviates from that of the ideal solution with an increasing concentration. Properties characteristic of nondilute solutions will be examined in detail in Chapter 4. 

If we glance back over the various branches of application of thermodynamics to chemical problems detailed in the preceding sections of this book, looking more especially at the historical sequence, we shall find that the physical chemists have, until recently, focussed their attention on the theory of dilute solutions. This preference is due to the great stimulus given by the... 

Such an electrochemical arrangement can also be used to transport oxygen from one electrode to the other by the imposition of an externally applied potential. This technique, known as coulometric titration , has been used to prepare flowing gas mixtures of oxygen/argon with a controlled oxygen partial pressure, to vary the non-stoichiometry of oxides, to study the thermodynamics of dilute oxygen solutions in metals, and to measure the kinetics of metal oxidation, as examples. 

As the laws of dilute solution are limiting laws, they may not provide an adequate approximation at finite concentrations. For a more satisfactory treatment of solutions of finite concentrations, for which deviations from the limiting laws become appreciable, the use of new functions, the activity function and excess thermodynamic functions, is described in the following chapters. 

Both modifications affect the analysis of dilute solution behavior, and it is difficult to judge how much the e/e0 term is actually needed. In any case, as the authors themselves point out (41), the e/e0 term makes an entirely negligible contribution to solvent activity in concentrated solutions. For example, simple calculations yield a contribution of approximately l%ina 10% solution of natural rubber in benzene at 30° C (M = 500000, [t/]g = 250, y=0.4). It is therefore clear that thermodynamic measurements can furnish no evidence for or against continued collapse in concentrated solutions. 

Defect thermodynamics, as outlined in this chapter, is to a large extent thermodynamics of dilute solutions. In this situation, the theoretical calculation of individual defect energies and defect entropies can be helpful. Numerical methods for their calculation are available, see [A. R. Allnatt, A. B. Lidiard (1993)]. If point defects interact, idealized models are necessary in order to find the relations between defect concentrations and thermodynamic variables, in particular the component potentials. We have briefly discussed the ideal pair (cluster) approach and its phenomenological extension by a series expansion formalism, which corresponds to the virial coefficient expansion for gases. 

Thermodynamic properties in dilute aqueous solutions are taken to be functions of ionic strength so that concentrations of reactants, rather than their activities can be used. This also means that pHc = — log[H+] has to be used in calculations, rather than pHa = — log a(H + ). When the ionic strength is different from zero, this means that pH values obtained in the laboratory using a glass electrode need to be adjusted for the ionic strength and temperature to obtain the pH that is used to discuss the thermodynamics of dilute aqueous solutions. Since pHa = — log-/(H + ) + pHc, the use of the extended Debye-Hiickel theory yields... 

R. A. Alberty, The role of water in the thermodynamics of dilute aqueous solutions, Biophys. Chem. 100, 183-192 (2003). 

The expressions of (9 In Ya/ x )p T,x and of their infinitely dilute limits are important for the thermodynamics of dilute ternary solutions [20,21], especially for dilute supercritical ternary solutions [9-12], The limiting expressions (17) and (20) were already derived in a different way by Jonah and Cochran [12] and Chailvo [11], In the next section of the paper, the above expressions will be applied to ternary supercritical solutions. 

In the previous two sections we have discussed deviations from ideal-gas and symmetrical ideal solutions. We have discussed deviations occurring at fixed temperature and pressure. There has not been much discussion of these ideal cases in systems at constant volume or of constant chemical potential. The case of dilute solutions is different. Both constant, T, P and constant T, pB (osmotic system), and somewhat less constant, T, V have been used. It is also of theoretical interest to see how deviations from dilute ideal (DI) behavior depends on the thermodynamic variable we hold fixed. Therefore in this section, we shall discuss all of these three cases. 

When a solvent is also a reactant, its concentration is so large compared with the extent of reaction that it does not change. (Compare the convention we had used, in studying the thermodynamics of aqueous solutions, of viewing water as always being in its standard state in all dilute solutions.) Thus, the dependence of the rate on the concentration of ethyl alcohol cannot be determined unless ethyl alcohol becomes a solute in some other solvent, so that its concentration can be varied. 

The pertinence of the Flory-Huggins theory (18) has been discussed in detail elsewhere (2), It continues to be used and is valuable in the study of thermodynamics of dilute solutions. It is of relatively little help in solving engineering and formulation problems. 

While at DuPont, Dr. Flory, like his mentor. Dr. Carothers, was involved in the controversies that existed among chemists, such as K.H. Meyer, and H. Staudinger. Carothers agreed with Staudinger that derivatives of cellulose did not form aggregates in dilute solutions. Meyer insisted that micelles of polymers existed in solutions but through an analysis of the thermodynamics of polymeric solutions, Flory proved that Meyer s views were incorrect. 

The treatment presented here devolves from the thermodynamics of dilute polymer solutions developed by Huggins [25, 26] and by Flory [27-29], now... 

The existence of dilute solutions of macromolecules was denied by many experts until the macromolecular hypothesis was largely accepted in the time period from 1930 to 1940. The dilute-solution state is still the basis for characterizing individual macromolecules and the interactions of pairs of macromolecules and the solvent. The structural, thermodynamic, and hydro-dynamic properties of polymer solutions are explained in terms of the random-coil model developed by Kuhn, Debye, Flory, Kirkwood, Yamakawa, and deGennes. While this subject alone could easily be the basis for a one-semester course, the topics are developed so that the material could be presented as part of a complete development of the subject. 

Jacobus Henricus van t Hoff (1852-1911) a Dutch physical and organic chemist (Nobel Prize in Chemistry in 1901). His research work concentrated on chemical kinetics, chemical equilibrium, osmotic pressure, and crystallography. He is one of the founders of the discipline of physical chemistry. He explained the phenomenon of optical activity by assuming that the chemical bonds between carbon atoms and their neighbors were directed towards the corners of a regular tetrahedron, applied the laws of thermodynamics to chemical equilibriums, showed similarities between the behavior of dilute solutions and gases, and worked on the theory of the dissociation of electrolytes. In 1878, he became professor of chemistry at the University of Amsterdam, and in 1896 he became professor at the Prussian Academy of Science at Berlin, where he worked until his death. 

Most aspects have not been worked out for the thermodynamic properties and equilibria of systems of macromolecules in the crystalline, glassy, or solution form. There is much uncertainty about the use of dilute solution reference states for supercritical components, particularly in multisolute, multisolvent solutions. 

As well as controlling chain dimensions, solvent quality affects the thermodynamics of dilute polymer solutions. This is because interactions between polymer chains are modified by the presence of solvent molecules. In particular, solvent molecules will change the excluded volume for a polymer coil, i.e. how much volume it takes up and prevents neighbouring chains from occupying. In a theta solvent, the excluded volume is zero (this holds for the excluded volume for a polymer segment or the whole coil). The solution is said to he ideal if the excluded volume vanishes. Deviations from ideality for polymer solutions are described in terms of a virial equation, just as deviations from ideal gas behaviour are. The virial equation for a polymer solution in terms of polymer concentration is given by Eq. (2.9). The second virial coefficient depends on interactions between pairs of molecules in particular it is proportional to the excluded volume. Therefore, in a theta solvent, = 0. If the solvent is good then Ai > 0, but if it is poor Ai < 0. If the solvent quality varies as a function of temperature and theta (0) conditions are attained, this occurs at the theta temperature. 

The principal question arising in investigating the thermodynamic behavior of systems containing chemically different polymers or copolymers concerns the role of interactions between unlike monomers. In this respect, the consideration of dilute solutions provides useful information as it elucidates the importance of chain conformations and monomer concentration fluctuations. In a dilute solution of two polymers A and B the osmotic pressure may be approximated by a virial expansion limited to the second order ... 

If the Flory theory is indisputably a reference for the thermodynamics of polymer solutions, it suffers from a lack of accuracy in its description of dilute polymer solutions as previously mentioned. Well suited to the case of concentrated solutions, this theory depicts the behavior of dilute solutions and describes the forces due to excluded volume as the result of a perturbation to random walk statistics for example, it does not account for the significant variations experienced by the density of segments in dilute media. Indeed, the replacement of the radial variation of this function (which describes the density of interaction in the medium) by an average value is not satisfactory. 

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