Theories for Concentrated Solutions

By this time, we should recognize one omnipresent assumption in all of the foregoing theories the assumption of a dilute solution. Restricting our arguments to dilute solution allows a focus on diffusion and a neglect of the convection that diffusion itself can generate. In terms of this book, the restriction to dilute solution uses the simple ideas in Chapter 2, not the more complex concepts in Chapter 3. 

The restriction to dilute solution is less serious than it might first seem. While correlations of mass transfer coefficients like those in Chapter 8 are often based on dilute solution experiments, these correlations can often be successfully used in concentrated solutions as well. For example, in distillation, the concentrations at the vapor-liquid interface may be large, but the large flux of the more volatile component into the vapor will almost exactly equal the large flux of the less volatile component out of the vapor. There is a lot of mass transfer, but not much diffusion-induced convection. Thus constant molar overflow in distillation implies a small volume average velocity normal to the interface, and mass transfer correlations based on dilute solution measurements should still work for these concentrated solutions. 

In a few cases, however, these simple ideas of mass transfer fail. This failure is most commonly noticed as a mass transfer coefficient k that depends on the driving force. In other words, we define as before 

These shortcomings lead to alternative definitions of mass transfer coefficients that include the effects of diffusion-induced convection. One such definition is 

We want to calculate this new coefficient, just as we calculated the dilute coefficient in earlier sections of this chapter. In general, we might expect to repeat the whole chapter, producing an entirely new series of equations for the film, penetration, and surface-renewal, and boundary-layer theories. However, these calculations not only would be difficult but also would retain the unknown parameters like film thickness and contact time. 

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For dilute solutions, Equations 4 and 5 reduce to the Bronsted-Guggenheim equations, and the parameters a23 and cu2 can be expressed in terms of the interaction parameters of tne Bronsted-Guggenheim theory. For concentrated solutions, Harned s rule is a simple empirical extension of the Brb nsted-Guggenheim theory. Thus, 1t 1s surprising how well the rule describes activity coefficients 1n highly concentrated solutions. 

Boussinesq and Cerruti made use of potential theory for the solution of contact problems at the surface of an elastic half space. One of the most important results is the solution to the displacement associated with a concentrated normal point load P applied to the surface of an elastic half space. As presented in Johnson [49]... 

The Gouy-Chapman theory for metal-solution interfaces predicts interfacial capacities which are too high for more concentrated electrolyte solutions. It has therefore been amended by introducing an ion-free layer, the so-called Helmholtz layer, in contract with the metal surface. Although the resulting model has been somewhat discredited [30], it has been transferred to liquid-liquid interfaces [31] by postulating a double layer of solvent molecules into which the ions cannot penetrate (see Fig. 17) this is known as the modified Verwey-Niessen model. Since the interfacial capacity of liquid-liquid interfaces is... 

Non-linear viscoelastic flow phenomena are one of the most characteristic features of polymeric liquids. A matter of very emphasised interest is the first normal stress difference. It is a well-accepted fact that the first normal stress difference Nj is similar to G, a measure of the amount of energy which can be stored reversibly in a viscoelastic fluid, whereas t12 is considered as the portion that is dissipated as viscous flow [49-51]. For concentrated solutions Lodge s theory [52] of an elastic network also predicts normal stresses, which should be associated with the entanglement density. 

The case of activity coefficients in solutions is easily but tediously implemented since well-constrained expressions exist, like those produced by the Debye-Hiickel theory for dilute solutions or the Pitzer expressions for concentrated solutions (brines). The interested reader may refer to Michard (1989) for a recent and still reasonably simple account. However simple to handle, activity coefficients introduce analytically cumbersome expressions incompatible with the size of a textbook. Real gas theory demands even more complicated developments. 

Instead of the dilute solution approach above, concentrated solution theory can also be used to model liquid-equilibrated membranes. As done by Weber and Newman, the equations for concentrated solution theory are the same for both the one-phase and two-phase cases (eqs 32 and 33) except that chemical potential is replaced by hydraulic pressure and the transport coefficient is related to the permeability through comparison to Darcy s law. Thus, eq 33 becomes... 

With reference to a solvent, this term is usually restricted to Brpnsted acids. If the solvent is water, the pH value of the solution is a good measure of the proton-donating ability of the solvent, provided that the concentration of the solute is not too high. For concentrated solutions or for mixtures of solvents, the acidity of the solvent is best indicated by use of an acidity function. See Degree of Dissociation Henderson-Hasselbalch Equation Acid-Base Equilibrium Constants Bronsted Theory Lewis Acid Acidity Function Leveling Effect... 

We here define our model and present a self-contained introduction to perturbation theory, deriving the Feynman graph representation of the cluster expansion. To deal with solutions of finite concentration we introduce the grand-canonical ensemble and resum the cluster expansion to construct the loop expansion. We Lhen show that without further insight the expansions can be applied only in the (9-region or for concentrated solutions since they diverge term by term in the excluded volume limit. 

Abstract Macromolecular coils are deformed in flow, while optically anisotropic parts (and segments) of the macromolecules are oriented by flow, so that polymers and their solutions become optically anisotropic. This is true for a macromolecule whether it is in a viscous liquid or is surrounded by other chains. The optical anisotropy of a system appears to be directly connected with the mean orientation of segments and, thus, it provides the most direct observation of the relaxation of the segments, both in dilute and in concentrated solutions of polymers. The results of the theory for dilute solutions provide an instrument for the investigation of the structure and properties of a macromolecule. In application to very concentrated solutions, the optical anisotropy provides the important means for the investigation of slow relaxation processes. The evidence can be decisive for understanding the mechanism of the relaxation. 

We might proceed by plotting versus m, drawing a smooth curve through the points, and constructing tangents to the curve at the desired concentrations in order to measure the slopes. However, for solutions of simple electrolytes, it has been found that many apparent molar quantities such as tp vary linearly with yfm, even up to moderate concentrations. This behavior is in agreement with the prediction of the Debye-Hiickel theory for dilute solutions. Since... 

The Rouse model is the earliest and simplest molecular model that predicts a nontrivial distribution of polymer relaxation times. As described below, real polymeric liquids do in fact show many relaxation modes. However, in most polymer liquids, the relaxation modes observed do not correspond very well to the mode distribution predicted by the Rouse theory. For polymer solutions that are dilute, there are hydrodynamic interactions that affect the viscoelastic properties of the solution and that are unaccounted for in the Rouse theory. These are discussed below in Section 3.6.1.2. In most concentrated solutions or melts, entanglements between long polymer molecules greatly slow polymer relaxation, and, again, this is not accounted for in the Rouse theory. Reptation theories for entangled... 

In fact, the above-mentioned equation is valid only for the perfectly ordered case, i.e., the rods are all aligned in parallel. This illustrates that the Flory theory works well for concentrated solutions. 

However, the problem was tackled a long time ago, and as early as 1942 Flory and Huggins independently presented a fairly simple theory for polymer solutions. This theory explained the large increase of osmotic pressure that is observed when the concentration goes up. Moreover, it predicted correctly that, at large concentrations, the properties of the solution become independent of the molecular mass of the polymers and then depend only on the mass concentration of solute. 

According to (4.44) in the Flory-Krigbaum theory for dilute solutions, when the solvent is the polymer (x 00), the molar volume of the solvent vy 00 too, then A2 —> 0. This implies that the bulk phase is the theta solvent of polymers. With this approach, Rory had already recognized the unperturbed chain conformation in concentrated polymer solutions (Flory 1953). 

This result can be only applied to very dilute solutions of strong monovalent electrolytes, which is the basis of the Debye-Hiickel theory. With concentrated solutions of higher valent ions, the linearization of the exponent function will not hold [9]. The exact formulation of the Poisson-Boltzmarm equation for a symmetric... 

In the Flory-Huggins theory for polymer solutions the difference of the chemical potentials of the solvent in the solution and pure state Ap reads, taking the base molar fraction of the solvent as the generalized concentration variable,... 

Even though the Flory-Huggins theory predicts an upper critical solution temperature and allows a qualitatively correct phase diagram to be calculated, it cannot predict the experimentally observed lower critical solution temperature observed for virtually all polymer solutions. The fundamentally incorrect assumption in the theory is that the volume of mixing of the solution is assumed to be zero. To remedy this problem and improve the predictive power of the theory of concentrated solutions, a full fteory for... 

The thermodynamic properties of concentrated polymer solutions were studied by Floryi and independently by Huggins. The Flory-Huggins theory of polymer solutions still forms the basis for much discussion of these solutions in industry and even in academic research. Understanding this model is important for making coimections to much of the literature. Flory also substantially improved this model to include compressible fluids. The Flory-OrwoU theory of polymer solutions is still transparent and easily applicable, predicting both upper and lower critical solution temperatures. More-empirically adequate theories of concentrated solutions do not lend themselves to simple lecture presentation and often require detailed computer calculations to obtain any results. Concentrated solutions also introduce the phenomenon of viscoelasticity. An extensive treatment of the full distribution of relaxation times necessary to imderstand the dynamic properties of polymers in concentrated solution is presented. 

Eq. 1 is no longer valid. To account for the nonuniform segment distribution, the general theory for dilute solutions, where the chemical potential of the solvent is expressed in virial form, has to be considered. Fortunately, it has been found that the change in chemical potential of the polymer with increasing dilution is so small that it does not have any appreciable effect on its equilibrium melting temperature [12]. For practical purposes, Eq. 1 is obeyed over the complete concentration range of dilutions. 

To model a porous electrode involves the material, energy, momentum, and charge-balance expressions where the species fluxes are accounted for by the transport equatirms. In addition, the equations require boundary conditions in order to solve, which can vary depending on the electrode. As a simple example, let s take a dilute solution of two cations (A, B) and an anion (M), and assume Darcy s law and electrrMieutrality hold. The modeling variables and equations including possible boundary cmiditions are shown in Table 1. It should be noted that for concentrated solutions the COTicentrated-solution theory equations would be needed as well as an additional equation for the solvent. 

The activity is related to the concentration (moles/ liter) of the compound (X) by Eq. 5.13, where y is the activity coefficient. Gamma is always less than one, and is an empirically determined factor that mediates the concentration of the compound to reflect the amount of compound free in the solution. The activity coefficient of ions can be estimated from Debye-Huckel theory. Consult any quantitative analysis textbook to get a thorough discussion of activities and activity coefficients. It is particularly important to use activities in place of concentrations for concentrated solutions in water, and for nonaqueous solvents, which are the next two topics. 

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