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Behavior of the Chemical Potential

The characterization of a DI solution can be carried out along different but equivalent routes. Here, we have chosen the Kirkwood-Buff theory to provide the basic relations from which we derive the limiting behavior of DI solutions. The appropriate relations [Pg.387]

Since we are interested in the limiting behavior - 0 we have separated the singular part, p, as a first term on the rhs. [Pg.388]

Note that the response of the chemical potential to variations in the density p is different for each set of thermodynamic variables. The three derivatives in (6.9.1)-(6.9.3) correspond to three different processes. The first corresponds to a process in which the chemical potential of the solvent is kept constant (the temperature being constant in all three cases) and therefore is useful in the study of osmotic experiments. This is the simplest expression of the three and it should be noted that if we simply drop the condition of constant, we get the appropriate derivative for the pure A component system. This is not an accidental result in fact, this is the case where strong resemblance exists between the behavior of the solute A im. solvent B under constant and a system in a vacuum which replaces the solvent. We return to this analogy in section 6.11. [Pg.388]

The second derivative, (6.9.2), is the most important one from the practical point of view, since it is concerned with a system under constant pressure. The third relation, (6.9.3), is concerned with a system under constant volume, which is rarely useful in practice. [Pg.388]

A common feature of all the derivatives (6.9.1)-(6.9.3) is the p divergence as p - 0 (which is the reason for the convenient form in which we have written them note also that we always assume here that all the Ga/s are finite quantities). [Pg.388]


To differentiate between the variety of phase equilibria that occur, Ehrenfest proposed a classification of phase transitions based upon the behavior of the chemical potential of the system as it passed through the phase transition. He introduced the notion of an th order transition as one in which the nth derivative of the chemical potential with respect to T or p showed a discontinuity at the transition temperature. While modern theories of phase transitions have shown that the classification scheme fails at orders higher than one, Ehrenfest s nomenclature is still widely used by many scientists. We will review it here and give a brief account of its limitations. [Pg.76]

Aside from this, a comparison with the equilibrated mixture results shows that we are now dealing with a situation in which the partial quenching alters the behavior of the chemical potential, and to a much lesser extent the internal energy. As the dipolar density is increased, the HNC equation of the electrolyte breaks down. This is very likely due to a demixing transition, as we could conclude from the stability analysis carried out following the prescriptions of Chen and Forstmann [24, 28], According to them, it is possible to analyze the stability of the grand potential functional for the case of a ion-dipole mixture with equal size particles. The fluctuations in this quantity for the present case can be cast in the form [24],... [Pg.325]

Once the limiting behavior (5.57) has been attained, we see that all three equations have the same formal form, i.e., a constant of integration, independent of pA, and a term of the form kT In pA. This is quite a remarkable observation, which holds only in this limiting case. This uniformity of the behavior of the chemical potential already disappears in the first-order deviation from a DI solution, a topic discussed in the next chapter. [Pg.152]

Care must be exercised when calculating VAS from the derivative of the chemical potential. We have used equation (0.1) to take the derivative with respect to P, then take the limit of pA — 0. If we first take the low-density limit of (0.1) and then differentiate, we get the wrong result. The limiting behavior of the chemical potential is... [Pg.366]

In Section 4.8, we considered the limiting behavior of the chemical potential as 0. We have seen that the formal appearance of the chemical potential is independent of the thermodynamic variables used to describe the system. In this section, we discuss first-order deviations from DI solutions. In fact, these nonideal cases are of foremost importance in practical applications. There exist formal statistical mechanical expressions for the higher-order deviations of DI behavior however, their practical value is questionable since they usually involve higher-order molecular distribution functions. As in the previous section, we derive all the necessary relations from the Kirkwood-Buff theory, and we will be mainly concerned with the behavior of the solute A,... [Pg.159]

The fraction activated material is given by r] = rN2- In the limit r 0, we have = 0 forX < 1 and r] = 1 - X if X > 1. Hence, CAEPs and TAEPs behave in quite similar ways (see Figure 4 and Figure 8). Indeed, Eqs. (14) and (15) also apply to CAEPs, signifying that CAEPs and TAEPs belong to a different universality class than EPs do. In fact, the formal limits r 0 and Xa 0 are equivalent in that the polymerization transition becomes a true (continuous) phase transition. That this must be so, can be inferred from the behavior of the chemical potential of the monomers nearXp = 1. For both systems the (dimensionless) chemical potential ln(l—A2 ) —N2 exhibits a discontinuity at the critical point Xp = 1. In addition, the heat capacities calculated within the given theoretical framework are typical of mean-field theories near a critical point, that is, their values jump at the critical point [29]. [Pg.95]

THE GIBBS-DUHEM EQUATION AND THE COUNTERINTUITIVE BEHAVIOR OF THE CHEMICAL POTENTIAL... [Pg.82]

Fugacity was invented to remedy the counterintuitive behavior of the chemical potential, which makes it approach minus infinity as the concentration approaches zero. For pure ideal gases the fugacity is the same as the pressure, and for ideal gas mixtures the fugacity of one species is equal to that species partial pressure. [Pg.104]

It is at this point that the environmental analyst has to identity the natme of the chemicals and their potential effects on the ecosystem(s) (Smith, 1999). Although petroleum itself and its various products are complex mixtures of many organic chemicals (Chapters 2 and 3), the predominance of one particular chemical or one particular class of chemicals may offer the enviromnental analyst or scientist an opportunity for the predictability of behavior of the chemical(s). [Pg.151]

In the preceding chapters we considered Raoult s law and Henry s law, which are laws that describe the thermodynamic behavior of dilute solutions of nonelectrolytes these laws are strictly valid only in the limit of infinite dilution. They led to a simple linear dependence of the chemical potential on the logarithm of the mole fraction of solvent and solute, as in Equations (14.6) (Raoult s law) and (15.5) (Heiuy s law) or on the logarithm of the molality of the solute, as in Equation (15.11) (Hemy s law). These equations are of the same form as the equation derived for the dependence of the chemical potential of an ideal gas on the pressure [Equation (10.15)]. [Pg.357]

To overcome this weakness, we are developing a quantitative structure-activity strategy that is conceptually applicable to all chemicals. To be applicable, at least three criteria are necessary. First, we must be able to calculate the descriptors or Independent variables directly from the chemical structure and, presumably, at a reasonable cost. Second, the ability to calculate the variables should be possible for any chemical. Finally, and most importantly, the variables must be related to a parameter of Interest so that the variables can be used to predict or classify the activity or behavior of the chemical (j ) One important area of research is the development of new variables or descriptors that quantitatively describe the structure of a chemical. The development of these indices has progressed into the mathematical areas of graph theory and topology and a large number of potentially valuable molecular descriptors have been described (7-9). Our objective is not concerned with the development of new descriptors, but alternatively to explore the potential applications of a group of descriptors known as molecular connectivity indices (10). [Pg.149]

In this section we would like to deal with the kinetics of the liquid-liquid phase separation in polymer mixtures and the reverse phenomenon, the isothermal phase dissolution. Let us consider a blend which exhibits LCST behavior and which is initially in the one-phase region. If the temperature is raised setting the initially homogeneous system into the two-phase region then concentration fluctuations become unstable and phase separation starts. The driving force for this process is provided by the gradient of the chemical potential. The kinetics of phase dissolution, on the other hand, can be studied when phase-separated structures are transferred into the one-phase region below the LCST. [Pg.54]

Let us check the conditions under which the derivative of the chemical potential is positive in a miniemulsion. Under the hypothesis of small concentrations of costabilizer in the droplet and of ideal behavior of the... [Pg.166]

Another application of atomistic simulations is reported by De Pablo, Laso, and Suter. Novel simulations for the calculation of the chemical potential and for the simulation of phase equilibrium in systems of chain molecules are reported. The methods are applied to simulate Henry s constants and solubility of linear alkanes in polyethylene. The results seem to be in good agreement with experiment. At moderate pressures, however, the solubility of an alkane in polyethylene exhibits strong deviations from ideal behavior. Henry s law becomes inapplicable in these cases. Solubility simulations reproduce the experimentally observed saturation of polyethylene by the alkane. For low concentrations of the solute, the simulations reveal the presence of pockets in the polymer in which solubility occurs preferentially. At higher concentrations, the distribution of the solute in the polymer becomes relatively uniform. [Pg.160]


See other pages where Behavior of the Chemical Potential is mentioned: [Pg.322]    [Pg.102]    [Pg.103]    [Pg.335]    [Pg.387]    [Pg.123]    [Pg.322]    [Pg.102]    [Pg.103]    [Pg.335]    [Pg.387]    [Pg.123]    [Pg.14]    [Pg.165]    [Pg.356]    [Pg.240]    [Pg.8]    [Pg.20]    [Pg.421]    [Pg.380]    [Pg.357]    [Pg.449]    [Pg.174]    [Pg.238]    [Pg.242]    [Pg.577]    [Pg.41]    [Pg.154]    [Pg.175]    [Pg.206]    [Pg.286]    [Pg.474]    [Pg.99]    [Pg.36]    [Pg.36]    [Pg.92]    [Pg.407]    [Pg.52]   


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