Maxwell thermodynamic equality

When it is experimentally feasible, the temperature dependence of AV can be compared with the pressure dependence of the entropy of activation. The equality of these terms is defined from extending the Maxwell thermodynamic equality ... 

A thermodynamic example may be illustrative. Consider Maxwell s model of the Gibbs USV surface for water (Fig. 1.1), as depicted schematically in Fig. 9.1. In this model, the physical (77, S, V) coordinates are associated with mutually perpendicular axes, and three chosen points on this surface form a triangle whose edges, angles, and area are as shown in Fig. 9.1a. However, the model might have been constructed (with equal thermodynamic justification) in a skewed /io/ orthogonal axis system (Fig. 9.1b) in which the... 

This equation shows that the change in flux of some quantity caused by changing the direct driving force for another is equal to the change in flux of the second quantity caused by changing the driving force for the first. These equations resemble the Maxwell relations from thermodynamics. 

Maxwell first noted the cross relations based on a property of the total differentials of the state functions. The cross differentiations of a total differential of the state function are equal to each other. Table 1.14 summarizes the total differentials and the corresponding Maxwell relations. The Maxwell relations may be used to construct important thermodynamic equations of states. 

It should be noted that the Maxwell-Stefan D calculated from Eq. 4.1.5 can be quite sensitive to the model used to compute T, an observation first made by Dullien (1971). One of the reasons for this sensitivity is that E involves the first derivative of the activity coefficient with respect to composition. Activity coefficient model parameters are fitted to vapor-liquid equilibrium (VLE) data (see, e.g., Prausnitz et al., 1980 Gmehling and Onken, 1977). Several models may provide estimates of In % that give equally good fits of the vapor-liquid equilibrium data but that does not mean that the first derivatives of In % (and, hence, E) will be all that close. To illustrate this fact we have calculated the thermodynamic factor, E, for the system ethanol-water with several different models of In %. The results are shown in Figure 4.3 a). The interaction parameters used in these calculations were fitted to one set of VLE data as identified in the figure caption. Similar illustrations for other systems are provided by Taylor and Kooijman (1991). 

As the mole fraction of either component in a binary mixture approaches unity, the thermodynamic factor E approaches unity and the Fick D and the Maxwell-Stefan D are equal. This result is shown clearly in Figures 4.1-4.3. The diffusion coefficients obtained under these conditions are the infinite dilution diffusion coefficients and given the symbol )°. 

Statistical theories of thermodynamics yield many correct and practical results. For example, they yield the canonical and grand canonical distributions for petit and grand systems, respectively these distributions, which were proposed by Gibbs, have been shown by innumerable comparisons with experiments to describe accurately the properties of quasistable states. Again, they predict the equality of temperatures of systems in mutual stable equilibrium, the Maxwell relations, and the Gibbs equation. 

This is the well known equal areas rule derived by Maxwell [3], who enthusiastically publicized van der Waal s equation (see figure A2.3.3. The critical exponents for van der Waals equation are t5q)ical mean-field exponents a 0, p = 1/2, y = 1 and 5 = 3. This follows from the assumption, common to van der Waals equation and other mean-field theories, that the critical point is an analytic point about which the free energy and other thermodynamic properties can be expanded in a Taylor series. 

Figure A3.3.5 Thermodynamic force as a function of the order parameter. Three equilibrium isotherms (full curves) are shown according to a mean field description. For T < T, the isotherm has a van der Waals loop, from which the use of the Maxwell equal area construction leads to the horizontal dashed line for the equilibrium isotherm. Associated coexistence curve (dotted curve) and spinodal curve (dashed line) are also shown. The spinodal curve is the locus of extrema of the various van der Waals loops for T

Prior to Harwood s work, the existence of a Bootstrap effect in copolymerization was considered but rejected after the failure of efforts to correlate polymer-solvent interaction parameters with observed solvent effects. Kamachi, for instance, estimated the interaction between polymer and solvent by calculating the difference between their solubility parameters. He found that while there was some correlation between polymer-solvent interaction parameters and observed solvent effects for methyl methacrylate, for vinyl acetate there was none. However, it should be noted that evidence for radical-solvent complexes in vinyl acetate systems is fairly strong (see Section 3), so a rejection of a generalized Bootstrap model on the basis of evidence from vinyl acetate polymerization is perhaps unwise. Kratochvil et al." investigated the possible influence of preferential solvation in copolymerizations and concluded that, for systems with weak non-specific interactions, such as STY-MMA, the effect of preferential solvation on kinetics was probably comparable to the experimental error in determining the rate of polymerization ( 5%). Later, Maxwell et al." also concluded that the origin of the Bootstrap effect was not likely to be bulk monomer-polymer thermodynamics since, for a variety of monomers, Flory-Huggins theory predicts that the monomer ratios in the monomer-polymer phase would be equal to that in the bulk phase. 

In the region occupied by the polymer chain, solvent molecules are mixed. Let A/ro be the chemical potential of the solvent molecule measured from the value in the pure solvent. From the thermodynamic condition A/xo = (9AF/dNo)n = —(< / )(9 AF/d4>)=0 that the chemical potential of a solvent molecule inside the region occupied by the polymer should be equal to that in the outside region, we can derive Maxwell s rule of equal area for the osmotic pressure in the form... 

This oscillatory behavior is counterintuitive and apparently nonphysical it seems to violate the laws of thermodynamics. In fact, it does. One reason is that our assumption of equal transit times for all molecules across the membrane requires a Maxwell demon to help molecules avoid the collisions that produce a distribution of transit times. 

Extended Stefan-Maxwell constitutive laws for diffusion Eq. 4 resolve a number of fundamental problems presented by the Nemst-Planck transport formulation Eq. 1. A thermodynamically proper pair of fluxes and driving forces is used, guaranteeing that all the entropy generated by transport is taken into account. The symmetric formulation of Eq. 4 makes it unnecessary to identify a particular species as a solvent - every species in a solution is a solute on equal footing. Use of velocity differences reflects the physical criterion that the forces driving diffusion of species i relative to species j be invariant with respect to the convective velocity. Finally, all possible binary solute/solute interactions are quantified by distinct transport coefficients each species i in the solution has a diffusivity or mobility relative to every other species j, Djj or up, respectively. 

The ternary diffusion coefficient strongly depends on the solution concentration. In order to calculate accurate mass transfer coefficients, experimental data of diffusion coefficients at the interest concentrations and temperatures are necessary. However, data are not available at concentrations and temperature used at the present study, it was assumed that the ternary diffusion coefficients were equal to the binary diffusion coefficients. The binary diffusion coefficients of the KDP-water pairs and the urea-water pairs were taken from literature (Mullin and Amatavivadhana, 1967 Cussler, 1997). The values were transformed into the Maxwell-Stefan diffiisivities using the thermodynamic correction factor. 

Thermometry is based on the principle that the temperatures of different bodies may be compared with a thermometer. For example, if you find by separate measurements with your thermometer that two bodies give the same reading, you know that within experimental error both have the same temperature. The significance of two bodies having the same temperature (on any scale) is that if they are placed in thermal contact with one another, they will prove to be in thermal equilibrium with one another as evidenced by the absence of any changes in their properties. This principle is sometimes called the zeroth law of thermodynamics, and was first stated as follows by J. C. Maxwell (1872) Bodies whose temperatures are equal to that of the same body have themselves equal temperatures. ... 

Also, it is importsnl to point out that the direct and convene piezoekctric coefficients are not equal [101], as is usually assumed. The difference tentk to be small. - 10%, in hard and brittle fenoekcirics, but as will be demonstrated later in this chapter, the correction term can dominate in soft, pliable polymcn. The relatioa between tlie direct and converse piezoelectric coefficient is typically determined by deriving a Maxwell relation from a thermodynamic potential. A thorough description of Maxwell relations and thermodynamic potentials is presented in a book by Cailen (90. ... 

The introductory Section 3.1.2.5 in Chapter 3 identifies the negative chemical potential gradient as the driver of targeted separation, and the relevant species flux expression is developed in Section 3.1.3.2 (see Example 3.1.9 also). Section 3.1.4 introduces molecular diffusion and convection and basic mass-transfer coefficient based flux expressions essential to studies of distillation and other phase equilibrium based separation processes. Section 3.1-5.1 introduces the Maxwell-Stefan equations forming the basis of the rate based approach of analyzing distillation column operation. After these fundamental transport considerations (which are also valid for other phase equilibrium based separation processes), we encounter Section 3.3.1, where the equality of chemical potential of a species in all phases at equilibrium is illustrated as the thermodynamic basis for phase equilibrium (Le. = /z ). Direct treatment of distillation then begins in Section 3.3.7.1, where Raouit s law is introduced. It is followed by Section 3.4.1.1, where individual phase based mass-transfer coefficients are reiated to an overall mass-transfer coefficient based on either the vapor or liquid phase. 

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