Gibbs-Duhem integration point

While the main driving force in [43, 44] was to avoid direct particle transfers, Escobedo and de Pablo [38] designed a pseudo-NPT method to avoid direct volume fluctuations which may be inefficient for polymeric systems, especially on lattices. Escobedo [45] extended the concept for bubble-point and dew-point calculations in a pseudo-Gibbs method and proposed extensions of the Gibbs-Duhem integration techniques for tracing coexistence lines in multicomponent systems [46]. 

Once a state point of coexistence is established, additional state points can be determined expeditiously through application of the Gibbs-Duhem integration method [48,85,86]. In this approach a differential equation for the coexistence line is used to guide the establishment of state points away from the known coexistence point. The most well known such formula is the Clapeyron equation [41]... 

The n= 12 soft sphere model is the high-temperature limit of the 12-6 Lennard-Jones (LJ) potential. Agrawal and Kofke [182] used this limit as the starting point for another Gibbs-Duhem integration, which proceeded to lower temperatures until reaching the solid-liquid-vapor triple point. The complete solid-fluid coexistence line, from infinite temperature to the triple point, can be conveniently represented by the empirical formula [182]... 

Interesting connections between many of the methods discussed in the present chapter have been pointed out by Escobedo [54,55]. In particular, Escobedo suggests that Gibbs-Duhem integration, pseudo-ensembles, and the NPT + test particle method can be considered as low-order approximations of a histogram reweighting approach. 

Figure 6 Two algorithms to conduct Gibbs-Duhem integrations. The pressure route (upper-left) corresponds to integration of Clapey-ron .s equation and entails iVPr-ensemble simulations of both phases (depicted here by the two boxes). The chemical potential route (upper-right) entails p.VT simulations. In both cases, a trapezoidal predictor-corrector integration. scheme is illustrated here the integration advances from point 0 to point 1...

An example drawn from Deitrick s work (Fig. 2) shows the chemical potential and the pressure of a Lennard-Jones fluid computed from molecular dynamics. The variance about the computed mean values is indicated in the figure by the small dots in the circles, which serve only to locate the dots. A test of the thermodynamic goodness of the molecular dynamics result is to compute the chemical potential from the simulated pressure by integrating the Gibbs-Duhem equation. The results of the test are also shown in Fig. 2. The point of the example is that accurate and affordable molecular simulations of thermodynamic, dynamic, and transport behavior of dense fluids can now be done. Currently, one can simulate realistic water, electrolytic solutions, and small polyatomic molecular fluids. Even some of the properties of micellar solutions and liquid crystals can be captured by idealized models [4, 5]. 

Two limitations are involved in the derivation of the above equation (1) the compositions of mixed solvents (points c and d) should be close enough to each other for the trapezoidal mle used to integrate the Gibbs-Duhem equation to be valid, (2) the solubility of the solid should be low enough for the activity coefficients of the solvent and cosolvent to be taken equal to those in a solute-free binary solvent mixture. In addition, the fugacity of the solid phase in Eq. (4) should remain the same for all mixed solvent compositions considered. 

It is possible, as a result of cancellation, to satisfy the integral test of Eq. 10.2-13 while violating the differential form of the Gibbs-Duhem equation, Eq. 9.3-15, on which Eq. 10.2-13 is based, at some or all data points. In this case the experimental data should be rejected as thermodynamically inconsistent. Thus, the integral consistency test is a necessar). but not sufficient, condition for accepting experimental data. 

These data can be studied in two ways. The first is to use the Gibbs-Duhem equation and numerical integration methods to calculate the vapor-phase mole fractions, as considered in Problem 10.2-6. A second method is to choose a liquid-phase activity coefficient model and determine the values of the parameters in the model that give the best fit of the experimental data. We have, from Eq. 10.2-2b, that at the jth experimental point... 

In Figure 1 b a different construction is described. In this situation two coexistence lines are being traced by the Gibbs-Duhem method, each line having one phase (a) in common. The point where the fi-ct coexistence coincides with the -a. coexistence again forms a triple point. Subsequent integration then follows coexistence between the and p phases. It might be... 

Intercepts and Common Tangents to Agmixing ts. Composition in Binary Mixtures. Euler s integral theorem and the Gibbs-Duhem equation provide the tools to obtain expressions for Agmixing and (9 Agmixing/9y2)r,/) in binary mixtures. This information allows one to evaluate the tangent at any mixture composition via the point-slope formula. For example, if i i = and p,2 = M2 when the mole fraction of component 2 is y, then equations (29-73) and (29-76) yield ... 

This equation is rarely used for testing the consistency of p, jr, and y data, the claim being made that it is difficult to obtain the gradients with sufficient accuracy except where there are a considerable number of points spread over the entire composition range. The most common method for testing consistency is to use an integrated form of the Gibbs-Duhem equation ... 

The measurement of polymer solutions with lower polymer concentrations requires very precise pressure instruments, beeause the difference in the pure solvent vapor pressure becomes very small with deereasing amount of polymer. At least, no one can really answer the question if real thermodynamie equilibrium is obtained or only a frozen non-equilibrium state. Non-equilibrium data ean be deteeted from unusual shifts of the %-function with some experience. Also, some kind of hysteresis in experimental data seems to point to non-equilibrium results. A eommon eonsisteney test on the basis of the integrated Gibbs-Duhem equation does not work for vapor pressure data of binary polymer solutions because the vapor phase is pure solvent vapor. Thus, absolute vapor pressure measurements need very careful handling, plenty of time, and an experienced experimentator. They are not the method of choiee for high-viseous polymer solutions. 

As has been pointed out, the equations are all integrations of the Gibbs-Duhem relationship. They consequently cannot be applied to systems which when treated in the ordinary fashion apparently do not follow this basic relation, as in the case of dissociation of electrolytes in solution. 

Applications of the Integrated Equations. The usefulness of the Gibbs-Duhem equation for establishing the thermodynamic consistency of, and for smoothing, data has been pointed out. The various integrated forms are probably most useful for extending limited data, sometimes from even single measurements, and it is these applications that are most important for present purposes. 

If the logarithms of the activity coefficients of a binary system are plotted against molar composition, Eq. (53) relates the slopes at any value of the x s. This is useful for testing the consistency of data. Such testing becomes easier if we have a systematic way to correlate and smooth data and extend them over the entire range of composition. Integrated forms of the Gibbs-Duhem relation allow us to do this. Theoretically, one reliable measurement of vapor-liquid equilibrium at any point can then be used to characterize a system. 

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