Pseudo-radial distribution function

Figure 4.1-11 The EXAFS data and pseudo-radial distribution functions of Co(ll) in (a) basic and (b) acidic chloroaluminate ionic liquid. Reproduced from reference 46 with permission.

Figure 4.1-13 Comparison of the experimental without (—) and with (—) triphenylphosphine at (solid line) and fitted (dashed line) (a) EXAFS 80 °C and in the presence of triphenylphosphine and (b) pseudo-radial distribution functions and reagents at 50 °C for 20 min (—). Repro-...

Figure Ic is the pseudo-radial distribution function obtained by Fourier transforming X(k). Exact positions of the peaks are shifted because of 4>(k) amplitudes are affected by the leading terms in the above equation.

Figure 4 shows then the role EXAFS can play in the in situ description of this calcination process It is composed of various pseudo-radial distribution functions, uncorrected from phase shifts, obtained for various loadings, various temperatures and different precursors. First of all, two successive steps are clearly evidenced ...

Baston et al. [60] studied the samples of ionic liquid after the anodization of uranium metal in [EMIMjCl using the U Lm-edge EXAFS to establish both the oxidation state and the speciation of uranium in the ionic liquid. This was part of an ongoing study to replace high-temperature melts, such as LiQ KQ [61], with ionic liquids. Although it was expected that, when anodized, the uranium would be in the +3 oxidation state, electrochemistry showed that the uranium is actually in a mixture of oxidation states. The EXAFS of the solution showed an edge jump at 17166.6 eV, indicating a mixture of uranium(IV) and uranium(VI). The EXAFS data and pseudo-radial distribution functions for the anodized uranium in [EMIMjCl are shown in Eig. 4.1-12. 

PRDF Pseudo Radial Distribution Function obtained from the EXAFS spectram and related to the radial distribution of atoms surrounding a particular atom with a shift caused by the scattering phase... 

The approach considered is that first proposed by Enskog himself (Chapman Cowling 1952), who suggested that a pseudo-radial distribution function g for a real gas, to replace the function for hard spheres at contact, and a consistent effective hard-sphere diameter could be obtained from the equation of state for the real gas. Specifically, he proposed that the radial distribution function could be obtained from the equation of state for the real gas by replacing the pressure of the real gas by the thermal pressure Pt through the equation... 

In the hrst a pseudo-radial distribution function, g , is identihed for each real, pure component at a particular temperature and the density of interest from measurements of the pure component viscosity for each fluid. This can be accomplished by writing equation (5.48) for a pure fluid and solving for the pseudo-radial distribution function at each density (DiPippo etal. 1977 Vesovic Wakeham 1989a,b). Hence,... 

There are two possible solutions of equation (5.53), gf and g , corresponding to the positive and negative roots of equation (5.53). Although both solutions for gj will reproduce the pure component viscosity, only one of the solutions is physically plausible at a given density. At low densities the pseudo-radial distribution function should tend monotonically to unity as the density tends to zero, while at high densities the pseudo-radial distribution function should monotonically increase. The first requirement is met by the g branch of the solution, while the second one is met only by the g branch. Thus, in order to construct the total pseudo-radial distribution function, gJ" is used at low densities up to some switchover density p, where the switching to the gf branch is performed. In order to ensure a continuous and smooth transition between the two branches, the switchover density must be chosen for each isotherm at the point where the roots gt and gf are equal, so that... 

It follows that from measurements of the viscosity of each pure fluid as a function of density at a particular temperature, it is possible to determine both an and the pseudo-radial distribution function g/. First, the minimum value of the group p) is... 

Fig. 5.11. The pseudo-radial distribution function deduced from the viscosity of nitrogen at T = 200 K switching density p =13,300 mol m. ...

The next step in the procedure of evaluation of the mixture properties is the evaluation of the pseudo-radial distribution functions for all i — j interactions in the mixture as well as the mean free-path parameter atj for the unlike interaction. It is consistent with the remainder of the procedure to estimate them from mixing rules based upon a rigid-sphere model. Among the many possible mixing rules for the radial distribution function one that has proved successful is based upon the Percus-Yevick equation for the radial distribution function of hard-sphere mixtures (Kestin Wakeham 1980 Vesovic ... 

If, now, the derived values of all of the pseudo-radial distribution functions g/y are used to replace the corresponding rigid-sphere quantities in equations (5.48)-(5.52), then it remains only to evaluate A j and for all of the binary interactions to permit a calculation of the dense gas mixture viscosity. These latter quantities are just those required for the evaluation of the dilute-gas mixture viscosity and discussed in Chapter 4, so that the methods proposed there for their evaluation can be employed again here. 

In these equations the pseudo-radial distribution functions for the real gas have been inserted in place of the radial distribution functions of hard sphoes at contact, by analogy with the treatment above for the viscosity. Also, different mean firee-path shortening parameters, y/y, have been introduced which are again related to the co-volume for molecules i and j. All other symbols have been previously defined, and it should be noted that may be related to the viscosity of compound i by means of equation (4.44) of Chapter 4. 

The method for predicting the thermal conductivity of a dense-gas mixmre (Mason et al. 1978 Kestin Wakeham 1980 Vesovic Wakeham 1991) is analogous to that for the viscosity, so that only its main features need be described here. The pseudo-radial distribution function for the thermal conductivity of the pure species is obtained by solving equation (5.60) for g,- at each temperature and the mixture molar density... 

The essence of the application of these equations to real fluids consists of three parts first the replacement of the hard-sphere results for the pure gas viscosity and the interaction viscosity by the values for the real fluid system second, the evaluation of a pseudo-radial distribution function for each binary interaction to replace the hard-sphere equivalent at contact and, finally, the selection of a molecular size parameter for each binary interaction to account for the mean free-path shortening in the dense gas. 

An alternative application of the Thome-Enskog equations is that en loyed by Veso-vic Wakeham (1989,1991). The procedure has been described in detail in Section 5.5, so that it is necessary here to describe only the sources of information employed. First the correlation of the viscosity of pure carbon dioxide has been taken from the work of Vesovic etal. (1990) and that of ethane from the workofHendl etal. (1994). For each temperature at which it was desired to evaluate the mixture viscosity these correlations were used in equation (5.53) to evaluate the pseudo-radial distribution functions g,- for each pure component, while equation (5.56) was used to determine a consistent value of the mean free-path shortening parameter a,-, for i-i interactions. Figure 15.3 contains a plot of the pseudo-radial distribution function determined in this way for the two pure gases at one temperature. 

Fig. 15.3. Pseudo-radial distribution functions for ethane and carbon dioxide at a temperature of 350K.

The evaluation of the excess contribution to the thermal conductivity of carbon dioxide-ethane mixtures has been performed by a route parallel to that employed for the viscosity. Again, that route has been described in detail in Section 5.5.1.2 so that here it is merely recorded that for pure carbon dioxide and pure ethane the thermal conductivity has been taken from the work of Vesovic et al. (1990, 1994). All other quantities required for the calculation are the same as those employed for the zero-density thermal conductivity. In this context it should be noted that the Thome-Enskog equations employed for these calculations have, as their zero-density limit, the Hirschfelder-Eucken result in the form of equation (15.5). Figure 15.3 contains the pseudo-radial distribution functions for carbon dioxide and ethane determined... 

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