Single-Parameter Expressions

The simplest single-parameter expression for the molar excess Gibbs free energy of a binary system is the two-suffix Margules equation used already in Section 12.10  

1 is applicable to systems where the activity coefficients of the two components are symmetrical. It can thus correlate successfully only systems similar in structure and molar volumes, such as benzene-cyclohexane. 

In 1969 this author proposed a modification of the Wilson equation, which leads to a single-parameter expression. To this purpose, in Eq. 13.12.2- which according to Wilson is a constant proportional to the energy of interaction between molecules of component - is evaluated from  

Evaluation of Xf, and Ajj in the original expression through Eqs 13.14.3 and 13.14.4, leaves only one adjustable parameter, Ajj, for the pair of components / and J. This, in turn, reduces to half the number of parameters needed to describe a system. Thus for a ternary mixture only three parameters are needed, as compared to six for the original Wilson equation. 

The expression is very useful in cases where only limited data are available. An interesting such application is the case where only one infinite dilution activity coefficient for a binary system - obtained usually through gas-liquid chromatography (GLC) - is available. A combination of such GLC measurements and the single-parameter Wilson equation in predicting the solubility of SO2 as a function of its partial pressure in several solvents is demonstrated by Bogeatzes(1973). 

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The methods of state counting were further refined by Rabinovitch and his collaborators. By comparing variants of the semi-classical equation with computed, exact counts, it was discovered that accurate sums can be found using a single parameter expression. 

Correlation of binary vapor-liquid equilibrium data, and prediction of multicomponent VLE behavior from binary ones, using this single-parameter expression is discussed by this author (1969 1971) and by Hankinson et al (1972). Typical multicomponent predictions are shown in Figure 13.12. In general, the results obtained are reliable and close to those obtained by the two-parameter expression, even for four- and five-com-... 

Single-parameter expressions for the NRTL, LEMF, and UNIQUAC equations have also been developed. Krummins et al (1980) evaluated several of them and concluded that ... 

Discuss the advantages and disadvantages in using single-parameter expressions. 

Since A Aw is the change in interaction energy per 1,2 pair, it can be expressed as some multiple of kT per pair or of RT per mole of pairs. It is also conventional to consolidate the lattice coordination number and x into a single parameter x, since z and x are not measured separately. With this change of notation Az Aw is replaced by its equivalent xRT, and Eq. (8.41) becomes... 

In order to simplify the expression for G, one has to employ a sufficiently simple model for the vibrational modes of the system. In the present case, the solvent contribution to the rate constant is expressed by a single parameter E, the solvent reorganization energy. In addition, frequency changes between the initial and final states are neglected and it is assumed that only a single internal mode with frequency co and with the displacement Ar is contributing to G. Thus the expression for G reduces to [124] ... 

The ratio between the bed and particle diameters and the Reynolds number based on bed diameter, superficial velocity, and solid density appear only in the modified drag expression, in which they are combined, see Eq. (40). These parameters form a single parameter, as discussed by Glicksman (1988) and other investigators. The set of independent parameters controlling viscous dominated flow are then... 

Whereas the latter expression must be solved numerically for low temperatures, the entropy at high temperatures can be derived by a series expansion [4], For the Debye or Einstein models the entropy is essentially given in terms of a single parameter at high temperature ... 

After these preliminary remarks, the term polarity appears to be used loosely to express the complex interplay of all types of solute-solvent interactions, i.e. nonspecific dielectric solute-solvent interactions and possible specific interactions such as hydrogen bonding. Therefore, polarity cannot be characterized by a single parameter, although the polarity of a solvent (or a microenvironment) is often associated with the static dielectric constant e (macroscopic quantity) or the dipole moment p of the solvent molecules (microscopic quantity). Such an oversimplification is unsatisfactory. 

DR. MEYER The critical point to recognize is the factoring. The energy expressions contain a AE term which has its own solvent dependence, and the electron transfer part has its own solvent dependence. When you start breaking the problem down into its component parts, instead of attempting to derive a single parameter fit, it may be possible to treat the problem in a general way. 

The statistical treatment of a hemiisotactic polymer can be made on the basis of a single parameter a the corresponding formulas are reported in Table 4, last column. For extreme values of a the polymer is no longer hemiisotactic but syndiotactic (for a = 0) or isotactic (for a = 1). The particular distribution existing in the hemiisotactic polymer is not reproducible with either the Bernoulli or the Markov processes expressed in m/r terms. 

One widely used smearing method was developed by Methfessel and Paxton. Their method uses expressions for the smearing functions that are more complicated than the simple example above but are still characterized by a single parameter, ct (see Further Reading). 

Two points have to be stressed before considering the measurement of morphology. The first point to make in discussing filler morphology is that, except for rare instances such as monomodal glass spheres, the morphology of filler particles is complex and they will have a distribution of shapes and sizes which cannot be expressed as a single parameter. 

In a 1991 study by van Reis et al. (5), a filtration operation as applied to harvest of animal cells was optimized by the use of dimensional analysis. The fluid dynamic variables used in the scale-up work were the length of the fibers (L, per stage), the fiber diameter (D), the number of fibers per cartridge (k), the density of the culture (p), and the viscosity of the culture (p). From these variables, scale-up parameters such as wall shear rate (y ) and its effect on flux (L/m /h) were derived. Based on these calculations, an optimum wall shear rate for membrane utilization, operating time, and flux was found. However, because there is no single mathematical expression relating all of these parameters simultaneously, the optimal solution required additional experimental research. 

A simplified approach to assess MU is the JUriess-for-purpose approach, defining a single parameter called the fitness function. This fitness function has the form of an algebraic expression u=f(c) and describes the relationship between the MU and the concentration of the analyte. For example, = 0.05c means that the MU is 5% of the concentration. Calculation of the MU will hereby rely on data obtained by evaluating individual method performance characteristics, mainly on repeatability and reproducibility precision, and preferably also on bias [21,40,41]. This approach can more or less be seen as a simplification of the step-by-step protocol for testing the MU, as described by Eurachem [14]. 

Here L is the linearized part of the operator appearing in the right-hand side of equation (1), and h = 0( x 2) stands for the contribution of the nonlinear terms. Suppose that we control a single parameter, and let Xc be a bifurcation point. We place ourselves in the vicinity of this point, a fact that we express as follows ... 

The distribution of the cis and turns double bonds in a given polymer chain may be expressed in terms of the ratios rt = (tt)/(tc) and rc = (cc)/(ct). If the probability of formation of a cis double bond is independent of the configuration of the previous double bond, the distribution will be random (Bernoullian) and characterized by a single parameter rt = 1 /rc. This is the case for polymers of norbornene with less than 35% cis content, but for polymers with more than 50% cis content the distribution is generally somewhat... 

The subject of the forthcoming sections will be the activity of the Ni— SiO-2 catalyst systems. Since we will be discussing the experimental results on the basis of one single kinetic expression and with a special interpretation of the parameters of this expression, it is necessary to give some consideration to this subject. 

The factor FHA in equation (94) can be replaced by F a if it is permissible to assume that Gibbs free energies of transfer are linear functions of n. With this simplification the consequences of the transfer effect are reduced to the inclusion of a single parameter in the expression for the solvent isotope effect. 

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