Equations group contribution

Equations (2 -(4) clearly illustrate the increase in distance in the interactions between atoms X and Y in going from the additivity of atomic, to bond, and further to group contributions. 

Correlation methods discussed include basic mathematical and numerical techniques, and approaches based on reference substances, empirical equations, nomographs, group contributions, linear solvation energy relationships, molecular connectivity indexes, and graph theory. Chemical data correlation foundations in classical, molecular, and statistical thermodynamics are introduced. 

A relatively simple example of a group contribution technique is the method for estimating Hquid and soHd heat capacities (159). This method is a modification of Kopp s rule (160,161) which was originally proposed in 1864. Kopp s rule states that, at room temperature, the heat capacity of a soHd compound is approximately equal to a stoichiometric summation of the heat capacities of its atoms (elements). The Hurst-Harrison modified equation is as follows ... 

Molecular Connectivity Indexes and Graph Theory. Perhaps the chief obstacle to developing a general theory for quantification of physical properties is not so much in the understanding of the underlying physical laws, but rather the inabiUty to solve the requisite equations. The plethora of assumptions and simplifications in the statistical mechanics and group contribution sections of this article provide examples of this. Computational procedures are simplified when the number of parameters used to describe the saUent features of a problem is reduced. Because many properties of molecules correlate well with stmctures, parameters have been developed which grossly quantify molecular stmctural characteristics. These parameters, or coimectivity indexes, are usually based on the numbers and orientations of atoms and bonds in the molecule. 

Liquid Viscosity The viscosity of both pure hydrocarbon and pure nonhydrocarbon hquids are most accurately predicted by the method of van Velzen et al. The basic equation (2-112) depends on group contributions which are dependent on stnic tiire for the calculation of compound-specific constants B and To-... 

This topological approach with connectivity indices has been extended by Bicerano [58] to a point where many of the physical properties of the polymer can be estimated from empirical predictive equations. Bicerano correlated connectivity indices with group contribution values in order to develop a model equation for a specific property [58]. The usefulness of Bicerano s equations is that they can be extended to predict the same property values for new interested polymers. 

The UNIQUAC equation developed by Abrams and Prausnitz is usually preferred to the NRTL equation in the computer aided design of separation processes. It is suitable for miscible and immiscible systems, and so can be used for vapour-liquid and liquid-liquid systems. As with the Wilson and NRTL equations, the equilibrium compositions for a multicomponent mixture can be predicted from experimental data for the binary pairs that comprise the mixture. Also, in the absence of experimental data for the binary pairs, the coefficients for use in the UNIQUAC equation can be predicted by a group contribution method UNIFAC, described below. 

Moreover, the objective function obtained by minimizing the square of the difference between the mole fractions calculated by UNIQUAC model and the experimental data. Furthermore, he UNIQUAC structural parameters r and q were carried out from group contribution data that has been previously reported [14-15], The values of r and q used in the UNIQUAC equation are presented in table 4. The goodness of fit, between the observed and calculated mole fractions, was calculated in terms RMSD [1], The RMSD values were calculated according to the equation of percentage root mean square deviations (RMSD%) ... 

Since most synthetic and natural gas systems will contain some amount (however small) of heavy undefined components, we have been searching for improved methods of predicting critical properties and an equation of state which does not use critical constants (or quasi critical constants) to determine the parameters for the equation. Development of improved critical property prediction methods appears to be a waste of time. Wilson and Cunningham (6) have presented an equation—the Parameters From Group Contributions (PFGC) equation of state which satisfies our needs. As the name implies, the parameters in this equation of state are estimated by group contribution rather... 

The basic equations used to predict the thermodynamic properties of systems for the SRK and PFGC-MES are given in Tables I and II, respectively. As can be seen, the PFGC-MES equation of state relies only on group contributions--critical properties etc., are not required. Conversely, the SRK, as all Redlich-Kwong based equations of states, relies on using the critical properties to estimate the parameters required for solution. 

A set of equations has been proposed by Van Krevelen (1990) for the calculation of the solubility parameter components using the molar attraction by a group contribution methodology ... 

The procedure followed in the use of the tables of Andersen et al. [1], and Yoneda [4] is illustrated below for the estimation of standard entropies. These tables also include columns of base structure and group contributions for estimating fHm,298.i5K> thc Standard enthalpy of formation of a compound, as well as columns for a, b, and c, the constants in the heat capacity equations that are quadratic in the temperature. Thus it is possible to estimate AfGm gg.isK by appropriate summations of group contributions to Af7/ 298.i5K and to 5m,298.i5K- Then, if information is required at some other temperature, the constants of the heat capacity equations can be inserted into the appropriate equations for AG, as a function of temperature and AGm can be evaluated at any desired temperature (see Equation 7.68 and the relation between AG and In K). 

Since a does correlate with V (equation 10) and molecular volumes can be treated as summations over atomic and/or group contributions , it might be inferred that an analogous approximation (equation 12) could be applied for a ... 

The superscripts h and p refer to group contributions due to hydrocarbon and polar groups, respectively. The polar term is often found to be small while the shape of the cavity occupied by the solute molecule is irregular and requires a shape factor. Equation (4.9) then becomes ... 

The relative retention a=k/ki is a measure of the separation selectivity for two compounds i and j with retention factors ki and kj, respectively, differing by one repeat structural unit, An=l.p in Equation 5.16 is the end-group contribution to the retention factor. The conventional theory describes adequately the retention of oligomers and lower homopolymers and copolymers up to the molar masses 10,000-30,000Da for higher polymers the accuracy of the determination of retention model parameters is too low [95]. 

Reaction 2. The thermodynamic and kinetic quantities characterizing this reaction depend on the nature of the C—H bond broken. AH has been calculated, for each of the reactions having the general Equation 2, using data from Refs. 10, 33, 40, 50. The value of AS2 has been estimated by the method of partial group contributions (9). K2 has thus been calculated. 

The critical properties, that are essential basic data if a cubic equation of state is used, can be evaluated using group contribution methods but the numerical values obtained depend on the method used. In particular, this fact represents a problem for multifunctional components that are generally involved in processing natural products and/or pharmaceuticals. As an example, depending on the prediction method used, a critical temperature ranging from 817.8 to 1254.0 K can be obtained for cholesterol [60]. 

Values of the activity coefficients are deduced from experimental data of vapor-liquid equilibria and correlated or extended by any one of several available equations. Values also may be calculated approximately from structural group contributions by methods called UNIFAC and ASOG. For more than two components, the correlating equations favored nowadays are the Wilson, the NRTL, and UNIQUAC, and for some applications a solubility parameter method. The fust and last of these are given in Table 13.2. Calculations from measured equilibrium compositions are made with the rearranged equation... 

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