Polynomial activity coefficients

The Calculation of Activity Coefficients in the Ternary System. The polynomial with 10 constants was used to describe the experimental quantity A(m 1,7712) in the whole concentration range... 

Eight different temperatures (coded X data below) were used. The resulting activities are given as Y. Determine the polynomial regression coefficients. 

Since the degree of coupling is directly proportional to the product Q (D/k)in, the error level of the predictions of q is mainly related to the reported error levels of Q values. The polynomial fits to the thermal conductivity, mass diifusivity, and heat of transport for the alkanes in chloroform and in carbon tetrachloride are given in Tables C1-C6 in Appendix C. The thermal conductivity for the hexane-carbon tetrachloride mixture has been predicted by the local composition model NRTL. The various activity coefficient models with the data given in DECHEMA series may be used to estimate the thermodynamic factors. However, it should be noted that the thermodynamic factors obtained from various molecular models as well as from two sets of parameters of the same model might be different. 

The isothermal compressibilities have been calculated with eq 5, using for the isothermal compressibilities of the pure substances the data from refs 53—56 (only the value for 2-butanol was taken as that for isobutanol). The data have been fitted using the Redlich—Kister equation.The values of D have been obtained from the activity coefficients or total pressure data by the sliding polynomials method. To check the accuracy of our calculations, the D values have been... 

The treatment of the solubility data in the paper for the determination of the standard enthalpy of dissolution could not be fully understood and the following treatment was resorted to by the evaluator. The data in the temperature interval 283 to 313 K were selected. Approximate activity coefficients were taken from the data for MgS04 in [50HAR/OWE]. A second order polynomial was fitted to these data and mean activity coefficients for CaSe04 in the saturated solutions obtained by interpolation. No attempt was made to correct for the temperature variation of the activity coefficient. 

Most other approaches to the calculation of activity coefficients for solution components, including solid and gaseous as well as liquid solutions, have used some form of a virial equation as a starting point. A virial equation is simply an equation for the ideal state (e.g., the ideal gas equation) followed by an ascending polynomial in one of the state variables. It seems to work well as a basis for activity coefficients because the form of the equation has a basis in statistical mechanics. 

Several Correlative Liquid Mixture Activity Coefficient Models 431 The simplest polynomial representation of G satisfying these criteria is... 

The data m, y and cj) of NiS04 required in the plots of Figure A-5 and Figure A-6 were taken from Table 16, Appendix 8.10 of this textbook. The activity coefficient (sat) has been calculated by a third order polynomial of In y vs. 4m, as depicted in Figure A-5. For the activity of water, uw(sat), (f> was extrapolated to the saturated solution molality with a fourth order polynomial in w, as depicted in Figure A-6. 

Figure A-5 Extrapolation of activity coefficients to saturated solution at 25°C. Solid curve calculated with a third order polynomial in 4m In = A + + Brm +...

Use the data in Examnie 12.6 to fit the excess Gibbs free energy of the system ethanol/acetonitrile to a Redlich-Kister polynomial with two parameters, obtain equations for the activity coefficients. Use these equations to obtain the activity coefficients at infinite dilution and to construct the Pxy graph of this system at 40 "C. 

The Margules equation models the excess Gibbs free energy by a two-parameter Redlich-Kister polynomial. The excess Gibbs energy and the activity coefficients are given by the following equations ... 

You may have noticed that the Margules model is equivalent to the two-parameter Redlich-Kister polynomial used in Examples 12.6 and 12.7. This maybe confirmed by setting A12 = Oo andA = Oo + Oi to the above equations (see Example 12.8. below). In the form given here, the expressions for the activity coefficients are symmetric in the two components such that each expression is obtained from each other by switching the subscripts 1 and 2. 

Notice that in this model the excess Gibbs free energy is not a polynomial in the mol fraction. The van Laar equation is a two-parameter model and the constants and Ba are related to the activity coefficients at infinite dilution ... 

Hirata (47) also presented an equation for the calculation of activity coefficients for this quaternary system. The equation was based on the Margules equation and consisted of 64 constants rearranged as a polynomial series in terms of mole fractions. It was tested by Pilavakis and the results found inconsistent even for binary mixtures. 

For weak adds and bases with double and triple dissociations. Equations 2 4b and 2-4c are used, respectively, in place of Equation 2-4a in Equation 2-4h. Appendix F has a FORTRAN subroutine listing that finds the pH that satisfies the charge balance for i adds and bases with single, double, or triple dissodations by interval halving, The pKj, and pK coeffidents and the solution density should be corrected for process temperature and composition. The effect of changes in the activity coefficients with ionic strength should also be factored in [Ref. 2.6]. The reader is directed to References 2.4 and 2.8 for a more detailed discussion on the effect of ion concentrations on activity coeffidents. The polynomial equation for acid and sodium ion error at die end of the subroutine should be replaced with one that fits the data from the glass manufacturer. 

Tables 3 and 4 contain values of the log water activity and log sulfuric acid activity in molarity units. These can be obtained at any temperature by using the polynomial coefficients supplied by Zeleznik,45 which are based on all of the preexisting thermodynamic data obtained for this medium. The numbers were converted to the molarity scale using the conversion formula given in Robinson and Stokes 46 Molarity-based water activities are given for HCIO4 in Tables 5 and 6. These are calculated from data obtained at 25°C by Pearce and Nelson,17...

The surface tension of the system KF-KBF4 decreases with the increasing content of KBF4, obviously due to the covalent character of the bonds in the BFJ complex anions, which are surface active and concentrate on the melt surface. Similar values as well as the shape of the surface adsorption curve were found when it was calculated from the polynomial coefficients and from the excess Gibbs energy of mixing in the liquid phase. Even both the calculated interaction parameters B are relatively close. 

Under these conditions system (9.1) still admits a unique steady state, but linear stability analysis shows that the latter is always stable (Goldbeter Dupont, 1990) this rules out the occurrence of sustained oscillations around a nonequilibrium unstable steady state. This result holds with previous studies of two-variable systems governed by polynomial kinetics these studies indicated that a nonlinearity higher than quadratic is needed for limit cycle oscillations in such systems (Tyson, 1973 Nicolis Prigogine, 1977). Thus, in system (9.1), it is essential for the development of Ca oscillations that the kinetics of pumping or activation be at least of the Michaelian type. Experimental data in fact indicate that these processes are characterized by positive cooperativity associated with values of the respective Hill coefficients well above unity, thus favouring the occurrence of oscillatory behaviour. 

No attempt will be made here to compute further force constants instead the reader is referred to the papers of Crawford and Miller cited above. However, as an example of the calculation of frequencies from a set of known constants, the Eiu. factor (infrared active) will be worked out, using Crawford and Miller s constants. Although a prolilcm of this size (three frequencies) can be carried through by direct expansion to the polynomial form, we shall not use such a method, but shall instead illustrate some of the matrix numerical methods which were described in Sec. 9-7. This method will also yield the transformation coefficients between symmetry and normal coordinates. 

The Tail equation is a four-parameter representation of the P-V-T behavior of polymers. The four parameters are the zero pressure isotherm, Vq. Tait parameter, B, thermal expansion coefficient, a, and activation energy, In some industrial systems polynomial expressions are used for the zero-pressure isotherm and Tait parameter as follows ... 

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