Thermodynamics polynomials

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 thermodynamic properties of single-component condensed phases are traditionally given in tabulated form in large data monographs. Separate tables are given for each solid phase as well as for the liquid and for the gas. In recent years analytical representations have been increasingly used to ease the implementation of the data in computations. These polynomial representations typically describe the thermodynamic properties above room temperature (or 200 K) only. 

Thermodynamic representation of transitions often represents a challenge. First-order phase transitions are more easily handled numerically than second-order transitions. The enthalpy and entropy of first-order phase transitions can be calculated at any temperature using the heat capacity of the two phases and the enthalpy and entropy of transition at the equilibrium transition temperature. Small pre-tran-sitional contributions to the heat capacity, often observed experimentally, are most often not included in the polynomial representations since the contribution to the... 

Simple polynomial expressions constitute the most common analytical model for partial or integral thermodynamic properties of solutions ... 

The same type of polynomial formalism may also be applied to the partial molar enthalpy and entropy of the solute and converted into integral thermodynamic properties through use of the Gibbs-Duhem equation see Section 3.5. 

Burcat [ Thermochemical Data for Combustion Calculations, in Combustion Chemistry. (W. C. Gardiner, Jr., ed.), Chapter 8. John Wiley Sons, New York, 1984] discusses in detail the various sources of thermochemical data and their adaptation for computer usage. Examples of thermochemical data tit to polynomials for use in computer calculations are reported by McBride, B. J Gordon, S., and Reno, M. A., Coefficients for Calculating Thermodynamic and Transport Properties of Individual Species, NASA, NASA Langley, VA, NASA Technical Memorandum 4513, 1993, and by Kee, R. J., Rupley, F. M and Miller, J. A., The Chemkin Thermodynamic Data Base, Sandia National Laboratories, Livermore, CA, Sandia Technical Report SAND87-8215B, 1987. 

FITDAT Kee, R. J., Rupley, F. and Miller, J. A. Sandia National Laboratories, Livermore, CA 94550. A Fortran computer code (fitdat.f) that is part of the CHEMKIN package for fitting of species thermodynamic data (cp, h, s) to polynomials in NASA format for usage in computer programs. 

MECHMOD A utility program written by Turanyi, T. (Eotvos University, Budapest, Hungary) that manipulates reaction mechanisms to convert rate parameters from one unit to another, to calculate reverse rate parameters from the forward rate constant parameters and thermodynamic data, or to systematically eliminate select species from the mechanism. Thermodynamic data can be printed at the beginning of the mechanism, and the room-temperature heat of formation and entropy data may be modified in the NASA polynomials. MECHMOD requires the usage of either CHEMK1N-TT or CHEMKIN-III software. Details of the software may be obtained at either of two websites http //www.chem.leeds.ac.uk/Combustion/Combustion.html or http //garfield. chem.elte.hu/Combustion/Combustion. html. 

Langmuir model, 38 164-166 polynomial model, 38 167 theoretical background, 38 150-175 thermodynamics, 38 150-163 mobile, 26 360 modes, hydrogenolysis, 30 44 monolayer dispersion, 37 33-34 of NHj, 34 171... 

Equilibrium combustion product compositions and properties may be readily calculated using thermochemical computer codes which minimize the Gibbs free energy and use thermodynamic databases containing polynomial curve-fits of physical properties. Two widely used versions are those developed at NASA Lewis (Gordon and McBride, NASA SP-273, 1971) and at Stanford University (Reynolds, STANJAN Chemical Equilibrium Solver, Stanford University, 1987). 

Indeed, there are two approaches to the theory of binding phenomena. The first, the older, and the more common approach is the thermodynamic or the phenomenological approach. The central quantity of this approach is the binding polynomial (BP). This polynomial can easily be obtained for any binding system by viewing each step of the binding process as a chemical reaction. The mass action... 

The thermodynamic tables of Robie et al. (1978) use the Haas-Fisher polynomial (eq. 3.53.1). Helgeson et al. (1978) use the Maier-Kelley expansion, changing sign at the third term ... 

Equation (5.21) assumes ternary interactions are small in comparison to those which arise from the binary terms. This may not always be the case and where evidence for higher-order interactions is evident these can be taken into account by a further term of the type Gijit = x< xj Xk Lijk, where Lijk is an excess ternary interaction parameter. There is little evidence for the need for interaction terms of any higher order than this and prediction of the thermodynamic properties of substitutional solution phases in multi-component alloys is usually based on an assessment of binary and ternary terms. Various other polynomial expressions for the excess term have been considered, see for example the reviews by Ansara (1979) and Hillert (1980). All are, however, based on predicting the properties of... 

The thermodynamical average 5 (a) over the Legendre polynomials P occur in the expressions for the susceptibilities, the specific heat, and the dipolar fields in Section II.B. For uniaxial anisotropy these averages read... 

Multiple solutions to equations occur whenever they have sufficient nonlinearity. A familiar example is equHihrium composition calculations for other than A B. The reaction composition in the reaction A i B yields a cubic polynomial that has three roots, although all but one give nonphysical concentrations because thermodynamic equilihrium (the solution for a reactor with f —> co or T — co) is unique. 

For any more comphcated rate expressions the equations become polynomials in coverages and pressures, and the general solution is uninstructive. As we saw for any multistep reaction with intermediates whose concentrations we do not know and that may be small (now we have surface densities rather than fiee radical concentrations), we want to find approximations to eliminate these concentrations from our final expression. The preceding solution was for steady state (a very good approximation for a steady-state reactor), but the expression becomes even simpler assuming thermodynamic equilibrium. 

In this chapter, we will try to answer the next obvious question can we find an explicit reaction rate equation for the general non-linear reaction mechanism, at least for its thermodynamic branch, which goes through the equilibrium. Applying the kinetic polynomial concept, we introduce the new explicit form of reaction rate equation in terms of hypergeometric series. 

This means that we could have a situation when more than one root of kinetic polynomial vanishes at the thermodynamic equilibrium. However, only one of these roots would be feasible. 

We have found recently the topological interpretation of property (34). The stoichiometric constraints (24) can be interpreted in terms of the topological object, the circuit. Existence of the circuit "explains" the appearance of the cyclic characteristic in the constant term of kinetic polynomial. Thus, we can say that in some sense the correspondence between the detailed mechanism and thermodynamics is governed by pure topology. 

The cyclic characteristic C is small in the vicinity of thermodynamic equilibrium. We can find the overall reaction rate approximation in the vicinity of equilibrium either directly from kinetic polynomial or by expanding the reaction rate in power series by the small parameter C. The explicit expression for the first term is presented by Lazman and Yablonskii (1988, 1991). It is written as follows ... 

Note that both Equations (56) and (60) result in the same series for the root of kinetic polynomial corresponding the "thermodynamic branch" (see Appendix 4 for the proof). 

Eley-Rideal mechanism. Kinetic polynomial here is quadratic in R (see Equation (48)). There is only one feasible solution (49) here. The feasible branch should vanish at the thermodynamic equilibrium. Thus, the only candidate for the feasible branch expansion is R = — [Bq/Bi] because the second branch expansion is R — —B2/Bi+[Bq/Bi] and it does not vanish at equilibrium. First terms of series for reaction rate generated by formula (55) at = 1 are... 

We know from Proposition 1 that the constant term Bq C vanishes at the thermodynamic equilibrium. Some features of Equation (67) similar to the known LHHW-kinetic equation. There is a "potential term" Bq responsible for thermodynamic equilibrium, there is a "denominator" of the polynomial type. However there is also a big difference. Equation (67) includes the term D, which is generated by the non-linear steps. 

In the case of the quadratic equation, the convergence condition for the "thermodynamic branch" series is simply positive discriminant (Passare and Tsikh, 2004). For kinetic polynomial (48) this discriminant is always positive for feasible values of parameters (see Equation (49)). This explains the convergence pattern for this series, in which the addition of new terms extended the convergence domain. 

Applying "kinetic polynomial" approach we found the analytical representation for the "thermodynamic branch" of the overall reaction rate of the complex reaction with no traditional assumptions on the rate limiting and "fast" equilibrium of steps. 

Por Keq > 0, condition (A4.6) could be satisfied only, if j = 0. There is a bijection between solution (A4.1) and condition (A4.6), and the case 7 = 0 corresponds to the only feasible solution (A4.2) (see Proposition A4.1). However, when p = 1, there is only one branch of solutions of kinetic polynomial vanishing at the equilibrium. As the thermodynamic branch satisfies the equilibrium condition (A4.0) and there are no other branches vanishing at the equilibrium (we proved in Appendix 3 that Bi O at the equilibrium (see also Lazman and Yablonskii, 1991), this branch should be feasible. By continuity, this property should be valid in some vicinity of equilibrium. 

Let now p> 1. In this case, we have p branches of kinetic polynomial zeros vanishing at the equilibrium. Which one corresponds to the thermodynamic branch ... 

D.B. Marsland, Data fitting by orthogonal polynomials, in. CACHE Thermodynamics (ed. R.V. Jelinek), Sterling Swift, Manchaca TX, 1971. 

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