Hypergeometric series

The resulting series is a particular solution to Eq. (153) known as the hypergeometric series. It converges for x] < 1. It is usually denoted as... 

International Tables for Crystallography 1992). The function <]/> for Slater-type radial functions can be expressed in terms of a hypergeometric series (Stewart 1980), or in closed form (Avery and Watson 1977, Su and Coppens 1990). The latter are listed in appendix G. As an example, for a first-row atom quadrupolar function (/ = 2) with n, = 2, the integral over the nonnormalized Slater function is... 

Overall Reaction Rate Equation of Single-Route Complex Catalytic Reaction in Terms of Hypergeometric Series... 

As a result, it was obtained an analytical expression of the reaction rate in terms of hypergeometric series with no classical simplifications about the "limiting step" or the "vicinity of the equilibrium". The obtained explicit equation, "four-term equation", can be presented as follows in the Equation (77) ... 

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. 

Finally, we present the results of the case studies for Eley-Rideal and LH reaction mechanisms illustrating the practical aspects (i.e. convergence, relation to classic approximations) of application of this new form of reaction rate equation. One of surprising observations here is the fact that hypergeometric series provides the good fit to the exact solution not only in the vicinity of thermodynamic equilibrium but also far from equilibrium. Unlike classical approximations, the approximation with truncated series has non-local features. For instance, our examples show that approximation with the truncated hypergeometric series may supersede the conventional rate-limiting step equations. For thermodynamic branch, we may think of the domain of applicability of reaction rate series as the domain, in which the reaction rate is relatively small. 

We consider below the possibilities for simplification of overall reaction rate equations and introduce the main result of this chapter — the hypergeometric series for reaction rate. 

Recently Passare and Tsikh (2004) provided the detailed description of the domains of convergence of multi-dimensional hypergeometric series representing the roots of algebraic equations. They have relied on results of Birkeland obtained in 1920s. Birkeland found the Taylor series solutions to algebraic equations of the type... 

Figure 3 Overall reaction rate and its approximations by first 1, 2,..., 5 terms of hypergeometric series (Eley-Rideal mechanism). Parameters 62 = 0-71, t) = 0.2 and r2 = 7.

Figures 4 and 5 compare the exact solutions of the kinetic polynomial (51) (i.e. the quasi-steady-state values of reaction rate) to their approximations. Even the first term of series (65) gives the satisfactory approximation of all four branches of the solution (see Figure 4). Figure 5, in which "brackets" from Appendix 2 are compared to the exact solution, illustrates the existence of the region (in this case, the interval of parameter /2), where hypergeometric series converge for each root.

Passare, M., and Tsikh, A., Algebraic equations and hypergeometric series, in "Legacy of Niels Henrik Abel The Abel Bicentennial", Oslo, Springer, June 3-8, 653-672 (2002). 

C.J. van Duijn, Andro Mikelic, I.S. Pop, and Carole Rosier, Effective Dispersion Equations for Reactive Flows with Dominant Peclet and Damkohler Numbers Mark Z. Lazman and Gregory S. Yablonsky, Overall Reaction Rate Equation of Single-Route Complex Catalytic Reaction in Terms of Hypergeometric Series A.N. Gorban and O. Radulescu, Dynamic and Static Limitation in Multiscale Reaction Networks, Revisited... 

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