Zenkevich, Kranicz: GENERAL CHOICE OF NON-LINEAR REGRESSION FUNCTIONS...

GENERAL CHOICE OF NON-LINEAR REGRESSION FUNCTIONS FOR THE APPROXIMATION OF VARIOUS PHYSICO-CHEMICAL CONSTANTS WITHIN SERIES OF HOMOLOGUES

Igor G. Zenkevich¹, Balázs Kránicz²

¹Chemical Research Institute of St. Petersburg State University,
Universitetsky pr., 2, St. Petersburg 198504, Russia; E-mail igor@IZ6246.spb.edu
²University of Veszprem, Department of Image Processing and Neurocomputing,
Egyetem u., 10, P.O.Box 158, H-8200, Veszprem, Hungary; E-mail kraniczb@almos.vein.hu

The general method of the choice of optimal regression functions for the approximation of various physico-chemical constants of organic compounds can be based on the selection of best function f(n_C) not for their real values, but for their numerical derivatives fⁿ(n_C) of different order (n) followed by multi-step integration of these functions. This approach was proposed first for boiling points of homologues with f''(n_C) = a/n_C, that gives the target equation T_b = a log n_C + b n_C log n_C + c [ 1,2] . More precise choice of approximating function for n-order derivatives permits us to generalize this method for other properties.

The following formula determines the numerical derivatives of any equidistant data set: f''= (f_k-1 + f_k+1 – 2f_k) / h², where h denotes the step between two neighboring data points (n_C = 1). The regression functioncan be fit onto the data representing the second derivatives of reciprocal boiling points (x = 1/T_b) or absolute values of relative densities (d₄²⁰) and refractive indices (n_D²⁰). After twice integrating this function we get:

(1)

Using another fitting procedure the parameters d and e of this function can be determined. It is noteworthy that i. the parameter d seems to be neglectable, that corresponds to the existence of limiting theoretical values of physico-chemical constants at
n_C®

, namely 1/T_b® 0, d₄²⁰®0.857± 0.008, n_D²⁰ ®1.475± 0.003 [ 3] and ii. surprisingly the values of parameter c for various properties of different organic compounds are quite close to the negative Euler-number, i.e. - 2.71828… (!).
The table illustrates the precision of the different properties (P) approximation by method proposed.

Table. Some results (partially) of boiling points, relative densities,
and refractive indices of n-alkylbenzenes approximation with eq. (1)

Compound	MW	*T_b, ⁰C*	*\|T_b-T_b,precalc\|*	*d₄²⁰*	*d₄²⁰_,precalc*	*n_D²⁰*	*n_D²⁰_,precalc*
Toluene	92	110.62	0.15	0.8669	Eliminated	1.49693	Eliminated
Ethylbenzene	106	136.19	0.24	0.8670	Eliminated	1.49588	1.49583
Propylbenzene	120	159.22	0.93	0.8620	0.8621	1.49202	1.49215
Butylbenzene	134	183.27	0.13	0.8601	0.8599	1.48979	1.48965
Pentylbenzene	148	205.4	0.38	0.8585	0.8585	1.48780	1.48784
Hexylbenzene	162	226.1	0.27	0.8575	0.8575	1.48640	1.48647
. . . . .	. . .	. . .	. . . . .	. . .	. . .	. . . .	. . . .
Heptadecylbenzene	316	397	0.09	0.8546	0.8546	1.48110	1.48106
Octadecylbenzene	330	408	0.27	0.8546	0.8546	1.48090	1.48089
Nonadecylbenzene	344	419	0.17	0.8545	0.8546	1.48070	1.48075
Eicosylbenzene	358	429	0.21	0.8545	0.8546	1.48050	1.48063
Average differences \|P_experim - P_precalc\|			0.3 K	0.00006		0.00006

References:
1. Zenkevich I.G. Zh. Phys. Khim. (Rus). 72 (1998) 1286-1291.
2. Zenkevich I.G. Proc. 20^th Int. Symp. on Capil. Chromatogr. Italy, 1998. Rep. A.03 (CD-ROM).
3. Ioffe B.V., Zenkevich I.G. Zh. Phys. Khim. (Rus). 74 (2000) 2101-2106.