Use of recurrence relations for approximating properties of any homologs of organic compounds

Mathematical properties of recurrence relations, making them applicable to approximating various physicochemical constants of organic compounds (A) in homologous series, are discussed. The relationship A(n + 1) = aA(n) + b is shown to be valid not only for single-chain linear homologs, but also for compounds of the same series with branched alkyl substituents and in groups of homologs of other types (multichain, cyclic, insertion). Equations relating the coefficients a and b of the recurrence equations for multichain and normal linear groups of homologs are derived.     

The use of recurrent equations in chromatography

Recurrent equations A(x + k) = aA(x) + b were shown to be applicable to the approximation not only of virtually arbitrary properties of organic compounds (A) in homologous series (A = n _C, k = 1 or 2) but also of the dependences of chromatographic retention parameters on the number of carbon atoms in homologue molecules (A = t _R). The same equations described the temperature dependences of retention times of arbitrary compounds under isothermal separation conditions in gas chromatography (x = T, k = ΔT = const) and the dependences of retention times on the concentration of an organic solvent as an eluent component (x = C, k = ΔC = const) under isocratic separation conditions in high performance liquid chromatography.     

Recurrence calculations of organic compound boiling temperatures from data on preceding homologues

A new approach to calculating the temperatures of boiling at atmospheric pressure (T _b) of organic compounds from arbitrary homologous series is suggested. The approach is based on the linear dependence of these values on T _b for the preceding homologues, T _b(n) = aT_b(n ? 1) + b. This dependence, revealed for the first time, was used to obtain a recurrence relation for calculating T _b of organic compounds within any series from the data on three simpler homologues of the same series. The mean a and b values can be used to estimate T _b of an arbitrary organic compound from T _b for one preceding homologue with an accuracy not inferior to that provided by the modern ACD software. Correlations of the general form P(n) = aP(n ? 1) + b are observed not only for the boiling points of organic compounds but also for their other properties P (refractive indexes, relative densities, and ionization energies). This opens up the possibility of creating unified algorithms for calculating various physicochemical constants of organic compounds instead of particular algorithms for every particular property known earlier.     

Approximation of any physicochemical constants of homologues with the use of recurrect functions

Dependencies of various physicochemical constants of organic compounds (A) versus number of carbon atoms in the molecule within different homologous series [A = f(n _C )] usually are non-linear. The simplest recurrent equation A(n + 1) = a A(n) + b, connecting A-values for homologues (n + 1 carbon atoms) with the values of the same constants for previous members of series (n carbon atoms), indicates practically “ideal” linear character for most properties of organic compounds. It is the reasonable basis for approximation (or extrapolation) any physicochemical constants within any homologous series using the standard approach without special selection of appropriate algebraic functions. Principal mathematical properties of the function A(n + 1) = aA(n) + b and some of its chemical applications are considered.     

Recurrent equations of physicochemical constants for homologs substantiated with numerical sequences

The applicability of the unified first order recurrent equations A(n + 1) = aA(n) + b to the approximation of the physicochemical constants of various organic compounds (not only members of homologous series), previously found for normal boiling points, was extended to the dielectric constant (ε). This tendency is equivalent to the general procedure for calculating the ε of any compound from the data for preceding homologs with an accuracy comparable to the average interlaboratory reproducibility of the results of ε measurements. To substantiate this universal character of recurrent relations we consider the analogy between the constants of successive homologs and the recursive numerical sequences (Fibonacci and Padovan sequences versus Lucas and Perrin sequences, respectively).     

General relations holding in variation of physical properties of organic compounds within homologous series

The patterns of variation of different parameters (A) of organic compounds within a homologous series (such as boiling point under atmospheric pressure, critical temperature, critical pressure, refractive index, relative density, viscosity, surface tension, saturated vapor pressure, dielectric constant, first adiabatic ionization energies, etc.) are identical, and they can be described in terms of a single linear recurrent equation, A(n+1)=aA(n)+b. This equation relates a property of any member of a homologous series to the corresponding parameters of preceding homologs. The largest deviations from the proposed relation were revealed only in some homologous series for a few (1 or 2) simplest representatives which are characterized by deviations of all parameters.     

Application of recurrence relations to melting points of homologs

The previously established recurrence equations A(n + 1) = aA(n) + b characterizing variations in the majority of physicochemical constants of organic compounds in homologous series and relating the properties of adjacent homologs with n + 1 and n carbon atoms are applicable in the modified form [T _m(n + 2) = aT _m + b] to the melting points. This second-order recurrence is observed with a high accuracy, despite significant alternations in the melting points of even and odd homologs. The method for calculating melting points, based on this recurrence, has no analogs among the previously suggested methods and makes it possible, in particular, to reveal errors in reference data, originating in most cases from the low purity of the samples dealt with.     

Description of the even-odd alternation of the clearing temperatures over homologous series of mesogens by second-order differential equations

It is shown for liquid crystals of various chemical classes that the nematic-isotropic transition temperatures in homologous series of compounds undergoing enantiotropic transitions are described by secondorder inhomogeneous recurrent equations with constant coefficients: T(n + 2) = aT(n + 1) + bT(n) + c (n is the number of alkyl carbon atoms). Solutions of the corresponding differential equations as T(n) functions allow reproduction of the even-odd alternation of the clearing temperatures.     

Calculating the acidity constants of homologues and isomers of organic acids by means of recurrence relations

It is noted that the pK _a values of organic acids can be calculated using the unique recurrence relation pK _a(n + 1) = apK _a(n) + b from the pK _a values of other (usually the simplest and, consequently, better characterized) homologues of the same series. It is shown that this relation is valid within two taxonomic groups: insertion homologues of the ω-substituted acids X(CH₂) _n CO₂H (n ≥ 1) and isomers that differ in the position of substituents X in their alkyl fragments, k-X(C _n H_2n )CO₂H (n ≥ 1, 1 ≤ k ≤ n + 1). It is concluded that this algorithm is a consequence of the unique mathematical properties of recurrence relations.     

Reciprocally Unambiguous Conformity Between GC Retention Indices and Boiling Points within Two- and Multidimensional Taxonomic Groups of Organic Compounds

It was shown that reciprocally unambiguous conformity between GC retention indices (at least for the commonly used standard nonpolar polydimethylsiloxane liquid phases) and boiling points of organic compounds is typical not only within one-dimensional taxonomic groups (homologous series and/or groups of congeners), but also within two- and multidimensional taxonomic groups (with simultaneous variations of some structural fragments). In all cases, this conformity is described by three-parameter non-linear equations log RI = a log T_b + b (n₁ + Σ k_i n_i) + c, where n₁ is the serial number of homologue within corresponding series and n_i is the number of other structural fragments in the molecules. The coefficients k_i in this equation reflect the relative alterations of molecular polarizabilities and may be estimated as ratios of refractions k_i = R_D(X)/R_D(CH₂), where X are variable structural fragments within a group of congeners, R_D(CH₂) = 4.647 cm³mol⁻¹. The approach under discussion permits precalculation of the retention indices of any organic compounds with known boiling points. The precision of proposed method of RI precalculation is comparable with the contemporary level of interlaboratory reproducibility of experimental RI determination with standard nonpolar liquid phases (5–10 i.u.).     

Calculation of gas chromatographic retention indices of organic compounds from the boiling points of their structural analogs

The possibility of calculating gas chromatographic retention indices (1) is discussed. The indices are important parameters for chromatospectral identification of organic compounds from the boiling points of their structural analogs (T _b ^* ) using the linear logarithmic equation log I = a log T _b ^* + bA + c, where A are the structural parameters reflecting the one- to- one position of the compounds being compared in the corresponding taxonomic groups, including homologous series.     

Chemical shifts of the protons of the primary amine group and its structure

In the NMR spectra of a series of amine-containing organic compounds, a linear dependence has been found to exist between the chemical shifts of the signals for the primary amino group protons, δ_NH2 values, and the s-character (b²) of N-H bonds. The points for the amines (aliphatic and aromatic) lie on a straight line of equationS(%) = 2.1δ_NH2 + 18.4The corresponding correlation for amides is a straight line parallel to the abscissa, on a level of (b²) = 0.33, i.e. the value characterizing the pure sp² hybrid state on the N atom.     

Optically active cyclic poly(ether sulfone)s based on chiral 1,1′-bi-2-naphthol

Optically active cyclic poly(ether sulfone)s are prepared from 4-fluorophenyl sulfone and (R)- or (S)-1,1′-bi-2-naphthol. The measurement of MALDI-TOF MS shows that the product is composed of a series of cyclic and linear oligomers where the repeating unit number (n) is from 2 to 12, from which the cyclic dimers (n=2), and cyclic trimer (n=3) have been separated from their homologous compounds by TLC successfully. Specific optical rotation [α]_D²⁵ is −583.0 for (R)-cyclic dimer, +588.0 for (S)-cyclic dimer, +22.7 for (R)-cyclic trimer, and −20.3 for (S)-cyclic trimer. Their properties are also determined by other methods, such as ¹H NMR and CD etc.     

Estimation of boiling points of organic compounds at various pressures using recurrent relations

A new approach is proposed for the estimation of boiling points (T _b) of organic compounds at reduced pressure from their values at atmospheric pressure based on the application of a recurrent relation: T _b (log P + Δlog P) = aT _b (log P) + b. Estimation of coefficients in this relation for the compounds different by their chemical nature gives the following average values: a = 1.126, b = ?41.7. Successive application of this relation with Δlog P = 1 (that corresponds to 10-fold decrease in pressure) allows estimation of the T _b values at the pressure values of 100, 10 and 1 torr from the value of T _b (760 torr) by simple arithmetic calculation with an average accuracy about 8°C.     

Ermittlung kinetischer Parameter aus der Kurvenform für Reaktionen unter Aufheizung

Form indexes for DTA or TG curves (S _T orS _x) must be treated separately. Only forS _x can clear relations to be developed for the order of reactionn. In the rational range ofn between 0.5 and 3.0 we found for linear, exponential and hyperbolic programmes these functions were found to be of the type $$S_x = an^{0,5} + b$$ Since a low dependence on frequency factork ₀ has been established, for a linear programme the simultaneous determination ofn andk ₀ may be performed.     

The solubility of fullerene C70 in monocarboxylic acids C n − 1H2n − 1COOH (n = 1–9) over the temperature range 20–80°C

The isothermal (20°C) solubility of fullerene C₇₀ in solvents of the homologous series of monocarboxylic acids C _{n ? 1}H_{2n ? 1}COOH (n = 1–9) and polythermal solubility over the temperature range 20–80°C of fullerene C₇₀ in solvents of the homologous series of monocarboxylic acids C _{n ? 1}H_{2n ? 1}COOH (n = 4–9) were studied. The corresponding solubility diagrams were obtained and characterized.     

A method for the calculation of heats of solution (ΔH s ) for organic compounds from the number of constituent carbon atoms and the functional groups

The heat of solution (ΔH _s ) of several homologous series of linear and monofunctional organic compounds (n-alkanes,n-alcohols,n-aldehyds and esters) in polar stationary phase (Carbowax 1540) can be expressed by ΔH _s =a+b·n _R +c·C _G , wherea is a constant,n _R is the number of carbon atoms not belonging to the functional group in the molecule andC _G represents the contribution of each functional group to the ΔH _s value. The accuracy of the ΔH _s values calculated by this equation is sufficient for all the compounds tested (average relative error=1.92%).C _G can be calculated from the plots ΔH _s versusn _R .     

Physicochemical properties of compounds of alkyl sulfates and cationic copper(II) complexes with some organic reagents

Compounds of alkyl sulfates and cationic copper(II) complexes with some organic reagents (pyridine, 1,10-phenanthroline, 2,2'-dipyridyl) were prepared, and their physicochemical characteristics (composition, thermal stability, solubility) were determined. The synthesized compounds are poorly soluble (solubility product K_S = n×10^–20–n×10^–22) and thermally stable (90–260°С). An effect of the hydrophobilicity of alkyl sulfates on the UV characteristics of the copper(II)–organic reagent systems and on the solubility of the studied compounds was evaluated.     

Violation of the linearity principle of additivity of sorption energy in chromatography. A universal equation describing retention behavior

Deviations from the additivity of energy contributions to substance sorption energies, determined on the basis of thermodynamic studies of GC behavior of homologous series of organic compounds fall into two categories: one for n-alkanes, the other for homologous series containing a functional group. A previously derived equation is proposed for homologous series, describing the deviation from the linear dependence of retention parameters with a propagating homolog n-alkyl chain. The equation permits calculation of retention parameters of homologs starting from the first member in gas-liquid, gas-solid, and liquid-liquid systems. The results prove its universal applicability.     

Rectangular source integral and recurrence relations

In this paper Hubbell's rectangular source integral H′(a,b), which is a double integral, is expressed as a series of many converging single integrals I_n (a,b). Recurrence relations relate these integrals. Once one integral I₁ is computed, recurrence relations are used to compute other integrals. I₁(a,b) can be computed analytically. H′(a,b) is approximated by considering the first seven terms in the series and the results are found to give good results for various values of a and b. Results are presented for the values of a and b (0.1 to 20 and to 2), respectively. The rate of convergence depends on the values of a and b.