Recurrent relations for the approximation of the physicochemical constants of homologues

In addition to the earlier revealed physicochemical constants of homologues whose changes in arbitrary series obey the simplest linear recurrent relations A (n + k) = aA(n) + b, such equations are shown to be applicable to the approximation of the solubility of organic compounds in water (k = 1), temperature dependences of the solubility of organic and inorganic compounds in water (k = ΔT), and nematic-isotropic phase transition temperatures for liquid crystals (k = 2). The a and b coefficients of linear recurrent relations are only determined by the nature of the homologous difference, and, if the homologous difference is the same, they are close for different series. This enables various properties of virtually arbitrary organic compounds to be described by unified recurrent equations, which is equivalent to the existence of a general method for their calculation. For continuous properties (for the example of the temperature dependence of solubility), a method for solving recurrent equations with nonintegral or nonequidistant argument values is suggested.     

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.     

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.     

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.     

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.     

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.     

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.).     

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.     

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.     

Computing boiling temperatures and vapor pressures in homologous series of perfluorinated oligomers

Correlation relationships are proposed for computing boiling temperatures of T _b oligomers with the general formula R₁(CF₂) _n R₂ at normal pressure, where R₁ and R₂ are arbitrary end groups and the increment of the CF₂ fragment is 20.4 K. The dependences of coefficients A and B in the Clausius-Clapeyron equation (ln P = A ? B/T) on the length of the oligomer chain are determined by computing critical temperatures and pressures using an additive scheme.     

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.     

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.     

Line-width temperature dependence of selected R-branch transitions in the ν3 fundamental of 14N216O between 135 and 295 K

The temperature dependence of N₂ foreign gas broadening coefficients for eight R-branch transitions in the ν₃ fundamental of ¹⁴N₂¹⁶O have been measured at 135, 176, 210 and 295 K by tunable diode laser spectroscopy. In approximate agreement with the theoretical value of 0.75 for n in the relation γ⁰_L(T)/γ⁰_L(T₀) = (T₀/T)ⁿ for quadrupole—quadrupole collisional interactions, n ranged from 0.66 to 0.71. We observed a distinct dependence between J and the exponent n in the range between R(15) and R(43).     

Interaction of 3,5-dimethylpyrazole with cobalt(II) carboxylates containing coordinated 1,10-phenanthroline or 2,2′-dipyridyl

The reaction of Co₃(??-OOCBu ^t )₆(NEt₃)₂ with 1,10-phenanthroline or 2,2??-dipyridyl in benzene at room temperature yields L₂Co₂(??-OH₂)(??-OOCBu ^t )₄ complexes (L₂ = Phen (1a) and Dipy (1b)). The reaction of n1a (1b) with 3,5-dimethylpyrazole gives a mixture of L₂Co(Hdmpz)₂(OOCBu ^t )₂ complexes (2a, 2b) and L₂Co(Hdmpz)(OOCBu ^t )₂ (3a, 3b), and their yield is determined by the ratio of the initial reagents. As distinct from pivalates, for cobalt(II) phenanthroline-benzoate, only the (Phen)Co(Hdmpz)₂(OOCPh)₂ complex (4) has been isolated. The structures of the synthesized compounds have been determined by X-ray crystallography.     

High-temperature heat capacity of YbAl3(BO3)4

The isobaric heat capacity C _p (T) of YbAl₃(BO₃)₄ grown by spontaneous crystallization from solution (100 ? n) wt % (Bi₂Mo₃O₁₂ + 2.5% B₂O₃ + 0.75% Li₂MoO₄) + n wt % YbAl₃(BO₃)₄ is studied experimentally in the region of 344–1016 K. It is established that there are no extrema on the C _p (T) dependence, and the obtained data can be described using the Berman-Brown polynomial. The temperature variations of enthalpy and entropy are calculated from the C _p (T) dependence.     

Prediction of molecular weight and density of n-alkanes (C6-C44)

Heavy n-alkanes and their mixtures were characterized by high temperature-simulated distillation using gas chromatography with a capillary column. In this work, the atmospheric boiling point is determined by the HT-SimDis GC method. In this study, molecular weights and density of n-alkanes were evaluated with this method by using retention times and normal boiling points as input data. ASTM D2887 calibration mixture containing 17 n-alkanes in the C₆-C₄₄ range were used for qualitative analyses. Retention times (t_R) of n-alkanes were measured with this method. The other input data that normal boiling points (T_b) and molecular weight (M) had been taken in the literature. Experimental densities (at 20 °C) of n-alkanes were obtained from API Research Projects. Empirical molecular weight and density correlations were developed by using the nonlinear and multiple regressions with correlation coefficients. The results of calculations were compared with experimental data. Normal boiling point predictions were obtained as an average absolute deviation of 1.07%. Molecular weight and density results were evaluated as average absolute deviations of 0.68% and 0.21%, respectively.     

Application of the modified van der waals equation for unsaturated vapour and liquid states

The Law-Lielmezs (L-L) modification of the Van der Waals equation of state: P = RT/(V-b)-a(T)/V² where: a(T) = a(T_c)·a(T_c·a(T¹) and: a(T¹) = 1 + pT^1q has been extended to include unsaturated states in terms of a correcting function C_f(such that the α(T¹) term becomes: a(T¹) = 1 + _pC_fT^1q The proposed extension has been compared with the results obtained by the use of the original Van der Waals equation of state.     

Synthesis and crystal structures of yttrium sulfates Y(OH)(SO4), Y(SO4)F,YNi(OH)3(SO4)-II and Y2Cu(OH)3(SO4)2F·H2O

Y(OH)(SO₄), Y(SO₄)F, YNi(OH)₃(SO₄)-II and Y₂Cu(OH)₃(SO₄)₂F·H₂O are obtained from hydrothermal reactions at 380°C under a pressure of 210 MPa. Their crystal structures were refined from single-crystal X-ray diffraction data. The four compounds have the following space groups and unit cells: Y(OH)(SO₄), P2₁/n, a=7.9498(6), b=10.9530(9), c=8.1447(6) Å, β=93.764(1)°; Y(SO₄)F, Pnma, a=8.3128(9), b=6.9255(7), c=6.3905(7) Å; YNi(OH)₃(SO₄)-II, Pnma, a=6.9695(8), b=7.2615(8), c=10.292(1) Å; Y₂Cu(OH)₃(SO₄)₂F·H₂O, P2₁/n, a=11.6889(7), b=6.8660(4), c=12.5280(8) Å, β=97.092(1)°. The coordination environments of the yttrium atoms in the four structures vary from highly irregular 6+2, 6+3, 7+1 coordination polyhedra to relatively regular dodecahedra.