Formulas of neighbour members in homologous series are known to differ in the same fragment. For n-alkanes having formula CnH2n+2 and increment Δn = 1, it is the so called homologous difference CH2. One of important properties of organic compounds is the normal boiling temperature (T). Its dependence on the number, n, of carbon atoms in the chain length is assumed to be given by the equation
(1)
where the parameters C0, C1, C2, k1 and k2 are determined using the temperature data of series. This equation is a solution of the inhomogeneous linear difference equation of second order
(2)
where the constant coefficients a, b, c are calculated as the least squares estimates defined on the basis of some restricted number of the experimental data. If z1 and z2 are the positive roots of the quadratic equation z2 + az + b = 0, it follows that k1 = ln(z1)/Δn, k2 = ln(z2)/Δn. Additionally, we can use the obvious identity: C0 = c/(1 – a – b). In an effort to use equation (1) for interpolations we deal with the T(n) data under condition that Δn > 1.
For the n-alkanes, precise experimental data are available [1], providing determination of coefficients a, b, c, as well as of the parameters k1 and k2. In particularly, for the n-alkanes with n ∈ [3÷18] Δn = 3 we obtain k1 = 0.054 ± 0.028, k2 = 0.3556 ±0.0053. The mean absolute difference between interpolated and experimental values of boiling temperatures does not exceed 0.0610 with standard deviation ±0.0500. An accuracy of the boiling temperatures of the n-alkanoles (n ∈ [4÷14] and Δn = 2), the alkancarboxilic acids (n ∈ [4÷10] and Δn = 2) and the alkyl arenas (n ∈ [6÷14] and Δn = 2) [2] is lower as compared to that of the n-alkanes. Nevertheless, numerical analysis of these series revealed that equation (1) is also suitable for approximation of the T(n) data, the above-mentioned values of the parameters k1 and k2 being applied. In all these cases, the mean absolute difference between interpolated and experimental values of boiling temperatures does not exceed 1.00 with standard deviation ±0.70.
Results demonstrate, that the parameters k1 and k2 may be treated as latent invariants which are identical for the T(n) data within the homologous series under consideration.
References:
1. Zwolinsky B.J., Wilhoit R.C. Handbook of vapor pressures and heats of vaporization of hydrocarbons and related compounds. Thermodynamics Research Center, Department of Chemistry Texas A&M University, College Station, Texas 77843 Publication N101 (1971), 329 p.
2. Zenkevich I.G. J. Gen. Chem. (Russ.). 2006, V. 76, N11. 1821–1833.