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Abstract— <p>First order recurrence relations <i>A</i>(<i>n</i><sub>C</sub> + 1) = <i>aA</i>(<i>n</i><sub>C</sub>) + <i>b</i> are used to approximate variations in different properties of organic compounds (<i>A</i>) in homologous series, depending on number <i>n</i><sub>C</sub> of carbon atoms in the molecules. However, they cannot be used to approximate melting temperatures (<i>T</i><sub>melt</sub>), due to their values alternating in homologs with even and odd numbers of carbon atoms in molecules of many series. It is confirmed that this problem can be solved using second-order recurrence relations <i>A</i>(<i>n</i><sub>C</sub> + 2) = <i>aA</i>(<i>n</i><sub>C</sub>) + <i>b</i>. The algebraic non-recurrent solution to this recurrent equation contains factor (−<InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11504_2025_6170_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">\(\sqrt a \)</EquationSource> <!--PhysChA2570067Zenkevich-m1--> </InlineEquation>)<sup><i>n</i>C</sup>, the sign of which is a function of the parity or oddness of the number of carbon atoms in the molecule. This type of function for characterizing alternating properties of homologs is proposed for the first time. It is shown that preliminary statistical processing of the values is needed to achieve the highest accuracy of approximating <i>T</i><sub>melt</sub> from different sources using calculated average values and their standard deviations, and rejecting unreliable data. An example of such processing is considered in detail using <i>T</i><sub>melt</sub> of <i>n</i>-alkanecarboxylic acids C<sub><i>n</i></sub>H<sub>2<i>n</i>+1</sub>–CO<sub>2</sub>H with 0 ≤ <i>n</i><sub>C</sub> ≤ 35. Recursive errors in <i>T</i><sub>melt</sub> homologs characterized via approximation do not exceed the standard deviations of randomized interlaboratory data.</p>

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Using Second-Order Recurrence Relations to Approximate Alternating Melting Points of Homologs

  • 期刊论文

  • 发表日期：

    2025年6月15日

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  • I. G. Zenkevich

摘要

Abstract—

First order recurrence relations A(nC + 1) = aA(nC) + b are used to approximate variations in different properties of organic compounds (A) in homologous series, depending on number nC of carbon atoms in the molecules. However, they cannot be used to approximate melting temperatures (Tmelt), due to their values alternating in homologs with even and odd numbers of carbon atoms in molecules of many series. It is confirmed that this problem can be solved using second-order recurrence relations A(nC + 2) = aA(nC) + b. The algebraic non-recurrent solution to this recurrent equation contains factor (− \(\sqrt a \) )nC, the sign of which is a function of the parity or oddness of the number of carbon atoms in the molecule. This type of function for characterizing alternating properties of homologs is proposed for the first time. It is shown that preliminary statistical processing of the values is needed to achieve the highest accuracy of approximating Tmelt from different sources using calculated average values and their standard deviations, and rejecting unreliable data. An example of such processing is considered in detail using Tmelt of n-alkanecarboxylic acids CnH2n+1–CO2H with 0 ≤ nC ≤ 35. Recursive errors in Tmelt homologs characterized via approximation do not exceed the standard deviations of randomized interlaboratory data.