12199 IUPAC Gold Book - mean-square unperturbed end-to-end distance 10.1351/goldbook.12199 12199 current 1 Mean-square end-to-end distance of a chain having unperturbed dimensions. \(\lt\!r_{\rm{o}}^{2}\! \gt\) of a hypothetical, sufficiently long, linear chain of identical single skeletal bonds having rotational isomeric states that are independent of each other is given by the equation \[\lt\!r_{\rm{o}}^{2}\!\gt = nl^{2} \frac{(1 + \cos\theta)}{(1 - \cos \theta)} {\cdot} \frac{(1 + \lt\!\cos \phi\!\gt )}{(1 - \lt\!\cos \phi\!\gt)}\] where \(n\) is the number of skeletal bonds of length \(l\) and valence-angle supplement \(\theta\), and \(\lt\!\cos \phi\!\gt\) is the average of the cosine of the dihedral angles \(\phi\) defined over sequences of three contiguous skeletal bonds. This equation is useful for understanding the significance of characteristic ratio, steric factor, and effective bond-length. effective bond-length https://goldbook.iupac.org//terms/view/12202 rotational isomeric states https://goldbook.iupac.org//terms/view/12187 PAC, 2015, 87, 71. 'Definitions of terms relating to individual macromolecules, macromolecular assemblies, polymer solutions, and amorphous bulk polymers (IUPAC Recommendations 2014)' on page 76 (https://doi.org/10.1515/pac-2013-0201) https://goldbook.iupac.org/terms/view/12199/html https://goldbook.iupac.org/terms/view/12199/json https://goldbook.iupac.org/terms/view/12199/plain Citation: 'mean-square unperturbed end-to-end distance' in IUPAC Compendium of Chemical Terminology, 5th ed. International Union of Pure and Applied Chemistry; 2025. Online version 5.0.0, 2025. 10.1351/goldbook.12199 The IUPAC Gold Book is licensed under Creative Commons Attribution-ShareAlike CC BY-SA 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) for individual terms. If you are interested in licensing the Gold Book for commercial use, please contact the IUPAC Executive Director at executivedirector@iupac.org . The International Union of Pure and Applied Chemistry (IUPAC) is continuously reviewing and, where needed, updating terms in the Compendium of Chemical Terminology (the IUPAC Gold Book). Users of these terms are encouraged to include the version of a term with its use and to check regularly for updates to term definitions that you are using. 2026-06-07T03:06:52+00:00