12265 IUPAC Gold Book - Flory–Fox equation 10.1351/goldbook.12265 12265 current 1 Equation relating intrinsic viscosity, \(\eta\), and molar mass, \(M\), to the mean-square radius of gyration, with \[[\eta]M = \mathit{\Phi}^{\prime}\!\lt\!s^{2}\!\gt^{3/2}\] where \(\mathit{\Phi}^{\prime}\) is a parameter the value of which depends on the molar-mass distribution, macromolecular constitution, and chain expansion. \([\eta]M/N_{\rm{A}}\), where \(N_{\rm{A}}\) is the Avogadro constant, is the equivalent hydrodynamic volume in viscous flow, \(V_{\eta}\), with \(V_{\eta} = 4\uppi r_{\eta}^{3}/3\), Hence, the Flory–Fox equation is consistent with the Kirkwood–Riseman theory and the Flory–Fox assumption. (See Notes to Kirkwood–Riseman theory and Flory–Fox assumption.) For a solution in the theta state, \(\mathit{\Phi}^{\prime}\) is denoted \(\mathit{\Phi}_{\uptheta}^{\prime}\) and its value is given by the Kirkwood–Riseman theory, with \(\mathit{\Phi}_{\uptheta}^{\prime} = \pu{4.22E22 mol-1}\). The Flory–Fox equation is sometimes written in terms of \(\lt\!r^{2}\!\gt\), the mean-square end-to-end distance, instead of \(\lt\!s^{2}\!\gt\), with \[[\eta]M = \mathit{\Phi} \lt\!r^{2}\!\gt^{3/2}\] where \(\mathit{\Phi} = \mathit{\Phi}^{\prime}/6^{3/2}\). The latter equality assumes that \(\lt\!r^{2}\!\gt = 6\!\lt\!s^{2}\!\gt\), which is only exactly true in the theta state. In this case, \[[\eta]M = \mathit{\Phi}_{\theta}\!\lt\!r^{2}\!\gt^{3/2}\] where \(\mathit{\Phi}_{\theta} = \mathit{\Phi}_{\theta}/6^{3/2} = \pu{2.87E21 mol-1}\). \(\mathit{\Phi}\) is known as the viscosity function or the Flory function. Flory–Fox assumption https://goldbook.iupac.org//terms/view/12264 Kirkwood–Riseman theory https://goldbook.iupac.org//terms/view/12263 equivalent hydrodynamic volume https://goldbook.iupac.org//terms/view/12257 mean-square end-to-end distance https://goldbook.iupac.org//terms/view/12197 mean-square radius of gyration https://goldbook.iupac.org//terms/view/12192 PAC, 2015, 87, 71. 'Definitions of terms relating to individual macromolecules, macromolecular assemblies, polymer solutions, and amorphous bulk polymers (IUPAC Recommendations 2014)' on page 104 (https://doi.org/10.1515/pac-2013-0201) https://goldbook.iupac.org/terms/view/12265/html https://goldbook.iupac.org/terms/view/12265/json https://goldbook.iupac.org/terms/view/12265/plain Citation: 'Flory–Fox equation' 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.12265 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. 2026-04-18T08:21:35+00:00