symbols: $\lt\!s_{i}^{2}\!\gt$, $R_{\rm{g}}$; units: $\pu{nm}$, $\pu{m}$
https://doi.org/10.1351/goldbook.12191
For a flexible macromolecule composed of \(n\) mass elements, of masses \(m_{i}\), \(i = 1, 2, \ldots, n\), located at statistical-mechanical mean-square distances \(\lt\!s_{i}^{2}\!\gt\) from the centre of mass, the root-mean-square radius of gyration is the square-root of the mass average of \(\lt\!s_{i}^{2}\!\gt\) over all mass elements, i.e., \[\lt\!s^{2}\!\gt^{1/2}\;= \left(\sum\limits_{i=1}^{n} m_{i} \lt\!s_{i}^{2}\!\gt\!/\sum\limits_{i=1}^{n} m_{i} \right)^{1/2}\]
Notes:
- The mass elements are usually taken as the masses of the skeletal groups constituting the macromolecule, e.g., \(\ce{-CH2-}\) in poly(methylene).
- \(\lt\!s_{i}^{2}\!\gt\) is the statistical mechanical average of \(s_{i}^{2}\) over all chain conformations.
- The radius of gyration is a parameter characterizing the size of a macromolecule as a time-averaged spherical domain in laboratory coordinates.