End-to-End Distance:

Since a polymer chain is not a regular object and because it is subject to dynamic structural equilibrium that involves motion and further, because polymers display polydispersity in size, it is necessary to consider a statistical measure of a chain size.  For a random chain the mean size is 0, <R> = 0, since it is equally likely to have positive and negative deviations from a chain path that leads back to the origin.  Then the lowest order moment with a non-zero value reflecting the size of a polymer chain is the second order moment about the mean chain position or the radius of gyration.  This is identical to the standard deviation about the mean for the chain length distribution.

Where the summation is over all segments of a chain structure.  A related size is the radius of gyration, R_g, which is essentially the moment of inertia for a structure.  The moment of inertia is the distance from the center of mass where all the mass could be placed to produce the same momentum on rotation of a structure about the center of mass.  For a random linear chain structure (d_f = 2) the radius of gyration is given by,

where N is the number of linear "steps" in a random polymer chain and l is the length of a step.  (The direction of each step is random.)  The Flory radius, R_F, is defined by the random walk law,

The Flory radius applies, for the most part, in the melt.   When a polymer is put in solution with an organic solvent, such as polystyrene in gasoline, the coil expands to a larger size than the size reflected by equation 3.  The expanded coil radius is given by,

where d_f is called the mass fractal dimension and GS stands for good solvent.  For a random coil d_f is 2 and for an expanded coil d_f is 1.67.

Equation (3) can be obtained directly from the Gaussian distribution function. 

Gaussian Distribution Function:

The Gaussian distribution function describes the probability of finding a value in a random distribution with a large number of samples. 

For a random coil the standard deviation, s, is given by,

Integration of equation (5) times R² yields equation (6).