Measuring by scattering at small angles

One important property of the form factor in dilute solutions is that at low scattering angle qRg 1) it becomes independent of any assumption about the shape of the molecule. Using the Taylor series expansion, 

In the final relation, Eq. (2.47) was used for R and the average ( ) is over different polymer conformations contributing to scattering. 

Recall from Chapter I that it is convenient to plot the reciprocal Rayleigh ratio times concentration and optical constant K [Eq. (1.96)] to determine the weight-average molar mass from the zero concentration limit. In Chapter 1, we considered the Rayleigh ratio in the zero wavevector 

Note that the plus sign in front of the (i g) term arises because (1 - 1 + X for small values of x. Thus, extrapolation of the ratio 

Kc/Re to zero concentration plotted as a function of sin ( /2) allows determination of the radius of gyration of the polymer (or any other scattering object) from the slope for low scattering angles (qRg 1) and the mass of the object from the -intercept. For polydisperse samples, this 

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