# How to calculate the radius of gyration (Rg) and the molar mass from light scattering data (Rθ and θ/degree)?

I have a homework problem that I don’t have a solution for, so I would like understand how to solve it and to know if my suggested way of solving it is on the right track. The problem is as following:

A polysaccharide was analyzed by light scattering at several angles ($$\theta$$), and the following data were at hand:

1. the system constant $$K = 2 \times 10^{-8}$$,
2. the concentration is $$2.5~\mathrm{mg/mL}$$,
3. the effective wavelength $$\lambda = 400~\mathrm{nm}$$,
4. the second virial coefficient is assumed to be $$0$$ (meaning that no correction is needed for the concentration dependence).

Calculate the molar mass and the radius of gyration for the polysaccharide.

We are also given the following information, where I think $$R_\theta$$ is the measured Rayleigh ratio, and $$\theta$$ is the scattering angle in degrees:

$$\begin{array}{rr} R_{\theta}& \theta\\\hline 5 & 60\\ 4 & 90\\ 3.33&120 \end{array}$$

From the lectures I was given the following equation to find the molar mass and $$R_\mathrm{g}$$:

$$\frac{K \times C}{R_\theta} = \frac{1}{M} \left( 1 + \frac{16\pi^2}{3\lambda} R_\mathrm{g} \times \sin \left(\frac{\theta}{2}\right)^2 \right)$$

I am unsure if the way I think is correct — and I don’t really understand all the steps. But here is what I think I have to do, or at least how to starts:

1. extrapolate to zero concentration to account for large molecules (polysaccharide) and non-ideal solution. To do so I think I have to plot $$\frac{K \times C}{R_\theta}$$ against $$\sin\left(\frac{\theta}{2}\right)^2$$.

2. From this the plot I get the linear equation $$y = ax + b$$ which I can use to:

• Find molar mass because: $$y - \text{intercept} = \frac{1}{M}$$
• Find $$R_\mathrm{g}$$ because: $$\frac{\text{slope}}{y - \text{intercept}} = R_\mathrm{g}$$
3. I read somewhere that I should make some sort of a second graph from the first one, in order to remove the $$\frac{1}{M}$$ part of the equation, and thus, I can somehow calculate $$R_\mathrm{g}$$, but I don't understand this.

My Questions:

• Am I generally on the right track here?
• What units do I get the molecular mass in from the above equation? Is it KDa?
• I think I get $$R_\mathrm{g}$$ value in nm?