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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?
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