# DOSY-NMR: How to calculate the volume of the molecule from the diffusion coefficient?

DOSY-NMR allows for the determination of the diffusion coefficient of the analyte molecule. How do I calculate the volume/radius of the molecule from the diffusion coefficient?

You can try and use the Stokes-Einstein equation for the diffusion coefficient $$D=\frac{k_\mathrm{B}T}{\zeta}$$ where the friction term is $$\zeta = 6\pi\eta r$$ and $k_\mathrm{B}$ is the Boltzmann constant, $\eta$ the solvent viscosity $(1\ \mathrm{cP} = 10^{-3}\ \mathrm{kg~m^{-1}~s^{-1}} = 10^{-3}\ \mathrm{Pa~s})$ and r the molecule's radius . You only have one number in diffusion constant so you can only get radius or volume assuming the molecule is a sphere.

The model assumes that the molecules are larger than the solvent molecules, i.e. solvent is effectively continuous not molecular. There are slightly different expressions for the friction, 4 instead of 6 for example depending on whether 'slip' or 'stick' limits are assumed.

• Are there other ways to calculate the volume without assuming the molecules to be a sphere. Can I calculate the diffusion constants using e.g. MM or DFT? Oct 29 '16 at 12:54
• @ketbra What exactly are you after? Don't you know what substance you are measureing?
– Karl
Oct 29 '16 at 21:37
• @Karl I know the substance but whant to show wheather its a monomer/dimer/tetramer and want to argue with the size of the molecule. Oct 30 '16 at 8:01
• If you only have one measurement then you can only assume a sphere, you do not have enough information to assume a prolate or oblate spheroid. You should be able to distinguish monomer etc just from the diffusion coef as the molecules probably coil up or cluster in some way . You can certainly get size & shape from calculation. If you could measure rotational diffusion, say by fluorescence anisotropy, then this gives a better idea of shape. Oct 30 '16 at 8:56
• @ketbra You can surely differentiate betwen monomers, dimers and trimers spectroscopically. After that the diffusion cofficients are distinctive.
– Karl
Oct 30 '16 at 20:42