Question

The translational diffusion coefficient of the enzyme aspartate transcarbamylase in dilute aqueous solution was found to be $3.75\times {10}^{-7}c{m}^2{s}^{-1}$at $20°C$($\eta =1.002cP$). What is the hydrodynamic diameter of the particles? What does the hydrodynamic diameter mean if the particles are nonspherical?

Short answer

The hydrodynamic diameter of a flexible chain-containing enzyme particle is calculated to be $11.4nm$, and if the particle is nonspherical, its diameter would be the same as the diameter of a hypothetical sphere with the same translational diffusion coefficient.

Step-by-step solution

Given information 

Coefficient of translational diffusion$=D_T=3.75\times {10}^{-7}c{m}^2{s}^{-1}=3.75\times {10}^{-11}{m}^2{s}^{-1}$

Absolute temperature$=T=20°C=273.15+20=293.15K$

Viscosity$=1.002cP=1.002\times {10}^{-3}Pas$

Calculation

The hydrodynamic diameter can be calculated from the Stokes-Einstein relation given below:

$d_h=\frac{kT}{3\pi \eta D_T}$

Where,

$k=$Boltzmann's constant$=1.38066\times {10}^{-23}J{K}^{-1}$

$T=$absolute temperature

$\eta =$viscosity

$D_T=$coefficient of translational diffusion

Now, by substituting the given values in the above equation, we get,

localid="1646223978277" role="math" $$\begin{gathered} d_h=\frac{1.38066\times {10}^{-23}\times 293.15}{3\times 3.14\times 1.002\times {10}^{-3}\times 3.75\times {10}^{-11}} \\ {d}_h=11.4\times {10}^{-9}m=11.4nm \end{gathered}$$

Final answer

The hydrodynamic diameter can be calculated as $11.4nm$. Also, if the particles are nonspherical, the hydrodynamic diameter is equal to the diameter of a hypothetical sphere with the same coefficient of translational diffusion.