Please help with this diffusion and mass transfer problem Pl

Please help with this diffusion and mass transfer problem. Please include all equations and method used. Thank you.

Find the relationship for mean square distance (displacement) , moved in time t in one dimension (its in Datta) and use it to calculate the time required for insulin (MW 6,000, D_insulin-water = 8.2 times 10^-7 cm^2 sce^-1) to: Cross a red blood cell whose diameter is 7 times 10^-4 cm. State any assumptions you make. Traverse the length of a 10 cm long nerve cell. State any assumptions you make. Comment briefly on your results as they impact insulin delivery in type 1 diabetes.

Solution

Solution:

Mean square displacement is defined as the square of the difference between the position at time t, x(t) and the initial position, x_o. MSD is equal to the product of the diffusion rate and the time taken.

MSD= [x(t)-x_o]² = Dt

a). Diameter of RBC = 7 x 10^-4

Assumption- let the insulin molecule is placed at the edge of the RBC which is also the origin.

Therefore, x_o= 0 cm

Object at time t, x(t) = Diameter of rbc = 7 x 10^-4 cm

D = 8.2 x 10⁷ cm²sec^-1

MSD = (7 x 10^-4 cm)² = 8.2 x 10⁷ cm²sec^-1 x t

OR t= 7 x 10^-8 cm)² / 8.2 x 10⁷ cm²sec^-1

=> t= 0.853 x 10^-15 sec

b). Length of nerve cell: 10 cm

Assumption- let the insulin molecule is placed at one end of the nerve cell, which is also the origin.

Therefore, x_o= 0 cm

Object at time t, x(t) = 10 cm

D = 8.2 x 10⁷ cm²sec^-1

MSD = (10 cm)² = 8.2 x 10⁷ cm²sec^-1 x t

OR t= 100 cm² / 8.2 x 10⁷ cm²sec^-1

=> t= 12.195 x 10^-7 sec

c). Insulin delivery is directly proportional to its diffusion rate across the cells, esp. blood cells. The high diffusion rate of insulin in rbc would ensure prompt control of blood sugar in Type 1 diabetes patients.