Enzyme is immobilized in spherical beads (radius=R). Substrate
diffuses from the surface to the center of the beads. Assuming
substrate concentration is relatively small compared to enzyme Km.
We also define a=r/R, c=S/Ss
a) Derive an expression of c as a function of
a.

b) Derive an expression of substrate concentration
(S/Ss) at the center of the beads
(a=0) as a function of Thiele modulus φ. And calculate
S/Ss at the center of the beads when φ equals to 1; 2; 4.
c) You are working for a biotechnology company. Harry, the head of
your department, asks you to make sure S/Ss (a=0) is higher than
0.3 for earning money. What Thiele modulus φ can you choose to meet
this requirement?

d) For the real situation, because the
substrate concentration inside of the sphere is lower than the
surface substrate concentration, the reaction rate is also lower
inside of the sphere.

The ideal situation, which we are dreaming of,
but we can not achieve it, would be the substrate concentration
equals to Ss everywhere, regardless of the position inside of the
sphere.

The effectiveness factor is defined as
(the real rate*volume)/(the ideal rate*volume).
Derive an expression of the effectiveness factor
as a function of Thiele modulus φ.