Problem $6$

Mitochondria are the centers for energy conversion in the body. Enzymes attached to the internal membranes oxidize sugars to produce energy. To get an idea about mitochondria performance, the system shown in the figure below was developed. The substrate diffuses through the liquid phase until it reaches the membrane surface where it reacts. The rate of reaction is governed by Michaelis$-$Menton kinetics:

$-r_s[\frac{\text{m}\text{o}\text{l}}{s}]=\frac{k_s^{\text{'}\text{'}}{c}_s}{1+K_m{c}_s}$

a$)$ Develop an expression for the concentration profile, mass flux through the liquid, and mass transfer coefficient to the solid phase.

b$)$ What happens when When $K_m{c}_s$ is $>>1?$