Ethyl acetate is used as a solvent for the extraction of caffeine from coffee beans in a membrane extraction unit. A key parameter that determines the peformance of such a unit is the rate of transport of caffeine across the membrane. In a laboratory set up $($see Figure$),$

caffeine solution and ethyl acetate are separated by a porous membrane. Caffeine diffuses through the mebrane pores and is dissolved in ethyl acetate. Therefore, concentration of caffeine in water $(C_F)$ decreases, while caffeine concentration in ethyl acerate $(CP)$ increases with time. The rate of diffusion of caffeine across the membrane $(R)$ can be expressed by:

$R=KA(C_F-C_P)$

Where $K$ is the overall mass transfer coefficient and $A$ is the membrane area. The volume of water phase $(V_F)$ and the volume of ethyl acetate phase $(V_P)$ remain constant during the extraction process and both phases are fully mixed. Initially $(att=0),$ the concentration of caffeine in water is $C_{Fo}$ and ethyl acetate has no caffeine $(C_{Po}=0)$ :

a$)$ Write caffeine balances for the water phase $($Feed$)$ and for the ethyl acetate phase $($Permeate$)$ and show that the differential equations describing this system are given by:

$\frac{d{C}_F}{dt}=(C_F-C_P)$

$\frac{d{C}_P}{dt}=(C_F-C_P)$

Where $$ and $$ are constants that depend on known parameters, ie $A,K,V_P,$ and $V_F.$

b$)$ Using D$-$operator, solve the above system of ODEs to obtain an expression for the concentration of caffeine in permeate as a function of time, $C_P(t).$