Pharmacokinetic models are used to describe the distribution of a drug molecule throughout the body flowing into and out of organs. In this example we will look at the decomposition of a drug molecule as it flows into an organ where it undergoes a chemical reaction. The flow rate of blood into and out of the organ is given by $v(\frac{mL}{\text{m}\text{i}\text{n}}).$ The drug enters at a concentration $C_{D,o}(\frac{mo{l}_D}{mL})$ and goes through a deactivation reaction, leaving at a concentration $C_D(\frac{mo{l}_D}{mL}).$ The deactivation reaction rate is given by the following

expression: $r(\frac{m_D{l}_D}{\text{m}\text{i}\text{n}})=kV{C}_D.$ We will assume that the organ is well mixed and the concentration of drug throughout the organ is $C_D.$ The organ has a volume of $V(mL).$

A$)$ Is the process continuous, batch, or semi$-$batch? Is it transient or at steady state? Why?

B$)$ Draw and label a block flow diagram of the process.

C$)$ What would you expect $C_D$ to be if the reaction rate constant was small, $k0?$

What would you $C_D$ to be if the reaction rate constant was large, $k?$

D$)$ Using a differential material balance, identify the terms for the rate accumulation, input, output, consumption, and generation for the drug $($some of these maybe $0).$ Solve this equation to give a function that calculates $C_D$ based on the other values in the problem.

E$)$ Using the equation in part D$),$ show that your predictions from C$)$ are correct.