III. $7$ Develop the linearized models for the following processes.

$($a$)$ Packed absorption column with $N$ ideal plates. The mass balance of component $i$ at the $n$th ideal plate is given by

$H\frac{d{x}_n}{dt}=L{x}_{n+1}-(L+mV)x_n+mV{x}_{n-1}$

where $H=$ holdup of liquid solvent at the $n$th plate

$L=$ flow rate of the liquid solvent

$V=$ flow rate of the gaseous stream

$x_n=$ mole fraction of component $i$ at the $n$th plate

$m=$ slope of the equilibrium curve for component $i$