Often in the theory of chromatography we assume local equilibrium at all points in the column between the adsorbent particles and the adjacent fluid. Under these conditions, we can express \(c_{p a}\) as a function of \(c_{a}\) and substitute into the differential equation of the previous problem.

a. Assume \(c_{p a}=f\left(c_{a}\right)\) and show that the differential equation describing chromatography can be written as:

\[\left[\varepsilon+(1-\varepsilon) \frac{\partial}{\partial t} f\left(c_{a}\right)\right] \frac{\partial c_{a}}{\partial t}+\varepsilon v_{i} \frac{\partial c_{a}}{\partial z}=0\]

b. The equation in part

(a) can be solved using the Method of Characteristics [8]. We solve this equation simultaneously with the definition of the total derivative:

\[d c_{a}=\left(\frac{\partial c_{a}}{\partial z}\right) d z+\left(\frac{\partial c_{a}}{\partial t}\right) d t\]

to obtain \(\partial c_{d} \partial z\) and \(\partial c_{d} \partial t\). Solve these equations and show that there are certain characteristic velocities at which a particular concentration moves through the bed. (Hint: solve the equations via matrices and calculate the determinant to get the velocity. \(d c_{a}=0\) within the velocity wave.)

c. How does this wave velocity depend upon the adsorption isotherm and, in particular, the Langmuir isotherm given by:

\[c_{p a}=c_{p o} \frac{K c_{a}}{1+K c_{a}}=f\left(c_{a}\right)\]