Problem $1(50$ points$).$ Two commonly expressions to model the adsorption of proteins on ion exchange resins are the Langmuir isotherm and the steric mass action model. The Langmuir model is

$q=\frac{q-KC}{1+KC}$

Where $q_m$ is the monolayer capacity and $K$ is an equilibrium constant. Both am and $K$ are often a function of salt concentration when the model is used to fit ion exchange adsorption data. A different presentation of Langmuir model is$,$

$\frac{q}{q_{\text{r}\text{a}\text{f}}}=\frac{\frac{C}{C_{\text{r}\text{a}\text{f}}}}{R+\frac{(1-R)C}{C_{\text{r}\text{a}\text{f}}}}$

Where

$R=\frac{1}{1+K{C}_{\text{r}\text{a}\text{f}}}$

Is the separation factor and $C_{ne}$ is a reference concentration usually the feed concentration. When $R$ goes to $1$ the isothermal is linear and when $R$ goes to zero the isothermal is rectangular. The SMA model is

$C=\frac{q{C}_z^z}{K_{}[q_0-(z+)q{]}^z}$

$q_0$ is the total ion exchange capacity $C$ is the counterions concentration, $z$ is the protein effective charge and $$ is a steric hindered factor.

a$)$ Show that when $C$ is low and $C$ is large then $q=q-$dots$=\frac{q_2}{z+}$

b$)$ Show that when Capproaches to zero, the model approaches the linear limit$,$

$q=K_i(\frac{q_q}{c_i}{)}^{2}c=KC$

c$)$ The following data was obtained for Lysozyme adsorbed on SP$-$Sepharose in $10$mMNa,HPO, at pH $6\mathrm{.}5,$

$\backslash$table$[[C_{vg}=120mM,C_v=50mM,$can$=20mM$