# 1.7: Sorption and Chromatography


## Adsorption, Ion Exchange, and Chromatography

$$c_i$$ = concentration of species i in the mobile phase (mass volume-1) or (mole volume-1)

$$k_i$$ = empirical constant for species i for isotherms (units vary)

$$K_i$$ = adsorption equilibrium constant for species i

$$n_i$$= internal parameter for isotherms (units vary)

$$p_i$$= partial pressure of species i (pressure)

$$q_i$$= amount of species i adsorbed per unit mass of adsorbent at equilibrium (mass mass-1) or (mole mass-1)

$$q_{m_i}$$ = amount of species i adsorbed per unit mass of adsorbent at maximum loading, where maximum loading corresponds to complete surface coverage (mass mass-1) or (mole mass-1)

linear isotherm:

$q_i=k_ip_i \tag{31.1}$

Freundlich isotherm:

$q_i=k_ip_i^{1/n_i} \tag{31.2}$

Langmuir isotherm:

$q_i=\frac{K_iq_{m_i}p_i}{1+K_ip_i} \tag{31.3}$

chromatography equilibrium:

$K_i=\frac{q_i}{c_i} \tag{31.4}$

## Modeling Differential Chromatography

$$\alpha_i$$ = average partitioning of species i between the bulk fluid and sorbent (unitless)

$$\epsilon_b$$ = sorbent porosity, ranges from 0 to 1 (unitless)

$$\epsilon^*_{p,i}$$ = inclusion porosity, accounts for accessibility of sorbent pores to species i (unitless)

$$\tau_f$$ = sorbent tortuosity factor, usually approximately 1.4 (unitless)

$$\omega_i$$ = fraction of solute in the mobile phase, relative to sorbed solute, at equilibrium (unitless)

$$A$$ = cross-sectional area of the column (area)

$$c_{f,i}$$ = concentration of species i in the mobile phase (mass volume-1) or (mol volume-1)

$$D_{e,i}$$ = effective diffusivity of species i within the sorbent pores (length2 time-1)

$$E_i$$ = coefficient that accounts for axial diffusion of species i and non-uniformities of flow (length2 time-1)

$$H_i$$ = height of theoretical chromatographic plate for species i (length)

$$k_{a,i}$$ = kinetic rate constant of adsorption of species i to the sorbent (time-1)

$$k_{c,i}$$ = mass transfer coefficient of species i in the mobile phase (length time-1)

$$k_{c,i,tot}$$ = overall mass transfer coefficient of species i (length time-1)

$$K_{d,i}$$ = equilibrium distribution coefficient of species i between the mobile phase and sorbent (unitless)

$$L$$ = length of column (length)

$$m_{0_i}$$ = amount of solute i fed to column (mass) or (mol)

$$R_{1,2}$$ = resolution of species 1 and 2 in the proposed operating condition (unitless)

$$R_p$$ = radius of sorbent particles (length)

$$s_i$$ = variance of the Gaussian peak of the distribution of species i along the column length (time)

$$t$$ = elapsed time since loading of the column (time)

$${\overline t}_i$$ = mean residence time of species i in the column (time)

$$u$$ = actual fluid velocity through the bed (length time-1)

$$u_s$$ = superficial fluid velocity through the bed (length time-1)

$$z$$ = position along the length of the column, in the direction of flow (length)

$$z_{0,i}$$ = mean position of species i along the length of the column as a function of time (length)

$z_{0,i}(t)=\omega_iut \tag{32.1}$

$\omega_i=\frac{1}{1+\frac{1-\epsilon_b}{\epsilon_b\alpha_i}} \tag{32.2}$

$\alpha_i=\frac{1}{\epsilon_{p,i}^*(1+K_{d,i})} \tag{32.3}$

$u=u_s/\epsilon_b \tag{32.4}$

$\overline t_i=\frac{L}{\omega_iu} \tag{32.5}$

$c_{f,i}(z,t)=\frac{m_{0_i}\omega_i}{A\epsilon_b(2\pi H_iz_0)^{0.5}}\textrm {exp}\left(\frac{-(z-z_0)^2}{2H_iz_0}\right) \tag{32.6}$

$H_i=2\left[\frac{E_i}{u}+\frac{\omega_i(1-\omega_i)R_pu}{3\alpha_ik_{ci,tot}}\right] \tag{32.7}$

$N_{{\rm Pe},i}=N_{\rm Re}N_{{\rm Sc},i}=\frac{2R_pu\epsilon_b}{D_i} \tag{32.8}$

if $$N_{{\rm Pe},i}<<1$$

$E_i=\frac{D_i}{\tau_f} \tag{32.9}$

else

$E_i=\dfrac{2R_pu \epsilon_b}{N_{ Pe,E,i}} \tag{32.10}$

$$N_{{\rm Pe},E,i}$$ calculated by 15-61 or 15-62, Seader

$\frac{1}{k_{ci,tot}}=\frac{1}{k_{c,i}}+\frac{R_p}{5\epsilon_{p,i}^*D_{e,i}}+\frac{3}{R_pk_{a,i}\epsilon_{p,i}^*}\left[\frac{K_{d,i}}{1+K_{d,i}}\right]^2 \tag{32.11}$

$s_i^2=\frac{\overline t_iH_i}{\omega_iu} \tag{32.12}$

$R_{1,2}=\frac{\textrm {abs}(\overline t_1-\overline t_2)}{2(s_1+s_2)} \tag{32.13}$

##### Example

1.0 g of species A is added to a chromatography column of cross-sectional area 1.0 m2 and length 1.0 m. Mobile phase is added at a flowrate of $$4.0 \times 10^{-3}$$ m3/s. Species A has a mass transfer coefficient of $$2.0 \times 10^{-5}$$ m/s in this solvent. The selected sorbent has a porosity of 0.40 m and average particle radius of $$5.0\times 10^{-6}$$ m. For species A in this sorbent, the inclusion porosity is 0.80, $$K_d = 50$$, $$E = 2.0\times 10^{-8}$$ m2/s, $$k_a = 100$$ s-1 and the effective diffusivity is $$3.5\times 10^{-12}$$ m2/s.

(a) When is mean expected elution time for species A?

(b) Plot the concentration profile for species A at 0.05 m increments along the column length in 10-minute increments, until all of the solute has eluted.

(c) Find the variance of the peak for species A in the proposed operating condition.

(d) The column feed also contains 1.0 g of species B. Species B has a mass transfer coefficient of $$1.0\times 10^{-5}$$ m/s in the mobile phase, inclusion porosity of 0.50, $$K_d = 60$$, $$E = 3.0\times 10^{-8}$$ m2/s, effective diffusivity of $$4\times 10^{-12}$$ m2/s and $$k_a = 200$$ s-1. What is the resolution of these two species in the proposed operating condition?

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