# 7 Sorption and Chromatography

## Adsorption, Ion Exchange, and Chromatography = concentration of species i in the mobile phase (mass volume-1) or (mole volume-1) = empirical constant for species i for isotherms (units vary) = adsorption equilibrium constant for species i = internal parameter for isotherms (units vary) = partial pressure of species i (pressure) = amount of species i adsorbed per unit mass of adsorbent at equilibrium (mass mass-1) or (mole mass-1) = 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:

(31.1) Freundlich isotherm:

(31.2) Langmuir isotherm:

(31.3) chromatography equilibrium:

(31.4) ## Modeling Differential Chromatography = average partitioning of species i between the bulk fluid and sorbent (unitless) = sorbent porosity, ranges from 0 to 1 (unitless) = inclusion porosity, accounts for accessibility of sorbent pores to species i (unitless) = sorbent tortuosity factor, usually approximately 1.4 (unitless) = fraction of solute in the mobile phase, relative to sorbed solute, at equilibrium (unitless) = cross-sectional area of the column (area) = concentration of species i in the mobile phase (mass volume-1) or (mol volume-1) = effective diffusivity of species i within the sorbent pores (length2 time-1) = coefficient that accounts for axial diffusion of species i and non-uniformities of flow (length2 time-1) = height of theoretical chromatographic plate for species i (length) = kinetic rate constant of adsorption of species i to the sorbent (time-1) = mass transfer coefficient of species i in the mobile phase (length time-1) = overall mass transfer coefficient of species i (length time-1) = equilibrium distribution coefficient of species i between the mobile phase and sorbent (unitless) = length of column (length) = amount of solute i fed to column (mass) or (mol) = resolution of species 1 and 2 in the proposed operating condition (unitless) = radius of sorbent particles (length) = variance of the Gaussian peak of the distribution of species i along the column length (time) = elapsed time since loading of the column (time) = mean residence time of species i in the column (time) = actual fluid velocity through the bed (length time-1) = superficial fluid velocity through the bed (length time-1) = position along the length of the column, in the direction of flow (length) = mean position of species i along the length of the column as a function of time (length)

(32.1) (32.2) (32.3) (32.4) (32.5) (32.6) (32.7) (32.8) if (32.9) else

(32.10)  calculated by 15-61 or 15-62, Seader

(32.11) (32.12) (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 m3/s. Species A has a mass transfer coefficient of m/s in this solvent. The selected sorbent has a porosity of 0.40 m and average particle radius of m. For species A in this sorbent, the inclusion porosity is 0.80, , m2/s, s-1 and the effective diffusivity is 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 m/s in the mobile phase, inclusion porosity of 0.50, , m2/s, effective diffusivity of m2/s and s-1. What is the resolution of these two species in the proposed operating condition? 