# 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)

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)

= 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?