Variation of peak shape and peak tailing in chromatography

Presenters: Raffi Manjikian, Nicole Charles, Antonio Macaluso
Department of Chemistry and Biochemistry
Seton Hall University
South Orange, NJ 07079

Abstract
We have developed novel finite difference software for modeling chromatographic
peak shape when both partition and adsorption simultaneously control the
distribution on the column. This software uses four dimensionless input
parameters (the mobile phase fraction, the surface area constant, the adsorption
constant, and the amount of theoretical plates) to compute the numerical
representation of the peak. Analysis performed upon each peak yields its
retention factor and its USP tailing factor (defined as the ratio of two extensions
from the maximum point of a peak measured at 5% of the peak height). Variation
of these input parameters allows the output to be varied between partition and
adsorption and any combination between these extremes. This presentation
focuses on the finite difference simulation both in the partition extreme and
under adsorptive control when high values of the adsorptive mass balance
parameter, ΓoA/CoVm, produce peak shapes indistinguishable from partition
peaks. The results of this work help to confirm the validity of the original
simulation and assist in the identification of the experimental conditions that
require both partition and adsorption considerations to describe peak shape more
accurately while also improving our understanding of various column materials.

Introduction
We have recently described novel finite difference software to model
adsorption effects in partition chromatography. Written as an Excel®
VBA, this program predicts behavior when both partition between
mobile and stationary phases and adsorption on the stationary surface
control the chromatographic distribution.

Using 4 dimensionless input parameters representing
(1) the mobile phase partitioning fraction (X);
(2) the adsorption equilibrium constant (KadCo);
(3) the area-volume ratio constant (ΓoA)/(CoVm); and
(4) the number of theoretical plates (No);
the VBA generates a numerical representation of each peak and
computes its chromatographic characteristics such as tR, σ2, k’, and the
USP peak tailing factor.

Partition – Adsorption Mechanism

Separation & Transport in the
Partition-Adsorption Model

Computing the Mobile Phase
Fraction x(n,r)
x(n,r) = ½ { ξ + * ξ2 + 4X/(KadCo)∙ft(n,r)]½ } where
ξ = X – 1/(KadCo)ft(n, r) – (ΓoA/CoVm)∙X/ft(n,r)
X = [1 + Kp(Vs/Vm)]-1 = fixed partitioning fraction
(KadCo) = dimensionless adsorption constant
(ΓoA/CoVm) = area-volume ratio constant
This computation is performed at each plate during
each iteration – in theory, (nmax)(rmax) times during the
simulation of a single peak.

Computing the Mobile Phase
Fraction x(n,r)
This computation is performed at each plate during each
iteration – in theory, (nmax)(rmax) times during the
simulation of a single peak.
In practice, the computation of x(n,r) is performed
many fewer times than (nmax)(rmax) by using an
algorithm that limits it to those cells where f(n,r) > 0.
Mass balance is performed at the conclusion of the
simulation of each peak to assure that this algorithm
has produced valid results.

Fundamental Plate Count
Expressions
N = tR2/σ2 (1)
N’ = tR(tR-to)/σ2 (2)
No = defined number of theoretical plates (3)
In partition chromatography N’ agrees with No
while N does not.
Direct rearrangement of eq. 2 yields:
tR = N’(σ2/tR) + to (4)
Eq. 4 is of the form y = mx + b

Linear Regression Diagnostic: tR vs. (σ2/tR)
tR = N’(σ2/tR) + to (4)
Eq. 4 is of the form y = mx + b with
y  tR & x  (σ2/tR) and m  N’ & b  to
so that a plot of tR vs (σ2/tR) will have a predicted
slope N’ and intercept to.

This linear regression is employed diagnostically in
all of the work that follows.

Other Regression Diagnostics
N’= tR(tR-to)/σ2 to give tR= N’(σ2/tR)+ to

For well behaved partition character, N’→No and
to→No. Therefore, the slope is equal to N’ and
the intercept is equal to to.

Since there are No transfers to get to the leading
edge to the detector plate, No/b= to/b= 1.0 in
well behaved region.

More Diagnostics for Partition
Chromatography
Since N/N’= (tR2/σ2)/((tR2-tRto)/σ2)= 1/(1-(to/tR))

N/N’= (1-(to/tR))-1 = (1-(1/(k’+1)))-1

N/N’= (k’/(k’+1))-1 = (1+ k’-1)

Thus, (N/m)/(1+ k’-1) = 1 for well behaved
partition chromatographs.

Typical Simulation Output
Results shown in the following figure were obtained
by setting No = 1000; X = 0.5; ΓoA/CoVm = 1.0; and
KadCo = 1.5, 4.0, 6.5 and 9.0, respectively.
The computed tailing factors of the resulting peaks
were 1.150, 1.254, 1.304, and 1.335, respectively.
Linear regression (inset) on a plot of tR vs σ2/tR gave
a slope m = 952 (corresponding to N’) and an
intercept b = 1116 (corresponding to to) with R2 =
1.0.

Typical Partition - Adsorption
Output

Achieving Partition Behavior
The results shown above exhibit significant tailing,
as expected when adsorption is operative.
We have discovered two ways to achieve partition-
like behavior using this VBA:
(1) Set ΓoA/CoVm = 0 and allow the retention factors
to be determined by the input values of X and No.
(2) Set X = 0.9999 and allow the retention factors to
be determined by ΓoA/CoVm, KadCo, and No.

No=1000 X=0.9999 and ΓoA/CoVm=0

No=1000 X=0.9999 and ΓoA/CoVm=0.1

No=1000 X=0.9999 and ΓoA/CoVm=0.316

No=1000 X=0.9999 and ΓoA/CoVm=3.16

No=1000 X=0.9999 and ΓoA/CoVm=31.6

No=1000 X=0.9999 and ΓoA/CoVm=100

Conclusions
In the course of this work, where partition was
excluded by setting X = 0.9999 to prohibit extraction
in the stationary phase, partition-like results were
obtained by employing values of ΓoA/CoVm > 1.0.
In general, it was observed that larger values of
ΓoA/CoVm produced adsorption peaks that were
indistinguishable from partition peaks.

What does this finding mean?
The results obtained by applying the partition
diagnostic to adsorption peaks clearly indicate
that when [ΓoA/CoVm] is large, adsorption is
indistinguishable from partition. These
observations suggest that the adsorption model
we have developed using these finite difference
methods may be generally more applicable to
understanding chromatographic processes than
traditional methodology based solely on partition.

Future Work
We are developing a theoretical expression for retention factor (k’) in
partition–adsorption chromatography.

k’ = ko’ + (KadCo) ∙ [(ΓoA)/(VmCo)] ∙ (1-θ)
where ko’ is the retention factor for partition in the absence of
adsorption. Given k’ as simulation output for each peak and all else
but θ as simulation input, this equation allows θ, the average fraction
of occupied adsorption sites to be computed for each peak.

When X=0.9999, ko’=0 and as (ΓoA)/(VmCo) approaches infinity, θ
approximately equals zero.

Therefore, k’= (KadCo) ∙ [(ΓoA)/(VmCo)] which leads to

k’= Kad(ΓoA)/Vm

References
1. Pittcon11, #860-6
2. EAS11, #294
3. Pittcon12, #810-5P

Variation of peak shape and peak tailing in chromatography

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