Simulation of Elution Curves for Chromatography Columns with a Low

Chapter 19

Downloaded by NORTH CAROLINA STATE UNIV on August 8, 2012 | http://pubs.acs.org Publication Date: November 16, 2006 | doi: 10.1021/bk-2007-0945.ch019

Simulation of Elution Curves for Chromatography Columns with a Low Number of Theoretical Plates Willy Brüchle Gesellschaft für Schwerionenforschung, Planckstrasse 1, 64291 Darmstadt, Germany

A new description of chromatographic peaks is derived, showing that Poisson curves are good fits to describe the position and the shape of elution peaks as a function of distribution coefficients and the number of theoretical plates.

Introduction Chromatographic techniques are among the most effective chemical methods to separate and characterize closely related chemical substances or ions. More than half a century ago the main theoretical foundations for these methods were developed; even a Nobel Price was awarded to Martin and Synge (/) for their theoretical and practical methodic developments. Meanwhile dozens of journals publish thousands of articles dealing with chromatographic separations, e.g., with gas chromatography, H P L C , countercurrent chromatography, affinity chromatography, to mention only some popular methods. There exist a large number of pioneering theoretical descriptions of the peak position and the width, often based on the molecular dynamic theory of chromatography, originally © 2007 American Chemical Society

In Applied Modeling and Computations in Nuclear Science; Semkow, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2006.

269


270 developed by Giddings and Eyring (2). Their model represents the progress of a single molecule through the column as a chain of elementary adsorptiondesorption stochastic processes. The majority of these papers have started with partial differential equations leading to Bessel functions. The asymptotic expansion of the Bessel functions weighted by the probability of the initial state of the molecule should yield the distribution curve for the elution. Rather complicated calculations using the characteristic function method are described in Ref. (5). A relatively easy way to describe the chromatographic process is the "theoretical plate" concept, originating from distillation. It is an imaginary section of a column in which a complete equilibrium step takes place. The model was successfully used in Ref. (7) for liquid-solid chromatography. Usually textbooks dealing with chromatography present the two following formulae without derivation: K = (T -T )F/m d

r

0

= (V -Vo)/m,

or

p

V = K m+ p

d

V, 0

(1)

where T = retention time [min] T = column hold-up time due to the free column volume [min] F = flow rate of the mobile phase [mL/min] m = mass of the stationary phase (ion exchanger, reversed phase material) [g] K = partition coefficient (distribution coefficient) V = volume eluted until the peak maximum [mL] VQ = free column volume (dead volume) [mL]. The number of theoretical plates is determined empirically from the chromatogram as: r

0

d

P

2

N = (T /a) = r

2

16(7>/w) ,

(2)

where w is the baseline width of the peak, determined graphically by drawing straight lines along the edges of the linear peak diagram, σ is the half width at the inflection point (square root of the variance). T is the retention time until the peak maximum is reached. Typically a Gaussian distribution profile is assumed for an ideal elution peak. To compare the performance of columns of different length, the term height equivalent to a theoretical plate or plate height is used: r

Η = L/N, where L is the column length and Ν is the number of plates.


(3)

271


During a series of fast automated separations on rather small chromatographic columns (1.6 mm i.d., 8 mm long; and 1 mm i.d., 3.5 mm long) with the A R C A and the A I D A systems (4,5), it was noticed that the peak shapes differed from a Gaussian; in addition it was often not possible to measure complete elution curves, but only two fractions. Still the distribution coefficient Kd should be determined. The knowledge of the peak shapes is indispensable for the multi-column technique also (6). In Ref. (5), the Glueckauf equation of chromatography was used (7): 2

A(V) = A

max

cxp(-N(V -V) l2V V), p

p

(4)

where A , N, and V are the maximum peak height, the number of theoretical plates, and the peak volume, respectively. There are interesting aspects in Ref. (7) about peak broadening when the original band entering the separation column is already broad. But it is nowadays rather simple to put this into a simulation program. max

p

New Results Is it possible to derive the peak position and the peak shape using simplifying assumptions and not too complicated mathematics? Yes, it is. This chapter will try to show how this can be achieved. In a real experiment any amount of liquid pumped onto the top of the incompressible chromatography column continuously flows through the column. So, all exchange processes in different layers of the column will proceed simultaneously. In a simulation this behavior has to be modeled stepwise. Instead of using successive infinitesimal volumes of the mobile phase passing the column (1), the "theoretical plate" concept (1,8) was used in this work. According to it, the volume corresponding to a theoretical plate was used for a discontinuous flow treatment (8). As in the treatment of a separating funnel, the partition coefficient K = C /Ci d

s

(5)

describes the ratio of the concentrations of the separated species in the solid and in the liquid phase, for each theoretical plate. The underlying assumption is that the distribution coefficient does not change with the change in concentration of the solute, and the diffusion from one plate to another is negligible. The activity transported from plate / to plate / + 1 is:


272 Δα(/) = 0(Ο (\-K l{K +\)) d

,

d

(6)

where α(ί) denotes the total (solid plus liquid) activity in plate i. This is valid for the case of equal volumes of solid and liquid phase. If the volumes are not equal, then Eq. 7 is valid: Aa(i) = a(i)(\-K /(K


d

d

+ r )).

(7)

v

In this formula, r is the ratio o f the free column volume to the mass of the exchanger material Vç/m. The amount Aa(i) corresponds to the activity in the liquid phase of plate /. The small volume of the plate, containing the activity Δα(ι) is transported to the following plate, where the next equilibrium is established with the activity retained on the solid phase. It is easy to check the validity of these formulae. Dividing the activity of the solid phase, a(i) - Δα(/), v

by ΔΛ(0 results in K /r , consistent with Eq. 5. Mayer and Tompkins (#) used this attempt, but their material balance does not fit. In their Eq. 2, they assume that the activity in plate / is composed of the activity eluted from plate / - 1 and the activity that was in the plate in the step before; the simultaneous elution to plate / + 1 is not considered. The formulae they give would allow some activity to break through the column even with the first small volume of one theoretical d

v

plate, especially in the case of small K values. The same happened in Ref. (2), d

where it is explicitly mentioned two times, that molecules with no reaction (K = d

0) will be eluted in exactly the time / = 0. This is not correct: even with K = 0 at least the dead volume of the column must be eluted before the substance will break through the column. How to solve this problem? Instead of developing the activity distribution from plate 1 to plate Ν (the last plate of the column), one simply starts the simulation from the last plate, N, and goes back to plate 1. Now it is natural, that in the first step (with the volume of one plate) the substance will only reach plate 2, and one needs at least Ν - 1 steps until some activity will leave the column, when the elution starts with all activity in plate 1. In Fig. 1 and Fig. 2, the wrong treatment, leaving to early break-through for low K values is compared to the calculations in reverse order, where a "dead volume" results. Two different K values (K = 1 in Fig. 1, and K = 0.1 in Fig. 2) are assumed for a simulation with 5 theoretical plates (N = 5). A rather extreme example is shown in Fig. 2. After the first step with the volume of only one plate, the activity would be smeared out over the entire column when the computing is started with the first plate and the eluted activity is transported to plate N. The correct sequence for this simulation is to start with plate N let the activity from plate Ν - 1 be distributed between both, and go on d

d

d

d

d

9


273 in reverse order to plate 1. Now a sharp front is passing through the column, and only after Ν - 1 plate volumes (the dead volume V of the column) the activity will break through. This is in agreement with experimental observations. 0

Kd = 1.0 eluted volume:

O.ZVo 0.4Vo O.6V0 O.8V0

IVo


calculation top -> bottom

eluted: 3.1%

10.9% 22.7% 36.3% 50.0%

calculation in reverse order

eluted: 0.0%

0.0%

0.0%

0.0%

3.1%

Figure 1. Comparison of two different approaches (upper and lower panel) to describe the elution for a column with 5 theoretical plates and K = 1.0. d

K =0.1 d

eluted volume:

0.2Vo

0.4Vo

O.6V0

O.8V0

IVo

calculation top -> bottom

eluted: 62.1% 90.3% 98.0% 99.6%

99.9%

calculation in reverse order

eluted: 0.0%

0.0%

0.0%

0.0%

62.1%

Figure 2. Comparison of two different approaches (upper and lower panel) to describe the elution for a column with 5 theoretical plates and K = 0.1. d


274 The difference between Eqs. 6 and 7 results only in different dead volumes of the column. The shape of the elution curve is not influenced by the coefficient r (see Fig. 3). This figure demonstrates, how the dead volume is influenced by the relation of the free column volume to the mass of immobile exchanger.


v

Figure 3. Comparison of different r , resulting only in a shift of the elution curve due to different dead volumes. v

A l l following pictures were calculated with the mass of exchanger = 10 mg, and a dead volume of the column = 10 / / L (corresponding to r = 1). A small B A S I C program, given in the Appendix, is able to describe realistic elution curves. The integral eluted activity is collected in cell a(N + 1). It is easy to display the activity distribution along the column, or the elution curves. v

Figures 4 and 5 show some calculated curves for different K -values. The difference between Figs. 4 and 5 is the number of theoretical plates. One can see d

that the maximum of the elution curves is determined by K and N, while the shape depends on TV only. A similarity with the Poisson curves from Réf. (P) is obvious; a numerical comparison shows that the elution curves can very well be described by: d

N

A(V) = A

maxM

l

~ exp(- )/(N-\)\ M

, with μ = {Υ- V )NI(mK ) . 0

d

(8)

Here, V is the eluted volume, m is the mass of the exchanger material, and V is the dead volume of the column. The maximum of the Poisson curve is at μ = Ν - 1 ; the median (P) (50 % of the activity is below, and 50 % above this point) is located at μ + 0.67. This is one difference between the symmetric Gaussians, and the skewed Poisson curves. For the Gaussians, the maximum and the median are at the same position. 0

ίη

η


275

14-,

αIOO L M

12

10

Elution curves for 10 theoretical plates, 10mg exchanger material

*> "S

Κ =10, Κ =50, Κ =100 d a d


(Ο

Ii 460μΙ_ U910ML

2000

Figure 4. Elution curves for N=10 and different K values. d

J

4.0

ΙΜ09μΙ_

3.5

3.0

25

Elution curves for 100 theoretical plates, 10mg exchanger material

Η

Κ =10, Κ =50, Κ.=100 d d d

2.0

1-54 U 505μί

1.0

ΙΗΟΟΟμί.

0.5

r=—'—ι—>—ι—·—ι— —ι 1

0

200

400

600

800

1000

1200

1400

1600

1800

2000

ML Figure 5. Elution curves for N=100 and different Kj values.


276 One can determine the K curve using:

d

value from the peak maximum of the Poisson

K =(V-V )N/(N-\)m. d

(9)

0


The difference from the classical Eq. 1 is the factor N/(N- 1). For large N, this factor approximates unity and can be neglected. In Fig. 6, one can see the influence of the number of theoretical plates TV on the shape and position of the elution peaks, when the K value and the amount of exchanger material are kept constant. The "classical" peak would be at 110 / / L . For Ν = 5, 10, and 100 one gets 90 / / L , 100 / / L , and 109 / / L , respectively. If one d

has to determine K values from the peak position for Ν > 100, the corrections will normally be less than the experimental errors. For a better separation between similar substances, the Ν in nowadays analytical or gas chromatography columns is usually > 100000. However, this is not the case in fast radiochemical separations taking less than 5 s, using column dimensions of, e.g., 1.6 mm i.d. χ 8 mm long. d

μΙFigure 6. Simulation of different peak shapes with different numbers of theoretical plates N.



277

Figure 7. Comparison of different fit curves; Glueckaufs formula is illustrated with and without dead volume. Upper part: K = 10, Ν = 10, 10 mg ion exchanger material. Lower part: K = 20, Ν = 10, 10 mg ion exchanger material. d

d

Figure 7 shows a comparison of the stepwise calculation by the program presented in this work, the Poisson-curve fit, and a fit by the Glueckauf equation, shifted by V . Glueckauf (7) did not use the "free column volume" and he thought it was only a result of the "discontinuous" treatment. To get a reasonable fit, however, one has to consider V in continuous treatments as well 0

0


278 as in discontinuous calculations (see Fig. 3). The higher the Kj values, the better the fits will match the exact calculation. The fits for higher K values and larger Ν are not easy to distinguish from each other. d


Summary It was shown that the chromatographic elution peaks can be well described by the Poisson curves. The position of the maximum as well as the peak width and shape is given by a single formula as a function of K4 value and the number of theoretical plates. In the limit of large numbers of theoretical plates, the formula is identical to the classical one describing elution peaks. The advantage of the new method is that it can also describe the elutions from short ion exchanger columns. For this case, the classical formulae give significant deviations when determining K values from the maximum of elution curves. d

Acknowledgements I would like to thank my colleagues: V . Pershina and M . Schàdel for their careful reading of the manuscript and for their helpful comments. Last not least, I would like to express my appreciation for the authors of G F A BASIC, a program that has always satisfied my programming needs.

Appendix Main Part of a BASIC Program to Simulate Elution Peaks DIM

a(10000)

INPUT

" K d ,

INPUT

"mg

INPUT

" r a t i o

a(l)

=

REM

a l l

o f

t h e o r .

o f

phases

p l a t e s " ; k d , η " ; m g

l / s " ; r v

1 a c t i v i t y

on_column k d i v

number

E x c h a n g e r m a t e r i a l

=

=

i n

1

-

kd

/

(kd

mg

*

r v

/

η

v_p

=

REM

v _ p

f i r s t

p l a t e

a t

s t a r t

time

1

volume

o f

+

one

rv) p l a t e

DO FOR

i

REM

s t a r t

=

η

DOWNTO from

1

l a s t

p l a t e ,

g o i n g

up


279 d i f

=

a ( i ) a ( i


NEXT

a ( i ) -=

+

*

k d i v

d i f

1)

+= d i f

i

REM

o n e p l a t e

INC

n _ p o f

volume

REM

n r .

v o l

=

REM

m i c r o l i t e r s

REM

PLOT

REM

p l o t

a c t i v i t y

REM

PLOT

needs

LOOP

+

1)

UNTIL

t h e

column

volumes

v _ p p a s s i n g

t h e column

v o l , a ( n + l )

on_column a(n

p l a t e

n _ p *

h a s p a s s e d

-= =

l e a v i n g

some

t h e

column

s c a l i n g

a(n+l)

0

o n column

< . 0 0 1

References 1. 2. 3. 4. 5. 6. 7. 8. 9.

Martin, A . J. P.; Synge, R. L .M.;Biochem J. 1941, 35, 1358-1368 Giddings, J. C.; Eyring, H . J. Phys. Chem. 1955, 59, 416-421. Cavazzini, Α.; Remelli, M . ; Dondi, F. J. Micro Sep. 1997, 9, 295-302. Schädel, M . Radiochim. Acta 2001, 89, 721 -728, and references therein. Haba, H . ; et al. J. Am. Chem. Soc. 2004, 126, 5219-5224. Kronenberg, Α.; et al. Radiochim. Acta 2004, 92, 379-386. Glueckauf, E.; Trans. Faraday Soc. 1955, 51, 34-44. Mayer, S. W.; Tompkins, E . R. J. Am. Chem. Soc. 1947, 69, 2866-2874. Brüchle, W. Radiochim. Acta 2003, 91, 71-80.


Simulation of Elution Curves for Chromatography Columns with a Low

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