Dispersion In

8/4/2019 Dispersion In

1/88

Chrom-Ed Book Series

Raymond P. W. Scott

Dispersion in

ChromatographyColumns

This eBook is protected by Copyright law and International Treaties. This is a PROOFREADERS only copy, do not reproduce.

11

1

2

3

4

5

6

7

8

9

10

11

1213

2

3


2/88

COPYRIGHT @2003 by LIBRARY4SCIENCE, LLCALL RIGHTS RESERVED

Neither this book or any part may be reduced or transmitted in anyform or by any means, electronic or mechanical , includingphotocopying, microfilming, and recording or by any informationstorage and retrieved system without permission in writing from thepublisher except as permitted by the in-user license agreement.

World Wide Webhttp://www.library4science.com/

This eBook is protected by Copyright law and International Treaties. All rights are reserved. This book is covered by an End User Licensee

Agreement (EULA). The full EULA may be seen athttp://www.library4science.com/eula.html.

24

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

5

6

7
http://www.library4science.com/eula.htmlhttp://www.library4science.com/eula.htmlhttp://www.library4science.com/eula.htmlhttp://www.library4science.com/eula.html


3/88

Contents

Introduction to the Rate Theory..............................................................1The Summation of Variances..................................................................2

The Alternative Axes of a Chromatogram..........................................9The Random Walk Model.................................................................13Dispersion Processes that take Place in an LC Column...................22

The Multipath Process...................................................................23

Longitudinal Diffusion..................................................................24The Diffusion Process...................................................................24The Resistance to Mass Transfer in The Mobile Phase................29The Resistance to Mass Transfer in the stationary Phase.............30Resistance to Mass Transfer Dispersion.......................................31Diffusion Controlled Dispersion in the Stationary Phase............34

Diffusion Controlled Dispersion in the Mobile Phase.................35Effect of Mobile Phase Compressibility On the HETP Equation for aPacked GC Column...........................................................................37The Van Deemter Equation...............................................................43

Alternative Equations for Peak Dispersion ..........................................55The Giddings Equation.....................................................................56The Huber Equation..........................................................................57The Knox Equation...........................................................................59The Horvath and Lin Equation..........................................................61The Golay Equation..........................................................................61

Experimental Validation of the Van Deemter Equation...................66The Effect of the Function of (k') on Peak Dispersion.....................82

Summary................................................................................................84

References ............................................................................................86


38

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

3334

35

36

9

10


4/88


5/88

Introduction to the Rate Theory

The separation of a solute pair in a chromatographic system depends on

moving the peaks apart in the column and constricting their dispersion

so that the two solutes are eluted discretely. The factors that control

retention have been discussed in Book 7 and in this book the processes

of peak dispersion will be considered together with the means by which

peak dispersion can be minimized.

Solute equilibrium between the mobile and stationary phases is never

achieved in the chromatographic column except possibly at the

maximum of a peak. To circumvent this non equilibrium condition and

allow a simple mathematical treatment of the chromatographic process,

Martin and Synge (1) borrowed the plate concept from distillation

theory and considered the column consisted of a series of theoretical

plates in which equilibrium could be assumed to occur. In fact each

plate represented a 'dwell time' for the solute to achieve equilibrium atthat point in the column and the process of distribution could be

considered as incremental. This approach has been discussed in Book

6.

Employing this concept an equation for the elution curve can be easily

obtained and, from that basic equation, others can be developed that

describe the various properties of a chromatogram. Such equations

have permitted the calculation of efficiency, the number of theoretical

plates required to achieve a specific separation and among manyapplications, elucidate the function of the heat of absorption detector.

The Plate Theory, however, does little to explain how the efficiency of

a column may be changed or, what causes peak dispersion in a column

in the first place. It does not tell us how dispersion is related to column

geometry, properties of the packing, mobile phase flow-rate, or the


111

1

2

3

4

5

6

7

8

9

10

11

12

1314

15

16

17

18

19

20

21

22

23

24

2526

27

28

29

30

31

3233

34

35

36

37

12

13


6/88

physical properties of the distribution system. Nevertheless, it was not

so much the limitations of the Plate Theory that provoked Van Deemter

et al (2) (who were chemical engineers and mathematicians) to

develop, what is now termed the Rate Theory for chromatographic

dispersion, but more to explore an alternative mathematical approach toexplain the chromatographic process. Virtually all basic

chromatography theory evolved over the twenty five years between

1940 and 1965 and it was in the middle of this period that Van Deemter

and his colleagues presented their Rate Theory concept in (1956).

Since that time, other Rate Theories have been presented, together with

accompanying dispersion equations and in due course each will be

discussed, but most were very similar in form to that of Van Deemter

et al. It is interesting to note, however, that, even after thirty five years

of chromatography development, the equation that best describes band

dispersion in practice is still the Van Deemter equation. This is

particularly true for columns operated around the mobile phase

optimum velocity where the maximum column efficiency is obtained.

The purpose of the Rate Theory is to help understand the processes that

cause dispersion in a chromatographic column and to identify those

properties of the chromatographic system that control it. Such

information will allow the best column to be designed to effect a given

separation in the most efficient way. However, before discussing the

Rate Theory some basic concepts must be introduced and illustrated.

The Summation of Variances

The width of the band of an eluted solute relative to its proximity to its

nearest neighbor determines whether two solutes are resolved or not.

The ultimate band width as sensed by the detector is the result of a

number of individual dispersion processes taking place in the

chromatographic system, some of which take place in the column itself

and some in the sample valve, connecting tubes and detector (see book

10). In order to determine the ultimate dispersion of the solute band it

is necessary to be able to calculate the final peak variance. This is

achieved by taking into account all the individual dispersion processes

that take place in a chromatographic system. It is not possible to sum



214

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

1718

19

20

21

22

23

2425

2627

28

29

30

31

32

33

34

35

36

37

15

16

17


7/88

the band widths (standard deviation or ()) resulting from eachindividual dispersion process to obtain the final band width, but it is

possible to sum all the respective variances. However, the summation

of all the variances resulting from each process is only possible if each

process is non-interacting and random in nature. That is to say, theextent to which one dispersion process progresses is independent of the

development and progress of any other dispersion process.

Thus, assuming there are (n) non-interacting, random dispersive

processes occurring in the chromatographic system, then any process

(p) acting alone will produce a Gaussian curve having a variance p

2

,


318

1

2

3

4

5

6

78

9

10

11

12

19

20


8/88

Hence,



421

1

22

23

24


9/88


525

1

26

27


10/88

where, (



628

1

2

29

30

30


11/88


732

33

34


12/88

) is the variance of the solute band as sensed by the detector.

The above equation is the algebraic enunciation of the principle of the

summation of variances and is fundamentally important. If the

individual dispersion processes that are taking place in a column can be

identified, and an expression for the variance arising from each

dispersion process evaluated, then the variance of the final band can be

calculated from the sum of all the individual variances. This is how the

Rate Theory provides an equation for the final variance of the peak

leaving the column.

The Alternative Axes of a Chromatogram

An elution curve of a chromatogram can be expressed using parameters

other than the volume flow of mobile phase as the independent variable.The Plate theory provides an equation that expresses the retention and

standard deviation of a peak in terms volume flow of mobile phase.

However, instead of using milliliters of mobile phase, as the

independent variable, solute concentration in the mobile phase can be

related, time, or distance traveled by the solute band along the column

and proportionally the same chromatogram will be obtained. This is

illustrated in figure (1)

As the curves are describing the same chromatogram, by proportion,

the ratio of the variance to the square of the retention, in the respective

units in which the independent variables are defined, will all be equal.

Consequently, v

2

V r2

= x

2

l 2=

t2

t r2

where v, x and t are the standard deviations of the elution curveswhen related to the volume flow of mobile

phase, the distance traveled by the solute along

the column and time, respectively.and Vr, l and tr refer to the retention volume, column length

and retention time, respectively.



835

1

2

3

4

5

6

7

8

9

1011

1213

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

36

37

37


13/88

2 v

v

2 x

x

2 t

tF l o w o f M o b i l e P h a s e

i n P l a t e V o l u m e s

R e t e n t i o n T i m e D i s t a n c e T r a v e l l e d

a l o n g C o l u m n

Figure 1 Alternative Axes of a Chromatogram

Now, from the Plate Theory (see book 6) it has been shown that,


938

1

2

34

5

6

7

8

9

39

40


14/88

Thus,



1041

1

42

43

43


15/88

and

Therefore,l

n=

x2

l

The ratio, (l

n), (the column length divided by the number of theoretical

plates in the column) has, for obvious reasons, become termed the

Height Equivalent to the Theoretical Plate (HETP) and has been given

the symbol (H). However, it is seen that (H) is numerically equal to, ,

which is, in fact, the variance per unit length of the column. Thus, the

function, , is the variance that the Rate Theory will provide an explicit

equation to define and can be experimentally calculated for any column

from its length and column efficiency. It follows that the equations that

give a value for, (H), the variance per unit length of the column, have

been termed HETP equations.

To develop an HETP equation it is necessary to first identify the

dispersion processes that occur in a column and then determine the

variance that will result from each process per unit length of column.

The sum of all these variances will be (H), the Height of the

Theoretical Plate, or the total variance per unit column length. There

are a number of methods used to arrive at an expression for thevariance resulting from each dispersion process and these can be

obtained from the various references provided. However, as an

example, the Random-Walk Model introduced by Giddings (3) will be

employed here to illustrate the procedure. The theory of the Random-

Walk processes itself can be found in any appropriate textbook on

probability (4) and will not be given here but the consequential

equation will be used.

The Random Walk Model

The random-walk model consists of a series of step-like movements for

each molecule which may be positive or negative the direction being

completely random. After (p) steps, each step having a length (s) the

average of the molecules will have moved some distance from the


1145

1

2

3

4

5

6

7

8

9

10

11

12

13

14

1516

17

18

19

20

21

22

23

24

25

26

27

28

2930

31

32

33

34

35

36

46

47


16/88

starting position and will form a Gaussian type distribution curve with

a variance of s2 .

Now according to the random-walk model,

= s p (1)

Equation (1) can be used in a general way to determine the variance

resulting from the different dispersion processes that occur in an

chromatography column. The application of equation (1) is simple, the

problem that often arises is the identification of the average step and

sometimes the total number of steps associated with the particular

process being considered. As an illustration of its use it will be used to

the problem of obtain an expression for the radial dispersion of a

sample when it is placed on a packed column in the manner of Horne et

al. (5).

When a stream of mobile phase carrying a solute impinges upon a

particle, the stream divides and flows around the particle. Part of the

divided stream then joins other split streams from neighboring

particles, impinges on another and divides again. When a sample is

placed on the column at the center of the packing, initially it is in acondition of non-radial equilibrium, but as a result of this process the

sample spreads across the column during passage through the column

and eventually achieves radial equilibrium (the concentration of solute

is constant across a cross section of the column. Early work in liquid

chromatography, used relatively low inlet pressures and, thus, samples

could be injected directly onto the column with a syringe through an

appropriate septum device as in gas chromatography. This method of

injection often resulted in radial equilibrium never being achieved by

the solutes before they were eluted. The introduction of the sample

valve, however, aids in establishing radial equilibrium early in the

separation but unless some special spreading device is employed at the

front of the column, it will not necessarily occur at the point of

injection. The stream splitting process is depicted in figure (2). If a



1248

1

23

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

2223

24

25

26

27

28

29

30

31

32

33

34

35

49

50

50


17/88

particular molecule is considered passing round a particle, it will suffer

a lateral movement that can be seen to be given by,

Lateral Movement/Particle =dp

2cos

d p

e

Figure 2. The Mechanism of Radial Dispersion

It follows, that the average lateral step will be,dp

2

cos

2

+

2

d = dpEmploying the random walk function, the radial variance will be given

by:- 2 = Num ber of Steps ) X Step Length ) 2 (2)

Now assuming one lateral step is taken by a molecule for everydistance (jdp) that it moves axially, then, (n) the number of steps is


1351

1

2

3

4

5

6

7

8

9

10

11

12

13

52

53


18/88

given by:-



1454

1

55

56

56


19/88

where (l) is the distance traveled axially by the solute band.

Thus, substituting for (n) in equation (2),


1558

1

2

3

4

5

59

60


20/88



1661

1

62

63

63


21/88

In practice the value of (j) will lie between 0.5 and 1.0, but, for

simplicity the value of (j) will be taken as unity. This implies that one

lateral step will be taken by a given molecule for every step traveled

axially equivalent to one particle diameter .

Thus,

or, (3)

Consider a sample injected precisely at the center of a 4 mm diameter

LC column. Employing equation (3) allows the distance traveled

axially by the solute band before the radial standard deviation of the

sample of solute is numerically equal to the column radius to be

calculated. That is, the band has now spread evenly across the column

and the solute is in radial equilibrium.

For the conditions given above R = 0.2.

Thus, substituting this value in equation (3),

or, l =0 . 2

( )2

d p(4)

The distance that a solute band must pass along a column before a

sample, injected at the center of the packing, is evenly spread across its

diameter, can be calculated for columns packed with different sized

particles using equation (4). The results are shown as a graph relating

length against particle diameter in figure (3). The particle diameter

range normally employed in liquid chromatography is about 2-25

micron and so from figure 3, it is clear that radial equilibrium wouldnever be achieved for those column lengths commonly in use in LC.

Despite the lack of radial equilibrium, however, if the column packing

is completely homogeneous throughout the column length, then the

column efficiency should not be impaired. Unfortunately, ideal packing


1765

12

3

4

5

67

8

9

10

11

12

13

14

15

16

1718

19

20

21

22

2324

25

26

27

28

29

30

31

32

33

34

35

36

37

38

66

67


22/88

conditions are not always achieved and channeling often occurs, under

which circumstances lack of radial equilibrium could result in the

column efficiency being reduced with consequent loss in resolution. To

ensure radial equilibrium, it must either be achieved on injection (using

sample distribution device) or by employing narrow bore columnswhere radial equilibrium is more quickly reached. The latter

alternative, however, will depend on the resolution required and the

nature of the sample.

0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0

2 5 0 0

C

o

lu

m

n

L

en

g

th

T

ra

v

ersed

b

y

S

o

lu

te

(cm

)

0 1 0 2 0 3 0

P a r t i c l e D i a m e t e r ( m )

Figure 3. Graph of Column Length Traveled by the Solute Before

Radial Equilibrium is Achieved against Particle Diameter

Dispersion Processes that take Place in an LC Column

There are four basic dispersion processes that can occur in a packed

column that will account for the final band variance. They are,

Multipath dispersion, dispersion from Longitudinal Diffusion,This eBook is protected by Copyright law and International Treaties. All rights are reserved. This book is covered by an End User Licensee


1868

1

2

3

4

5

6

7

8

9

1011

12

1314

15

1617

18

19

20

69

70

70


23/88

dispersion from the Resistance to Mass Transfer in the Mobile Phase

and dispersion from the Resistance to Mass Transfer in the Stationary

Phase. All these processes are random and essentially non-interacting

and, therefore, provide individual contributions of variance that can be

summed to produce the final band variance.

The Multipath Process

In a packed column the solute molecules will describe a tortuous path

through the interstices between the particles and obviously some will

travel shorter paths than the average and some longer paths.

Consequently, some molecules will move ahead of the average and

some will lag behind thus causing band dispersion. The multipath

effect is illustrated in figure (4).

d L

Figure 4. Dispersion by the Multipath Effect

The Multipath effect can be applied to the Random Walk Model. The

average path length is equivalent to the mean diameter of the particle(dp) and thus the number of steps will be equivalent to the column

length divided by the average step i.e. .

Thus, applying equation (1),

or,

Dividing by the column length, (l), the variance per unit length or of

the multipath dispersion will be obtained and, thus, the multipath

contribution (HM) to the overall height of the theoretical plate (H),


1971

1

2

3

4

56

7

8

9

10

11

12

13

14

15

16

1718

1920

21

22

23

24

25

26

2728

29

30

31

32

3334

72

73


24/88

In fact, Van Deemter introduced a constant (2) to account for theinhomogeneity of packing so his expression for the Multipath

contribution became,

HM = 2 d p

For an ideally packed column (l) will be 0.5, and under which

circumstances the value fro (HM) reverts to,

.Longitudinal Diffusion

If a local concentration of solute is placed at the midpoint of a tube

filled with either a liquid or a gas the solute will slowly diffuse to

either end of the tube. It will first produce a Gaussian distribution with

a maximum concentration at the center and, finally, when the solutereaches the end of the tube, 'end' effects occur and the solute will

continue to diffuse until there is a constant concentration throughout

the length of the tube. The process is illustrated in Figure 5.

M o b i l e P h a s e

S a m p l e

Figure 5. Longitudinal Diffusion

Before dispersion due to longitudinal dispersion is discussed some

basic principles of diffusion need to be considered.

The Diffusion Process

Diffusion processes play important parts in peak dispersion. The

process not only contributes to dispersion directly (i.e., longitudinal

diffusion), but also helps to reduce the dispersion that results from



2074

1

2

3

4

5

6

7

8

910

11

12

13

14

15

16

17

18

1920

21

22

2324

2526

27

28

29

75

76

76


25/88

solute transfer between the two phases. Consider the situation depicted

in figure 6.

d x

D d c

d xm x =

A

B

D d c

d xm

x + d x = +D d c

d x( )d

d xd x

d c

d t

1 c m

1 c m

S a m p l e I n j e c t e d

x

D i f f u s i o n

C

Figure 6. The Diffusion Process

Consider a sample of solute is introduced in plane (A), (plane (A)

having unit cross-sectional area). Solute will diffuse according to

Fick's law in both directions ( x) and, at a point (x) from the sample

point, according, the mass of solute transported across unit area in unittime (mx) according to Fick will be given by,

(5)

where (Dm) is the Diffusivity of the solute in the fluid.

and is the concentration gradient at (x).

Now, mass of solute leaving the slice (dx) thick, at (x+dx), i.e.,

(mx+dx), is,

Thus, the net change in mass per unit time in the slice (dx) thick will be


2177

1

23

4

56

7

8

9

10

11

12

1314

15

16

17

18

19

20

21

2223

24

25

26

78

79


26/88

or

Now as ,

then

or,d c

d t= D m

d 2 c

d x 2(6)

Now, this is a standard differential equation and one solution to this

equation, which can be proved by appropriate differentiation, takes the

Gaussian form as follows:

Now, from the Plate Theory (see book 6),

,

where (n) is the variance of the Gaussian curve.

Now, (n) is the volume variance of the Gaussian curve ( i.e., ), then, bycomparison, (2Dmt) will be the length variance of the concentration

curve where (t) is the elapsed time. Consequently, if a differential

equation of the form is derived that describes some form of dispersion

that arises from a random diffusion process, then the solution will be a

Gaussian function and, more important from the point of view of theRate Theory, the Gaussian curve will have a variance given by (2Dmt).

Thus, if , the solution of the equation is a Gaussian function, and, for

that equation, 2 = 2 D m t (7)

Ordinary diffusion is the result of random molecular movement in first

one direction and then another and thus, resembles the Random Walk

Model. Uhlenbeck and Ornstein (8), derived the following expression

for the overall standard deviation () arising from diffusion process,

= 2 D m t( )0 . 5

where (t) is the time period over which the process occurs,



2280

1

2

34

56

7

8

9

10

11

1213

1415

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

81

82

82


27/88

and Dm is the Diffusivity of the solute in the mobile phase.

Actually, (t) is the time the solute spends in the moving phase and thus,

is given by, t =l

u, where (l) and (u) is the length of the column and the

linear mobile phase velocity respectively.

Thus,

and

Therefore, dividing by l to obtain the variance per unit length,

2

l= H L =

2 D mu

In the same manner as the constant () introduced by Van Deemter intothe function for multipath dispersion, Van Deemter also introduced a

constant () into the Longitudinal Dispersion contribution to varianceto account for some packing inhomogeneity. As a consequence, the

expression for the Diffusion contribution to the variance per unit length

of the column became,

HL

=2 D m

u(8)

To be precise, there should be a second longitudinal diffusion

contribution to the overall variance that would arise from the

stationary phase. The same method of derivation can be used but the

time the solute spends in the stationary phase is now, is now a function

of the capacity ratio of the column, (k').

Thus,

where (t"o) is the kinetic dead time, (see book 6)(k") is the kinetic capacity ratio.

The kinetic dead time and the kinetic capacity factors are calculated

from the kinetic dead volume as opposed to the thermodynamic dead

volume

Then from (7),


2383

1

2

3

4

5

6

78

9

1011

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

3031

32

33

3435

36

84

85


28/88

where (Ds) is now the solute Diffusivity in the stationary phase.

However, in this case, over the time period (tr), a fraction of time is

spent by the solute in the stationary phase and, thus,

Now, again noting that ,

Noting, again that, (HD(s)), the contribution to the variance per unit

length, will be . Then, the contribution to the total variance per unit

length for the column from longitudinal diffusion in the stationary

phase will be

i.e., . (9)

Introducing a packing factor the magnitude of which will depend on

the quality of the packing, the contribution to the variance per unit

length from diffusion in the stationary phase (HL(S))is given by,

HL ( S )

=2 2 k ' D S

uwhere, DSis the Diffusivity of the solute in the stationary phase

The total contribution to (H) from longitudinal diffusion will thus be:-

or, H L =2 1 D m

u1 + k '( ) (10)

where,

It is seen from equation (10) that the longitudinal diffusion term is a

function of (k') the capacity factor of the solute. While this could be

significant in an LC capillary column systems, where the film of



2486

1

2

34

5

67

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

2930

31

32

3334

35

36

37

87

88

88


29/88

stationary phase could be continuous along the length of the column, it

will not be so in a packed column. The stationary phase in a packed

column is broken into segments between each particles and between

each pore in each particle so free continuous diffusion in the stationary

phase would be impossible. It follows that the longitudinal diffusionterm for packed columns should be independent of the k' of the solute

or, very nearly so, and in practice the effect of longitudinal diffusion in

the stationary phase is ignored.

The Resistance to Mass Transfer in The Mobile Phase

As a solute band progresses along a column, the solute molecules are

continually transferring from the mobile phase into the stationary phase

and back from the stationary phase into the mobile phase. This transfer

process is not instantaneous, because a finite time is required for the

molecules to traverse (by diffusion) through the mobile phase in order

to reach, and enter the stationary phase. Thus, those molecules close to

the stationary phase will enter it almost immediately, whereas those

molecules some distance away from the stationary phase will find their

way to it a significant interval of time later. However, as the mobile

phase is moving, during this time interval while they are diffusing

towards the stationary phase boundary, they will be swept along the

column and thus dispersed away from those molecules that were closeand entered it rapidly. The dispersion resulting from the resistance to

mass transfer in the mobile phase is depicted in figure 7.


2589

1

2

3

4

5

6

7

89

1011

12

13

14

15

16

17

18

19

20

21

22

23

24

2526

90

91


30/88

1 2

1 2 3 4

5 6

1 2 3 4 5 6

1 2 35

3 45 6

S t a t i o n a r y P h a s e








46

Figure 7. Resistance to Mas Transfer in the Mobile Phase

The diagram shows 6 solute molecules in the mobile phase and those

closest to the surface (1 and 2) enter the stationary phase immediately.

During the period, while molecules 3 and 4 diffuse through the mobile

phase to the interface, the mobile phase moves on. Thus, when

molecules 3 and 4 reach the interface they will enter the stationary

phase some distance ahead of the first two. Finally, while molecules 5

and 6 diffuse to the interface the mobile phase moves even further

down the column until molecules 5 and 6 enter the stationary phase

further ahead of molecules 3 and 4. Thus, the 6 molecules, originally

relatively close together, are now spread out in the stationary phase.

This explanation is a little over-simplified, but gives a correct

description of the mechanism of mass transfer dispersion.

The Resistance to Mass Transfer in the stationary Phase

The resistance to mass transfer in the Stationary phase is depicted infigure 8.



2692

12

34

5

6

7

8

9

10

11

12

13

14

15

1617

18

1920

21

22

93

94

94


31/88

2 1

4 3 2 1

5 6

1 2 3 45 6

3 4 5 6










6 5 4 3 2 1

Figure 8. Resistance to Mass Transfer in the Stationary Phase

The dispersion resulting from the resistance to mass transfer in the

stationary phase can be described in the same way as that in the mobile

phase. Molecules close to the surface of the stationary phase, will leave

and enter the mobile phase before those that have diffused farther into

the stationary phase and, thus, have further to diffuse back to the

surface. Consequently, during the period required for the solute

molecules to diffuse to the stationary phase surface, those moleculesthat were close to the surface will be swept along by the moving phase

and dispersed from those molecules still diffusing to the surface. In

figure 6,molecules 1 and 2, (the two closest to the surface) will enter

the mobile phase and begin moving with the mobile phase along the

column. This process will continue while molecules 3 and 4 diffuse to

the interface at which time they, also, will enter the mobile phase and

start following molecules 1 and 2 down the column. All four molecules

will continue their journey down the column while molecules 5 and 6

diffuse to the mobile phase/stationary phase interface. By the time

molecules 5 and 6 enter the mobile phase, the other four molecules will

have been smeared along the column and the original 6 molecules will

have suffered dispersion.

Resistance to Mass Transfer Dispersion


2795

12

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

2425

2627

96

97


32/88

Both the resistance to mass transfer in the mobile phase and that in the

stationary phase, can be treated quantitatively using the random walk

model. Recalling that if each molecule of a group takes a series of

steplike movements, (positive or negative) the direction being

completely random, then after (p) steps have been taken, each stephaving an average length (s), the average of the molecules will have

moved some distance from the starting position and will form a

Gaussian type distribution curve with a variance of2,where

or

In the first instance, consider that the distribution is energy controlled

and not diffusion controlled, thus, a solute molecule will desorb from

the stationary phase when it randomly has sufficient kinetic energy tobreak its association with a molecule of stationary phase (see book 1).

Similarly, a molecule will be absorbed under the same conditions.If (kd) is the desorption rateconstant then the mean desorption time

(td) for the adsorbed molecule will be . Correspondingly, if the

adsorption rate constant is (ka), then the mean adsorption time for a

free molecule in the mobile phase will be .

Consider a peak moving down a column. During this migration

process, adsorption and desorption steps will constantly and frequently

occur and each occurrence will be a random event. Now a desorption

step will be a random movement forward as it releases a molecule into

the mobile phase. Conversely, an adsorption step will be a random

movement backward, as it is a period of immobility for the molecule

while it resides in the stationary phase and the rest of the zone moves

forward. The total number of random steps, taken as the solute mean

position moves a distance along the column, is the number of forward

steps plus the number of backward steps. Now the distribution of the

solute is dynamic and is also an equilibrium system, consequently, eachdesorption step must be followed by an adsorption step. It follows that

the total number of steps will be twice the number of adsorption steps

that take place in the migration period.

On average, a molecule will remain in the mobile phase a time (ta)

before it is adsorbed. During this time, it will be moving at the meanThis eBook is protected by Copyright law and International Treaties. All rights are reserved. This book is covered by an End User Licensee


2898

1

2

3

4

5

6

7

8

9

1011

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

3435

36

37

99

100

100


33/88

velocity of the mobile phase (u) and will, thus, move a distance (uta).

Thus, in moving a distance , the total number of adsorptions will be

and the total number of steps including the adsorption and desorption

steps will be . It also follows, from the Random Walk Model that,

where

It is now necessary to determine the average step length (s) to obtain an

expression for (H).

The step length is that length moved by the molecule relative to that ofthe zone center and, while the molecule has move (vta) during time

(ta), the zone center has also moved. Now, it was shown in the Plate

theory that the zone velocity is where (k") is the dynamic capacity

ratio of the solute.

Consequently, .

and, ,

i.e.,

In practice, it is more convenient to express (H) in terms of (t d) as

opposed to (ta). The ratio of the mean phase residence times is the time

the solute spends in the mobile phase divided by the time spent in the

stationary phase,

Thus,

and H R M T =2 ( k " ) 2

1 + k "( )2

ut dk "

=2 k "

1 + k "( )2

u t d (11)

Equation (11), is derived from the approximate random walk theory,

however, it is rigorously correct and applies to heterogeneous surfaces

containing wide variations in properties as well as to perfectly uniform

surfaces. It can also be used as the starting point for the random walk

treatment of diffusion controlled mass transfer similar to that which

takes place in the stationary phase in GC and LC columns.


29101

1

2

3

45

67

8

9

10

11

12

13

14

1516

17

1819

2021

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

102

103


34/88

Diffusion Controlled Dispersion in the Stationary Phase

The difference between diffusion controlled dispersion and dispersion

resulting from adsorption and desorption processes is that the transfer

process is concentration controlled. Reiterating equation (7),

.

Thus, during solute transfer between the phases, (t) is now the average

diffusion time (tD) and () is the mean distance through which thesolute diffuses, i.e., the depth or thickness of the film of stationaryphase (df).

Thus, or (12)

where (DS) is the Diffusivity of the solute in the stationary phase.

Substituting (tD) for (td) from (12) in (11),

H M T S =q k "

1 + k "( ) 2d f

2

D Su (13)

where (q) is a configuration factor.

The constant (q) accounts for the precise shape of the 'pool' of

stationary phase which, for a uniformly coated GC capillary column, (q

= 2/3). Diffusion in 'rod' shaped and 'sphere' shaped bodies, (e.g.,

paperchromatography and LC) (q=1/2 and 2/15), respectively (7).

For a GC capillary column,

(14)

and a close approximation for an LC or GC packed column, (HMTS)

would be given by

H M T S =2 k "

1 5 1 + k "( ) 2d f

2

D Su . (15)



30104

1

23

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

2324

25

26

27

2829

3031

32

33

34

35

36

37

105

106

106


35/88

Diffusion Controlled Dispersion in the Mobile Phase

Dispersion processes in the mobile phase will also be diffusion

controlled, thus, again reiterating equation (7),

While in the mobile phase a solute molecule must traverse from

localities of high velocity (near the center of a capillary column or the

center of an inter-particle channel) to that of low velocity (the interface

between the two phases at the capillary column walls or, in the packed

column, the surface of the particles). Let the "exchange time' between

the two extreme velocities be (tm). Now, the distance between the

extremes of velocity will depend on the geometry of the column

system, but in general it can be assumed a molecule must diffuse adistance (mdm) to move from one velocity extreme to the other.Depending on the column (tubular or packed), and on the homogeneity

of the packing, (particle size and shape) the value of (m) may rangewidely from much less than unity to significantly greater than unity.

Now, from equation (7),

(=mdp) and (t=tm)

Thus,

or t m = m

2

2 D md p

2. (16)

Now, in the equation from the Random walk concept,

or

In addition, = tmu andor . (17)

Substituting for (tm) from (16) in (17)


31107

12

3

45

67

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

2829

3031

32

33

34

35

36

3738

108

109


36/88

Now,

Thus, H M T M = 2

l=

m2

2 D md p

2 u (18)

Equation (21) applies to all types of columns each requiring a different

constant (m) which is partly determined by the geometry of thedistribution system and partly by the capacity ratio of the solute. A

function for (m) has not been developed for packed LC or GCcolumns, but due to the geometric simplicity Golay (8) was able to

develop the function for the capillary column ,viz.

and thus, for a capillary column,

H M T M =1 + 6 k " + 1 1 k " 2

2 4 1 + k " 2

2r 2

2 D mu . (19)

The argument used to develop the function describing (tm) for a

capillary column is similar to that for the packed column but (r), the

column radius, replaces (dp) the particle diameter.

Due to the varying physical nature of the different packings, it appears

column, but it was suggested by Klinkenberg and Purnell that as no one

has developed a specific function for () for a packed columns,

that the function,

(developed by Golay for the capillary column) could also be used for

packed GC and LC columns. However, in a packed column. the flow

patterns in interstices between the particles is very complex and the

eddies and pseudo turbulence that are generated between the particles

greatly increases the effective diffusivity of the solute in the mobile

phase. In fact, the magnitude of equation (18) can be so reduced by thelarge increase in (Dm) that the overall contribution to the peak

dispersion can become small enough to be ignored.



32110

1

2

3

4

5

6

7

8

910

11

12

13

14

15

16

17

1819

20

21

22

2324

25

26

27

28

29

30

31

32

33

111

112

112


37/88

Summarizing,, the rate theory provides the following equations that

describe the variance per unit length (H) for four different columns.

1. The Open Tubular GC Column,

(20)

2. The Packed GC Column

(21)

This equation is basically that derived by Van Deemter et al. in 1956.

3. The Packed LC Column

(22)4. The Open Tubular LC Column

(23)

It should be noted that all the equations assume that the mobile phase

is incompressible which will not be true for equations (23) and (24). It

follows that equations (23) and (24) will require modification in order

to be applicable to practical situations.

Effect of Mobile Phase Compressibility On the HETP Equation for a

Packed GC Column

As the pressure falls along the column length, the velocity changes

and, as the solute diffusivity depends on the pressure, the diffusivity of

the solute will also change. The multi-path term, which contains no

velocity or gas pressure dependent parameters, will be unaffected and

the expression that describes it the same. The other terms in the HETP

equation, however, all contain parameters that are affected by gas

pressure (solute diffusivity and mobile phase velocity) and, therefore,

need to be modified to accommodate the compressibility of the mobile

phase.

Reiterating the HETP equation for a packed column,


33113

1

23

4

5

6

7

8

9

1011

1213

14

1516

1718

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

114

115


38/88

where f1(k") = and f2(k") =

Consider a point (x) along the column,

The HETP equation, as derived by Van Deemter, is seen to apply only

to a particular point distance (x) from the inlet of the column where the

pressure is constant.

Now, applying the gas laws Px u x = Pou o or

where (Px) is the pressure at point (x) along the column,

(ux) is the linear velocity of the mobile phase at point (x),

(Po) is the pressure at the column exit,

and (uo) is the linear velocity of the mobile phase at the column

exit.

Now from the kinetic theory of gases, it can be shown that thediffusivity of a solute in a gas is inversely proportional to the pressure.

Thus, DxPx = DoPo or

where (Do) is the solute diffusivity at the end of the column at (Po)

and (Dx) is the solute diffusivity at point (x) and pressure (Px)

Thus, and

Substituting for and in the HETP equation,

H x = 2 d p +2 D m ( o )

u o+

f1 ( k " ) d p2

D m ( o )u o +

f 2 ( k " ) d f2 P 0

D S P xu o



34116

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

117

118

118


39/88

It is now seen that if the mobile phase velocity is taken as that at the

column outlet, only the resistance to the mass transfer term for the

stationary phase is position dependent. All the other terms can be used

as developed by Van Deemter, providing the diffusivities are also

measured at the outlet pressure (atmospheric).

The resistance to the mass transfer term for the stationary phase must

be considered in isolation. The experimentally observed plate height

(variance per unit length) resulting from a particular dispersion process[e.g., (hs), the resistance to mass transfer in the stationary phase] will

be the sum of the local plate height contributions (h'); i.e.,

Consequently, substituting for (h') the expression for the resistance to

mass transfer in the stationary phase will be,

or (24)

Now, it can be shown that, (see equation 23 Book 6),

or (25)

Substituting for (Px) from equation (25) in equation (24),

(26)

Let w =

Then or

Furthermore, when x=0, then and when w = L, then .

Substituting for

and (dx) in equation (26) and inserting the new limits

and integrating,


35119

1

2

3

4

5

6

7

8

9

10

11

1213

14

1516

17

1819

20

21

22

23

24

25

26

27

28

29

30

3132

33

34

3536

37

38

39

40

120

121


40/88

or

Thus, (27)

Thus, the complete HETP equation for a packed GC column, that takes

into account the compressibility of the carrier gas, will be

H = 2 d p +2 D m ( o )

u o+

f1 ( k " ) d p2

D m ( o )u o + 2

f 2 ( k " ) d f2

D S + 1( )u o (28)

Now from D'Arcy's Law

or

Thus,

Integrating from x=0 to x = L and Px = Pi to Po,

or

Solving for (), (29)

Substituting for () in equation (27) from equation (29),

Thus the HETP equation given by equation (28) becomes

H = 2 d p +2 D m ( o )

u o+

f1 ( k " ) d p2

D m ( o )u o + 2

f2 ( k " ) d f2

D Su 0 L

K P o+ 1

0 . 5

+ 1

u o(30)



36122

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

1617

18

19

20

21

22

23

24

25

26

27

28

29

30

31

3233

123

124

124


41/88

Equation (30) gives the variance per unit length of a GC column in

terms of the outlet pressure (atmospheric); the outlet velocity; and

physical and physicochemical properties of the column, packing, and

phases and is independent of the inlet pressure. However, equation (28)

is the recommended form for HETP measurements as the inlet pressureof a column is usually known, (and consequently (), the inlet/outletpressure ratio is also known) and the equation is less complex and

easier to use. The important aspect of this development is that the

resistance to mass transfer in the stationary phase is seen to be a

function of the inlet-outlet pressure ratio ().

Regrettably, the average velocity is the variable that is almost

universally used in constructing HETP curves in both GC and LC. This

is largely because it is simple to calculate from the ratio of the column

length to the dead time. Unfortunately, in GC, the use of the average

velocity provides very erroneous data and, for accurate column

evaluation and column design, the exit velocity must be employed

together in conjunction with the inlet-outlet pressure ratio.

An example of the errors that can occur from the use of the average

velocity, as opposed to the exitvelocity, is shown in figure 9 from data

obtained for a capillary column.

H E T P C u r v e E m p l o y i n gt h e E x i t V e l o c i t y

H E T P C u r v e E m p l o y i n gt h e A v e r a g e V e l o c i t y

M o b i l e P h a s e V e l o c i t y ( c m / s e c . )

P

la

te

H

eig

h

t


37125

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

126

127


42/88

Figure 9. HETP Curves for the Same Column and Solute Using the

Average Mobile Phase Velocity and the Exit Velocity

The two curves are clearly quite different and, if the results are to befitted to the HETP equation, only the data obtained using the exit

velocity will give meaningful values for the exclusive dispersion

processes. This problem is further emphasized in the graphs shown in

figure 10. In figure 10, the individual contributions from the different

dispersion processes are obtained by deconvoluting the HETP curve

obtained using the average velocity data.

It is seen that using the average velocitydata, the extracted value for

the multi-path term is negative, which is physically impossible (for acapillary column should be zero or very close to zero). In contrast, the

values obtained from data involving the exit velocity give small

positive, but realistic values for the multi-path term.

In all aspects of column evaluation and column design in GC, the

compressibility of the mobile phase must be taken into account or

serious errors will be incurred.



38128

1

2

3

45

6

7

8

9

10

11

1213

14

15

16

17

18

19

20

21

22

23

129

130

130


43/88

L o n g i t u d i n a l D i f f u s i o n B / u

H E T P C u r v e ( H )

R e s i s t a n c e t oM a s s T r a n s f e r C u

0

M u l t i - p a t h T e r m A

A v e r a g e V e l o c i t y ( c m / s e c . )

H

(cm

)

Figure 10. De-Convolution of the HETP Curve Obtained Using the

Average Mobile Phase Velocity

The Van Deemter Equation

The Van Deemter equation (9) was derived as long ago as 1956 and

was the first rate equation to be developed. There are, however, a

number of alternative rate equations that have been reported, but when

subjected to experimental test, the Van Deemter equation has been

shown to be the most appropriate equation for the accurate prediction

of dispersion in chromatographic systems. The Van Deemter equation

is particularly pertinent at mobile phase velocities around the optimum

velocity (a concept that will shortly be explained). Consequently, as all

columns should be operated at, or close to, the optimum velocity formaximum efficiency, the Van Deemter equation is particularly

important in column design. Restating the Van Deemter equation,

where f1(k") = and f2(k") =


39131

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

1819

2021

22

23

132

133


44/88

Note, this equation ignores the second order effect of any longitudinal

diffusion that might be present in the stationary phase. In fact, in the

original form, the equation was introduced by Van Deemterfor packed

GCcolumns and consequently, the longitudinal diffusion term for the

liquid phase was not included and the function 2 Dmu

1+ k' , was

replaced by,2 Dm

u. This was because the diffusivity of the solute in a

gas is four to five orders of magnitude greater than in a liquid.

Thus, in the expression for longitudinal diffusion,

the second function would not be significant.

In addition, Van Deemter considered the resistance to mass transfer in

the mobile phase to be negligible. This was because the eddies that

form between the particles causes a turbulence or secondary flow that

greatly increases the diffusivity of the solute between the pores. As a

result, the function,, was also not included. Consequently, the equation

actually developed by Van Deemter took the form,

The form taken by f2(k') was considered by Van Deemter to be,

and thus, the complete HETP equation became,

(31)

Equation (31), however, was developed for a GC column and in the

case of an LC column, the resistance to mass transfer in the mobile

phase should be included. Van Deemter et al. did not derive an

expression for f1(k') for the mobile phase but Purnell (10) suggested

that the function of (k'), employed by Golay (8) for the resistance to

mass transfer in the mobile phase in his rate equation for capillaryThis eBook is protected by Copyright law and International Treaties. All rights are reserved. This book is covered by an End User Licensee


40134

1

2

3

4

5

6

78

9

1011

12

1314

15

16

17

18

19

20

21

22

23

24

25

26

2728

29

30

31

32

33

34

35

36

37

38

135

136

136


45/88

columns, would also be appropriate for a packed column. The form of

f1(k') derived by Golay was as follows,

Thus, Van Deemter's equation for LC becomes,

(32)

Equation (32 ) can put in a simplified form as follows,

(33)

where, A=2d p, B=2 D m and

Equation (33) is a hyperbolic function which has a minimum value of

(H) for a particular value of (u). Thus, a maximum efficiency will

obtained at a particular linear mobile phase velocity.

An example of an HETP curve obtained in practice showing this

hyperbolic relationship is given in figure 11.

-0.000 0.100 0.200 0.300 0.400 0.5000.600

u ( c m / s e c )

-0.000

0.001

0.002

0.003

C u r v e F i t

M u l t i p a t h T e r m

R e s i s t a n c e t oM a s s T r a n s f e r

L o n g i t u d i n a lD i f f u s i o n


41137

1

2

3

4

5

6

78

9

10

11

12

1314

15

16

17

18

19

20

2122

23

2425

2627

138

139


46/88

Figure 11. HETP Curve for Hexamethylbenzene

The HETP curve is the result of a curve fitting procedure to the

experimental points shown. The column was 25 cm long, 9 mm in

diameter and packed with 8.5 micron (nominal 10 micron) Partisil

silica gel. The mobile phase was a solution of 4.8%w/v ethyl acetate in

n-decane. The hyperbolic form of the curve is confirmed, the minimum

is clearly indicated and the fit of the points to the curve is good. From

the curve fitting procedure the values of the Van Deemter constants

could be determined and the separate contributions to the curve from

the multipath dispersion, longitudinal dispersion and the resistance to

mass transfer calculated and included in the figure.

It is seen that the major contribution to dispersion at the optimum

velocity (where the value of (H) is a minimum) is the multipath effect.

Only at much lower velocities, does the longitudinal diffusion effect

become significant. Conversely, the mobile phase velocity must be

increased to about 0.2 cm/sec before the dispersion due to the

resistance to mass transfer begins to become relatively significant

(compared to that of the multipath effect).

The values for the Van Deemter constants were found to be,

A=0.00117B=0.0000175 cm2/sec

and C=0.00250 sec.

If the mean particle size was 0.00085 cm, then as, A = 2 dp

Then, 0.00117 = 2 0.00085 or, = 0.69

Giddings (11) determined theoretically that, for a well packed column

() should take a value of about 0.5 and thus the column used was

reasonably well packed.In a similar manner, as B = 2Dm

Then, 0.0000175 = 2gDm



42140

12

3

4

5

6

7

8

9

10

11

1213

14

15

16

17

18

19

2021

2223

24

25

2627

28

29

3031

32

33

3435

36

37

38

39

141

142

142


47/88

Katz et al (12) determined the diffusivity of hexamethylbenzene in

4.83%w/v ethyl acetate in n-decane and found it to be 1.17x10-5

cm2/sec,

Consequently, = 0.747

Giddings (11) also determined that, for a well packed column, ()should be about 0.6 so again the dispersion from longitudinal diffusion

as measured. confirms that the column was reasonably well packed.

For a packed column, the particle size has a profound effect on the

minimum value of the HETP of a column and thus the maximum

efficiency attainable. It would also indicate that the highest column

efficiency would be obtained from the smallest particles. However, this

assumes that an unlimited pressure is available and the apparatus cantolerate such pressures. Within practical limits of pressure, the smaller

the particle diameter, the smaller will be the minimum HETP and thus,

the larger the number of plates per unit length obtainable from the

column. However, the total number of theoretical plates that can be

obtained, will depend on the length of the column which, in turn, must

take into account the available inlet pressure.

The optimum mobile phase velocity can be obtained by differentiating

equation (33) with respect to (u) and equating to zero, thus,

and

Equating to zerodH

du= 0 and consequently,

B

u2

+ C = 0

Thus, (34)

Substituting for (B) and (C),

and letting Ds = Dm,


43143

1

2

3

4

5

6

7

89

10

11

12

13

14

15

16

17

18

19

2021

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

144

145


48/88

(35)

It is seen from equation (35) that the optimum velocity is, directly

proportional to the diffusivity of the solute in the mobile phase. To a

lesser extent it also appears to be inversely dependent on the particlediameter of the packing and the film thickness of the stationary phase.

The film thickness of the stationary phase is determined by the physical

form of the packing, that is, in the case of silica gel, the nature of the

surface and in the case of a reverse phase, on the bonding chemistry.

Now, if it is assumed that df > 1, that is, for well retained peaks, and taking

() as 0.6

u o p t = D md p

4 . 4 2 [ ] 0 . 5 2 . 1 D md p

1 . 6 3 D md p

(36).

Under these circumstances, it is seen that the optimum velocity is

directly proportional to the solute diffusivity in the mobile phase and

inversely proportional to the particle diameter of the packing. The

quality of the packing, also plays some part, but to a very less

significant extent. It is now possible to determine the factors that

control the magnitude of Hmin. Substituting the function for uopt from

equation (34) in equation (33) an expression for Hmin is obtained.

and (37)

Substituting for A, B, and C,

(38)



44146

1

2

3

4

56

7

8

910

11

12

13

14

15

16

17

1819

20

21

22

23

24

25

26

27

28

29

3031

3233

3435

147

148

148


49/88

Noting that Ds= Dm, and simplifying,

(39)

Again, assuming that, df


50/88

or, n P d p

2

2 . 4 D m (43)

It is seen from equation (43) that, if an LC column is operated at its

optimum linear velocity, the maximum efficiency obtainable for well

retained peaks will be directly proportional to the inlet pressure

available (P) and the square of the particle diameter of the packing.

Thus, the larger the particle diameter, the greater efficiency attainable

at a given pressure. This is because, as the particle diameter is

increased the column permeability is also increased allowing a longer

column to be used. The permeability increases as the square of the

particle diameter but the variance per unit length only increases

linearly with the particle diameter. Thus, doubling the particle diameterwill allow a column four times the length to be used but the number of

plates per unit length will be halved. Consequently, the column

efficiency will be increased by a factor of two. It is also seen that the

higher efficiencies will be obtained with mobile phases of low

viscosity and for solutes of low diffusivity. Solvent viscosity and solute

diffusivity tend to be inversely proportional to each other and so the

sensitivity of the maximum obtainable efficiency to either solvent

viscosity or solute diffusivity will generally not be large. The

approximate length of a column that will provide the maximum column

efficiency when operated at optimum velocity is given by, l = nHmin.

Thus, =0 . 6 2 P d p

3

D m (44)

It is seen that the column length varies inversely as the product of the

solute diffusivity in the mobile phase and the mobile phase viscosity in

much the same way as the column efficiency does when operating atthe optimum velocity. As would be expected the column length is

directly proportional to the inlet pressure but, less obviously is also

proportional to the cube of the particle diameter.

The analysis time for a solute mixture in which the last peak is eluted at

a capacity ratio of k'F is given by,



46152

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

2728

29

3031

32

33

34

153

154

154


51/88

t = (1 + k'F )t 0 = (1 + k'F )l

u opt

Substituting for (uopt) and (l) from equations (36) and (44),

or, t = ( 1 + k ' F )0 . 4 9 P d p

4

D m2

(45)

It is seen from equation (45) that the analysis time is proportional to the

fourth power of the particle diameter and inversely proportional to the

square of the diffusivity in the mobile phase. In a similar manner to

column length, the analysis time is also directly proportional to the

applied inlet pressure and inversely proportional to the mobile phase

viscosity.

Summarizing, if a column is operated at its optimum velocity and the

solute concerned is eluted at a relatively high k' value, and assuming

the film thickness of the stationary phase is small compared with the

particle diameter (a condition that is met in almost all LC separations)

then,

u o p t 1 . 2 6 D m

dp

(36)

Hmin. 1.48dp (46)

n P d p

2

2 . 4 D m (43)

l 0 . 6 1 7 P d p

3

D m (44)

t (1+ k'F )0 .38 Pd

p

4

Dm

2

(45)


47155

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

156

157


52/88

As already stated, these equations are insufficiently precise to be used

for accurate column design but, they can give a general indication of

the more important performance specifications that could be expected

from an approximately optimized chromatographic system.

It would be of interest to use the equations to calculate the maximum

efficiency, column length and analysis time that would be obtained

from columns packed with particles of different diameter under

conditions that commonly occur in LC analyses.

The diffusivity of solutes having molecular weights ranging from about

60 to 600 vary from about 1x10-5 to 4x10-5 cm2 /sec.(13) Thus, an

average value of 2x10-5 cm2 /sec will be taken in the following

calculations. In a similar way a typical viscosity value for the mobile

phase will be assumed as that for acetonitrile (14), i.e. 0.375x10-2

poises. The inlet pressure taken will be 3,000 p.s.i. or about 200

atmospheres. Many LC pumps available today will operate at 6,000

p.s.i. or even 10,000 p.s.i., however, it is not the pump that determines

the average operating pressure of the chromatograph, but the sample

valve.

Sample valves can, for a limited time, operate very satisfactorily at

very high pressures but their lifetime at maximum pressure issignificantly reduced. Most valves, designed for high pressure

operation will have extensive lifetimes when operated at 3,000 p.s.i.

and so this is the pressure that will be employed in the using the above

equations. The value assumed for the d'Arcy constant,() is 35 when(P) is measured in p.s.i. and has been determined experimentally (14).

The capacity factor of the last eluted peak was taken as 5 for

calculating the elution time. Employing the above values, equation (16)

was used to calculate maximum efficiency attainable for columns

packed with particles of different diameters. The results obtained are

shown in figure 12. It is seen from figure 12 that changing the particle

diameter from 1 to 20 micron results in an efficiency change from

about 3500 theoretical plates to nearly 1.5 million theoretical plates

and furthermore, this very high efficiency is achieved at an inlet

pressure of only 3000 p.s.i..This eBook is protected by Copyright law and International Treaties. All rights are reserved. This book is covered by an End User Licensee


48158

1

2

3

45

6

7

8

910

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

159

160

160


53/88

0 10 20 30

3

4

5

6

7

P a r t i c l e D i a m e t e r ( m i c r o n )

Figure 12. Graph of Log Maximum Efficiency against Particle

Diameter

It is also seen that the maximum available efficiency increases as the

particle diameter increases. This is because, as already discussed, if the

pressure is limited, in order to increase the column length to provide

more theoretical plates, the permeability of the column must be

increased to allow the optimum mobile phase velocity to be realized. It

is possible to increase the inlet pressure to some extent, but ultimately

the pressure will be limited and the effect of particle diameter will be

the same but at higher efficiency levels. The column lengths necessary

to achieve these efficiencies can be calculated employing equation

(44). The results obtained are shown as curves relating maximum

column length to particle diameter in figure 13.

It is seen from

Documents

Dispersion In