On the statistical independence of various column contributions to band broadening. Part 3: Independent treatment of longitudinal diffusion in the plate model

On the Statistical Independence of Various Column Contributions to Band Broadening Part 3: Independent Treatment of Longitudinal Diffusion in the Plate Model

Olle Nilsson Department of Technical Analytical Chemistry, Lund Institute of Technology, P. 0. Box 740, S-220 07 Lund 7, Sweden

Key Words:

Chromatography theory, GC and LC Chromatography plate concept, “theoretical” and

“discontinuous” Partial differential contributions Variance addition rule Theoretical band shapes

Summary

It is shown theorectically that the classical formula for calcula- ting the theoretical plate number from squared first central moment, fkg, and second central moment, u2, according to fitheor = f&du2, is valid only when the capacity ratio, E, approaches infinity. The general relation between the classical experimental HETP value, H = L/jitheor, and the underlying true theoretical plate height, AL, is found to be

L(u’)2 2& - 2 4 E 6 l S

ii 0 l + i i H’= - + k - + A L -

when (u’p is the total column contribution to band broadening,L isthecolumnlength,& istheaveragediffusioncoefficient ofthe sample component in the mobile phase, 4 is its value in the stationary phase, and ii is the average linear velocity of the mobile phase. The mobile phase displacement, as well as the mass exchange process, is assumed to be continuous, but the application of the plate concept conditions leads to a mass balance equation that can be interpreted as belonging to a modified discontinuous plate model. The contributions 2&,/0 and k 2 &/ii from longitudinal sample diffusion in the mobile and stationary phases, respectively, are consistent with the assumption that the processes are statistically independent, although the common solution technique of the differential equations does not take full account of this independence.

1 Introduction

In the first application of the theoretical plate concept to chromatography Martin and Synge [ 11 used approximations that are valid only when the number of eluted plate volumes (n in their paper) is much larger than the plate number of the column (r in their paper). Thus, there is a risk that the validity range of the derived plate number formula (r = 4 t2/? in ref. 121) is limited to cases when the average time ts spent by the sample in the stationary phase is much larger than the average time fm spent in the mobile phase, i. e. when the capacity ratio, i? = cs/Frn, approaches infinity. Martin and Synge [I] were also aware of how the value obtained depends on molecular diffusivity at various

mobile flow rates and particle sizes, and that the number hence cannot be a substance-independent column speci- fication. For these reasons their formula is regarded merely as a definition of the theoretical plate number of a selected substance on a column. With our nomenclature it is most exactly written as

where fms (= fs + tm) is the first central moment, equal to the center of gravity time of the band, and u2 is the second central moment, with u equal to the standard deviation width of a Gaussian-shaped band. For convenience many chromatographers prefer to measure retention time at the peak maximum, tms = t, + t,, and width at half height, b0,5, in which case we can define

The principally discontinuous plate model of Martin and Synge was later treated in a mathematically more rigorous way by Mayer and Tompkins [3], who deduced relations which imply that the retention time should be partially corrected for the dead time of the column, f,, to give formulas corresponding to

(3)

Simpson and Wheaton [4], following a suggestion of Matheson [4], also deduced explicitly that the correct formula is equivalent to

(4)

A quite independent approach was made by Golay[a, who by applying the “telegrapher‘s equation” also arrived at the discontinuous t,,t, formula. He repeated his result in a

0 1982 Dr. Alfred Huethig Publishers 0344-71 38l821060311-10$01 .OO Journal of HRC & CC 31 1

Statistical Independence of Column Contributions to Band Broadening

following paper [6] but nevertheless adopted an equation of the ntheor type as the definition of the plate number. A similar inconsistent approach is noted for Kaiser[-/l, who in one of his books (p. 12 in ref. [ A ) states that a formula of the discontinuous type is “besser”, but uses the ntheor definition in the rest of his books. There was obviously a strong opinion among leading chromatographers at that time in favor of the original ntheOr formula of Martin and Synge. The nomenclature committee of the first chromatography symposium in London in 1956 also recommended [8] the use of a formula equivalent to eq. (2) above. The clearest literature support found in favor of their decision is given in a paper by Glueckauf [9], who claimed that formulas of types (1) and (2) are consistent with the continuous nature of the chromatographic process, while formulas of types (3) and (4) are in error because they are obtained upon discontinuous treatments of the process.

The original discontinuous model of Martin and Synge [l] provides a satisfactory description of peakshape and peak width as a function of column length. It does not make any distinction between band broadenings caused by longitudinal and transversal diffusion, respectively, and hence cannot account for the well-known HETP minimum obtained at a certain mobile phase flow rate. As early as 1965, Giddings (p. 20 in ref. [lo]) regarded the plate model as obsolete for current use because of its “nearly total failure in describing the physical and molecular events occurring in chromatography”. It is the aim of the present author to show, firstly, that Glueckauf‘s grounds for a recommenda- tion of the ntheor formula were not actually valid and, secondly, that a slightly modified plate model is capable of making a distinction between the HETP contribution due to longitudinal diffusion and that due to non-equilibrium. In doing so we will re-examine the plate concept in a way that roughly follows the treatment of Glueckauf in outline, so that the limitations of the older theories can be recognized easily.

2 Modified Plate Model Consider a column of total length L composed of constant, length-independent fractions of mobile and stationary phase, F, and F,, respectively. Suppose that the movement of the mobile phase with the local linear velocity

Al At

u = -

causes a delay of the sample concentration in the stationary phase, C,, with respect to the equilibrium concentration in the mobile phase, Ceq = CJK. (For the sake of simplicity we shall drop the subscript in Cm used previously for the description of the sample concentration in the mobile phase.) The delay is equally well described with the

actual sample concentration in the mobile phase as C and the delayed equilibrium concentration in the stationary phase as C,, eq = KC, in accordance with the two equivalent definitions of the concentration-based equilibrium constant

Furthermore, let us assume that C, represents the mean composition of a given Al-segment and is delayed the local length increment, fAl, where the fraction f (-1h) depends

Figure 1

Relative f distance to the mean concentration of a plate at various local concentration cunratures.

on the local concentration profile [Figure 11. Putting the capacity ratio, E, equal to the mass ratio, q, we find in the conventional way, with total cross-sectional area equal to Ac, that [ 1 11

Here dsl-fAl,t is the elementary mass of the sample in the stationary phase of length dl and located around I-fAl at time t, while dml,t is the elementary sample mass in the mobile phase segment of the same length and located aroun,d I at the time t. The definition of a theoretical plate requires [l] that the mobile phase composition leaving at I, C,,, is in equilibrium with the mean stationary phase composition of the Al-segment, C,, located at I-fAl, so Al is equal to the local plate height.

31 2 VOL. 5, JUNE 1982 Journal of High Resolution Chromatography & Chromatography Communications


Supposing that the column can be divided into a row of successive plates of this varying local length Al, we get a similar equilibrium condition after a continuous displacement of the mobile phase with A l during At. It is implied that the change of the local plate height is small during such a small step, and the same ratio is expected for the new masses dSl-fAl,t+At and dml,t+At present at the old positions after the time interval At. (Cf. Part 2 of this series [l I]). This gives

i; = dsl-fAl,t+At dml,t+At

and

for the new sum of the equilibrated masses.

For a given mobile phase element the new equilibrium condition is reached immediately after the displacement of the element with the local plate length Al, i.e. after the time At = Alh. During this time the potential mass increase in the mobile phase at time t, dml-Al,t - dml,t, associated with a negative concentration profile in the mobile phase, gives rise to the mass increase in the mobile phase at I, dml,t+At - dml,t, together with a corresponding increase in the stationary phase at I-fAl, dsl-fAl,t+At - dsI-fAl,t, which latter increase is equal to iT(dml,t+At - dml,t) according to the equilibrium conditions (7) and (8). This gives the principal mass balance equation of the plate concept as

Rewriting this eq. (1 Oa) as

reveals that the difference between the potential mobile phase content at I-AI$ dml-Al,t, and the value actually obtained at I, dml,t+At, is equal to the mass increase in the stationary phase at the intermediate position I-fAl during the time At.

Another way of writing this relation is

which form is similar to that derived by Glueckauf [9] from the discontinuous plate model, although the interpretation is different. In our case the equilibrated masses have to be regarded as composed of the elementary mobile phase quantity dml-Al.t, moved a distance AI during the time At,

plus the old stationary phase content dsl-fAl,t located at about the middle of the region traversed by the elementary mobile phase segment. The classical discontinuous model on the other hand considers the masses of the whole AI- segments, the moving sample quantity ml-Al,t being added to the stationary quantity sl,t located at the end of the traversed Al-distance. Note, however, that in this work we do not really regard the process as discontinuous. The discontinuous character of the mass balance equations is rather due to the fact that the equilibrium conditions of a row of successive theoretical plates have to be repeated discontinuously in length and time. The mobile phase content at I-AI cannot influence the content at I until the time At = AI/u has elapsed.

3 Contributions Due to Longitudinal Diffusion

In this work we also add to the movement-induced increase (dml - Al,t-dml,J the increase of mass A(dml)diff, caused by the net longitudinal diffusion into the actual - mobile phase segment dl at I, plus the equilibrated quantity k A (dml)din obtained in the stationary phase at I-fAl due to this longitudinal diffusion process in the mobile phase. The corresponding mass change (dSl-fAl)diff, caused by longitudinal sample diffusion in the stationary phase at I - fAl, is negligible in GC but generally has to be added in LC. Its equilibrated contribution in the mobile phase is l / E times this contribution at I -fAl. Assuming that the changes during the step are independent of one another, we get the total mass balance equation

(I+@ (dmi,t+At - dmi,t) = (dmi-Ai,t - d q t ) + f (I+hA(dmi)diff + (1+s)A(dSI-fAddiff (1 1)

Most of these sample masses are referred to the same volume increment of the mobile phase, so we prefer to express the mass balance equation with the corresponding concentration contributions, C, Cdiff, and Cs,diff, according to

1

k

In order to diminish the discontinuous character of this equation we apply limited Taylordevelopments. This yields

forthe total concentration change in the mobile phase, and

Journal of High Resolution Chromatography & Chromatography Communications VOL. 5, JUNE 1982 31 3


This yields the total diffusion contribution of eq. (12) as for the concentration profile-dependent mobile phase contribution, associated with the mere movement of the phase.

The diffusion contributions are more difficult to develop. If we assume that the integrated concentration increase caused by longitudinal sample diffusion in the mobile phase A(Cl)diff, can be represented by the independent quantity (Cl,t+A,t - Cl,J)diff, where the latter term actually is zero, limited Taylor series expansion furnishes

when neglecting terms of higher degree.

Together with the relation u = AVAt these developments in eqs. (13), (14), and (20) change the total mass balance eq. (12) to

Here the first time differential can be developed according to the diffusion law [ 121

In order to estimate the second time differential in this eq. (21) we proceed as done previously [ 1 13 by approximating eq. (21) to where Dm/(l+E) is the effective diffusion coefficient under

the given local conditions (p. 230 in ref. [lo]). The longitudinal diffusion in the stationary phase at I-fAl causes a change of the sample concentration in that phase, A(Cs,l-fAl)diff = (CS,i-m,t+At - cs,I-fAl,t)diff, which can similarly be developed according to

fi (E) + u a t I

and

These equations do not contain the displacement A l and hence are approximately valid also in the equilibrium case. They simply describe the fact that the sample band is pro- pagating with the velocity u/( 1+T?). Differentiating these equations we get

In this case the first term of the Taylor expansion is related to the local diffusion coefficient of the sample in the stationary phase, D,, and the I-based local concentration curvatures, ( a2Cs,l-fAl/a12)t and (a2C/ a12)t, according to the diffusion law [ 121

and

Fs( 1 +T? )

where the last step follows from the equilibrium condition Thus, the second time differential in eq. (21) can be estimated as 1111



which simplifies the mass balance equation to

- 2D, uAl l+k

The pressure dependence of Dm and u are approximately the same, so the ratio Dm/u can be regarded as a constant represented either by the outlet quantity Dm,oluo at the pressure Po or by the quantity bm/G at the average pressure P. We prefer using the latter ratio, since U = L/ fm and b, = Dm, POIF can be estimated easily [13]. Assuming also constant u = U, Ds = 4, and AI = AL we obtain the more easily solvable differential equation

- 26 k - x [ S + k - + A L - - ] = O

U U l + k

4 Comparison with the Solution of Glueckauf

Apart from the diffusion term our eq. (26) is equivalent to eq. (10) in the work of Glueckauf [9]. The factor a/a obtained by Glueckauf in his discontinuous treatment can be shown to be equal to 1/( 1 +a from his own definitions. His time variable, the elution volume v, corresponds to FmAc u t in our case and his displacement A v is equal to our FmAc u At. After application of u = Al/At we have

and from eq. (7)

Now, Al in eq. (28), as well as every other quantity involved in eq. (7), is defined for the authentic, continuous chromatography case [ll]. Hence eq. (29) also has to be valid for that case. It is therefore impossible to make the description “continuous” by making Av infinitesimally small, as done by Glueckauf, without treating Al in the same way.

When AI is made small eq. (26) simplifies to

+k - )=O - 2Ds u

i.e. to the case that the band broadening is governed by longitudinal diffusion only. From the physical point of view this is possible only when the stationary phase fraction F, is zero and k = 0 according to eq. (29). If we want to main- tain has to approach infinity because of the physically necessary delay, Al > 0, when F, > 0. Consequently, we can expect that the results obtained by Glueckauffor Av 4 0 (without Al + 0) are valid for k = 0 and k + w only.

This does not prevent us from following Glueckauf in our solution of the general eq. (27). If we define the constant quantity

> 0 when Av - 0, then

we get an equation

which is of the same form as eq. (1 1) in ref. [9]. In agreement with Glueckauf we also define

which together with

Ahheor = ($ ) Mt (34)

transforms eq. (32) into



For the boundary conditions

implicitly assuming that the starting concentration Co at t = 0 is, like C, moving together with an equilibrated conter- part in the s phase, Glueckauf presents the solution as a difference between two normal curves of error according to

where by definition

x - 2 1 (38) 1 AE{x}= - erf {x} = __ e 1 dx

2 6 0

In the further development of this solution Glueckauf uses two contradictory approximations; firstly, that the number of theoretical plates No occupied by the injected sample is much smaller than [NI -Mt] and, secondly, that [NI -Mt] is small compared to Mt. The latter approximation is based on the fact that according to eq. (22) the band is propagat- ing with the velocity i / ( I + k ) , which means I/t = U/(l+E) in the vicinity of a peak, and hence NI - Mt according to eqs. (31) and (33). It holds better the narrower the band is relative to NI and Mt, i.e. the closerthe I/t ratio is to i/( l+E). Consequently, there is a risk that Glueckauf's final solution is restricted to long columns, as pointed out by Said [14].

The modified development in Appendix A avoids this diffi- culty and shows that for No 5 m 4 eq. (37) can be approximated within 1 percent to

For constant length quantitites, I = L+l0/2 and NI = NL+No/2, at the end of the column the time variable Mt is derived from eqs. (31) and (33) to be

since lo/L=No/NLand U/(l+E)=L/tms.Aftersome algebra the time-based C/Co function then becomes

This function is equivalent to that obtained by Glueckauf (eq. (24) in ref. [9]), despite the fact that we have not used his approximation I Nl-Mtl Q Mt - NI. His ax' corresponds to our net time, tms, since he measures the sample retention volume, V = ax', from the center of the injection at Noi2. Similarly, his N'/C corresponds to our NL/Fms, since he defines N' as equal to our net NL as well. Thus, his expression like our eq. (41) should describe the curvature profile within 1 percent for the main part of the band (at Ix I 5 2) even for short columns, provided No 5 -/4 according to (A6), i.e. when

Glueckauf states that the injection ratio has to be lo/L < l/=without specifying the maximum error then obtained.

5 Comparison with the Continuous Non- equilibrium Model

Defining the squared quantity of according to

t t ms NL

(43)

and the full injection time to as the time needed by the sample band for traversing the distance 10 according to

we can simplify the time-based funtion (41) to

(44)

(45)

This slightly skew function corresponds to the fully symmetrical

predicted from the differential equations of the continuous non-equilibrium model of Lapidus and Arnundson [I 51


Statistical independence of Column Contributions to Band Broadening

(474

and

when solved in the way described by van Deernter et al. [16] and Karnbara et al. [17,18]. Here amlFm denotes the rate constant controlling the equilibrium delay in the mobile phase of volume fraction Fm, when K is the equilibrium constant for partitioning of the sample between the stationary phase (of volume fraction FJ and the mobile phase. These authors [16,18] also assume that the mean second moment obtained as the value of of at the mean elution time, t = tms, i.e.

is a composite of the non-equilibrium contribution, 02(,), and the contribution from longitudinal diffusion, 02(~),

added according to the variance addition rule

for statistically independent processes. The predicted HETP expression [16,18]

is of the general van Deernter type [ 16,111.

This agreement between the elution functions of the two models becomes evident upon the following development of the Lapidus-Arnundson equations. By differentiating eq. (47a) with respect to time t an expression for (aCs/at)l is obtained, and the delay term can be eliminated to give the combined net equation

when neglecting the third derivative of the diffusion term. Applying the same type of estimations as in eqs. (23) and (29, valid for this eq. (51) as well, we get

In the preceding paper of this series i l l] , it is shown in a continuous way that

(53)

so this eq. (52) of the continuous non-equilibrium model is in fact equivalent to our main eq. (27) of the “discontinuous” plate model above.

At a given time t the second moment contribution of the column can be written

(54)

where Eltheor is the experimental theoretical plate number of the column defined by

(55)

analogous to formula (1) for the total experimental plate number of the equipment. Upon a comparison of eqs. (55) and (48) we hence see that this net number of the column, E’theor, is identical to NL, provided the experimental Fms and d values can be identified with the theoretical ones considered in functions (45) and (46). It means

which for I = L + Id2 and NI = NL + No/2 in eqn. (31) immediately gives

(57)

in agreement with the general van Deemter form of eq. (50). This expression (57) is also predicted by the relaxation-time and theoretical plate models [ I 11 assuming that the variance addition rule (49) is applicable.

Hence existing solutions to partial differential equations of the present type and the similar equations of Lapidus and Arnundson give results consistent with the assumption of

Journal of High Resolution Chromatography & Chromatography Communications VOL. 5. JUNE 1982 31 7


statistical independence of the various broadening processes, as expected from the independent derivations of the partial differential contributions added in the equations (cf. also eq. (59) below). Nevertheless, this common solution technique leading to the general elution functions (45) and (46) is subject to two types of restrictions which are of fundamental importance from other points of view.

6 Two Theoretically Important Restrictions of the Solutions

A first approximation involved is easily recognized upon a comparison of the predicted functions (45) and (46) with the two simultaneous sample distribution functions present in Figure 2. The relaxation-time and plate models

Sample mass (arbitrary units)

0.5 1 .o I'IL 1.5 Figure 2

Mass distributions in the mobile (m) and stationary (s) phases as a func- tionof relativelength(Y/L).Thelengthvariable, I' = 1-10/2, iscompared to ;he mean distance traveled by the band center in the stationary phase, I; = L. The delay is (AL/2)/L = 1/15 with E = 0.5 and NL = 22.5.

require [l 11 that the length-based sample concentration curve in the stationary phase is on average delayed by the distance AL12 with respect to the equally shaped curve in the mobile phase. Glueckauf's differential eq. (32) does not take account of this fact, nor do his boundary conditions (36). Consequently, the shape of the time-based sample concentration curve in the mobile phase can agree with function (45) only when the two sample bands are still pro- pagating on the column or AL is negligible at the moment of their combined elution. Function (45) obtained at a given length, I = L, simply represents the time-based version of the length-based, fully symmetrical [12], Gaussian distribution at a given time, t = tms,

present in the mobile phase of a column where I' = 1-10/2 can be sufficiently long compared to L. (See Appendix 6.)

However, in the case of a low number of AL plates on the column, this requirement of AL being small compared to the band width may not be fulfilled, which upon combined detection of both phases, associated with the common elution in the mobile phase only, for k # 1 leads to tailing peaks of different skews as evident from a summation of the two Gaussian distributions in Figure 2. The combined elution profile follows more closely [20] that based on the modified Bessel function of the more exact solution to the delay-dependent Lapidus-Arnundson equations, derived by Karnbara [17] for the case of no longitudinal diffusion and also predicted by the stochastic theory of Giddings and Eyring [21]. It is implied from this work that the discontinuous plate model is as good as the others and, in fact, the simple step and equilibrium procedure [l] gener- ates practically the same elution band shapes as those calculated by Denizot and Delaage [22] for the stochastic theory [19]. The 5onnections between the latter theory and the plate concept will be discussed further in another part of this series [20].

At very high ndiscvalueS normally used in practical chromatography this change of skewness is very small, however. The fact that the spreading developed in one of the phases is predicted to be strictly Gaussian is on the other hand of great importance for the theoretically correct application of the distillation plate concept to chromatography [20].

The second restriction of the solution technique mention- ed is due to the transformation of variables normally made. The common procedure [9,16,1T] is to treat the band broadening process in a coordinate system proceed- ing with the band center velocity U/( 1 +E), as meant e.g. by introducing the dimensionless, relative time variable Mt of eq. (33) above. It leads to a variation of C with relative time, (aC/aMd~, in eq. (35), which is formally independent of the mean front velocity U/( 1 +E) and hence depends on the front dispersion processes only. As before [l I] this dispersion-dependent variation is made up from contributions from both longitudinal diffusion and non-equilibrium, according to

but suffers in this form from afundamental restriction of its applicability. The variation of C with true time t in the original eq. (27) accounts for the dispersion due to longitudinal diffusion present even when U = 0 and there is no contribution due to non-equilibrium. The variation of C with relative time Mt in the transformed version (35), however, is not allowed to account for this physically possible case consistent with a statistical independence of the two



dispersion processes. The relative variables NI and Mt defining x = ( N I - M t ) / a according to eqs. (37) and (38) are allowed to vary independently upon solving eq. (35), but the variation of C with Mt obtained at constant NI is not the one associated with the absolute constancy of NI obtained upon an interruption of the movement of the mobile phase. Hence our solution of eq. (35) strictly accounts for a dispersion associated with the movement of the mobile phase, that is the contribution due to non-equilibrium only, as meant by Glueckauf.

The second-derivative term in eq. (33, (a2C/aNI2)Mt, is also assumed to be made up from two corresponding contributions, but the (aC/ aNI)Mt term is strictly coupled to the non-equilibrium contribution only. Note also that the last boundary condition in (36) above, C = 0 when NL = 0 and Mt > 0, forlxl< 2 requires that the mean band propagation velocity is more than twice the mean band broadening one, which may not be fulfilled in a practical case with low band velocity and considerable broadening contribution from longitudinal diffusion. The situation is similar upon the separate derivation of the Gaussian band profile associated with the band broadening due to longitudinal diffusion alone, as performed in the first part of this series [12]. In that case the ( aC/aNI)Mt term is missing in eq. (35) and the remaining terms are entirely coupled to the longitudinal diffusion process. The boundary conditions are also different, of course.

The solutions to the Lapidus-Amundson eq. (51) are impaired in a similar way. For example, Kambara and Ohzeki [18], like van Deemter et al. [16], deduce that the second moment to be used in the elution function (46) contains a contribution from longitudinal diffusion, a(&, added according to the variance addition rule (49). They make no transformation of variables, but use the approximation that forlxl< 2 the time variable t is equal to tms = = (l+K)tm, where Tm = L/G is a function of the column length L. Similarly they claim that the remaining composite variables can be treated as functions of the length variable I only, so that one can put X / a t = 0 in the solution of eq. (51). However, this elimination of the time dependence is strictly possible only for the non-equilibrium-coupled longitudinal transport contribution

ac

The independent contribution from longitudinal diffusion, (aC/at)diff, is left and equal to the corresponding Em( a2C/aI2)t term. Hence these contributions cancel in eqn. (51), they constitute a separate equation, and should therefore be excluded in the actual solution of eq. (51).The correctness of this statement is very clearly seen from the fact that, if the longitudinal diffusion term were the

dominating contribution in the Lapidus-Amundson eq. (51), a correct solution should predict a Gaussian distribution upon elution, while according to Kambara [17] the most exact solution to the non-equilibrium part is the modified Bessel function [20]

(61)

instead of (45).

Consequently the obtained total column contributions of eqs. (50) and (57) cannot be regarded as quite genuine results of the solution techniques generally applied to the mass balance equations. One may argue that this second restriction is of mathematical interest only, since the resulting second moments are in agreement with those expected [ 1 1,16,18] upon statistical independence. However, from a re-examination of the distillation plate theory it is found [20] that making such a distinction between the genuine solution of the non-equilibrium contribution and the non-genuine one of the diffusion part indeed provides a means for simple explanation of hitherto unexplained common chromatographic behavior.

Appendix A

Limited application of Taylor's theorem to eq. (37) yields

where x = (NI - Mt ) /KS ta r t i ng from eq. (38) the deriva- tives are

and

which leads to



For the restrictions

No 5 J6Mt/100- = 0.245 - 4 1 4 (A6)

this eq. (A5) can be approximated to

with an error less than 1 percent in the interesting range I x I = I (NI -Mt) I I q 6 2. Rewriting the exponential function as

and recalling x = (NI - M t ) / K we get eq. (39)

since the factor (1 + Ng18M3 varies only slightly, between 1 and 1.0075 for the restriction given in (A6).

Appendix B

A length-based version of ClCo distribution (39) is obtained by putting t = Fms in (33), which together with (31) and the relations

NI - No12 Nl-No12 N; - NL I 1012 r-10/2 I‘ L (B1) _ - - -

yields the constant relative time value

analogous to (40). Eqn. (39) then gives directly

and, after ?pplication of appropriate parts of (Bl),

From eqs. (44) and (48) we have lo/L = to/Fms and NL = = T ~ $ ( U ‘ ) ~ , respectively, which for I’ = I - 1012 leads to e.g. version (58)

(85)

By analogy with eqs. (43) and (46) we can also use the length-based second moment

ct _ to - _ _ _ _ _

corresponding to the more usual version

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[2]

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[lo] J. C. Giddings, “DynarnicsofChromatography”, 1. “Principles and Theory”, M. Dekker, N. Y. 1965.

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[16] J. J. van Deemter, F. J. Zuidenweg and A. Klinkenberg, Chem. Eng. Sci. 5 (1956) 271.

[17] T. Kambara, J. Chromatog. 19 (1965) 478.

[18] T. Kambara and K. Ohzeki, J. Chromatog. 21 (1966) 383.

[19] 0. Nilsson, unpublished results.

[20] 0. Nilsson, to be published.

[21] J. C. Giddings and H. Eyring, J. Phys. Chem. 59 (1 955) 41 6.

[22] F. C. Denizot and M. A. Delaage, Proc. Nat. Acad. Sci. USA 72

MS received: January 21,1981 Accepted by REK

A. T. James and A. J. P. Martin, Biochem. J. 50 (1952) 679.

(1 975) 4840.

320 VOL. 5, JUNE 1982 Journal of High Resolution Chromatography & Chromatography Communications

Documents

On the statistical independence of various column contributions to band broadening. Part 3: Independent treatment of longitudinal diffusion in the plate model