Copyright Joe Preston Foley. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.

EQUATIONS FOR THE CALCULATION OF
CHROMATOGRAPHIC FIGURES OF MERIT

By

JOE PRESTON FOLEY

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

1983

ACKNOWLEDGMENTS

First and foremost, I would like to thank my research director,

John G. Dorsey, not only for helping me with various research projects,

but for being a special friend--for always finding the time to listen

and for always doing the "extra things."

Second, I want to express my gratitude to Thomas J. Buckley and

Sharon G. Lias of the National Bureau of Standards for their help and

the use of their facilities in preparing this document.

Finally, I want to thank all the friends I made in Gainesville for

making my graduate education at the University of Florida the happiest

and most satisfying time of my life.

iii

To Mom and Dad, for their love, support, and encouragement;

4 CLARIFICATION OF THE LIMIT OF DETECTION IN CHROMATOGRAPHY.... 56

Introduction.............................. ................ 56
Identifying Current Problems.............................. 56
Literature Survey Results............................... 56
Numerical Example..................................... 58
Solving the Problems..................................... 60
Eliminating Mistaken Identities........................ 60
Choosing a Model......................................... 63
Using the Correct Units.................................. 66
Converting to Chromatographic Reference Conditions....... 70
The Numerical Example Revisited........................ 79
Conclusion................................................ 81

5 SUGGESTIONS FOR FUTURE WORK................................. 82

APPENDICES

A DISCUSSION AND LISTING OF THE BASIC PROGRAM, EMG-U........... 84

B UNIVERSAL EMG DATA AT a = 0.05, 0.10, 0.30, and 0.50......... 87

C DERIVATION OF V 95
C DERIVATION OF Vinj,max....................................... 95

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

EQUATIONS FOR THE CALCULATION OF
CHROMATOGRAPHIC FIGURES OF MERIT

By

JOE PRESTON FOLEY

December 1983

Chairman: Dr. John G. Dorsey
Major Department: Chemistry

The measurement and interpretation of several chromatographic

concepts and parameters, hereafter referred to as chromatographic

figures of merit (CFOMs), are improved via equations and concepts

developed in this work.

The previous uses of the exponentially modified Gaussian (EMG)

model in chromatography are briefly reviewed. A method for evaluating

the EMG function and a set of algorithms for obtaining universal data

are presented and are shown to be simpler and easier to use than those

previously reported. The corresponding BASIC program, EMG-U, is also

briefly discussed.

By use of the exponentially modified Gaussian (EMG) as the skewed

peak model, empirical equations based solely on the graphically

measurable retention time, tR, peak width at 10% peak height, W0.1, and

vi

the empirical asymmetry factor, B/A, have been developed for the

accurate and precise calculation of CFOMs characterizing both ideal

(Gaussian) and skewed peaks. These CFOMs include the observed

efficiency (number of theoretical plates), Nsys; the maximum efficiency
sys
attainable if all asymmetry is eliminated, Nmax; the EMG peak

parameters, tG, oG, and T; the first through fourth statistical moments;

the peak skew and peak excess, YS and YE; and two new CFOMs--the

relative system efficiency, RSE, and the relative plate loss, RPL.

Equations for the number of theoretical plates and the variance (second

central moment) are accurate to within + 1.5% for 1.00 < B/A < 2.76.

Width and B/A at 10% peak height are recommended.

The current problems with the LOD concept in chromatography are

reviewed. They include confusing the LOD with other concepts in trace

analysis; the use of arbitrary, unjustified models; the use of

concentration units instead of units of amount; and the failure to

account for differences in chromatographic conditions (bandwidths) when

comparing LODs.

Two models are proposed for calculating the chromatographic LOD. A

new concept, the standardized chromatographic LOD, is introduced to

account for differences in chromatographic bandwidths of experimentally

measured LODs. The standardized chromatographic LOD is shown to be a

more reliable CFOM than the conventional (non-standardized)

chromatographic LOD.

vii

CHAPTER 1
INTRODUCTION

Overview

Chromatography is a well-known method for the separation and

quantitation of chemical moieties from a (sample) mixture. Over the

years several concepts and parameters, hereafter referred to as

chromatographic figures of merit (CFOMs), have been introduced to

characterize the separation and quantitation. Unfortunately, some of

the CFOMs are often difficult to estimate [those which characterize

chromatographic peaks]; others are ambiguous [e.g., the limit of

detection (LOD)]. The goal of the present work, which is introduced in

more detail in the following two sections, is the improvement of the

measurement and interpretation of these chromatographic figures of

merit.

It is beyond the scope of this work to introduce or review the

development of these CFOMs from either a historical or theoretical point

of view. Such discussions and references to additional discussions may

be found elsewhere (1-10).

Chromatographic Peak Characterization

In recent years there has been considerable interest in the

characterization of experimental chromatographic peaks. Presented in

Table 1.1 are the names, symbols, and general expressions that have

evolved for the parameters used in chromatographic peak

1

Table 1.1. Common Chromatographic Figures of Merit

parameter symbol general expression

theoretical plates N (retention time) /variance

retention time tR peak maximum
M1 peak centroid
tm peak median

empirical asymmetry factor B/A see Figure 1.1

peak height hp see Figure 1.1

peak width at specified height Wa a denotes peak height fraction

peak width at base Wb tangents drawn from inflection points

peak shape function h(t)

statistical moments and related quantities

zeroth (peak area) MO h(t) dt

first (peak centroid) M1 t h(t) dt/MO

2nd central moment (variance) M2 (t M1)2 h(t) dt/MO

nth central moment Mn (t M1)n h(t) dt/MO

peak skew YE M /M23/2

peak excess YS Mn/M22 3

3

characterization. The graphical chromatographic parameters are

illustrated in Figure 1.1.

These CFOMs have been estimated either manually using graphical

measurements made directly from the chromatogram or by a computer

following data acquisition. Both methods have advantages and

disadvantages.

Manual methods were used exclusively at first and are employed

quite extensively today. For arbitrary peak shapes, they are accurate

for only five CFOMs: tR, B/A, hp, Wb, and Wa. If a Gaussian peak shape

is assumed, however, then M1 = tR, and M2 is only a function of Wb, Wa,

or MO and N may subsequently be calculated. Except for higher

even central moments, the remaining CFOMs are zero for Gaussian peaks.

For real chromatographic peaks, it is almost always a mistake to

assume a Gaussian peak shape. Experimentally these ideal, symmetric

peaks are rarely, if ever, observed due to various intracolumn and

extracolumn sources of asymmetry (5,11-23). Kirkland et al. have shown

that the plate count can be overestimated by as much or more than 100%

if any of the three most common Gaussian-based equations are

employed (23).

Computer estimation methods are more accurate than common manual

methods for a given CFOM but are not available to every chromatographer.

The general approach taken has been one of peak statistical moment

analysis (6,11,22-24). Via relatively simple algorithms all the CFOMs

may be determined quite accurately, though the precision of the second

and higher central moments is seriously affected by baseline noise (25).

The failure of the Gaussian function as a peak shape model for real

chromatographic peaks led to the search for a more accurate model and

1.0

F--

0U

-t--
N

SW",=A+B

0.0
O .<

tA tR tB
TIME

Figure 1.1. Graphical chromatographic parameters shown at peak height
fraction a = 0.10. Except for A, B, and B/A at a = 0.10,
all width related measurements are subscripted with the
value of a to prevent ambiguity.

5

the eventual acceptance of the exponentially modified Gaussian (EMG), a

function obtained via the convolution of a Gaussian function and an

exponential decay function which provides an asymmetric peak profile.

The development, characterization, and theoretical and experimental

justification of this model have been thoroughly reviewed (21,22,26,27).

Previous chromatographic studies (11,12,14,15,17-23,25-31) involving the

EMG function, summarized in Table 1.2, demonstrate the utility of this

skewed peak model.

Adoption of the EMG peak-shape model has improved the estimation of

the CFOMs. A new algorithm for the computer-based peak moment analysis

has been derived (25) and tested (22) which is less sensitive to

baseline noise and the uncertainty of peak start/stop assignments. More

recently, Barber and Carr described a manual method for CFOM

quantitation which requires the graphically measurable retention time

tR, peak width Wa, empirical asymmetry factor B/A, and successive

interpolations from three large-scale universal calibration curves

(31,32).

The primary objective of this part of the present study is the

development, using the EMG model, of accurate equations for CFOM

calculation dependent solely on tR, W0.1, and B/A. The need for

computerized data acquisition is thus circumvented, and, in addition,

CFOM calculation via these equations is expected to be faster and more

precise than the other accurate manual method since no graphical

interpolation is required.

The previously reported methods for evaluating the EMG function

(14,18,19,27) and obtaining chromatographic peak data (19,26,31) were

too inaccurate or too unwieldy to use in the present study, which

Table 1.2. Chromatographic Studies Using the Exponentially Modified Gaussian (EMG) Model

Year Subject in Which EMG Model Employed Reference

1959 effects of detector response time and flow sensitivity in GC 14

1959 determination of column efficiency in GLC 15

1966 extracolumn contributions to band broadening 11

1968 effect of detector volume on chromatographic peak shape 17

1969 resolution of overlapped GC peaks via least squares curve fitting 18

1969 effect of relaxation times on chromatographic peak shape 19

1969 effect of recorder response time on chromatographic peak shape 20

1970 least squares approximation of chromatograms 28

1970 resolution of unresolved GC peaks via Newton-Raphson curve fitting 29

1971 analysis of instrumentally distorted, overlapping curves 30

1972 dead volume effects on efficiency in GC 21

1972 theoretical investigation of strongly overlapped chromatographic peaks 27

1977 improved characterization of chromatographic band broadening 25

1977 effects of dead volume and flow rate on peak shape 22

1977 comparison of retention, separation measurement methods 26

1977 extracolumn effects in LC; effect of peak skew on plate count calcs 23

Table 1,2--extended.

1978 instrumental peak distortion (review) 12

1981 graphical deconvolution of EMG chromatographic peaks 31

1983 equations for chromatographic figures of merit Chapter 3, this work

8

employed an Apple II Plus microcomputer. This necessitated, therefore,

the development of a simpler method for evaluating the EMG function and

the incorporation of a simpler, more accurate, and more general set of

algorithms for obtaining the EMG data of interest.

Limit of Detection

The limit of detection (LOD) is generally defined as the smallest

concentration or amount of analyte that can be detected with reasonable

certainty for a given analytical procedure. Though arguably the most

important figure of merit in trace analysis, the LOD remains an

ambiguous quantity in the field of chromatography. Detection limits

differing by orders of magnitude are frequently reported for very

similar (sometimes identical!) chromatographic systems. Such huge

discrepancies raise serious questions about the validity of the LOD

concept in chromatography.

The primary objective of this part of the present work is to

restore the integrity of the LOD concept, to make the chromatographic

LOD a reliable, meaningful figure of merit. This will be accomplished

in two steps: First, the major sources of the discrepancies in

chromatographic detection limits, i.e., the current problems with the

LOD concept, will be identified. Second, each problem will be addressed

and eliminated (or circumvented).

CHAPTER 2
GENERATION OF THE EXPONENTIALLY MODIFIED GAUSSIAN (EMG) FUNCTION
AND RELATED DATA

Introduction

This chapter describes the improvements achieved by this study in

evaluating the EMG function and in obtaining the EMG data of interest.

The results of the present work are compared with those previously

obtained. In addition, the corresponding BASIC computer program, EMG-U,

is listed and discussed briefly in Appendix A. Universal EMG data are

tabulated in Appendix B. It is hoped that this will facilitate the use

of the EMG function, whenever applicable, in modeling studies in

chromatography or any other area.

EMG Evaluation

Background

Description of the EMG function. It is beyond our scope to derive

the EMG function from first principles. Those who so desire should see

the treatments given by Sternberg (11, pp. 250-253) or Kissinger et al.

(12, pp. 159-162). Their results are shown below in eq 2.1.

q = (1 + pz)-1 and p,bl,...,b5 are constants given in Table 2.2 (33).

Comparison. The values obtained for 14 are compared to the true

values (34,35) in Table 2.3 from z = -10 to z > 3.9. In addition, since

14 = I1/(7)1/2 (2.5)

they can be compared to values obtained for I1 via an asymptotic

14

Table 2.2. Constants in the Polynomial Approximation
for 14 in eq 2.4a

p = 0.2316419 b3 = 1.781477937

b, = 0.319381530 b4 = -1.821255978

b2 = -0.356563782 b5 = 1.330274429

aSee text immediately after eq 2.4 or reference 33.

Table 2.3. Accuracy of the Polynomial Approximation for 14 in Equation 2.4a

z 14, true (34) 14, this work 14, previous workb % REc

-12 d 1.83 x 10-33 1.78 x 10-33 + 3.1

-10 7.6 x 10-24 7.77 x 10-24 7.62 x 10-24 + 2.0

- 6.2 x 10-16 6.28 x 10-16 6.22 x 10-16 + 1.0

- 6 9.9 x 10-10 9.90 x 10-10 9.87 x 10-10 + 0.4

- 4 3.2 x 10-5 3.169 x 10-5 3.166 x 10-5 + 0.09

- 2 0.0227 0.0227 --- < +0.01

0 0.5000 0.5000 --- < +0.01

2 0.9772 (35) 0.9772 --- < +0.01

>3.9 1.0000 (35) 1.0000 --- < +0.01

aSee eqs 2.2, 2.4, and 2.1 for expressions for z, I4, and II, respectively.

bSee reference 27 and note that 14 = I/()1/2.

CPercentage error in 14, this work compared to 14, true for z > -2, and

compared to 1/(r) /2, previous work for z < -4.

Not reported for z < -10.

16

series (27), the most accurate method reported for z < -3. (Comparison

with values for 11 obtained via error function techniques when z > -3

(I4 > 10-3) is not illustrative since the maximum absolute error in the

polynomial approximation I4(z) is estimated to be 7.5 x 10-8.)

Table 2.3 shows that our method for evaluating 14 is exceptionally

accurate for moderate values of z. Moreover, it compares favorably with

the asymptotic series method for evaluating Ii, except for z < -8 where

the latter method is somewhat better. Re-examination of Table 2.1

shows, however, that for all practical purposes z > -8 whenever T/OG >

0.2. Thus our method for evaluation of 14 can be used to evaluate the

EMG function to within 1% or less for T/OG 2 0.2.

While being slightly less accurate than the most accurate previous

method, the new technique for evaluating the EMG function is much more

convenient than any of the previous methods. Only one simple subroutine

requiring just a few programming lines (see lines 2990-3160 in Appendix

A) is needed, whereas the other methods require at least two

subroutines, if implemented on any computer without a built-in error

function routine (i.e., nearly all microcomputers and many

minicomputers).

Obtaining Universal EMG Data

Background

Using the polynomial approximation for the integral in eq 2.4, the

EMG function can be evaluated over the entire practical time range.

Depending on the data required for the modeling process of interest, the

EMG peaks could be generated "on the fly" as needed. Alternatively, the

necessary EMG data could be generated (and stored) in advance and

accessed when needed. Though the storage requirements may seem

17

prohibitive for the latter, given a T/OG value and a peak height

fraction, a, three quantities completely specify an EMG peak (31).

Figure 1.1 shows an EMG peak with its pertinent graphical parameters.

Regardless of the retention time, tG, and standard deviation, oG of the

(unconvoluted) parent Gaussian peak (not shown), (B/A)a, Wa/aG, and (tR-

tG)/aG are universal constants so long as T/OG and a remain fixed.

Recent work has utilized these universal data sets almost exclusively

(31).

Experimentally, three parameters must be determined in order to

calculate the universal data: tR, tA, and tB (see Figure 1.1). Note

that t. must be obtained before tA or tB because hp = hEMG(tR) is needed

for the latter.

Measurement of the Pertinent Peak Parameters

Previous methods. In the past, tR has been determined by one or

more of the following methods: a) peak displacement data and knowledge

of tG (19); b) differentiation of hEMG(t) and solving for roots (26);

and c) least squares fitting of the top of the peak with a quadratic

gram polynomial (26,31).

Once tR has been found, tA and tB can be located. In the only

method reported previously, two points [tl, hEMG(t), [t2, hEMG(t2)

are found so that ti < tA < t2 (or t < tB < t2). Linear interpolation

yields the approximation for tA (or tB).

This work. The approach for finding these quantities is based on

iterative search mechanisms, as the flowchart in Figure 2.1 shows. In

the case of tR, initial time limits are easily found using the fact that

tR is always greater than tG. The EMG function is then evaluated from

18

Find initial time limits

hE(t)
hEJMt)

Y

Calculate new time limits

dt < d Y M Calculate value of t (and hp),
<4 tA or t .

N

Decrease value of dt

Figure 2.1. Simplified flowchart for locating tR (and hp), tA, or tB
The EMG peak parameters must be input before beginning tRis
search.

19

the lower time limit to the upper time limit in increments of dt. When

the maximum is found, new lower and upper time limits [given by

t(current) 2dt and t(current), respectively] closer to the peak

maximum are set, the time increment is decreased, and the search is

begun again. The retention time, tR, is approximated by t(last) -

dtmin*

The algorithm for estimating tA (or tB) is similar to the tA (or

tB) search algorithm previously discussed in that the time limits are

analogous. Since tA < tR < tB, initial values for tI and t2 are easily

determined. To locate tA (or tB), the EMG function is then evaluated

from t = t2 to tI in decrements of dt (or from ti to t2 in increments of

dt) until hEMG(t) < ahp. New values for tI and t2 [given by t(current)

and t(current) + dt, respectively] closer to tA (or tB) are then set,

the time decrement (increment) is decreased, and the search is repeated.

This is continued until tI and t2 are known to the desired precision; tA

(or tg) is then given by (tI + t2)/2.

The maximum error in the values of tR, tA, tB and related universal

EMG quantities obtained via these search algorithms is presented in

Table 2.4. In all cases the error is dependent on the smallest (most

precise) time increment (or decrement), dtmin, used in the last

iteration of each search. In theory, dtmin could be as small as

desired. Due to the finite precision of computers, however, the time

increment, dt, would ultimately be reduced to such a low value that

hEMG(t) = hEMG(t+dt). (2.6)

Henceforth the algorithms would cease to function accurately, if at all.

20

Table 2.4. Maximum Errors in the Universal
EMG Data and Selected Component Parameters

Parametera Maximum Error

R dtmin

tA tB + 1/2 dtmin

A = tR-tA B = tB-tR 3/2 dtmin

Wa/oG = (tB-tA)/CG dtmin/oG

(tRtG)/aG dtmin/G

aA, B, tR, tA, and tB defined in Figure 1.1;

oG' tG, and dtmin described in text.

21

As dt is decreased, the tR search algorithm will fail first, since

the slope of the EMG function is smallest in the region of tR. Somewhat

smaller (more precise) dtmin's could be employed in the searches for tA

and tB before the algorithm breakdown described by eq 2.6 would occur.

The increase in precision of tA and tB is probably not worth the effort,

however, since two out of the three universal EMG data expressions are

dependent on the least precise quantity, tR.

The minimum usable value of dt depends on the precision of the

computer employed and on the value of GG chosen. In this study, the

experimentally measured minimum ratio of dt/aG was 0.0002 for 0.1 < T/GG

< 3 using single precision arithmetic. Multiple precision capabilities

would allow a still lower dt/aG ratio to be used.

Comparison. The algorithms for tR, tA, and tB may be compared as

follows:

1. With the exception of the quadratic least squares method for finding
t,, all of the methods for obtaining tR, tA, and tB are designed for
simulated data (essentially no noise).

2. The two algorithms for calculating tA (or tB) are quite similar.
Both require two points which closely bracket the desired peak height
fraction, a, and both are relatively unbiased. The subsequent
interpolation performed in the previously described algorithm is
potentially more precise than the averaging of the final time limits in
the proposed search algorithm. If tA and tB are already known as
precisely as or more precisely than tR, however, additional improvements
in their precision, even if realized, will not yield significant
increases in the precision of B/A and (tR-tG)/cG.

3. Our approach for determining tR, though crude, is superior to the
other three methods previously discussed for the following reasons:
i) It is a general algorithm. Whereas methods a and b are specific to
the EMG model, our search mechanisms will work for that and other peak
models as well.
ii) It is accurate and unbiased. In contrast, the quadratic least
squares fitting method, though general, suffers from a small, but
nevertheless observable bias (due to a determinate error) which
increases with increasing peak asymmetry (26).
iii) It is easy to understand and implement. The other methods are
unnecessarily complex, though they admittedly have the potential
for greater precision.

22

4. The proposed search algorithms for determining tR, tA, and tB are
superior to the previous methods because they can be de-bugged more
easily. By having the time and the value of the EMG function printed
every time the EMG function is evaluated, the programmer can literally
watch the computer perform the search. Since the search logic is so
simple, programming errors are easily detected. Upon elimination of the
errors, the print statement may be removed.

Comparison of Universal EMG Data

Table 2.5 shows representative sets of universal EMG data obtained

from this study and from a previous work (32) which utilized the

quadratic least squares method and the interpolation method for the

location of tR and tA (or tB), respectively. The precision is reported

for our data (in terms of maximum errors) and is assumed to be no worse

than + 1 in the least significant digit of the previously reported data.

Several points should be noted:

1. Although this difference is slight or non-existent at high
asymmetries, the previously obtained universal data are somewhat more
precise. This is expected since the algorithms used in locating tR, tA,
and tB are potentially more precise than those developed here.

2. The data sets are in excellent agreement for all three universal EMG
quantities at low asymmetries (T/oG I 0.5). This agreement is
especially significant at T/OG = 0.1, because it shows that the moderate
errors introduced by the polynomial approximation for 14 in eq 2.4 when
-12 < z < -8 (see Table 2.3) are not transmitted to the universal EMG
data.

3. Whereas the Wa/aG data reported previously are consistent with the
corresponding data of this study over the entire range of T/aG, the
remaining data sets are discordant for T/aG 2 1.0. Relative to the
current data sets, the previous ones for (tR-tG)/aG and (B/A)a appear to
be slightly overestimated and underestimated, respectively. This
discrepancy is due to the use of a least squares fitting method in the
previous measurement of tR which overestimates this quantity for EMG
peaks (26) and other types of skewed peaks. This bias increases from an
insignificant value at low T/aG to an observable one at T/OG 2 1.0.

4. Despite the differences noted above, the general interlaboratory
agreement is quite good. Though the current data are more accurate,
either set of EMG data can be used with confidence for modeling studies.

Table 2.5. Comparison of Universal Exponentially Modified Gaussian (EMG) Data
Obtained from this Study and a Previous Study (32) at a = 0.10

T/oG B/A W/OG (tR-tG)/G
-----------------------------------------------------------------------------------------
This Previous This Previous This Previous
Studya Work Difference Studyb Work Difference Studyb Work Difference

14a YE = M4(eq 12a)/[M2(eq 2a)]2 3 1.09 2.76 -1.5, +14.0c f

14b YE = M4(eq 12b)/[M2(eq 2b)]2 3 1.09 2.76 -0.5, +19.5e f

aBased on tR, W0.1, and B/A measurements depicted in Figure 1.1. B/A measurements should be made to two
decimal places. Numerical coefficients should not be rounded further.
bError limits are -4.0%, -1.0% for 1.00 < B/A < 1.09.
CError limits are + 2.0% or less for 1.19 < B/A < 2.76.
dError limits reported for tG/oG = 20. For tG/oG > 20, the error limits will be smaller than those reported.
eError limits are between + 2.0% and + 5.0% for 1.19 < B/A < 2.76.
fPrecision of this CFOM is a function of peak shape. See plot of % RSD versus B/A in Figure 3.1.

33

Except for eqs 11b, 13b, and 14b which had biases of +0.976%,

+0.820%, and +1.111%, respectively, the bias for every equation given in

Tables 3.2, 3.3, and 3.4 was less than 0.6%.

Precision Summary. For the equations in Table 3.2, the estimated

relative standard (RSD) limits obtained via propagation of error theory

for Nsys, M2, aG, Nmax, and RSE were all less than or equal to 4.5%.

RSD limits for tG and M1 were + 0.2%.

The precision of the EMG equations in Table 3.2, the width-based

Gaussian equations, and the calibration curve method of Barber and Carr

(31) is compared for Nsys, M2, and oG in Table 3.5. The results shown

for the Gaussian equations are valid for 50%, 30%, and 10% width

measurements because RSD(W0.5) = RSD(W0.3) = RSD(W.1). The slightly

greater imprecision observed for the EMG equations is due to uncertainty

in the B/A measurement not required for the Gaussian equations. The

somewhat larger %RSDs for the method of Barber and Carr are probably due

to interpolation uncertainties (from the calibration curves) unique to

this method.

The precision of the remaining CFOMs in Table 3.2 was found to be

highly dependent on the peak shape. Rather than reporting RSD limits,

the RSDs for several CFOMs or groups of CFOMs have been plotted vs B/A

in Figure 3.1.

Other CFOM Equations

Listed in Tables 3.3 and 3.4 are smaller sets of CFOM equations

developed only for use in determining if a real chromatographic peak is

well-modeled by an EMG peak and should not be used to routinely

calculate any CFOM (including Nsys) since they are usually less

accurate, less precise, and more complex than the analogous equations in

Table 3.3. Equations for Use Only in Peak Modelinga
% rel %RSD
equation error limitsb limitsb

aBased on tR, W03, and (B/A)0.3 measurements. (B/A)o.3 measurements should be made to two
decimal places. Numerical coefficients should not be rounded further.
bValid for 1.06 < (B/A)0.3 < 2.08, which corresponds to 1.09 < (B/A) .I 2.76.

CError limits are -0.5%, +2.0% for 1.19 < (B/A)0.1 < 2.76.

dprecision ranges from + 13.9% at (B/A)0.1 = 1.09 to + 2.0% at (B/A)o.I = 2.76 in a manner similar
to the T, T/OG curve shown in Figure 3.1.

eError limits are -1.5%, +4.0% for 1.19 < (B/A)0. <2.76.

Error limits reported for tG/oG = 20. For tG/OG > 20, the error limits will be smaller than those
reported.

Table 3.4. Equations for Use Only in Peak Modelinga
% rel %RSD
equation error limits limits

aBased on tR, W0.5, and (B/A)0.5 measurements. (B/A)0.5 measurements should be made to two
decimal places. Numerical coefficients should not be rounded further.

bValid for 1.04 < (B/A)0.5 < 1.70, which corresponds to 1.09 < (B/A)0. 2.76.

CError limits are -1.0$, +2.0% for 1.19 (B/A)0.1 < 2.76.

dprecision ranges from + 24.5% at (B/A)0.1 = 1.09 to 3.2% at (B/A)0.1 = 2.76 in a manner similar
to the T, T/OG curve shown in Figure 3.1.

eError limits reported for tG/OG = 20. For tG/OG > 20, the error limits will be smaller than those
reported.

36

Table 3.5. Precisiona Comparison of Three Graphical Methods for
Estimating Nsys, M2, and "G

Nsys M2 aG
method

Gaussian eqs + 2.0 + 2.0 + 1.0

empirical eqs,
Table III, + 2.5 + 2.4 + 2.0
this work

aReported as percent relative standard deviation (%RSD).
Precision of equations estimated via error propagation, using
data of Table 3.6.

Figure 3.1. Dependence of precision on peak asymmetry for several
chromatographic figures of merit (CFOMs). The precision of all
other CFOMs (Ns s' M2, etc.) is essentially independent of peak
shape and is included in Tables 3.2-3.7.

v/8

8Q" g'9Z 3' 6'1 9'1 '1 I

7Idl

SX 'L2

-01
coo

-0
Cl)

gIZ

39

Table 3.2. However, the accuracy and precision for M2, G', tG and T at

a = 0.3 and 0.5 are still sufficient to permit peak modeling decisions

to be made.

Discussion

Detailed Discussion of Precision

Shown in Table 3.6 are the precision data for tR, Wa, and (B/A)a

used in this study. The RSD results were obtained by converting

previously reported raw precision data (32) to the form appropriate for

error propagation analysis for the conditions specified in Table 3.6.

Data excluded in the previous study for (B/A 1)a were also excluded in

this analysis. The RSDs of tR, Wa, and (B/A)a for individual peak

shapes (at a given peak height fraction) were averaged as done

previously, thereby implicitly assuming the independence of the RSDs on

peak shape.

The RSDs for tR, Wa, and (B/A 1)a for individual peak shapes were

originally reported relative to cG', G, and (B/A 1)a, respectively.

Multiplication by aG/tR, OG/Wa, and (B/A 1)a/(B/A)a converted them to

the appropriate form.

Since the RSDs of tR, Wa, and (B/A)a were assumed to be independent

of peak shape, intuitively it might seem that this should also be true

for the RSD of any calculated CFOM. This is not the case, however. For

one group of CFOMs (Nsys, M2, aG, tG, Nmax, RSE, and M)), a slight to

moderate variation in their RSDs with (B/A)a was observed. This can be

explained by examining the random error propagation in the general

empirical Nsys equation,

Nsys = Cl(tR/Wa)2/[(B/A) + C2] (3.9)

40

Table 3.6. Suggested Chromatographic Measurement Conditions and
the Resulting Precision (%RSD) Achieved for tR, Wa, and (B/A)a

Conditions

1. Chromatogram recording rate: 1 cm/G (W0.1 Z 4.3 cm)

2. Ruler resolution: + 0.2 mm

3. Minimum retention distance, (tR)min: 10 cm

4. Minimum peak height, (hp)min: 10 cm

Resultsa,b

CFOM (a = 0.1) (a = 0.3) (a = 0.5)

tR + 0.2 identical identical

Wa + 1.0 + 1.0 1.0

(B/A)a + 2.0 + 2.5 3.0

aData obtained from reference 32 and subsequently converted (see
Detailed Discussion of Precision) for a = 0.1, 0.5--results
interpolated for a = 0.3.

bRSD(tR) rounded to nearest 0.1%; %RSDs for Wa, (B/A)a rounded
to nearest 0.25%.

41

Assuming negligible covariances, RSD(Nsys) is given by

Even when terms 1-3 in eq 3.10 are constant, RSD(Nsys) will vary

somewhat with (B/A)a because of term 4. Clearly this variation will be

greatest for -(B/A)a < C2 < 0. Additionally, as C2 -(B/A)a,

RSD(Nsys) + -. For C2 = 0, RSD(Nsys) is essentially independent of

(B/A)a. Finally, for C2 > 0, a negligible to slight variation of

RSD(Nsys) with (B/A)a may be observed, depending on the magnitude of

terms 1 and 2 relative to the product of terms 3 and 4. Except for eqs

1 and 2(a,b) in Table 3.4, the RSD limits for Nsys, M2, OG' tG, Nmax'

RSE, and M1 calculated via equations in Tables 3.2, 3.3, and 3.4 varied

by less than 0.5% for 1.00 < B/A ( 2.76.

The remaining CFOMs (T, T/GG, RPL, M3, M4,YSYE) comprise a second

group whose RSDs are moderate to strong functions of peak shape as

Figure 3.1 shows. In every instance the imprecision is largest for the

least asymmetric peaks and smallest for the most highly skewed peaks.

Analysis of the error propagation equations show that one or more terms

within the equations get very large as the peak shapes become symmetric.

Why Measure at 10% Peak Height?

For reasons listed below, the recommended CFOM equations in

Table 3.2 are based on (in addition to tR) the measurement of W and B/A

at 10% peak height rather than at other peak height fractions such as

50%, 30%, or 5%:

42

1. Examination of Tables 3.2, 3.3, and 3.4 shows that many CFOM
equations at 10% are clearly superior to the corresponding ones at 30%
and 50% in terms of
a) precision (lowest RSD limits)
b) widest working range for equivalent accuracy (e.g., Nss)
c) simplicity for M2 (path a)

2. The Nsys equation at 10% peak height is more accurate for Gaussian
and near-Gaussian peaks than other Nsys equations developed at 50%, 30%,
or 5%. For example, at B/A = 1 the relative error was +0.6%, +10.0%,
+2.0%, and +2.5%, respectively.

3. In a previously reported graphical measurement study (31,32),
statistically significant positive and negative biases were detected in
the measurement of AO 5 and B0 5, respectively, resulting in a
consistent underestimation of (B/A0 5. No such biases were detected
for A0.1 and B0.1, and only a slight underestimation was observed for
(B/A)o.1.

4. It is likely that RSD(W05) > RSD(W0 1), since in going from W .1 to
WO0W the magnitude of the slope of the peak (on either side) decreases
much more rapidly than the peak width increases. Thus the precision for
the 5% CFOM equations would be poorer [assuming RSD(W0O05) contributes
substantially to the total uncertainty].

5. Superior resolution between overlapping peaks is required for
measurements at 5% peak height than at 10%.

6. As exemplified in Figure 3.2 for N Gaussian CFOM equations based
on width measurements at 10% are much less inaccurate (though still
exceedingly in error) for asymmetric peaks than the corresponding
Gaussian equations at 50% (shown) and 30% (not shown). That is, the
slope of the RE vs. (B/A)a (shown for a = 0.1) plots is smaller; thus
the approximate RE correction function in the denominator of eqs 3.2 and
3.5 (text) will be less sensitive to the measurement imprecision of
(B/A)a).

7. The sensitivity of the relative error (RE) correction functions to
the (B/A)a measurement imprecision is only slightly lower at 5% than at
10$ peak height (see Figure 3.2) and is insufficient to warrant CFOM
estimation at 5%.

8. As seen in Table 3.7, the RSE can be calculated much more accurately
using width measurements at 10% than at 50%, 30%, or 5%. In addition,
the precision is much better (lower RSD limits) at 10% than at 50% or
30%, and is comparable to that at 5%.

9. The empirical asymmetry factor measurement, B/A, was introduced at
10% peak height rather than at 50% or 30% because peak tailing is much
more apparent at 10%. Since then almost all empirical measurements of
asymmetry have been reported at this peak height fraction; these data
will be of little value in later years if the B/A peak height fraction
is redefined.

Figure 3.2. Relative error in plate count of three Gaussian equations (based on
the indicated peak width measurements) and the EMG-based Ns
equation (Table 3.2, eq 1) for ideal and skewed peaks.

9*Z z S91
Sv8
N ' ' -0
N 0

;O'OmN

N
o

N -0

sOO

'MN/ -00O

45

Table 3.7. Comparison of the Accuracy and Precision of the
RSE Equations at a = 0.05, 0.10, 0.30, and 0.50

equation %RE limits % RSD

RSE = 1.04 [(B/A)0.05 -200 -5.0, +4.5 + 4.0

RSE = 0.99 [(B/A)0. -224 + 2.1 + 4.5

RSE = 0.926[(B/A)0.30]-3"11 -7.5, +2.3 8.0

RSE = 0.913[(B/A)0 r50-433 -9.0, +4.5 + 13.0

46

10. It is easier to mentally compute 10% of an arbitrary peak height
than 50%, 30%, or 5%.

Taken collectively, the above arguments indicate that the best CFOM

estimation is obtained from graphical chromatographic measurements at

10% peak height.

General Aspects

Preliminary modeling of experimental peaks. Chromatographic peaks

should be examined for their resemblance to Gaussian, EMG, or other peak

shapes, first by visual inspection and then from the asymmetry factor

measurement. In the unlikely event that B/A = 1, the validity of the

Gaussian model can be checked by comparing the measured peak width

ratios W0.5 :W0.3 W0.1 to the theoretically predicted ratios 0.5487 :

0.7231 : 1. For B/A 2 1.09, the validity of the EMG model can be judged

by the agreement of values of G', M2 and/or T, and tG determined from

both B/A and Wa measurements at a = 0.1, 0.3, and 0.5 (see Tables 3.2,

3.3, and 3.4).

For slightly asymmetric chromatographic peaks, the assignment of

peak shape models may be ambiguous due to the imprecise measurement of

B/A (e.g., Is a peak with B/A = 1.03 0.02 Gaussian?). Insofar as

accuracy and precision are concerned, does it matter if EMG-based

equations are used with Gaussian peaks or vice-versa? As seen from

Table 3.2, the EMG based equations for Nsys, M2, and UG are accurate to

within 1.5%, + 1.5%, and + 4%, respectively, over the asymmetry range

1.00 < B/A < 1.09. Figure 3.3 shows the accuracy of the Gaussian based

equations (a = 0.1) over this same range. Clearly, little error in the

estimation of M2, Nsys, and aG will result from peak model

misassignments at low asymmetries (1.00 < B/A < 1.09) due to B/A

Figure 3.3. Relative error in the Gaussian standard deviation, plate count, and
second central moment for Gaussian equations based on tR, WO.1
measurements for ideal and slightly skewed peaks.

V/8
60' 1 90'1 01 001

-_*z-

-00

r-9z

O'O0

49

imprecision, though for B/A > 1.04 the EMG equations are more accurate.

Furthermore, as seen from Table 3.5, the change in the precision of CFOM

estimation resulting from peak model misassignment would be less

than 1%.

Double-checking the results. The universality of the relative

error approach (introduced in the Derivations section above) was checked

by the independent variation of tG and aG over the tG/OG range of 10 to

5000. The generality was confirmed by identical statistical results

(%RELs, etc.) over a given B/A range (e.g., 1.09 2.76) for all CFOMs

calculated from this approach.

As an additional check on the experimental work, M2 and Nsys were

calculated by statistical moment analysis (23), using the same tR search

algorithm as before. The agreement among the true, moment, and manual

values for both M2 and Nsys was within 1% for 1.00 < B/A < 2.05 with

very little bias present in either approximation. At higher asymmetries

(2.05 < B/A < 2.76), the manual values remained within + 1.5% of the

true values, but the corresponding moment values showed a significant

positive bias ranging from +1.5% to +5.5%. An even greater bias of

+4.4$ at B/A = 2.05 (T/aG = 2) reported elsewhere (23) was attributed to

arbitrary data truncation. This was probably the source of bias in our

statistical moment method as well, but regardless of the source of bias

the same conclusion may be drawn: at high asymmetries (2.05 < B/A <

2.76) the manual CFOM equation method is more accurate than the moment

method.

Working range of the equations. The EMG-based equations in

Table 3.2 were expected to be accurate over the asymmetry range used for

the least squares curve-fitting of the f(B/A) approximations. Thus,

50

except for eq 8, the accurate working range was thought to be 1.09 < B/A

< 2.76. Nevertheless, eqs 1, 2(a,b), 6, 7, and 10 allow accurate

estimation of Nss, M2, tG, Nmax, and M1, respectively, for Gaussian

shaped peaks (B/A = 1.00). Although somewhat surprising, this is

explained by the near convergence of these EMG equations to Gaussian

ones when the substitution B/A = 1.00 is made in the former. The N
sys
equation in Table 3.2, for example, becomes

Nsys =18.53(tR/W.1)2 (3.11)

which is within +0.6% of the Gaussian formula

NW0.1 = 18.42(tR/W0.1)2 (3.12)

CFOM equations could have been developed for asymmetries greater

than B/A = 2.76 (T/cG = 3), but a sacrifice of simplicity, accuracy, or

both would have been required. More importantly, however, it was felt

that nearly all peaks reported in the literature exhibit B/A's < 2.76.

Indeed, a chromatographic system producing peaks with B/A's > 2.76 is

operating at a relative system efficiency of less than 10%.

When the EMG Nsys equation was tested for asymmetries higher than

those for which it was developed, the %RE varied between -1.5% and -10%

for 2.77 < B/A < 4.00, compared to the %RE range of +70%, +110% for the

Gaussian 10% equation (N 0.1
0.1

Real versus ideal CFOMs; column characterization. Given that peak

asymmetry is (almost) always present in any real chromatographic system,

Nsys, M2, and tR represent the experimentally observed chromatographic

efficiency, peak variance, and retention time, respectively. The

51

corresponding CFOMs Nmax, G2, and tG represent idealized

chromatographic parameters which would describe the system if all

sources of asymmetry could be eliminated. If all or nearly all

asymmetry is extra-column in origin, then for a given set of conditions

Nmax, aG2, and tG are valid descriptors of the efficiency, band-

broadening, and retention characteristics of the column.

Pluralism of the method. As might be surmised from Table 3.2,

there is more than one way to calculate several of the CFOMs. The

variance, for example can be calculated via eq 2a from measurements of

0.1 and B/A or via eq 2b from tR and Nsys (eq 1). Generally, CFOM

estimates via the "b" equations are simpler, faster, equally precise,

but less accurate than estimates via the "a" equations. This trade-off

of accuracy for simplicity and speed is slight, however; in most cases

much time can be saved with little sacrifice in accuracy if the "b"

equations are employed.

Only the simplest and most accurate methods are given in Table 3.2.

Therefore, while Nsys could be calculated from its components tR, CG'

and T (see Nsys equation in Table 3.1), this method was not reported

since it would be much more time-consuming, tedious, and in all

likelihood less accurate and less precise than the Nsys equation in

Table 3.2.

Usefulness of RSE, RPL. The relative system efficiency (RSE) and

relative plate loss (RPL), two new parameters defined in Table 3.1, are

dualistic CFOMs. First, they can be interpreted intuitively as

sys/Nmax and (Nmax Nsys)/Nmax respectively, with a corrective

retention factor (tG/tR)2 applied. Alternatively, RSE and RPL can be

viewed as the relative contributions of symmetrical ( a2) and

52

asymmetrical (T2) band-broadening processes to the total system band-

broadening (M2, the total variance). If (t/tR)2 can be neglected

because of its nearness to unity, the former intuitive expressions for

RSE and RPL become particularly useful. For example, the best possible

efficiency, Nmax, can be related very simply to the true chromatographic

efficiency, Nys, by

Nmax = Nys/RSE (3.13)

In fact, this approximation is good to within + 2% with an RSD limit of

4000. Thus, eq 3.13 can serve as a useful estimation of Nmax for

moderate to high efficiency chromatographic systems.

Although B/A, RSE, and RPL are mutually interdependent (i.e., once

B/A has been measured RSE and RPL may be calculated), the specification

of RSE, RPL, or both in addition to the reporting of B/A greatly

enhances the qualitative description of a chromatographic system. Thus,

while B/A = 1.30 indicates that a given peak is asymmetric, the

corresponding RSE = 55% or RPL = 45% provides a clearer indication of

the actual efficiency and how much room for improvement exists.

Figure 3.4 shows the exponential-like relationship of RSE and RPL

with B/A. Chromatographic systems with asymmetries of 1.00 and 1.10 are

operating at much different relative efficiencies, while two systems

operating at B/A = 2.00 and 2.10 are realizing nearly the same relative

efficiencies.

Alternative derivations. Two other approaches for deriving CFOM

equations were not as successful as that already described. In the

first attempt, a modification of the Carr graphical method, three

universal calibration curves were approximated by linear or quadratic

53

100

RPL

75

I--
z
uJ
0 50
L.

25

RSE

0
I a I I i I a II
I 1.5 2 2.5

B/A

Figure 3.4. Relative system efficiency, RSE, and relative plate loss,
RPL, for ideal and skewed peaks.

54

polynomials. This approach yielded equations for aG, T tG, and the

remaining CFOMs in terms of W0.1, B/A, and tR, but was unsatisfactory

for three reasons: a) an accurate but simple approximation of T/CG in

terms of B/A cannot be obtained for the range 1.00 < B/A < 2.76 because

the relationship between them changes at B/A = 1.36 from a decidedly

nonlinear one to an almost perfectly linear one; b) the errors

introduced by the polynomial approximations tend to accumulate slightly

rather than cancel; and c) the equations derived for all the CFOMs

except T, OG, and tG are extremely unwieldly, e.g., eq 3.14 below,

Nsys = (tR/W0.1)2[g(B/A)/h(B/A)] (3.14)

where g and h are second degree polynomials of B/A. A variation of this

same approach in which B/A was substituted for the original abscissa

T/UG in the latter two calibration curves made little difference.

A second attempt, a variation of the relative error (RE) approach

utilizing (B A) as the approximation G in eq 3.5 for T, failed

because the simplest f(B/A) approximation required for sufficient

accuracy was too complex.

CFOM units. For ease of interlaboratory comparison, all non-

unitless CFOMs should be reported in time units; if units of length are

chosen instead, the recorder chart speed should be specified to permit

conversion to time units.

55

Conclusion

Superiority of EMG-based EQuations. Although Gaussian-based

equations are somewhat simpler than their EMG counterparts in Table 3.2

(compare eq 3.11, text with eq 1, Table 3.2), the EMG equations are

clearly superior because with comparable precision (see Table 3.5) they

are equally accurate for Gaussian (or near-Gaussian) peaks and

considerably more accurate for skewed (EMG) chromatographic peaks.

CHAPTER 4
CLARIFICATION OF THE LIMIT OF DETECTION IN CHROMATOGRAPHY

Introduction

In this chapter, the focus is on eliminating the confusion

surrounding the limit of detection (LOD) in chromatography. Some prior

knowledge of the LOD concept is assumed. This discussion primarily

applies to both gas and liquid chromatography using concentration or

mass sensitive detectors whose output is measured as peak height, though

two sections (Eliminating Mistaken Identities and Using the Correct Units)

apply to analyses using peak area as well. Ideal linear elution, i.e.,

Gaussian peak profiles, is assumed though moderate deviations can be

tolerated.

Identifying Current Problems

Literature Survey Results

Initially, to determine the sources of discrepancy in

chromatographic detection limits, we conducted a limited survey of

analytical textbooks, chromatographic monographs, and the primary

chromatographic literature. This survey revealed two mistakes of

omission, as well as four major sources of discrepancies. All six are

summarized in Table 4.1. The first two problems are blatant omissions

which discredit the work reported, at least to some degree. However,

they can be eliminated if more attention is given during manuscript

preparation, and thus may be dismissed without further discussion. The

56

Table 4.1. Problems with the limit of detection (LOD) concept in chromatograpy

mistakes of omission

1. Detection limits were not reported, though trace analysis was stressed.

2. Detection limits were reported, but no definition for the LOD was given.

sources of discrepancies

3. The LOD concept was confused with other concepts used in trace analysis.

4. Arbitrary LOD models were used with little or no justification.

5. Detection limits were reported in units of concentration rather than amount.

6. The effects that several chromatographic parameters have on the LOD were
unaccounted for.

58

remaining four problems (3-6 in Table 4.1) comprise the major sources of

discrepancy in chromatographic detection limits, and are the topics to

be addressed in this report.

Numerical Example

Before proceeding, however, a numerical example which incorporates

problems 4-6 (see Table 4.1) will be given. This example will

demonstrate to the reader the magnitude of these problems and will

facilitate later discussion. Though designed with liquid chromatography

in mind, the points made by this example apply equally well to gas

chromatography.

The initial assumptions, experimental conditions, and results of

this example are shown in Table 4.2. To avoid the confusion which it

would certainly have caused, problem #3 of Table 4.1 was not

incorporated into this example. If it had been, the results might have

been even more shocking. Nevertheless, as seen in the bottom row of

Table 4.2, the LODs for the two systems employing identical detectors

differ by three and one-half orders of magnitude!

Though a detailed explanation of this example is beyond the scope

of the present discussion, the huge discrepancy in the two LODs will be

reconciled after the last three problems in Table 4.2 are solved. This

example should awaken the reader to the seriousness of these problems

and demonstrate why they must be eliminated if the chromatographic LOD

is to be a meaningful figure of merit.

59

Table 4.2. Example of widely differing detection limits

Assumptions

1. Liquid chromatograph with UV absorbance detector

2. Beer's Law applies, i.e., A = ebc.

3. Analytical sensitivity, S = eb = 10,000 AU L mol"-

4. Peak to peak noise, Np = 2 x 10-5 AU

5. Root mean square noise, Nrms = 1/5 Npp

Variables Experiment A Experiment,B

Vinj 5 uL 20 uL

LOD defn 10 Npp/S 3 Nrms/S

VM(mL) 2.5 0.5

k 10 3

N (plates) 1000 10,000

Results

LODs reported 8.7 x 10-6 M 3.0 x 10-9 M

log (LODA/LODB) = 3.5 orders of magnitude difference!

60

Solving the Problems

Eliminating Mistaken Identities

The limit of detection has unfortunately been confused with three

other concepts--particularly the (minimum) detectability--which are also

used in characterizing chromatographic trace analyses. Table 4.3

includes symbols and definitions for all four of these concepts.

One reason that the LOD is confused with the other concepts in

Table 4.3, particularly the MD, is the redundancy in nomenclature of the

detection limit and the MD, as evidenced by the partial, but

representative list of (apparent) synonyms for these concepts (shown in

Table 4.3) which appear frequently in the literature. In general,

redundant terminology in science only serves to confuse. This is

especially true when the apparent synonyms for different concepts are

themselves quite similar. The use of these apparent synonyms should be

discontinued immediately.

Even if the confusion resulting from the redundant terminology

could be eliminated, the LOD might still be confused with the MD by the

apprentice chromatographer because their definitions, as shown in

Table 4.3 and in eqs 4.1 and 4.2, are so similar in appearance (cf.

meaning, however).

LOD = arbitrary detector signal level/S (4.1)

MD = arbitrary detector signal level/Sd (4.2)

Yet despite their similarities, the LOD and MD are distinct

concepts, as a closer scrutiny of Table 4.3 shows. The LOD is a general

concept characterizing any overall trace analytical procedure consisting

of one or more steps, whereas the MD is a specific term characterizing

Table 4.3. Trace analysis concepts in chromatography

1. limit of detection (LOD) smallest concentration or amount of analyte that can be detected
or detection limit with reasonable certainty for a given analytical procedure

2. (analytical) sensitivity slope of the calibration curve signal output per unit
(S) concentration or amount of analyte introduced in a given
analytical procedure

terms specific to chromatography

3. (minimum) detectability minimum concentration or mass flux of analyte passing through a
(MD) detector in unit time that can be discerned from the noise with
reasonable certainty

4. detector sensitivity (Sd) slope of the detector response curve signal output per unit
concentration (or per unit amount/unit time) of analyte passing
through a detector

aThe use of these apparent synonyms should be discontinued immediately.

62

one step in a chromatographic analysis: detection. For example, the

LOD must, by definition, include the chromatographic dilution of the

analyte, whereas the MD cannot. Furthermore, the LOD is measured

experimentally with a complete chromatographic system (including the

column) under the specific conditions of a given trace analysis; the MD,

in contrast, is measured without a column under conditions that may or

may not correspond to those of the analysis.

The two concepts may also be distinguished mathematically.

Assuming an analytical signal in terms of peak height, the relationship

for a concentration sensitive chromatographic system is (36)

LOD = (2)1/2 [VM(+k)/N1/2] b MD (4.3)

where VM represents the corrected gas holdup volume in GC and the column

void volume in LC; k is the capacity factor; N is the number of

theoretical plates; and b is a unitless parameter which permits the LOD

and the MD to be defined independently of one another. For a mass

sensitive chromatographic system, the relationship between the LOD and

the MD is (37)

LOD (2w)1/2 [tM(1+k)/N1/2] b MD (4.4)

where tM is the retention time of an unretained solute, corrected for

gas compressibility in GC.

One final difference should be noted: An arbitrary detector signal

level of twice the peak-to-peak detector noise has been universally

agreed upon for MD calculations (in practice, at least). No such

consensus exists for the LOD.

63

Choosing a Model

As discussed earlier, the LOD may be defined in terms of an

arbitrary signal-to-noise (S/N) level. In our survey of the

chromatographic trace analysis literature, we found a multitude of

arbitrary S/N levels used, ranging from 2 to 10. To further complicate

matters, neither the measures of the signal (peak height, peak area,

etc.) nor the noise (peak-to-peak, root mean square, etc.) were

specified in many instances.

These inconsistencies and ambiguities are not surprising since (to

our knowledge) no standard model for the LOD has ever been proposed,

much less adopted, by any recognized organization for the field of

chromatography! We note specifically the omission of an LOD definition

in chromatography by the American Society for Testing and Materials

(ASTM) and by IUPAC in their respective publications on gas and/or

liquid chromatography nomenclature (38-41). The omission by these and

other organizations is also substantiated in two reviews (42,43).

More importantly, however, the above inconsistencies and

ambiguities can be eliminated completely if a clearly stated LOD model

is adopted. Therefore the adoption of, with minor reinterpretation, the

IUPAC model for spectrochemical analysis (44) or a model based on first

order error propagation (10) is proposed. These models are given in

eqs 4.5a and 4.5b, respectively,

LOD = 3sB/S (4.5a)

(The model which the International Union of Pure and Applied
Chemistry (IUPAC) adopted in 1975 (44) was chosen specifically
for spectrochemical analysis. Though the ACS Subcommittee on
Environmental Chemistry reaffirmed this model in 1980 (45),
it did so for the area of environmental chemistry and not
specifically for the field of chromatography.)

64

LOD = 3[sB2 + si2 + (i/S)2ss2]1/2/S (4.5b)

where S, i, sS, and si are the analytical sensitivity (slope),

intercept, and their respective standard deviations of the calibration

curve obtained via linear regression; and sB is the standard deviation,

calculated from 20 or more measurements of the blank signal.

The factor of 3 in the numerator of the right hand expressions of

eqs 4.5a and 4.5b gives a practical confidence level of 90% to 99.7%,

depending on the probability distribution of the blank signal and the

accuracy of sB (10,44,46). Though smaller or larger factors could be

used instead of 3, the resulting confidence levels would be too low or

too high for practical use in most cases. Both the original proponents

of these models (10,44) and others (46) strongly recommend the use of

the factor 3. This author concurs.

Both models are proposed for adoption because it seems preferable

to let the chromatography community judge their respective merits.

Indeed, strong arguments can be made for each. The IUPAC model, on the

one hand, is computationally simpler and has already been employed,

though infrequently, in the chromatographic literature. On the other

hand, the error propagation model is not really all that complicated;

many pocket calculators with linear regression capability can be easily

programmed for the error propagation model. Furthermore, the error

propagation LOD model takes uncertainties of the slope and intercept of

the calibration curve into account, resulting in a more realistic

numerical estimate.

Interpretation. The IUPAC and error propagation LOD models were

developed originally for spectroscopic trace analysis. Nearly all the

65

associated concepts, [e.g., the calibration curve, the sample (analyte +

matrix), etc.] have identical, straightforward interpretations in

chromatography. One aspect, however, does not: the measurement of sB'

Intuitively it is clear that the chromatographic baseline is

somehow analogous to the blank signal in spectroscopy. We can now refer

to sg as the standard deviation of the chromatographic baseline (noise).

The measurement of sB remains unclear, however.

One possible procedure would be to estimate sB from 20 or more

measurements of only that portion of the baseline observed at the

analyte's retention time in the absence of the analyte (when a blank

solution is injected). This is directly analogous to the measurement of

the blank signal (at the analytical wavelength) in spectroscopy. Such a

literal procedure would require at least 20 injections of blank solution

(20 blank chromatograms!) and is obviously too impractical:

1. It would be much too time consuming!
2. It may require too much blank solution.
3. Variables which affect retention would require strict control.
4. The retention time of the analyte would need to be known very
precisely.

A much more practical procedure becomes apparent if one remembers

that the standard deviation (root mean square) of a random (periodic)

signal can be closely approximated by the quotient of the range (peak-

to-peak displacement) and a parameter, p, dependent on the type of

signal, i.e.,

asignal = range/p (4.6a)

For the measurement of sB, it should be noted that the range is

equivalent to the peak-to-peak noise of the baseline, Np_p, if the

latter is measured over a sufficiently wide region of the chromatogram

which includes the analyte peak. Additionally, the baseline usually

66

results from a normally distributed, random signal for which p = 5.

Thus in most cases the standard deviation of the chromatographic

baseline, SB, can be estimated from one-fifth of the peak-to-peak noise,

i.e.,

sB = Npp/5 (4.6b)

Two additional comments regarding the practical procedure for

measuring sB should be noted:

1. It is recommended that the "sufficiently wide region of the
chromatogram" be at least as wide as 20 base widths of the analyte peak.

2. If systematic fluctuations in the baseline are present, a value (of
p) less than 5 should be used in eq 4.6b. If, for example, a periodic
triangular baseline (possibly resulting from flow pulsations in the
detector cell due to an insufficiently dampened solvent delivery system)
is observed, then p = 3.5 and sB = N P_/3.5.

Using the Correct Units

A common misconception about chromatographic LODs is that they can

be reported in units of concentration rather than amount. The fallacy

of this assumption will be shown below.

Dimensional analysis. One way of deducing the correct units of the

chromatographic LOD is by dimensional analysis. Referring to the

definition of the MD in Table 4.2, it is clear that if the appropriate

units of concentration and mass flux are used for the MD in eqs 4.3 and

4.4 for the concentration sensitive and mass sensitive detector cases,

respectively, the units for the LOD in eqs 4.3 and 4.4 must be in terms

of an amount (e.g., moles, grams, or some multiple thereof).

Another approach via dimensional analysis is to consider the right

hand expression of eq 4.1. Given the definition for the analytical

sensitivity, it is clear that the units for this term (denominator of

67

eq 4.1) should be the quotient of the units of the measured signal and

the units of the independent variable. Therefore, since the units for

the noise expression (numerator of eq 4.1) are the same as those for the

measured signal, the units for the detection limit should be the same as

the units of the independent variable of the calibration curve. Thus,

to decide which units are correct for the LOD, we need only to identify

the units of the independent variable of the calibration curve, i.e., to

determine whether the chromatographic signal depends on the

concentration or amount of analyte injected.

Equations 4.7 and 4.8 (36,37) below show that the signal (peak

height, hp) is directly proportional to the maximum concentration,

Cmax,det' or the maximum flux, Fmax,det, of the chromatographic peak

flowing through the detector, which in turn are directly proportional to

the amount of analyte injected.

/2 ( '1/2
hp Cmax,det = inj N1/2 )1/ (4.7)

h Fmax,det inj N1/2 (r)1/2/tR (4.8)

Thus the same conclusion is reached once again: The chromatographic LOD

must be given in units of amount, i.e., in moles, grams, or some

fraction thereof.

Identifying faulty logic. Despite these convincing arguments, some

researchers insist on reporting their chromatographic LODs in

concentration units. Their rationale might be as follows:

1. The amount of analyte injected, qinj, is the product of the
concentration of analyte in the sample, Cinj, and the volume of sample
injected, Vinj.
inj = Cinj Vinj (4.9)

68

2. From eq 4.9 it is clear that the concentration of analyte injected,
Cinj, is directly proportional to the amount of analyte injected.

Cinj = injinj (4.10)

3. Therefore the relative LOD (in units of concentration), CL, is
proportional to the true, absolute LOD, qL (in units of amount).

CL = /Vinj (4.11)

Though reached in a straightforward manner, the conclusion stated

above is nevertheless false. The error in reasoning is best described

as an improper or incomplete analogy. In going from a true expression,

eq 4.10, to a false statement, eq 4.11, Cinj and qinj were replaced by

two limiting quantities CL and qL, respectively. No analogous

substitution was made for Vinj, however, and therein lies the error.

Vinj may vary continuously over 0 < Vinj < Vinj,max where Vinj,max is

some limiting, maximum injection volume to be discussed momentarily.

Unless Vinj Vinj,max eq 4.11 is false.

A numerical example will help demonstrate the absurdity of eq 4.11.

Suppose the true, absolute LOD (qL) for analyte X had been determined

independently by two scientists using the same LC system to be 1 x 10-12

mol. If the scientists had used different injection volumes of 5 uL and

50 uL, according to eq 4.11 the relative LODs (CL's) for the same

chromatographic system would be 2 x 10-6M and 2 x 10-7M, respectively.

Clearly eq 4.11 is inappropriate.

The correct expression, eq 4.12 below, is obtained by using

Vinj,max in place of Vinj. But this expression is of little value

because of the difficulty in obtaining a consistent, precise estimate of

inj,max'

CL = qL/Vinj,max (4.12)

69

Problems with estimating the maximum injection volume. Experimen-

tally, Vinj,max can be determined by increasing Vinj until column over-

loading or some other adverse phenomenon is observed. This operational

definition of Vinj,max is unsatisfactory, however, because the injection

volume at which these events occur is too dependent on the experimental

conditions, e.g., the sample matrix, the percent loading of the column.

Alternatively, numerous theoretical expressions are available for

the estimation of Vinj,max. Though many are overly specific, a few are

completely general. Perhaps the best is one which relates Vinj,max to
inj,max
the maximum tolerable degradation in resolution (5, p. 289). Our

extension of this expression is reported below as eq 4.13, and is

L,std = [at,ref/ct,exp qL,exp mass sensitive
detector

LC PGC OTGC

aV,ref(mL) 0.05 0.15 0.04

at,ref(min.) 0.05 0.015 0.026

79

The Numerical Example Revisited

We return to our hypothetical LOD example (Table 4.2 and associated

text) to reconcile the large differences in the reported detection

limits. Recall that since identical chromatographic detectors were

employed, the huge discrepancies were attributed solely to problems 4-6

of Table 4.1. By eliminating these problems one at a time, obtaining

identical detection limits at the conclusion, this claim will now be

proven.

The reconciliation is summarized in Table 4.6, though readers who

wish to perform the calculations will need to refer to conditions

specified in Table 4.2.

The progress of the LOD reconciliation may be noted by inspecting

either the LOD values themselves in the second and third columns, or

their ratio given in the fourth column, in orders of magnitude. The

degrees to which the given problems are responsible for the initial

discrepancy between the LODs are shown in the fifth column; they are

obtained by subtracting successive values in the adjacent column.

Three steps were performed in the reconciliation. First, the LOD

values reported incorrectly in units of concentration (mol L-) are

converted to the appropriate units of amount (mol) by multiplying by the

corresponding sample injection volumes (Vinj's). Second, these unit-

corrected LODs are then converted to values consistent with the IUPAC

model previously discussed (qL = 3 SB/S). For case A, since Np_p = 5sB

(as discussed earlier in Choosing a Model), the LOD must be reduced by a

factor of 50/3. For case B, since Nrms = sE, no adjustment is

necessary. Finally, the IUPAC consistent LOD values are converted to

80

Table 4.6. Reconciling the differences in the detection limits
from the numerical example in Table 4.2

Step LODA LODB log (LOD A log

0. Initial 8.7 x 10-6 M 3.0 x 10-9 M 3.5
0.6
1. Amount 44 pmol 60 fmol 2.9
1.3
2. IUPAC def'n 2.6 pmol 60 fmol 1.6
1.6

3. qL,std 150 fmol 150 fmol 0.0

81

standardized LODs, thereby adjusting for differences in the

experimental chromatographic conditions (bandwidths, or states).

As seen in Table 4.6, the discrepancy between the LODs is reduced

significantly with every successive stage. It should be noted that the

largest source of discrepancy is due to differences in the experimental

chromatographic conditions (bandwidths). This demonstrates the need for

a standardized chromatographic limit of detection. Finally, the LODs in

the bottom row of Table 4.6 are identical, indicating that the

reconciliation has been completely successful.

Conclusion

Meaningful chromatographic detection limits can be obtained only if

careful attention is paid to the application of the principles which

have been discussed:

1. The limit of detection (LOD) should not be confused with the
(analytical) sensitivity (S), the minimum detectability (MD), or the
detector sensitivity (Sd).

2. The experimentally observed LODs should be calculated using the IUPAC
and/or the error propagation model(s). The calibration curve should be
constructed from a plot of signal versus amount (not concentration!) of
analyte injected, thus insuring that the resulting LODs will be in units
of amount (not concentration!).

3. If obtained under non-standard conditions, the detection limits can
be standardized using the equations in Table 4.5. Standardized
chromatographic LODs are superior to their conventional (non-
standardized) counterparts for several reasons, in particular because
they permit the (valid) comparison of trace analysis chromatographic
systems in different areas (LC, PGC, and OTGC) and/or with different
types of detectors (mass and concentration sensitive).

CHAPTER 5
SUGGESTIONS FOR FUTURE WORK

The results of this work should facilitate and encourage the study

of a number of related and unrelated topics. A few examples are given

below.

With regard to the standardized chromatographic LOD concept

introduced in Chapter 4, if open tubular liquid chromatograpy ever comes

of age, it will be desirable to extend the LOD concept to this area by

defining two additional reference states (bandwidths).

The successful generation of the exponentially modified Gaussian

(EMG) function by microcomputer and subsequent measurement of the

associated universal data should encourage others to use this realistic

model in related or unrelated modeling studies. One project which has

already been suggested by Kirkland et al. (23) is the development of a

chromatographic resolution function (equation) which is applicable to

skewed peaks as well as to ideal ones.

The equations recommended in Chapter 3 for the characterization of

skewed (EMG) and Gaussian peaks could be tested on analomous peaks

resulting from column overloading, nonlinear distribution isotherms, and

other sources of analomous peak distortion.

Since the graphical deconvolution of the symmetric (aG2) and

asymmetric (r2) sources of band broadening is now possible via eqs 3 and

4 in Table 3.2, the next step would be the development of methods for

measuring the individual components of G2 and T2

82

83

And finally, the relative error (RE) approach used so successfully

in the derivation of the equations in Chapter 3 is completely general

and should be considered by other scientists for any semi-empirical

modeling in their fields.

APPENDIX A--DISCUSSION AND LISTING OF THE BASIC PROGRAM, EMG-U

EMG-U, the BASIC program which evaluates the EMG function and

obtains the universal data, is listed below. In addition to the

documentation supplied within the program itself and the flowcharts

given in Fig 2.1, the following should be noted:

1. By letting S = UG/T and Y = (t-tG)/aG, eq 2.4 can be written as

2. To the extent allowed by BASIC, the symbolism used in EMG-U is

consistent with that in eqs A.1 and A.2 and elsewhere in the text.

Greek symbols were spelled out partially or entirely.

3. EMG-U was developed on an Apple II Plus computer and optimized for

0.1 T/CG < 4 using tG = 100, G = 5, 0.001 < dtmin -. 1, and 0.001 a

< 1. Minor modifications may be required for optimum performance if

other computers or other values for the parameters are used.

4. EMG-U was designed to minimize execution time; major reductions are

not likely to be achieved unless a compiler is used.

5. The EMG evaluation subroutine (lines 2990-3160) may be used

independently of the other routines in EMG-U.

84

85

100 REM EMG-U EVALUATES THE EXPONENTIALLY MODIFIED GAUSSIAN (EMG)
FUNCTION
110 REM ALSO DETERMINES TR, HP, TA, TB, B/A, W/SIG, AND (TR-TG)/SIG
120 B$ = ADD LINE":C$ = 3150 PRINT T,HEMG":D$ = "TO HELP DEBUG."

130 ZERO = 0:PT5 = .5:WUN = 1:TWO = 2:RTTWOPI = SQR (TWO *
3.141592654):TEN = 10:K4 = 10000:K5 = 100000: REM COMMON NUMERICAL
CONSTANTS
140 P = .2316419:B1 = .31938153:B2 = .356563782:B3 = 1.781477937:B4
= 1.821255978:B5 = 1.330274429: REM CONSTANTS FOR 14 POLYNOMIAL
APPROX.
150 A = 1:TG = 100:SIG = 5: REM EMG PRE-EXPONENTIAL CONSTANTS

400 FOR R = R1 TO R2 STEP RSIZE
440 T = R: GOSUB 3300:R = T:S = WUN / R
480 GOSUB 1000: REM GO TO MAIN EMG DATA ROUTINE
500 BA = (TB TR) / (TR XTA):WS1G = (TB XTA) / SIG:RR = (TR TG)
/ SIG
540 BA = INT (BA K4 + PT5) / INT (K4 + PT5):T = RR: GOSUB 3300:RR
= T
600 PRINT R,BA: PRINT WSIG,RR: PRINT
700 TR = ZERO:HP = ZERO:XTA = ZERO:TB = ZERO
720 TI = ZERO:T2 = ZERO:DT = ZERO:HTEMP = ZERO
800 NEXT R

830 REM END OPTIONAL DISK STORAGE OF EMG DATA
850 END : REM PROGRAM HAS RUN TO COMPLETION

1000 T = TG: GOSUB 3000:HTEMP = HEMG:DT = WUN
1020 T = T + DT: GOSUB 3000
1040 IF HEMG < HTEMP THEN GOSUB 3300: GOSUB 3350: GOTO 1100
1060 HTEMP = HEMG: GOTO 1020
1080 GOSUB 3350
1100 IF DT < = TLR THEN 1160
1120 GOSUB 3450: GOSUB 1620
1140 GOTO 1100
1160 TR = T2 DT:T = TR: GOSUB 3000:HP = HEMG:HLOOK = HP ALPHA

1200 T = INT (TR + WUN + PT5)
1220 T = T WUN: GOSUB 3000
1240 IF HEMG > HLOOK THEN 1220
1260 DT = WUN: GOSUB 3400
1280 IF DT < = TLR THEN 1340
1300 GOSUB 3450: GOSUB 1830
1320 GOTO 1280
1340 GOSUB 3500:XTA = T: GOSUB 3000

86

1400 T = INT (TR + PT5) INT (SIG + PT5)
1420 T = T + INT (SIG + PT5): GOSUB 3000
1440 IF HEMG > HLOOK THEN 1420
1460 Ti = T INT (SIG + PT5):T2 = T:DT = WUN: GOSUB 2030
1480 IF DT < = TLR THEN 1540
1500 GOSUB 3450: GOSUB 2030
1520 GOTO 1480
1540 GOSUB 3500:TB = T: GOSUB 3000
1560 RETURN : REM END OF MAIN EMG DATA ROUTINE

1600 REM TR SEARCH LOOP BELOW THRU LINE 1720
1620 FOR T = Ti TO T2 + DT STEP DT
1640 GOSUB 3000
1660 IF HEMG < HTEMP THEN GOSUB 3300: GOSUB 3350: RETURN
1680 HTEMP = HEMG
1700 NEXT T
1720 A$ = "THE TR SEARCH HAS FAILED!": GOSUB 3550: END

1800 REM TA SEARCH LOOP BELOW THRU LINE 1950
1830 FOR T = T2 TO Tl DT STEP DT
1860 GOSUB 3000
1890 IF HEMG < HLOOK THEN GOSUB 3300: GOSUB 3400: RETURN
1920 NEXT T
1950 A$ = "THE TA SEARCH HAS FAILED!": GOSUB 3550: END

2000 REM TB SEARCH LOOP BELOW THRU LINE 2150
2030 FOR T = TI TO T2 + DT STEP DT
2060 GOSUB 3000
2090 IF HEMG < HLOOK THEN GOSUB 3300:T = T DT: GOSUB 3400: RETURN
2120 NEXT T
2150 A$ = "THE TB SEARCH HAS FAILED!": GOSUB 3550: END

2990 REM EMG EVALUATION SUBROUTINE BELOW THRU LINE 3160
3000 Y = (T TG) / SIG:E = S S / TWO S Y:Z = Y S:ZTEMP = ZERO
3020 IF Z > = ZERO THEN 3060
3040 ZTEMP = Z:Z = ABS (Z)
3060 NF = EXP ( Z Z / TWO) / RTTWOPI:Q = WUN / (WUN + P Z)
3080 PQ = Q (Bl + Q (B2 + Q (B3 + Q B4 + B5 Q Q)))
3100 I = NF PQ
3120 IF ZTEMP > = ZERO THEN I = WUN I
3140 HEMG = A S RTTWOPI EXP (E) I
3160 RETURN

3300 T = INT (T K5 + PT5) / INT (K5 + PT5): RETURN
3350 Ti = T TWO DT:T2 = T:HTEMP = ZERO: RETURN
3400 Ti = T:T2 = T + DT: RETURN
3450 DT = DT / TEN: RETURN
3500 T = (T1 + T2) / TWO: GOSUB 3300: RETURN
3550 PRINT CHR$ (7): PRINT A$;B$: PRINT C$: PRINT D$: RETURN

APPENDIX B--UNIVERSAL EMG DATA AT a = 0.05, 0.10, 0.30, 0.50

14
Table 2.2. Constants in the Polynomial Approximation
for Ijj in eq 2.4a
p = 0.2316419
b1 = 0.319361530
b2 = -0.356563762
b3 = 1.761477937
b4 = -1.821255978
be = 1.330274429
aSee text immediately after eq 2.4 or reference 33.

39
Table 3.2. However, the accuracy and precision for M2, Oq, tG and t at
a = 0.3 and 0.5 are still sufficient to permit peak modeling decisions
to be made.
Discussion
Detailed Discussion of Precision
Shown in Table 3.6 are the precision data for t, W_, and (B/A)
used in this study. The RSD results were obtained by converting
previously reported raw precision data (32) to the form appropriate for
error propagation analysis for the conditions specified in Table 3.6.
Data excluded in the previous study for (B/A 1) were also excluded in
this analysis. The RSDs of tR, Wa, and (B/A)a for individual peak
shapes (at a given peak height fraction) were averaged as done
previously, thereby implicitly assuming the independence of the RSDs on
peak shape.
The RSDs for tR, Wa, and (B/A 1) for individual peak shapes were
originally reported relative to aQ, aG, and (B/A 1) respectively.
Multiplication by Cg/tR, ancl (B/A 1)a/(B/A) converted them to
the appropriate form.
Since the RSDs of tR, WQ, and (B/A) were assumed to be independent
of peak shape, intuitively it might seem that this should also be true
for the RSD of any calculated CFOM. This is not the case, however. For
one group of CFOMs (Nsys, M2, G, tG, Nmax, RSE, and M^), a slight to
moderate variation in their RSDs with (B/A). was observed. This can be
explained by examining the random error propagation in the general
empirical N s equation,
Nsys = (tR/Wa)2/[(B/A)a + C2J
(3.9)

68
2. From eq 4.9 it is clear that the concentration of analyte injected,
Cinj, is directly proportional to the amount of analyte injected.
Cinj = qinj/Vinj
(4.10)
3. Therefore the relative LOD (in units of concentration), C^, is
proportional to the true, absolute LOD, q^ (in units of amount).
CL = qL/Vinj
(4.11)
Though reached in a straightforward manner, the conclusion stated
above is nevertheless false. The error in reasoning is best described
as an improper or incomplete analogy. In going from a true expression,
eq 4.10, to a false statement, eq 4.11, C^nj and q^nj were replaced by
two limiting quantities and q^, respectively. No analogous
substitution was made for V^nj, however, and therein lies the error.
Vinj W vary continuously over 0 < Vinj < Vinj>max, where Vinj>max is
some limiting, maximum injection volume to be discussed momentarily.
Unless V.nj = Vinj>max, eq 4.11 is false.
A numerical example will help demonstrate the absurdity of eq 4.11.
Suppose the true, absolute LOD (q^) for analyte X had been determined
1 P
independently by two scientists using the same LC system to be 1 x 10
mol. If the scientists had used different injection volumes of 5 uL and
50 uL, according to eq 4.11 the relative LODs (C^'s) for the same
chromatographic system would be 2 x 10^M and 2 x 10-^M, respectively.
Clearly eq 4.11 is inappropriate.
The correct expression, eq 4.12 below, is obtained by using
^inj,max -*-n Place of V n j But this expression is of little value
because of the difficulty in obtaining a consistent, precise estimate of
^inj,max*
CL ^L^inj ,max
(4.12)

PAGE 1

EQUATIONS FOR THE CALCULATION OF CHROMATOGRAPHIC FIGURES OF MERIT By JOE PRESTON FOLEY A DISSERTATION PRESENTED TO THE GRADUATE SCHOOI OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1983

PAGE 2

To Mom and Dad, for their love, support, and encouragement; and to my sister Barbara, for leading the way.

PAGE 3

ACKNOWLEDGMENTS First and foremost, I would like to thank my research director, John G. Dorsey, not only for helping me with various research projects, but for being a special friendÂ— for always finding the time to listen and for always doing the "extra things." Second, I want to express my gratitude to Thomas J. Buckley and Sharon G. Lias of the National Bureau of Standards for their help and the use of their facilities in preparing this document. Finally, I want to thank all the friends I made in Gainesville for making my graduate education at the University of Florida the happiest and most satisfying time of my life. 111

PAGE 4

TABLE OF CONTENTS CHAPTER PAGE ACKNOWLEDGMENTS iii ABSTRACT v i 1 INTRODUCTION 1 Overview Â„ 1 Chromatographic Peak Characterization 1 Limit of Detection 8 2 GENERATION OF THE EXPONENTIALLY MODIFIED GAUSSIAN (EMG) FUNCTION AND RELATED DATA 9 Introduction 9 EMG Evaluation 9 Background 9 Evaluation of the Integral Term 10 Obtaining Universal EMG Data 16 Background 16 Measurement of the Pertinent Peak Parameters Â„ 17 Comparison of Universal EMG Data 22 Conclusion Â„ 24 3 EQUATIONS FOR THE CALCULATION OF CHROMATOGRAPHIC FIGURES OF MERIT FOR IDEAL AND SKEWED PEAKS 25 Introduction 25 Derivations 25 Experimental 28 Apparatus 28 Procedure 28 Results 30 Recommended CFOM Equations 30 Other CFOM Equations 33 Discussion 39 Detailed Discussion of Precision 39 Why Measure At 10$ Peak Height? 41 General Aspects 46 Conclusion 55 IV

PAGE 5

4 CLARIFICATION OF THE LIMIT OF DETECTION IN CHROMATOGRAPHY 56 Introduction 5b Identifying Current Problems 56 Literature Survey Results 56 Numerical Example 58 Solving the Problems Â„ 60 Eliminating Mistaken Identities 60 Choosing a Model 63 Using the Correct Units 66 Converting to Chromatographic Reference Conditions 70 The Numerical Example Revisited 79 Conclusion 81 5 SUGGESTIONS FOR FUTURE WORK 82 APPENDICES A DISCUSSION AND LISTING OF THE BASIC PROGRAM, EMG-U 84 B UNIVERSAL EMG DATA AT a = 0.05, 0.10, 0.30, and 0.50 87 C DERIVATION OF V inj>max 95 REFERENCES 96 BIOGRAPHICAL SKETCH 100

PAGE 6

Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy EQUATIONS FOR THE CALCULATION OF CHROMATOGRAPHIC FIGURES OF MERIT By JOE PRESTON FOLEY December 1 983 Chairman: Dr. John G. Dorsey Major Department: Chemistry The measurement and interpretation of several chromatographic concepts and parameters, hereafter referred to as chromatographic figures of merit (CFOMs) are improved via equations and concepts developed in this work. The previous uses of the exponentially modified Gaussian (EMG) model in chromatography are briefly reviewed. A method for evaluating the EMG function and a set of algorithms for obtaining universal data are presented and are shown to be simpler and easier to use than those previously reported. The corresponding BASIC program, EMG-U, is also briefly discussed. By use of the exponentially modified Gaussian (EMG) as the skewed peak model, empirical equations based solely on the graphically measurable retention time, t fi peak width at 10% peak height, W Q ,, and vi

PAGE 7

the empirical asymmetry factor, B/A, have been developed for the accurate and precise calculation of CFOMs characterizing both ideal (Gaussian) and skewed peaks. These CFOMs include the observed efficiency (number of theoretical plates), N ; the maximum efficiency sys attainable if all asymmetry is eliminated, NÂ„ v ; the EMG peak parameters, t^, a-,, and t; the first through fourth statistical moments; the peak skew and peak excess, Yg and YÂ„; and two new CFOMs Â— the relative system efficiency, RSE, and the relative plate loss, RPL. Equations for the number of theoretical plates and the variance (second central moment) are accurate to within 1.5% for 1.00 <_ B/A Â£ 2.76. width and B/A at 10% peak height are recommended. The current problems with the LOD concept in chromatography are reviewed. They include confusing the LOD with other concepts in trace analysis; the use of arbitrary, unjustified models; the use of concentration units instead of units of amount; and the failure to account for differences in chromatographic conditions (bandwidths) when comparing LODs. Two models are proposed for calculating the chromatographic LOD. A new concept, the standardized chromatographic LOD, is introduced to account for differences in chromatographic bandwidths of experimentally measured LODs. The standardized chromatographic LOD is shown to be a more reliable CFOM than the conventional (non-standardized) chromatographic LOD. Vll

PAGE 8

CHAPTER 1 INTRODUCTION Overview Chromatography is a well-known method for the separation and quantitation of chemical moieties from a (sample) mixture. Over the years several concepts and parameters, hereafter referred to as chromatographic figures of merit (CFOMs), have been introduced to characterize the separation and quantitation. Unfortunately, some of the CFOMs are often difficult to estimate [those which characterize chromatographic peaks]; others are ambiguous [e.g., the limit of detection (LOD)]. The goal of the present work, which is introduced in more detail in the following two sections, is the improvement of the measurement and interpretation of these chromatographic figures of merit. It is beyond the scope of this work to introduce or review the development of these CFOMs from either a historical or theoretical point of view. Such discussions and references to additional discussions may be found elsewhere (1-10). Chromatographic Peak Characterization In recent years there has been considerable interest in the characterization of experimental chromatographic peaks. Presented in Table 1.1 are the names, symbols, and general expressions that have evolved for the parameters used in chromatographic peak

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characterization. The graphical chromatographic parameters are illustrated in Figure 1.1. These CFOMs have been estimated either manually using graphical measurements made directly from the chromatogram or by a computer following data acquisition. Both methods have advantages and disadvantages. Manual methods were used exclusively at first and are employed quite extensively today. For arbitrary peak shapes, they are accurate for only five CFOMs: t fi B/A, h p W b and W If a Gaussian peak shape is assumed, however, then M^ = t R and Mp is only a function of W,, W or Mq and N may subsequently be calculated. Except for higher even central moments, the remaining CFOMs are zero for Gaussian peaks. For real chromatographic peaks, it is almost always a mistake to assume a Gaussian peak shape. Experimentally these ideal, symmetric peaks are rarely, if ever, observed due to various intracolumn and extracolumn sources of asymmetry (5,11-23). Kirkland et al. have shown that the plate count can be overestimated by as much or more than 100% if any of the three most common Gaussian-based equations are employed (23). Computer estimation methods are more accurate than common manual methods for a given CFOM but are not available to every chromatographer. The general approach taken has been one of peak statistical moment analysis (6,11,22-24). Via relatively simple algorithms all the CFOMs may be determined quite accurately, though the precision of the second and higher central moments is seriously affected by baseline noise (25). The failure of the Gaussian function as a peak shape model for real chromatographic peaks led to the search for a more accurate model and

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H LJ < LJ CL Q LU N O Figure 1.1. Graphical chromatographic parameters shown at peak height fraction a = 0.10. Except for A, B, and B/A at a = 0.10, all width related measurements are subscripted with the value of a to prevent ambiguity.

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the eventual acceptance of the exponentially modified Gaussian (Â£MG), a function obtained via the convolution of a Gaussian function and an exponential decay function which provides an asymmetric peak profile. The development, characterization, and theoretical and experimental justification of this model have been thoroughly reviewed (21,22,26,27). Previous chromatographic studies (11,12,14,15,17-23,25-31) involving the EMG function, summarized in Table 1.2, demonstrate the utility of this skewed peak model. Adoption of the EMG peak-shape model has improved the estimation of the CFOMs. A new algorithm for the computer-based peak moment analysis has been derived (25) and tested (22) which is less sensitive to baseline noise and the uncertainty of peak start/stop assignments. More recently, Barber and Carr described a manual method for CFOM quantitation which requires the graphically measurable retention time tpi peak width W, empirical asymmetry factor B/A, and successive interpolations from three large-scale universal calibration curves (31,32). The primary objective of this part of the present study is the development, using the EMG model, of accurate equations for CFOM calculation dependent solely on tp, Wq and B/A. The need for computerized data acquisition is thus circumvented, and, in addition, CFOM calculation via these equations is expected to be faster and more precise than the other accurate manual method since no graphical interpolation is required. The previously reported methods for evaluating the EMG function (14,16,19,27) and obtaining chromatographic peak data (19,26,31) were too inaccurate or too unwieldy to use in the present study, which

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8 employed an Apple II Plus microcomputer. This necessitated, therefore, the development of a simpler method for evaluating the EMG function and the incorporation of a simpler, more accurate, and more general set of algorithms for obtaining the EMG data of interest. Limit of Detection The limit of detection (LOD) is generally defined as the smallest concentration or amount of analyte that can be detected with reasonable certainty for a given analytical procedure. Though arguably the most important figure of merit in trace analysis, the LOD remains an ambiguous quantity in the field of chromatography. Detection limits differing by orders of magnitude are frequently reported for very similar (sometimes identical!) chromatographic systems. Such huge discrepancies raise serious questions about the validity of the LOD concept in chromatography. The primary objective of this part of the present work is to restore the integrity of the LOD concept, to make the chromatographic LOD a reliable, meaningful figure of merit. This will be accomplished in two steps: First, the major sources of the discrepancies in chromatographic detection limits, i.e., the current problems with the LOD concept, will be identified. Second, each problem will be addressed and eliminated (or circumvented).

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CHAPTER 2 GENERATION OF THE EXPONENTIALLY MODIFIED GAUSSIAN (EMG) FUNCTION AND RELATED DATA Introduction This chapter describes the improvements achieved by this study in evaluating the EMG function and in obtaining the EMG data of interest. The results of the present work are compared with those previously obtained. In addition, the corresponding BASIC computer program, EMG-U, is listed and discussed briefly in Appendix A. Universal EMG data are tabulated in Appendix B. It is hoped that this will facilitate the use of the EMG function, whenever applicable, in modeling studies in chromatography or any other area. EMG Evaluation Background Description of the EMG function It is beyond our scope to derive the EMG function from first principles. Those who so desire should see the treatments given by Sternberg (11, pp. 250-253) or Kissinger et al. (12, pp. 159-162). Their results are shown below in eq 2.1. z/(2) 1/2 *EMG XW = *-i\UQ{Â£) --/ij expLu.muQ/ T ;_ ^ -o Â— -c G v h Â£MG (t) = [Atf G (2) 1/2 A] exp[0.5(a r /T) 2 (t-t r _)A] /exp(-x 2 )dx (2.1) /Â• PE 1 E 1 I 1 where z = (t t G )/a Q On/x (2.2) Equation 2.1 shows that the EMG function is defined by three parameters:

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10 the retention time, t Q and standard deviation, a Q of the parent Gaussian function; and the time constant, t ? f the exponential decay function. In addition, the quotient t/Oq is a fundamental measure of an EMG peak's asymmetry. The arbitrary constant A determines the amplitude of the function. Note that the right hand side of eq 2.1 can be broken into 3 parts: the pre-exponential term (PE.,), the exponential term (Â£.]), and the integral term (I,). Discrepancies. A literature survey which I conducted revealed three discrepancies in eq 2.1. In the first instance, a factor of 2 difference observed in the pre-exponential term, PE,, is relatively unimportant because this affects only the zeroth statistical moment (peak area). All other parameters, including higher order moments which are normalized by the zeroth moment, are unchanged by this factor of 2. The second and third discrepancies are serious, however. Both were observed in the denominator of the second quotient, (t t Q )/T, within the exponential term, E.,. In one case, an additional factor of 2 was present; in the other, a Q was added to this denominator and omitted from the numerator of the pre-exponential term, pÂ£,. These errors invalidate those expressions for the EMG function. Evaluation of the Integral Term Range of z. The methods used to evaluate eq 2.1 have frequently been omitted from the EMG literature. Since the evaluation of the first two terms is straightforward, the reported methods differ only in the manner in which the integral, Ij, is determined; the I-j approximations, in turn, depend on the value of z in the upper limit of I.. It is convenient to group the range of possible z values into three regions: a) z <. -3; b)-3 <. z < 4; and c) z >. 4 In region c, the definite

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11 integral equals the constant (tt) 1/2 to within 0.01? and is thus virtually independent of z. In regions a and b, however, the relative value of the definite integral is highly dependent on z. Numerous techniques permit the accurate evaluation of L in region b. In region a, however, as z becomes more negative, it becomes more difficult to approximate the integral to the same (high) relative accuracy. Thus it is pertinent to examine the practical minimum values of z for EMG peaks. As seen from eq 2.2, z depends on a) the normalized difference between the time of interest, t, and t Q ; and b) the reciprocal of the peak asymmetry, (T/a G )" 1 For a given peak shape (constant t/o g ), z will therefore be smallest (most negative) at the starting threshold of the peak, which may be conveniently defined as that time (t) on the leading edge of the EMG peak where the value of the function Ch Pun (t)] is a specified fraction, B, of its maximum value, i.e., h EMG (t)/h p = B. Moreover, the minimum z value will decrease as the starting threshold is decreased. This is shown in Table 2.1 where minimum z values are tabulated for EMG peaks with asymmetries ranging from 0.1 to 3. Another trend illustrated is that for a given starting threshold, z increases with increasing asymmetry (Va G ). Previous methods. Given this wide range of z values, how is I 1 (in eq 2.1) evaluated? Except for a vague reference to an unspecified polynomial approximation (30), all previous methods for calculating I 1 employ different techniques for different values of z. For moderate z values (e.g., regions b and c, above), some methods utilized the wellknown identity /. exp(-y 2 )dy = 0.5U) 1/2 [1 + erf(x)] (2.3)

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12 Table 2.1. Minimum z Values Needed to Evaluate EMG Peaks for Various Asymmetries (r/a Q ) and Starting Thresholds (B) a 0.001 0.01 0.1 T/0 G 0.10 b -12.9 -12.1 0.15 -10.3 9.6 8.7 0.20 8.6 7.9 7.0 0.25 7.5 6.9 6.0 0.30 6.8 6.2 5.3 1.00 4.4 3.6 2.7 3.00 3.6 2.8 1.9 a See eq 2.2 and text for description of z, T/a Q and B. Values for z have been rounded. b An underflow error occurs z < -13, thus preventing its measurement.

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13 (where erf is the error function) in order to take advantage of error function subroutines resident in the computers used (19,27). Others used eq 2.3, but approximated the error function by interpolation from a set of tabulated areas (14,18). For small z values (region a, above), the error function techniques were sufficiently inaccurate to warrant the use of other methods instead. In nearly every instance some type of asymptotic series was employed (14,19,27). In the lone exception, a Gaussian function was substituted for the EMG function for any part of the peak profile where very small z values were encountered (18). This work. The approach taken in this study is to transform the integral in eq 2.1 via change of variable [x = y/(2) 1/2 ] to (tt) 1/2 /exp(-y 2 )dy/(27r) 1/2 The EMG function can now be written as h EMG (t) = tAa G (27T) 1/2 / T ][E 1 (see eq 2.1)] / exp(-y 2 )dy/(2Tr) 1/2 (2.4) -/-co ?E 4 h The integral in eq 2.4 can be approximated by a polynomial approximation l 4 (z<0) = NF(z) P(q) and l 4 (z>.0) = 1l 4 (z<0), where NF(z) = exp(-z 2 /2)/(27r) 1/2 P(q) = ^q + b 2 q 2 + b 3 q 3 + b^q 4 + b 5 q 5 Â•1 q = (1 + pz)~ and p,b 1 ,...,b 5 are constants given in Table 2.2 (33). Comparison. The values obtained for 1^ are compared to the true values (34,35) in Table 2.3 from z = -10 to z > 3.9. In addition, since In = ^/(tt) 172 (2.5) they can be compared to values obtained for I* via an asymptotic

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14 Table 2.2. Constants in the Polynomial Approximation for lu in eq 2.4 a p = 0.2316419 t> 3 = 1.781477937 b 1 = 0.319381530 b 4 = -1.821255978 b 2 = -0.356563782 b 5 = 1.330274429 a See text immediately after eq 2.4 or reference 33.

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15 Cd t =T >l m CM o en r Â— Â— .. Â— Â— > T3 a Cd X X Xi X I i h1 O CD vO 1 1 1 Â„ Â•H G CO CM CM cÂ— X) ^ ^r cd S Q. tÂ— VO CM vO CO CTv oo G CM H O H Â•=r O N. 4J X H Cm ^~ v| O ^-v XJ G CO (s CD N & C Sh D. o \ cd G "* Â£4 M g O Cm rH r^ pr ^r vO o in to o rd o m CM i Â— 1Â— l CD 1] o ^ Â•H 3 8 I 1 o G G S en O o o TÂ— Q, St ^ O o CQ ^~ i Â— 1 Â— Â•cÂ— X IÂ—! u 3 c Â•H X X! X X X X cÂ— O i ^r ^r VO o **% Â— cd H N O m CM t Â— < Â— in m in a. l ^~ K cd Â— I I 1 i on en ^r cÂ— M [= G G o o o o Â— v_^ CM o O 3 d) 1 Â— Â— ^Â— CM G V, QÂ— r o 3 cÂ— o CM o G a G x> X X X X CM o fcÂ— o c a 0) M -a < 4-> CM o e'o OJ G Â• CM S-, bO JJ G oo =r cÂ— vO en ro O o o <Â— CD cd O Â• H CO Cm 4-5 a O, C\J cr CD CD c G CD G o ca rÂ— 1 cr. CD CD G Q, 4-5 rj CM O x> vO .=}CM o CM Â• CD 0) S o cd N T ~" Â— oo CO CO eu o H ] 1 1 Al cd X5 o a xT

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16 series (27), the most accurate method reported for z < -3. (Comparison with values for I 1 obtained via error function techniques when z > -3 (Ijj > 10") is not illustrative since the maximum absolute error in the polynomial approximation Iu(z) is estimated to be + 7.5 x 10~ 8 .) Table 2.3 shows that our method for evaluating 1^ is exceptionally accurate for moderate values of z. Moreover, it compares favorably with the asymptotic series method for evaluating I,, except for z < -8 where the latter method is somewhat better. Re-examination of Table 2.1 shows, however, that for all practical purposes z > -8 whenever T/a G J> 0.2. Thus our method for evaluation of 1^ can be used to evaluate the EMG function to within 1% or less for i/o n > 0.2. While being slightly less accurate than the most accurate previous method, the new technique for evaluating the EMG function is much more convenient than any of the previous methods. Only one simple subroutine requiring just a few programming lines (see lines 2990-3160 in Appendix A) is needed, whereas the other methods require at least two subroutines, if implemented on any computer without a built-in error function routine (i.e., nearly all microcomputers and many minicomputers) Obtaining Universal EMG Data Background Using the polynomial approximation for the integral in eq 2.4, the EMG function can be evaluated over the entire practical time range. Depending on the data required for the modeling process of interest, the EMG peaks could be generated "on the fly" as needed. Alternatively, the necessary EMG data could be generated (and stored) in advance and accessed when needed. Though the storage requirements may seem

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17 prohibitive for the latter, given a T/a Q value and a peak height fraction, a, three quantities completely specify an EMG peak (31). Figure t.1 shows an EMG peak with its pertinent graphical parameters. Regardless of the retention time, t Q and standard deviation, o of the (unconvoluted) parent Gaussian peak (not snown) (B/A) W_/a P and (t D ^G^ a G are universal constants so long as T / G and a remain fixed. Recent work has utilized these universal data sets almost exclusively (3D. Experimentally, three parameters must be determined in order to calculate the universal data: t R t, and t B (see Figure 1.1). Note that t R must be obtained before t. or t g because h p = h EMG (t R ) is needed for the latter. Measurement of the Pertinent Peak Parameters Previous methods In the past, t R nas been determined by one or more of the following methods: a) peak displacement data and knowledge of t G (19); b) differentiation of h Â£MG (t) and solving for roots (26); and c) least squares fitting of the top of the peak with a quadratic gram polynomial (26,31). Once t R has been found, t A and tg can be located. In the only method reported previously, two points [t.,, hg^tt^], [t 2 h EMG (t 2 )] are found so that t 1 < t A < t g (or t 1 < to < to). Linear interpolation yields the approximation for t (or t B ). This work The approach for finding these quantities is based on iterative search mechanisms, as the flowchart in Figure 2.1 shows. In the case of t R initial time limits are easily found using the fact that t R is always greater than t Q The EMG function is then evaluated from

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18 Find initial time limits Y J. Calculate new time limits N t = t + dt (t R t B only) t=t-dt (t A only) Y -> Calculate value of t (and h ) f A r V N _^L Decrease value of dt Figure 2.1. Simplified flowchart for locating t (and h ), t or t The El'IG peak parameters must be input before beginning this search.

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19 the lower time limit to the upper time limit in increments of dt. When the maximum is found, new lower and upper time limits [given by t( current) 2dt and t(current), respectively] closer to the peak maximum are set, the time increment is decreased, and the search is begun again. The retention time, t R is approximated by t(last) dt mm The algorithm for estimating t A (or tg) is similar to the t. (or t B ) search algorithm previously discussed in that the time limits are analogous. Since t A < t R < tg, initial values for t 1 and to are easily determined. To locate t A (or tg) the EMG function is then evaluated from t s tg to t^ in decrements of dt (or from t 1 to to in increments of dt) until h EMQ (t) < ahp. New values for t^ and to [given by t( current) and t(current) + dt, respectively] closer to t A (or tp) are then set, the time decrement (increment) is decreased, and the search is repeated. This is continued until t< and t~ are known to the desired precision; t (or tg) is then given by (t 1 + tg)/2. The maximum error in the values of t R t A tg and related universal EMG quantities obtained via these search algorithms is presented in Table 2.4. In all cases the error is dependent on the smallest (most precise) time increment (or decrement), dt ^ i used in the last iteration of each search. In theory, dt min could be as small as desired. Due to the finite precision of computers, however, the time increment, dt would ultimately be reduced to such a low value that h EMG (t ) ~ h EMG (t+dt). (2.6) Henceforth the algorithms would cease to function accurately, if at all.

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20 Table 2.4. Maximum Errors in the Universal EMG Data and Selected Component Parameters Parameter 3 Maximum Error H dt min *!Â• fc B 1/2 dt min A = t R -t A B = t B -t R 3/2 dt m n VG = (V t A> /a G dt min /a G ( W /ff G dt min /G G a A, B, to t., and tg defined in Figure 1.1; a,,, t P and dt .Â„ described in text. G' U' mm

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21 As dt is decreased, the t R search algorithm will fail first, since the slope of the EMG function is smallest in the region of t Q Somewhat smaller (more precise) dt min 's could be employed in the searches for t. and tg before the algorithm breakdown described by eq 2Â„6 would occur. The increase in precision of t A and t fi is probably not worth the effort, however, since two out of the three universal EMG data expressions are dependent on the least precise quantity, t R The minimum usable value of dt depends on the precision of the computer employed and on the value of o Q chosen. In this study, the experimentally measured minimum ratio of dt/ov, was 0.0002 for 0.1 < t/o q < 3 using single precision arithmetic. Multiple precision capabilities would allow a still lower dt/crÂ„ ratio to be used. Comparison The algorithms for t R tj,, and tg may be compared as follows : 1. With the exception of the quadratic least squares method for finding tg, all of the methods for obtaining t R t,, and tg are designed for simulated data (essentially no noise). 2. The two algorithms for calculating t. (or tg) are quite similar. Both require two points which closely bracket the desired peak height fraction, a, and both are relatively unbiased. The subsequent interpolation performed in the previously described algorithm is potentially more precise than the averaging of the final time limits in the proposed search algorithm. If t and tg are already known as precisely as or more precisely than t R however, additional improvements in their precision, even if realized, will not yield significant increases in the precision of B/A and (t R -t G )/ a G 3. Our approach for determining t R though crude, is superior to the other three methods previously discussed for the following reasons: i) It is a general algorithm. Whereas methods a and b are specific to the EMG model, our search mechanisms will work for that and other peak models as well. ii) It is accurate and unbiased. In contrast, the quadratic least squares fitting method, though general, suffers from a small, but nevertheless observable bias (due to a determinate error) which increases with increasing peak asymmetry (26). iii) It is easy to understand and implement. The other methods are unnecessarily complex, though they admittedly have the potential for greater precision.

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22 4. The proposed search algorithms for determining t R) t A and to are superior to the previous methods because they can be de-bugged more easily. By having the time and the value of the EHG function printed every time the EMG function is evaluated, the programmer can literally watch the computer perform the search. Since the search logic is so simple, programming errors are easily detected. Upon elimination of the errors, the print statement may be removed. Comparison of Universal EMG Data Table 2.5 shows representative sets of universal EMG data obtained from this study and from a previous work (32) which utilized the quadratic least squares method and the interpolation method for the location of t R and t A (or t g ), respectively. The precision is reported for our data (in terms of maximum errors) and is assumed to be no worse than + 1 in the least significant digit of the previously reported data. Several points should be noted: 1 Although this difference is slight or non-existent at high asymmetries, the previously obtained universal data are somewhat more precise. This is expected since the algorithms used in locating t R t., and t B are potentially more precise than those developed here. 2. The data sets are in excellent agreement for all three universal EMG quantities at low asymmetries (t/ctÂ„ <. 0.5). This agreement is especially significant at i/o Q =0.1, because it shows that the moderate errors introduced by the polynomial approximation for K in eq 2.4 when -12 < z < (see Table 2.3) are not transmitted to the universal EMG data. 3. Whereas the W a /a Q data reported previously are consistent with the corresponding data of this study over the entire range of T /cr p the remaining data sets are discordant for x/o Q 2 1.0. Relative to the current data sets, the previous ones for (t R -t G )/a Q and (B/A) appear to be slightly overestimated and underestimated, respectively. This discrepancy is due to the use of a least squares fitting method in the previous measurement of t R which overestimates this quantity for EMG peaks (26) and other types of skewed peaks. This bias increases from an insignificant value at low x/a Q to an observable one at r/a _> 1.0, 4. Despite the differences noted above, the general interlaboratory agreement is quite good. Though the current data are more accurate, either set of EMG data can be used with confidence for modeling studies.

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23 CD o T<*Â— tO rÂ— to 1 c o o o iÂ— XÂ— 1 CD o o O o o I Eh o o o o o I a) Â• Â• Â• Â• Â•i Cm o o o o o i Cm 3 + + + + 1 Â•H 1 1 Q CM a i co o co co C i 3 ^Â— cn to L o -P "s. I O Jrf cn (^n en I Â— o Â£ '-* 1 Â•H Eh ro CO ea^ cÂ— O 1 > o en CM cn i Â— l Â— Â•P 1

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24 Conclusion Though the exponentially modified Gaussian (EMG) model has already been employed in numerous studies in chromatography as Table 1.2 shows, its usage might have been still more extensive had it not been for some confusing discrepancies and for the overly complex methods used for its evaluation reported previously. Hopefully the clarification of these discrepancies and presentation of a simple method for evaluating the EMG function and obtaining universal data will encourage more scientists in all fields to use the EMG model, when appropriate.

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a CHAPTER 3 EQUATIONS FOR THE CALCULATION OF CHROMATOGRAPHIC FIGURES OF MERIT FOR IDEAL AND SKEWED PEAKS Introduction Adoption of the EMG model for chromatographic peak characterization results in a new set of chromatographic figures of merit (CFOMs) which re listed in Table 3.1. These CFOMs consist of fundamental and derived EMG parameters, the latter containing explicit expressions for the first through fourth statistical moments defined previously in Table 1.1. Included among these new CFOMs are the following: the retention time, t Q and standard deviation, a Q of the associated parent Gaussian peak from which the skewed peak is derived; the exponential modifier, t; the fundamental ratio, t/Oq, which characterizes peak asymmetry; the observed efficiency (number of theoretical plates) of a given (asymmetric) chromatographic system, N ; the maximum efficiency a given system could achieve, N max if all sources of asymmetry were eliminated; and finally, two CFOMs proposed originally in this work which demonstrate how peak asymmetry drastically reduces chromatographic efficiencyÂ— the relative system efficiency, RSE, and the relative plate loss, RPL. Derivations If the estimate G obtained from using a Gaussian peak shape equation is used to approximate the true value T of a CFOM for an asymmetric peak, the relative error RE which results is defined as 25

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26 Table 3.1. Chromatographic Figures of Merit Based on the Exponentially Modified Gaussian (EMG) Model Fundamental t o Qf t, x/a G Derived N sys = H 2 '(% 2 + t2 ) N max ~~ (V a G )Â£ RSE = ^ T sys /N max)
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27 RE = (G T)/T (3.1) which can be rearranged to give T = G/(fiE + 1) (3.2) Thus, the true value T and the Gaussian approximation G for the CFOM are related by the correction term (RE + 1) in the denominator of eq 3.2. Kirkland et al. have shown that RE = f(T/a Q ) (3.3) for the three popular Gaussian-based methods for determining plate counts of a system (23). Equation 3-3 should hold, in fact, for any CFOM for which a Gaussian approximation exists except t Q Since Barber and Carr have shown (31) that t/a G = f(8/A) (3.4) successive substitution of eqs 3-3 and 3.4 into eq 3.2 yields T = G/[f(B/A) + 1] (3.5) Although the exact form of f(B/A) is unknown, a least squares curve fitting of an RE versus B/A plot can give an excellent approximation. The above approach was used for calculation of N a rt and M sy s u c. Following this, t was calculated (see sixth equation, Table 3.1) by x [M 2 a Q 2]V2 (3<6) For the determination of t Q the universal relationship (t R t G )/ff Q = f(T/a Q ) (3.7) previously reported (31) was rearranged and combined with eq 3.4 to give fc G = H ~ a G f(B/A) (3.8)

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28 where f(B/A) was approximated by a least squares fit of (t R t Q )/a G vs. B/A. Since t R B/A, and W 0>1 are graphically measurable, all the remaining CFOMs can be calculated once the fundamental parameters oÂ„, t, and tn have been determined. Experimental Apparatus An Apple II Plus 48K RAM microcomputer was programmed in BASIC for EMG peak generation. A curve-fitting program available from Interactive Microware (P.O. Box 771, State College, Pa., 16801) with linear, geometric, exponential, and polynomial capabilities was used for the unweighted least squares fitting of various data sets. Procedure EMG Peak Generation. Except as otherwise noted, values of A = 1, t G = 100, and G = 5 were used in eq 2.4 for EMG peak generation. The x /q ratio was varied from 0.1 to 3 in 0.05 increments producing data equivalent to 59 peaks. The times for t R t A and tg (see Figure 1.1) at a = 0.1, 0.3, and 0.5 for each of the 59 peaks were determined to within 0.001, using the simple search algorithm described in Chapter 2; corresponding values of W and B/A were then computed. Development of the CFQM equations Textfiles of fiE(N M a r ) vs. (B/A) 0<1)0>3j0>5 (t R t G )/ G vs. (B/A) 0#1>0#3)0-5 and RSE vs. ^ B/A ^0.1 ,0.3,0.5 were made for 1 iT/cr G ^'3The Gaussian relations CT G = w 0. i/' 4 2 9 1932 and M 2 = (W Q# ^4 .291932) 2 were used in the Gaussian approximations for ^ sy3 a G and M 2 at a = 0.1. Similar equations were used for M 2 an d a Q at a .3, 0.5. The true values of N gys a Q M 2

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29 and RSE were computed from the known values of x, o, t Q and t R for a given peak. The CFOM equations were developed by unweighted least squares curve fitting. The relationship between several quantities [e.g., RE(N), (B/A) a to a / CT Q] and t/o q has been shown previously (23,31) to be nonlinear for < V G < 0.5 and nearly linear for 0.5 <. T /n <. 3. Because similar relationships were observed in some of the textfiles above, least squares fitting was limited to about the same t/cu range 0.5-3 (1.09 < B/A <_ 2.76) except for the RSE vs. B/A textfile where the complete set of values was used in the regression analysis. Although the least squares fittings of the various B/A textfiles were initially judged by visual inspection and the coefficient of determination (square root of the correlation coefficient), their final evaluation was based on the accuracy and simplicity of the resulting CFOM equations. For ease of use, the CFOM equations were first simplified algebraically and then by the successive rounding of numerical coefficients. The occasional nearness of the decimal coefficients to whole numbers was exploited. For example, if f(B/A) = 1.02(B/A) + 0.69, then for 1 .09 <. B/A i 2.76, the much simpler function f(B/A) = B/A + 0.72 is approximately the same (exactly if B/A = 1.5) and the accuracy of this function is not significantly affected. All CFOM equations were simplified as much as possible Â— any more rounding of the coefficients will result in appreciably greater error. Evaluation of the CFOM equations The accuracy of the CFOM equations was evaluated in terms of four parameters, listed in decreasing order of importance: the percent relative error limits

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30 ($RELs) which represent the maximum possible error of the CFO'M equations within the specified B/A range; the mean percent relative error or bias, J&RE = E%RE/n; the average magnitude of the percent relative error, Z|$RE|/n; and the standard deviation of the percent relative error. The precision of the empirical, EMG-based CFOM equations and three Gaussian CFOM equations was calculated via error propagation theory and is reported as percent relative standard deviation (S&RSD). The required precision estimates of t R W a and (B/A) (the graphically measurable quantities) were obtained using data from a previous study (32). Results Recommended CFOM Equations Listed in Table 3.2 are the empirical CFOM equations based on t R W 0.1' and B/ ^ A measurements (see Figure 1.1) which we recommend. Accuracy. Using equation 1 in Table 3.2, the true efficiency of a chromatographic system, N sys may be estimated to within + 1.5$ for both Gaussian and exponentially tailed peaks within the asymmetry range 1.00 1 B/A i. 2.76. Equations for Mg, t Q and M 1 are equally if not more accurate over this asymmetry range. All CFOMs except RPL, Mg, Y g> and Y E can be estimated to within 5% for 1.09 <. B/A <. 2.76, the asymmetry range over which most of the curve fitting was performed. All CFOM equations are accurate to within + 5% for 1.19 <. B/A <_ 2.76, and 18 out of 21 are accurate to within 2%.

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31 n uo CM ^r o a -p Â• Â• CO -H CM CM CM CM K S Â•H +1 +1 +1 +1 a* rH u u Cm CM o in Â• O CM St +1 -M +1 Cw m in jQ o o o -p o Â• Lfl IT\ LO LA o in o o LH H Â• T~" Â• Â• Â• Â• Â• Â• Â• Â• Â• B *1 o i Â— o PO LO ^r ,0 CM I o o CM 3 cr, o CM :1 O CÂ— CM cr, o JD CM 1 CT> O EÂ— CM O CM I O o CM I o CM I O O CM I CT% o CM Cm O CO h -H o H a 2 to o p cd S o o o Cm CQ C O Â•H -P cd cr Pd Â•a 73 C CD o o CD OS c\J ro CD rH -Q cd H C o Â•i-4 cd 3 cr CD in CM + CQ O 3S Â•P CO > w XI CM I CM CQ ^r iO o a CM 2 cd CM i Â— i i Â— i \ CM CM cq O CD ^^ e> G c D i_ Â— 1 \ CM /-% Â• Â•Â—V -Â• Â— v cd CO ro cd -O sr >> l CM CM CO "N, cr 3 CM cr cr CD \ Â— (1) CD ^-* CM Â• ^_^ ^_^ H IX o CM CM -P is as as i i i Â— i II it II ii ii C CM CD *v s D P H H in ^r o I << CM -O CM ? \ eq ^Â— ar m CM wP en Â• CJ V Â•Q X /-^ < ^=r 1 1 w^ V, w u D to cr D s. v^ CD o o> Â— p CT> P ^Â• UG O 11 -P 1! ii ]] tP X b cd ca \ C5 a CO H p a cc; Cxi co OS Qui CM cd J3 cd ITl -Q in oo

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32 CM o +1 Cm Cm Ch X3 o -H c\j o o o LA o + 0) o yo CM s CM I cr, o lo Â• CM CTv o m CM 1 cr. o O o in Â• o en 1 Â— + + o o Â• CM I CD in o o 0) CTi CM CTi O VO tÂ— CM I CTi O o I EÂ— CM I C^ o LO I M3 CM I o o I rCM 5 CT. o cr co So cd cr T3 co x> C CO -p X CD I i CM jo cti CO cd ^^ ^r ^ cr a) + N. Â• H a i Â— i -p eg cr a> CM cd CM cd CM U3 D on CM CM X3 =r CM on D CM CM on a3 CM CT 03 CM 2 CM X! CM m I on as cr a> m S3 CO >cd on cr CD CM cr CO CO n en cm CO CM O* CM X cd CM cr CO S cd CM X CM a* 0) CM 2 CM a" X o s HP o -p d) cd CD X! pa 3 bQ Â•H Cm C Â•H -o CO 4-5 O Â•H a a) -a ca 4J c cd a CD 53 ra cd CD j CQ Â•a c cd (35 -p a o -o CO CO rd cq cd S-, CO X! P S3 Cm T3 CO a e 3 o Sh CO X Â•P o a 3 O si CO CO P B tÂ— CM v| < pq v| M O d-, CO 00 CO M o O CM +1 CO U cd CO 4^ H S H H b O JM -a CD -P U o a co u 8 o X! -P cd X! -p Sh 0) CO Â•H a M -P Â•H s U3 u o u u <0 V CM o CM A cr> N. O -P a* o O O CM LO +1 -a a cd t> o Â•v. O CM + l a CO 4J 0) x> CO Ch cd w HP Â•H s r-t H O Jh Sh Sh O (w "O CO 4J Sh o a, 0) Sh 00 -p Â•H s Â•H Sh o Sh Sh m 0) Â•H C >H a) "v m CO CO Sh CO > co Cm SA Cm O V> o H Q, CO < Q) x: v. a) -p m w a> Q. CO .a CO a o, Cm o c o Â•H hP O a 3 ch CO Â•H o c% o M a G O Â•H CO H O Sh

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33 Except for eqs 11b, 13b, and 14b which had biases of +0.976$, +0.820$, and +1.111$, respectively, the bias for every equation given in Tables 3-2, 3-3, and 3.4 was less than + 0.6$. Precision Summary. For the equations in Table 3.2, the estimated relative standard (RSD) limits obtained via propagation of error theory for N sys' M 2' a G' N max' and RSE were a11 less than or equal to + 4.5$. RSD limits for t and H< were + 0.2$. The precision of the EMG equations in Table 3.2, the width-based Gaussian equations, and the calibration curve method of Barber and Carr (3D is compared for N sya M 2 and a Q in Table 3.5. The results shown for the Gaussian equations are valid for 50$, 30$, and 10$ width measurements because RSD(W 0>5 ) = RSD(W Q>3 ) = RSD(W Q 1 ). The slightly greater imprecision observed for the EMG equations is due to uncertainty in the B/A measurement not required for the Gaussian equations. The somewhat larger $RSDs for the method of Barber and Carr are probably due to interpolation uncertainties (from the calibration curves) unique to this method. The precision of the remaining CFOMs in Table 3.2 was found to be highly dependent on the peak shape. Rather than reporting RSD limits, the RSDs for several CFOMs or groups of CFOMs have been plotted vs B/A in Figure 31 Â• Other CFOM Equations Listed in Tables 3.3 and 3.4 are smaller sets of CFOM equations developed only for use in determining if a real chromatographic peak is well-modeled by an EMG peak and should not be used to routinely calculate any CFOM (including N ) since they are usually less accurate, less precise, and more complex than the analogous equations in

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34 J=) O c Tin a -p Â• Â• m Â• GO -H ^r -=f st OJ a S i +1 +1 +\ +1 co -p Â•H a H -r-l IS G G O g u O OJ o o +1 c LP. o I o CM +1 o 1 Â— + CM I o o to + in o I in Â• en + Cm Â• in m % Â• o ,-__ i +1 CO CO + CO o ^~v cd s bO *s c 03 Â•H ^-^ rH c^ o T3 sO -=T Â£ I CM JU fT) cd i Â— i Â• cu on O CL, o c \. Â•H 1 03 Â•> Â— > ro on H C CM B O o + o H #Â•*, m -P < on cu cti N, TO 3 33 o 3 cr x^ Â•Â— N CD UJ 1 cd -p V CO 3 Â— CM ss cr 00 Of") Â•\ W SO Â• OJ Â• O 05 Â• VO 3b -p m on 11 ii II CD rH CO ctf Â£3 a > 03 3g CM CM SI 33 in PO Â• o *-\ *i Ss pq Nw* CO in ( Â— 1 Â• 00 CM =t + o CM + on Â• on o Â• -"V o CM CM < y-*\ V, *N \ J) TÂ— i Â— 03 v. 1 1 1 1 en CM CM to Â— vP O Â• X) O O o Â• 1 Â— 1 CM 1 1 1 rP N. cd X3 en ~ i m C\J CM o 23 Â£ J2 i Â— i i Â— i -p o IS -p o -p s S rH Â•H B Â•H co G CD G C I cd B O -p -p g O 3 Â— vO CM a x: s~*. O cÂ— cÂ— -p eti G o Â• +1 "x -p CM CM M 03 O O ^^ JD CO V| -P v| CM -P G 1 o> XÂ— A Â• O O Â• o Â• CO G Q, o Â• o O -P TO *^-v vÂ— -<"> O G -a cu < < \ cu iÂ— i G \ II ^ o s 3 G 03 03 +J o O -^S K Â— G JG O G 3 TO v| o Â• v| O TO -C ^Â— V ^Ct, cd TO o o> =c Â• a. -P rH 1 Â— s on K (J a G JB Â• 03 Â• o 3 Â— w 1 CM en -H G Â• O Â•* G -P 3 G II o Â•H CO O Cd hO O **- Cm o Cm Â•rH Cm o a Cm Â• -tf*V Cl, io \ CM CT> S* x. Q3 o o Â• C O o ^Â—^ o V CM m -h Z -p TD H en + c + G C cd Â• +1 3 O Ctf o o o Â• Cm Â•H Â— Â•sS. a jc a. G < in O TO in a Â• x Â• G 1 G o s TO G O 3 v| 3 Q. G W3 O G K CO o cd S o cd G CC TO G D 00 TO -P o Â— -P N. -p 4J cd Â•H H rH Â•H G ^H G s G s a O Oh O H L ~ rH Â•H Cm rH rH H rH rH T> rH TO TD CO T3 G H G G -P CD a Â•H O o j.: O O G W Â•H rH G -p G G O cd o cd G G G G Q, m > a cx, o eg K) cd o a o T3 .p Cm G

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36 Table 3.5. Precision 3 Comparison of Three Graphical Methods for Estimating N M 2 and G method Gaussian eqs sys + 2.0 M+ 2.0 + 1.0 empirical eqs, Table III, this work + 2.5 + 2.4 2.0 calibration curve method, reported previously (3D + 5 + 3 3 a,. Reported as percent relative standard deviation (%RSD). Precision of equations estimated via error propagation, usins data of Table 3.6.

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rH r4 H a H CO u cti CD CD O, > CD <4-l CO O c Cm O o -P u Â•H a o CO a; Cm Â•H TD O c a) CD S-, a. >> Q. CD L, T3 P CD c CD n Â•H i Â£-i i >> > rH GO H ffl Â• rc! Â— Â•H ra -P 2: c Â• ij CD CÂ— C3 [& W Â• CD co m Q, ~Â— CD 1 C\J -P 03 Â• rl Â•h on C <-, o CD *-> CO 5 Â• CD r M P n c CD cvj o H H w co c r-l CM-H O !m Â•y^ L< O O b0 Cx, T3 a O c CD -P cd X) ci! a a m CD CD a> a. a. u X! cd cd a -P si a O CO
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38 ^ sjro Q_ ^ IS I01 CD cvi in C\J c\i o CD ro < CD O ro in CM _J__ O C\J in as a % in

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39 Table 3.2. However, the accuracy and precision for MU, a Q t Q and t at a = 0.3 and 0.5 are still sufficient to permit peak modeling decisions to be made. Discussion Detailed Discussion of Precision Shown in Table 3.6 are the precision data for t p W and (B/A) It EL 3, used in this study. The RSD results were obtained by converting previously reported raw precision data (32) to the form appropriate for error propagation analysis for the conditions specified in Table 3.6. Data excluded in the previous study for (B/A 1) were also excluded in CI this analysis. The RSDs of t R W & and (B/A) for individual peak shapes (at a given peak height fraction) were averaged as done previously, thereby implicitly assuming the independence of the RSDs on peak shape. The RSDs for t R W Qf and (B/A 1 ) for individual peak shapes were originally reported relative to a Q a Q and (B/A 1) respectively. Multiplication by o Q /t R Q /^ a and (B/A 1) /(B/A) converted them to the appropriate form. Since the RSDs of t R W & and (B/A) were assumed to be independent of peak shape, intuitively it might seem that this should also be true for the RSD of any calculated CFOM. This is not the case, however. For one group of CFOMs (N sys M Â£l o Q| t Q N max> RSE, and M,), a slight to moderate variation in their RSDs with (B/A),, was observed. This can be explained by examining the random error propagation in the general empirical N equation, N sys = C 1 (t R /W a ) 2 /[(B/A) a + C 2 ] (3.9)

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40 Table 3.6. Suggested Chromatographic Measurement Conditions and the Resulting Precision (%RSD) Achieved for tÂ„, W and (B/A) n a a Conditions 1. Chroma togram recording rate: 1 cm/o G (W Q < _> 4.3 cm) 2. Ruler resolution: +0.2 3. Minimum retention distance, (t R ) : 10 cm 4. Minimum peak height, (hp) min : 10 cm Results a b CFOM (a = 0.1) (a = 0.3) (a = 0.5) fe R +0.2 identical identical w a 1 1.0 + 1.0 (B/A) +2.0 +2.5 +3.0 Data obtained from reference 32 and subsequently converted (see Detailed Discussion of Precision) for a = 0.1, 0.5 Â— results interpolated for a = 0.3. %RSD(t R ) rounded to nearest 0.1%; %RSDs for W (B/A) rounded to nearest 0.25%.

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41 Assuming negligible covariances, RSD(N ) is given by sys RSD(N sys ) = {4[RSD(t R )] 2 + 4[RSD(W a )] 2 + 1 2 RSD[(B/A) a ] 2 [(B/A) a /((B/A) a + C 2 )] 2 } 1/2 (3.10) 3 4 Even when terms 1-3 in eq 3.10 are constant, RSD(N Â„) will vary sys somewhat with (B/A) a because of term 4. Clearly this variation will be greatest for -(B/A) a < C 2 < 0. Additionally, as C 2 + -(B/A) RSD(N gys ) -*Â• For C 2 =0, RSD(N sys ) is essentially independent of (B/A) a Finally, for C 2 > 0, a negligible to slight variation of RSD(N s ) with (B/A) a may be observed, depending on the magnitude of terms 1 and 2 relative to the product of terms 3 and 4. Except for eqs 1 and 2(a,b) in Table 3-4, the RSD limits for N M OJ a r t P N sys d h b max RSE, and M., calculated via equations in Tables 3.2, 3.3, and 3.4 varied by less than 0.5% for 1.00 <. B/A <. 2.76. The remaining CFOMs (t, t/o q RPL, M,, M 4 ,Y s ,Y e ) comprise a second group whose RSDs are moderate to strong functions of peak shape as Figure 3-1 shows. In every instance the imprecision is largest for the least asymmetric peaks and smallest for the most highly skewed peaks. Analysis of the error propagation equations show that one or more terms within the equations get very large as the peak shapes become symmetric. Why Measure at 10% Peak Height? For reasons listed below, the recommended CFOM equations in Table 3.2 are based on (in addition to t R ) the measurement of W and B/A at 10% peak height rather than at other peak height fractions such as 50%, 30%, or 5%: Â•--*--Â—-----

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42 1. Examination of Tables 3.2, 3-3, and 3.4 shows that many CFOM equations at 10$ are olearly superior to the corresponding ones at 30$ and 50$ in terms of a) precision (lowest RSD limits) b) widest working range for equivalent accuracy (e.g., N ) c) simplicity for M~ (path a) 2. The N s equation at 10$ peak height is more accurate for Gaussian and near-Gaussian peaks than other N equations developed at 50$, 30$, or 5$. For example, at B/A = 1 the relative error was +0.6$, +10.0$, +2.0$, and +2.5$, respectively. 3. In a previously reported graphical measurement study (31,32), statistically significant positive and negative biases were detected in the measurement of A Q 5 and B Q 5 respectively, resulting in a consistent underestimation of (B/a) q 5< No such biases were detected for Aq ^ and B Q and only a slight underestimation was observed for (B/A) Q> J. 4. It is likely that RSD(W 0#05 ) > RSD(W 0<1 ), since in going from W Q 1 to W 0.05 the ma S ni tude of the slope of the peak (on either side) decreases much more rapidly than the peak width increases. Thus the precision for the 5$ CFOM equations would be poorer [assuming RSD(Wq Q5 ) contributes substantially to the total uncertainty]. 5. Superior resolution between overlapping peaks is required for measurements at 5$ peak height than at 10$. 6. As exemplified in Figure 3.2 for N Gaussian CFOM equations based on width measurements at 10$ are much less inaccurate (though still exceedingly in error) for asymmetric peaks than the corresponding Gaussian equations at 50$ (shown) and 30$ (not shown). That is, the slope of the RE vs. (B/A) a (shown for a = 0.1) plots is smaller; thus the approximate RE correction function in the denominator of eqs 3.2 and 3.5 (text) will be less sensitive to the measurement imprecision of (B/A) a ). 7. The sensitivity of the relative error (RE) correction functions to the (B/A) a measurement imprecision is only slightly lower at 5$ than at 10$ peak height (see Figure 3.2) and is insufficient to warrant CFOM estimation at 5$. 6. As seen in Table 3.7, the RSE can be calculated much more accurately using width measurements at 10$ than at 50$, 30$, or 5$. In addition, the precision is much better (lower RSD limits) at 10$ than at 50$ or 30$, and is comparable to that at 5$. 9. The empirical asymmetry factor measurement, B/A, was introduced at 10$ peak height rather than at 50$ or 30$ because peak tailing is much more apparent at 10$. Since then almost all empirical measurements of asymmetry have been reported at this peak height fraction; these data will be of little value in later years if the B/A peak height fraction is redefined.

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45 Table 3.7. Comparison of the Accuracy and Precision of the fiSE Equations at a = 0.05, 0.10, 0.30, and 0.50 equation RSE = 1.04 [(B/A) 0#05 r 2 00 RSE = 0.99 C(B/A) Q#10 ] -2.24 RSE = 0.926[(B/A) 0>30 ]3 11 RSE = 0.913[(B/A) 0>5Q ]4 33 #RE limits % RSD -5.0, +4.5 + 4.0 + 2.1 + 4.5 -7.5, +2.3 + 8.0 -9.0, +4.5 + 13.0

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46 10. It is easier to mentally compute \Q% of an arbitrary peak height than 50%, 30%, or 5%. Taken collectively, the above arguments indicate that the best CFOM estimation is obtained from graphical chromatographic measurements at 10% peak height. General Aspects Preliminary mode ling of experimental peaks Chromatographic peaks should be examined for their resemblance to Gaussian, EMG, or other peak shapes, first by visual inspection and then from the asymmetry factor measurement. In the unlikely event that B/A = 1, the validity of the Gaussian model can be checked by comparing the measured peak width ratios W Q>5 :W q.3 :W 0.1 to the theoretically predicted ratios 0.5487 : 0.7231 : 1. For B/A 2 1.09, the validity of the EMG model can be judged by the agreement of values of o Q M 2 and/or x, and t Q determined from both B/A and W a measurements at a = 0.1, 0.3, and 0.5 (see Tables 3.2, 3.3, and 3.4). For slightly asymmetric chromatographic peaks, the assignment of peak shape models may be ambiguous due to the imprecise measurement of B/A (e.g., Is a peak with B/A = 1.03 0.02 Gaussian?). Insofar as accuracy and precision are concerned, does it matter if EMG-based equations are used with Gaussian peaks or vice-versa? As seen from Table 3.2, the EMG based equations for N M 2 and a Q are accurate to within 1.5%, 1.5%, and 4%, respectively, over the asymmetry range 1.00 < B/A < 1.09. Figure 3.3 shows the accuracy of the Gaussian based equations (a = 0.1) over this same range. Clearly, little error in the estimation of M 2 N sys and a Q will result from peak model misassignments at low asymmetries (1.00 < B/A < 1.09) due to B/A