Anal. Chem. 1987, 59, 102-109
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Optimization of Secondary Chemical Equilibria in Liquid Chromatography: Theory and Verification Joe P. Foley*’ and Willie E. May National Bureau of Standards, Center for Analytical Chemistry, Building 222, Room A113, Gaithersburg, Maryland 20899
A general theory for the optknlzatlon of secondary chemlcal equlllbrla In Hquld chromatography (SCE-LC) Is presented. Equations that predlct the opthnum moblle phase condltlons X. The are derived expllcltly for the equillbrlum AX + A selectlvtty of SCE-LC has been substantially underestlmated because MMopthlunmoMle phase condltkns were employed In previous selectlvlty estimates. An upper bound for the Selectivity Is given by cy I IOIb‘I, where ApK Is the dHference In the equilibrium constants of the two solutes. The deterself-selectlvlty, or ratio of capaclty factors, klAX/klA, mines the moblle phase optlmum and the overall chromatographic selectlvlty whlch can be achleved. The maximum and selectlvlty is predicted for several values of k’,/k’, can ApK. The opthnum concentratlon of equlllbrant, [XI, be predlcted from pX, = pK,, (1/2) log (k’Ax/k’A), where pX, = -log [XI, and pKA,, Is the average effecthre equlllbrium constant of the two solutes. The theory Is experlmentally verlfled for acid-base equlllbrla.
+
+
In liquid chromatography, the “primary” equilibrium or quasi-equilibrium is the distribution of the solute between the mobile phase and the stationary phase. Any other equilibria that occur in the mobile phase, stationary phase, or both are considered “secondary”. Analytes participating in secondary chemical equilibria (SCE) can exist in more than one form. In contrast to conventional LC where the solutes of interest are generally eluted in one form (by suppressing any SCE) and are separated via differences in their primary equilibrium constants, with SCE-LC the analytes are eluted in more than one form and are separated via differences in their secondary equilibrium constants. Some examples of secondary chemical equilibria (SCE) are shown in Table I. These single-step equilibria allow the analytes to exist in two forms. For example, given a certain mobile phase pH, a fraction of some weak acid will be present in the mobile phase as HA and the remainder will be present as A-. The power of SCE-LC is illustrated in Figure 1, where the retention of two hypothetical weak acids with different ionization constants is shown for a reversed-phase system. In contrast to conventional LC, which fails to provide sufficient selectivity to separate the acids either in their highly retained, protonated forms (pH 13) or in their less retained, anionic forms (pH 16),SCE-LC provides more than enough selectivity for the separation of the acids under intermediate conditions in which comparable amounts of both forms of each acid exist (3.5 < pH < 5.5). Furthermore, if the pK;s of the acids are known, the elution order can be predicted (stronger acid elutes first). The fact that SCE-LC may succeed when conventional LC fails is true not only for acid-base “secondary” equilibria but Present address: Department of Chemistry, Louisiana State
University, Baton Rouge, LA 70803.
Table I. General Form and Examples of Secondary Chemical Equilibria chemical eq
analyte
AX=A+Xa HA + A- + H+ BH+ + B + H+ ML + M + L IP==I+P SM==S+M
equilibrant
ion
X H+ H+ ligand or metal ion-pairing agent
solute
micelle
AX,A
weak acid weak base metal or ligand
’General form of secondary chemical equilibrium. for other SCE as well. Thus SCE-LC is a very powerful complement to conventional LC, and it is no surprise that SCE-LC has received considerable attention in the past (1-9). In particular the reader is referred to excellent reviews by Karger et al. ( 1 ) and Melander and Horvath (8). Although previous SCE-LC separations have been successful, conditions for optimum resolution have been found principally by trial-and-error or by empirical optimization strategies. Furthermore, despite the work performed in this area, no fundamental, theory-based optimization of any separation parameter has ever been reported. Until now, for example, there were no equations that could predict the pH that would give maximum selectivity for the separation of weak acids depicted in Figure 1. The purpose of the present paper is to report the development and verification of a fundamental theory for the optimization of selectivity in SCE-LC. Our approach is completely general and applies to any secondary equilibrium. The equations that we have derived predict the optimum mobile phase conditions, i.e., the pH giving maximum selectivity for the acids in Figure 1. Our equations also show that the separating power of SCE-LC has been substantially underestimated. In this paper we limit our discussion to isocratic elution and single-step equilibria. In the article that immediately follows we describe the effects of several variables on the self-selectivity ratio, k ’ m / k i , an important parameter in our SCE-LC optimization theory. We also discuss factors that influence retention and column efficiency in SCE-LC and summarize an approach for the optimization of resolution. Further reports will extend the theory to gradient SCE-LC and multistep equilibria.
THEORY BACKGROUND Universal Retention Model of SCE-LC. One of the advantages of SCE-LC over conventional LC is the availability of a simple, accurate, universal retention model. The general equation for the capacity factor of an analyte A which participates with the equilibrant X in the equilibria shown below is AX, + AXn-,
...
A
h’ = FAX(n)k’AX(n) + F A X ( n - l ) k i X ( n - l ) +
(1)
+ FAk’,
(2)
where FAX(,,),FAX(,-,), and F A are the stoichiometric fractions
This article not subject to US. Copyright. Published 1986 by the American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 59,
04 2
5
4
)E1 )Kz
5
6
PH Flgure 1. Retention as a function of pH for two hypothetical weak acMs (pK,’s = 4.4 and 4.6) on a reversed-phase column. I n contrast to what has previously been assumed, the best selectivity, a , is not obtained at a pH halfway between the pK,’s.
of the analyte in each of its forms and k’,,,), k’AX(n-l), ...,and k i are the limiting capacity factors of each form of the analyte. Stated another way, the observed capacity factor of an analyte will be the weighted average of all the analyte’s limiting capacity factors. Note that the equilibrant X is a species (e.g., H’) whose concentration can be controlled by a suitable buffer. Specific examples of equilibrants are given in Table I. The SCE-LC retention model has been experimentally verified for n = 1 and n = 2 previously by other workers (2, 5 ) and is also verified later in this report for n = 1. An equation obtained by using retention times instead of capacity factors has also been verified (3). SCE-LC Retention Model for a Single-Step Equilibrium. Of particular interest is the single-step equilibrium (n = 1case of eq 1 and 2) on which we focus in this report. The single-step equilibrium and corresponding capacity factor equation are given below
AX=A+X k ’ = Fmk’m
(3)
+ FAki
(4)
Classification of Analyte Retention i n SCE-LC. For a single-step equilibrium (eq 3), an analyte can exhibit one of three types of retention. 1. General Retention. Both forms AX and A of an analyte are retained, but to a differing degree, i.e., k k , = C1 # k‘A = Cz where C1, C2 > 0. 2. Partial Retention. Either form AX or A of an analyte is retained, but not both, i.e., either
NO. 1, JANUARY 1987
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taining k’in an optimum range, it provides little direction for optimizing selectivity. Because the selectivity term is generally regarded as the most important in liquid chromatography,we decided to base our theory on the optimization of this parameter. General Approach to Selectivity Optimization i n SCE-LC. It is important to note that the stoichiometric fractions FAX(n), etc., of eq 2 can be replaced by expressions dependent only on [XI, the concentration of equilibrant, and the n - 1 equilibrium constants. Furthermore, the equilibrium constants and the limiting capacity factors of eq 2 are fixed parameters for a given mobile phase and stationary phase, and thus the capacity factor of eq 2 is also only a function of [XI. Since selectivity, a , is just a quotient of capacity factors, the optimum mobile phase for maximum selectivity can be derived by differentiating selectivity with respect to [XI and solving for roots of the resulting equation! Symbolically = k’z/k’i = f([X])
da/d[X] = 0. Solve for [XIOPT
(6) (7)
Assumptions. 1. The equilibrium constants of the analytes are not the same, i.e., pK1 # pKz. This is a necessary condition for an SCE-LC separation (1). (We will later estimate the selectivity of SCLE-LC systems for a given difference in pK’s.) 2. The limiting capacity factors of the analytes are similar, Le., k’,(,) = k’AX(2) and k’,,,) = k i 5 z ) . This is a reasonable assumption because if either the k AX or klA values for the compounds of interest are drastically different, the compounds can be separated by suppressing the secondary equilibria (conventional LC). 3. The compounds exhibit either general or partial retention, but not degenerate retention. There is no optimum equilibrant concentration, [XIOPT, for compounds exhibiting degenerate retention because their retention is not a function of [XI. Expressing t h e SCE-LC Retention Model Explicitly in Terms of [XIand Kefp The SCE-LC retention model of eq 4 for the single-step equilibrium of eq 3 is given below explicitly in terms of the molar concentration of equilibrant, [XI, and the effective equilibrium constant, Keff= [A][X]/ [AX]. Equation 8 is obtained by expressing FAX and F A (from eq 4) in terms of Keffand [XI
k ’ = ([Xl/([Xl + Kedlk’AX + (Keff/([Xl + KedIkb, (8) Equation 8 can be rearranged (2) to yield
k’(1 + Keff/[XI) =
AX + W,ff/[XI) kb,
(84
k k X = C1and kb, = 0 or klAX = 0 and k i = C2
“Y” = intercept + “X” slope
3. Degenerate Retention. Both forms of an analyte are equally retained or are unretained, i.e.
A plot of k’(1 + K,,/[X]) vs. (K,ff/[X])for a few values of [XI permits the measurement of klAXand k i (2). The measurement of these parameters is discussed in the article that follows. Derivation of cyEvIAX for t h e General Retention Case. From eq 8, we can write the capacity factor for a pair of analytes
k i x = k’A 2 0 OPTIMIZATION What to Optimize. The main goal of every practicing chromatographeris the resolution of all the analytes of interest in a given sample. In terms of fundamental chromatographic parameters, the resolution, R,,between two adjacent peaks is given by (10) R, = (1/4)
N1‘* efficiency
((a- l ) / c ~ I
{k’/(l + k?)
selectivity
retention
(5)
As noted, the efficiency, selectivity, and retention terms are roughly independent of each other and R, may be maximized by individually optimizing each term. Although present chromatographic theory provides some guidelines for achieving efficient separations and for main-
k’l
=
+ Kl)Ik’AX(l)+ {Kl/([XI + Kl)Ik’A(l) (9)
k’z = {[XI/([Xl + KZ)Ikb,X(Z)+ (KZ/([Xl + K2)Iki(2, (10) Substitution of eq 9 and 10 into eq 4 yields a = (([XI + K,)/([XI
+ Kz)I W l k i X ( 2 ) + KZk’,(2))/([X]k’AX(l)+ Klki(l))\ (11)
Substitution of eq 11 into eq 7 permits the optimum mobile
ANALYTICAL CHEMISTRY, VOL. 59, NO. 1, JANUARY 1987
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phase for the SCE-LC general retention case to be derived. The result is A[XI20,
+ B[X]OpT + C = 0
‘”T
(12)
where
A =
(k’Ax(2) -
kIA(2))kIAX(l,KZ- ( k I A X ( 1 ) - kIA(l,)kIAX(Z)K1 (13a)
B = 2(k’AX(Z)k’,(l) - kaX(l)k’A(2))K1K2
(13b)
c = ((kLX(2)Kl+ kL(Z)K&k’A(l)- ( k ’ A X ( l ) K 2 +
W -20
k IA(l,Kl)kIA(2)1KlK2 (13c) Roots for [XIOPT in eq 12 can be obtained by use of the well-known quadratic formula. The equilibrium constants and limiting capacity factors needed to evaluate the coefficients A, B, and C in eq 13 either are known or can be measured. Equilibrium constants can be measured in a variety of ways and have been critically evaluated for thousands of compounds (11). Limiting capacity factors can be measured chromatographically, using eq 8a (2),and spectroscopically (12). Note that only one physically significant root will be obtained when solving eq 12. In addition, a selectivity maximum (rather than a minimum) is always obtained a t [XIom The latter can be shown by assuming values for the equilibrium constants and capacity factors of the solutes and then calculating the selectivity via eq 11 as [XI is varied in small increments. If we assume that the limiting capacity factors of the solutes are the same, i.e.
k ’.4X(Z)= k ’AX(1) and
i(2)
=
= (K1K2k’A/kkX)1’2
(14)
(15)
where PKAVG = (pK1+ pK2)/2, Le., a value halfway between the effective pK’s of the solutes. From this equation we see that pXopT depends on not only PKAVG, as was intuitively predicted ( I ) , but also on k i x / k i , a ratio of the limiting capacity factors. We will frequently refer to k i X / k l A as the self-selectivity. Whereas conventional selectivity, a , refers to a ratio of overall capacity factors of two different solutes (eq 6), the self-selectivity refers to a ratio of limiting capacity factors of one solute. From eq 11, 14, and 15 it is clear that in SCE-LC the conventional chromatographic selectivity depends on the solutes’ self-selectivities. Derivation of ahlAXfor the Partial Retention Case. The derivation is detailed in Appendix A. We only summarize the results here. In contrast to the general retention case, the first derivative approach of eq 5 for optimizing selectivity fails to yield a nontrivial root in the partial retention case. Instead, a result inconsistent with our assumptions is obtained. However, by taking the limit as [XI approaches 0 and in the SCE-LC selectivity equation, we have shown in Appendix A that a asymptotically reaches a maximum at either very low or very high values of [XI. Furthermore, this maximum is given by Ly
=
1OlAPKI
-as
00
os
IO
15
IO
@-pK(AV6)
Flgure 2. Comparison of general retention (A,W) and partial retention (A, 0)cases in secondary chemical equilibria liquid chromatography. Curves were constructed by using eq 11 with ApK = 0.1 and k’,lk‘, = 0 (A),0.01 (A),100 (W), and m (0). T
ALPHA
20
-!a
-10
-os
PO pX-pK(AV6)
os
IO
15
20
Effect of k‘,lk’, on the selectivity in SCE-LC. Curves were constructed by using eq 11 with ApK = 0.1 and k’,lk‘, = 3.16 (0), 10 (O), 31.6 (0),100 (a),and m (A).The optimum equilibrant concentration (px,,,) varies according to pXopT = pK,,, + (1/2) log
Taking the negative logarithm (base ten) of both sides of eq 14 yields
PXOPT= PKAVG + (1/2) log (k’,m/k’,)
-in
Flgure 3.
’A(1)
then eq 12 reduces to [XIOPT
-15
(16)
where ApK is the difference in the equilibrium constants of a pair of solutes. Comparison of the General and Partial Retention Cases. Figure 2 illustrates the similarities and differences
(k’A d k ’
A).
in the general and partial retention cases. For both cases the direction in which pXopT is shifted away from PKAVG is the same. For k f A X / k i< 1, pKopT < PKAVG and vice versa. In contrast, the values of pK0m and a m differ for the two cases. For the general retention case a reaches a somewhat smaller maximum a t an intermediate value of pX; for the partial retention case a asymptotically approaches a somewhat larger maximum a t extreme values of p X (0 or m).
EXPERIMENTAL SECTION Our equipment, chemical supplies, and procedures are detailed in the article that immediately follows ( 1 3 ) . The measurement of the limiting capacity factors, k ;IAand k 2-, is also described in detail.
RESULTS AND DISCUSSION THEORETICAL CONSIDERATIONS Due to the symmetry of the curves in Figure 2 about klAX/k/A= 1,we will only discuss some pertinent relationships for k i x / k $ > 1. Similar trends will also be observed for k’Ax/k’A < 1. Also, for simplicity of presentation we show trends for k’AX(2) = k ’ ~ x ( ~=) k’Ax , and k’,(,, = k’,,,) = k’, although this is not a requirement for SCE-LC separations. Effect of k’,/k’, and ApK on Selectivity in SCE-LC. Figures 3 and 4 illustrate the effect of k LXIk ’,and ApK on the overall chromatographic selectivity, a. For fixed values of ApK and k f A X / k 2> 1, both p X o p ~and a increase as k’,/k 2 increases, as shown in Figure 3. As k’,/k’, gets very large, ahlAxapproaches 10IAPa,the upper bound for the partial retention case. For fixed values of k’,/k’A, as ApK increases there is an increase in a and no change in p X o P ~as , shown in Figure 4. Comparison of Present and Previous SCE-LC Selectivity Estimates. In prior work it was intuitively assumed
ANALYTICAL CHEMISTRY, VOL. 59, NO. 1, JANUARY 1987
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Table 11.. Selectivity Comparison at pX = ~ K A V and G at pX = pX0p~" ApK = 0.05 ApK = 0.10 ApK = 0.20 PX = PKAVGPX = PXOFT PX = PKAVGPX = PXOPTPX = PKAVGPX = PXOPTPX = PKAVGPX = PXOPT ApK = 0.025
klAx/klA
1.010 1.017 1.024 1.026 1.028 1.029 1.029 1.029 1.029
2 4 10 20 50 100 300 1000 a b
1.010 1.019 1.030 1.037 1.044 1.048 1.053 1.056 1.059
1.019 1.035 1.048 1.053 1.057 1.058 1.059 1.059 1.059
1.020 1.039 1.062 1.076 1.090 1.099 1.108 1.114 1.122
1.04 1.07 1.10 1.11 1.12 1.12 1.12 1.12 1.12
1.08 1.15 1.21 1.23 1.25 1.25 1.26 1.26 1.26
1.04 1.08 1.13 1.16 1.19 1.21 1.23 1.24 1.26
1.08 1.17 1.27 1.34 1.41 1.46 1.51 1.54 1.58
'Selectivities calculated via eq 11 using [XI = 10'PHAvG or 1 O - P X O ~ . p x o p ~= P K A V G + (112) log (k'AX/klA). bSelectivitiescalculated via eq A4 using [XI = 10-PKAvGor via eq A8 using [XI = 10-PxO~in Appendix.
'"T
Table 111. Comparison of Percentage Changes in k'and a! for a pX Shift of fO.10 at pX = pKAVC and at pX = pXopTa,* PX = PKAVG k'filki 3.16 10 31.6 100 1000 10000 m
-10
-15
.10
4 5
00
05
IO
IS
f10 f6 f11 f12 f12 f12 f12
Aa/a, %
f0.5 f0.9 f1.2 f1.3 f1.3 f1.3 f1.3
Px = PxOPTc Ak'lk', % Aa/a, % f7 f13 f17 f21 f24 f25 i~26~
2. 1. Intercept Criterion. One of the data points must be reasonably close to the y axis, i.e., (K/[X])MINC ci. 2. Slope Criterion. The data must be spaced reasonably far apart, i.e., (K/[X]wx > K/[X]M,, + C,. When the above criteria were employed, consistent results for pHopT and all other parameters were obtained. Values of Ci = 20 and C, = 20 provided the best results in our studies. The range of pHopT values obtained by using the good twopoint combinations was 4.56-4.68, in good agreement with pHopT = 4.57 obtained when all 12 points were used in the regression. Thus accurate estimates of p X o p ~via eq 8a can be made from only two measurements/compound, although these estimates will be very sensitive to systematic errors. Ionic Strength Effects. In order for the SCE-LC retention model (eq 2 and 4) to be valid, the mobile phase ionic strength must be kept reasonably constant. This has been
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ANALYTICAL CHEMISTRY, VOL.
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0
t
5 IO TIME (min)
Flgure 7. Optimized, isocratic separation of methylated phenols via
= +0.85 V vs. Ag/AgCI): mobile phase, 30170 acetonltrlle/aqueous carbonate-bicarbonate buffer (pH 10.90, ionic strength 50 mM); stationary phase, 10 km pdystyrene-dMnylbenz;temperature, 25 OC; Row rate, 1.3mL/mln; solute identiflcatkmand effective pKa,phenol (A, 9.81), mtresol (B, 9.921, pcresol (C, 10.09),o-cresol (D, 10.09),2,5dlmethylphenol (E, 10.23),2,4dimethylphenol (F, 10.42),2,6-dimethylphenol (G, 10.45), and 2,3,5-trimethylphenoI(H, 10.50). SCE-LC with electrochemical detecdlon (Ew
noted previously by others (1,2,8) and was also unmistakably clear from our experiments. A more pertinent question is whether ionic strength effects on the equilibrium constants of the analytes and the buffer can be ignored, i.e., whether tabulated thermodynamic values can be used without correcting for ionic strength. Our experiments were performed at low to moderate ionic strengths (20-60 mM). The pHom,,, data reported in Table VI show that both the quadratic (I = 0) and linear ( I = 0) solutions are only shifted slightly higher by ignoring ionic strength effects. From these results we conclude that as long as ionic strength is held constant at low to moderate values, only a slight error will result from using uncorrected thermodynamic equilibrium constants in the calculations of the mobile phase optimum. Quadratic vs. Linear Optimization Equations. It is necessary to use the exact, more complex quadratic optimization equation (eq 12), or can the simpler, linear approximation (eq 15) be used instead? If the limiting capacity factors are the same, i.e., if k i x ( 2 ) = k’,,(,, and k’,,,, = k’,,,,, then the linear optimization equation, eq 15, is exact. As the capacity factors diverge, the linear optimization equation becomes less accurate. The decision of whether to use the exact quadratic (eq 12) or the linear approximation (eq 15) must then be based on two factors: (1) how similar the limiting capacity factors are and (2) how much accuracy is required. Application of the Theory to Multicomponent Samples. Although the optimization theory we have presented only applies directly to the separation of two-component mixtures, there are many ways that this theory can be utilized in the separation of multicomponent samples. In some instances the success of a multicomponentSCE-LC separation is limited by one critical pair of compounds that are difficult to separate. In this situation the theory can be applied directly to this critical pair. In most instances, however, there may be more than two compounds that are difficult to separate. Even under these circumstances the SCE-LC optimization theory can still be useful. If the limiting capacity factors of all the compounds
are sufficiently similar, i.e., the compounds cluster in a plot of k’Ax vs. Iti,the order of elution will not change over the entire range of equilibrant concentration. Under these conditions the optimum equilibrant concentration will vary systematically for adjacent pairs of compounds as described by our theory. A good separation can thus be achieved if the equilibrant concentration is varied with time, Le., a secondary equilibrium gradient (stepwise or continuous) is applied. Even if the limiting capacity factors of the compounds are not sufficiently similar for this direct optimization, the theory will still be helpful in providing useful starting conditions for such empirical optimization techniques as the window-diagram approach (15) and the simplex optimization method (3). An example of a multicomponent separation is shown in Figure 7 , where phenol, the cresols, and four other methylated phenols are separated isocratically on a polystyrene-divinylbenzene column using a 3:7 acetonitrile/40 mM aqueous carbonate-bicarbonate buffer and are then detected electrochemically. The mobile phase pH (10.90) was optimized for the separation of m- and p-cresol, since they were the least well-resolved. The SCE-LC separation in Figure 7 is noteworthy for two reasons: First, our separation of m- and p-cresol is, to our knowledge, the first separation using a reversed-phase stationary phase ever reported. Second, the elution order in Figure 7 depends primarily on the (secondary) equilibrium constants of the compounds, as SCE-LC theory predicts for solutes with similar limiting capacity factors. Future Work. The theory that we have developed and verified for SCE-LC separations, e.g., p X 0 p ~= PKAVG+ (1/2) log ( k ’ a / k ’ J , is a fundamental approach for the optimization of selectivity. Our theory was derived for single-step equilibria and isocratic mobile phases and shows that the selectivity of SCE-LC has been substantially underestimated, although there is an upper limit given by a = lOl*fl, where ApK is the difference in equilibrium constants of two solutes. In the article that immediately follows we describe the effects of several variables on the self-selectivity ratio, k ’m/k 2, an important parameter in our SCE-LC optimization theory. We also discuss factors that influence retention and column efficiency in SCE-LC. In future reports we plan to extend the theory to gradient SCE-LC and multistep equilibria.
NOMENCLATURE A, AX: two forms of a solute in a secondary chemical equilibrium (Table I) A, B, C: terms in the quadratic solution yielding the maximum selectivity (eq 12, 13) Ci, C,: intercept and slope criterion parameters C1,Cp: arbitrary constants F: stoichiometric fraction (eq 2, 4) HA, A-: weak acid, conjugate base (Table I) K, Keff: effective equilibrium constant for a solute Kth: thermodynamic equilibrium constant for a solute k’: solute capacity factor (eq 2, 4,8) k k , k A x : capacity factors of a solute in one form (A) or another (AX) k ’ , x / k k : self-selectivity (eq 15) ktHA,k 2.: capacity factors of a weak acid in its protonated or dissociated forms (Table IV) k‘,/kk-: self-selectivity of a weak acid (Table IV) N column plate count (eq 5) pH: -log [H’] pHom: mobile phase pH that provides the optimum (maximum) selectivity for two solutes p K -log K pKAVG:average of the pKs of two solutes, i.e., pKAvc = (pK, + pK2)/2 (eq 15, Table IV) ApK difference in pK values of two solutes ApKm: minimum ApK need to base line resolve two solutes PXOPT: -log [XIOPT (eq l 5 )
ANALYTICAL CHEMISTRY, VOL. 59, NO. 1, JANUARY 1987
R,: resolution (eq 5) SCE: secondary chemical equilibrium (equilibria) SCE-LC: mode of liquid chromatography which utilizes SCE S,: separation factor (eq 18) tR: solute retention time (eq 19) [XI: molar concentration of the equilibrant, the species (e.g., H+)that controls the SCE (relative proportions of A and AX) [XIopT: molar concentration of equilibrant that provides the optmum (maximum) selectivity for two solutes (eq 12,14) a: selectivity (eq 6, 11, 16, A4, A6)
APPENDIX: DERIVATION OF THE MAXIMUM SELECTIVITY (AMAX) FOR THE PARTIAL RETENTION CASE Assuming that k’= > 0 and k’, = 0, the capacity factor expression of eq 8 reduces to
k ’ = ([Xl/([Xl
+ K)lk’,x
(All
For a pair of analytes the capacity factors will be
a=
(A3)
< k‘2. The selectivity will be
(([XI + KJ/([Xl + Kz)) (k’Ax(z)/k’Ax(i)) (A4)
Using the approach of eq 7, one obtains da/dCXI = ( E X 3
+ K 2 ) - 2 ( k ‘ ~ ~ ( 2 ) / k ‘ ~ ( , ) ) ( K 2= -0K 1( A) S ) A
8
c
Since [XI, K z , and k’AX(2)> 0, terms A and B can never be zero. Term C can be zero, but only if K 2 = K1. This is inconsistent with assumption 1 (pKI # pKz) made earlier. Thus there are no nontrivial real roots of eq A5. The lack of succew in fiiding [XI, via differential calculus suggests that the condition for maximum (or minimum) selectivity may occur at an extreme value of [XI. Assuming k’AX(2)= /z’~(,), eq A4 becomes a=
(1x1+ K,)/([XI + Kz)
(A@
Now the limit of CY can be taken as [XI approaches infinity and zero, respectively lim a (eq A6) = 1
[XI-*
lim a (eq A6) = K,/Kz = 10IApxl Kz < K1 (A8) [XI+Thus from eq A7 and A8 we conclude that CY increases motonically as [XI decreases for partial retention SCE-LC systems. It should be realized, however, that although eq A8 predicts a maximum selectivity a t [XI = 0, this is not really true since we have assumed that the analytes are unretained under these conditions. A practical, near-maximum selectivity can be achieved by using a relatively small concentration of X at which the analytes are reasonably retained. If we assume instead that k‘,,,,, k’,,,,, k’, f 0, and k’,,,) = k‘U(,) = K’a= 0, then a new set of equations analogous to eqs A1-A8 is obtained. From this new set of equations one may conclude that CY increases monotonically as [XI increases. Although this new set of equations predicts a maximum selectivity a t [XI = m , this also is not really true since we have assumed that the analytes are unretained at [XI = m. A practical, near-maximum selectivity can be achieved by using a relatively large concentration of X at which the analytes are reasonably retained. LITERATURE CITED
k’l = ([XI/([Xl + K,)Jk’Ax(l) where K2 < K1 in order for k’,
109
(A7)
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RECEIVED for review April 22,1986. Accepted September 11, 1986. J.P.F. gratefully acknowledges support provided through a National Bureau of Standards/National Research Council Research Associateship.
