Anal. Chem. 1990, 62,923-932
(with deep penetration into the cavity) with these particular CSPs. Rather, there may be a looser association with partial or incomplete inclusion. However, because of the absence of competing solvent effects, this association is more than adequate to achieve resolution in many cases. The question as to whether there is less inclusion with these PMHP-CD phases than with lipophilic alkyl-CD phases is difficult to answer. Any differences may be simply of degree. The differences even could be related to the types of probe molecules used to evaluate selectivity. More work is being done to evaluate these possibilities.
ACKNOWLEDGMENT Preparative work of C. T. Rao, Y. Xia, and J. Ljanionz is gratefully acknowledged. LITERATURE CITED (1) GiI-Av, D.; Feibush, B.; Charles-Slgler, R. Tetrahedron Lett. 1988, 1009. (2) Westley, J. W.; Halpern, B.; Karger, B. L. Anal. Chem. 1988, 40, 2046. (3) Pollock, 0. E.;Kawauchi, A. H. Anal. Chem. 1988, 40, 1356. (4) Konig. W. A.; Nlckolson, G. J. Anal. Chem. 1975, 47, 951. (5) Felbush, B.; GII-Av, E. Tetrahedron 1970, 26, 1361. (6) Parr, W.; Howard, P. Y. J. Chromatogr. 1972, 71, 193. (7) Armstrong, D. W.; Hen, S. M. CRC Crit. Rev. AMI. Chem. 1988, 19, 175. (8) 01, N.; Kitohara, H.; Doi, T. J. Chromatogr. 1981, 269, 252. (9)Oi, N.; Kitohara, H.; Dol, T. J. Chromatogr. 1983, 254. 282. (10) Frank, H.; Nicholson, 0. J.; Bayer, E. J. Chromatogr. Sci. 1977, 15, 174. (11) Frank, H.; Nicholson, G. J.; Bayer, E. Angew. Chem. 1978, 90, 90. (12) Armstrong. D. W. Anal. Chem. 1987, 59, 84A. (13) Smolkova-Keulemansova, E. J. Chromatogr. 1982, 251, 17. (14) Smolkova-Keulemensova, E.;Kralova, H.; Krysl, S.; Feltl, L. J. Chromarogr. 1982, 241, 3.
923
(15) Mraz, J.; Feltl, L.; Smolkova-Keulemansova, E. J. Chromatogr. 1984, 286. 17. (16) Smolkova-Keulemansova, E.; Newmannova, E.; Feltl, L. J. Chromatogr. 1988, 365,27s. (17) Koscielski, T.; Sybiiska, D.; Jurczak, J. J. Chromatogr. 1983, 280, 131. (18) Armstrong, D. W.; DeMond, W. J. Chromatogr. Sci. 1984, 22, 411. (19) Armstrong, D. W.; Ward, T. J.; Armstrong, R. D.; Beesley, T. E. Sclence 1986, 232, 1132. (20) Armstrong, D. W.; DeMond, W.; Aiak, A.; Hlnze, W. L.; Riehl, T. E.; Bul, K. H. Anal. Chem. 1985, 57. 234. (21) Schurig, V.; Novotny, H.-P. J. Chromatogr. 1988, 441, 155. (22) Schurig, V.; Novotny, H.-P.; Schmaijing. D. Angew. Chem. 1989, 101, 785. (23) Smoikova-Keulemansova. E.; Pokorna, S.; Feltl, L.; Tesarik, K. HRC CC, J. High Resolut. Chromatogr. Chromatogr. Commun. 1988, 1 1 , 10764. (24) Konig, W. A.; Lutz, S.; Mlschnik-Lublecke, P.; Brassat, B.; Wenz, G. J. Chromatogr. 1988, 447, 193. (25) Konig, W. A.; Lutz, S.;Wenz, G.; von der Bey, E. HRC CC. J . High Resolut. Chromatogr. Chromatogr. Commun. 1988, 1 1 , 506. (26) Konig, W. A.; Lutz. S.;Colberg. C.; Schmidt, N.; Wenz, G.; von der Bey, E.; Mosandi, A.; Gunther, C.; Kustermann, A. HRC CC. J. High Resolut . Chromatogr . Chromatogr . Commun . 1988, 1 1 , 62 1. (27) Konig, W. A.; Lutz. S.; Wenz, G.; Gorgen, C.; Newmann, C.; Gabler, A.; Boiand, W. Angew Chem. 1989, 101, 351. (28) Armstrong, D. W. Pitt. Con. Tech Program Book 1989, 001. (29) Pitha, J.; Pitha, J. J. Pharm. Sci. 1985, 74, 987-990. (30) Pitha, J.; Rao, C. T.; Llndberg, B.; Seffers, P., unpublished results. (31) Macfarlane, R. D.; Torgerson, D. F. Science 1978, 191, 920. (32) Armstrong, D. W.; Sequin, R.; McNeal, C. J.; Macfarlane. R. D.; Fendler, J. H. J. Am. Chem. SOC. 1978, 100, 4605. (33) Hinze, W. L. Sep. furif. Methods 1981, 10, 159. (34) S@ma Chemical Company Catalog; Sigma Chemical Co.: St. Louis, MO, 1989; p 693. (35) Armstrong, D. W.; Faulkner, J. R., Jr.; Han, S. M. J. Chromatogr. 1988, 452, 323.
RECEIVED for review September 12,1989. Accepted December 8, 1989.
Theoretical Study of System Peaks in Linear Chromatography Sadroddin Golshan-Shirazi and Georges Guiochon*
Department of Chemistry, University of Tennessee, Knoxville, Tennessee 37996-1600,and Division of Analytical Chemistry, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831 -6120
A complete theory of system peaks in hear chromatography Is presented. This theory relates the posltlon and Importance of the peaks of sample components and additives observed to the competitive Isotherms of the compounds Involved. I n the case when the additive gives a detector response while the sample components do not, the theory permits the predlctlon of the response factors. The theory permlts the slmulation of the method of elution on a plateau for the determlnatlon of competitive Isotherms.
INTRODUCTION In a number of practical implementations of liquid chromatagraphy, the mobile phase is a solution of one or several additives or modifiers in a solvent, or it is a mixture of solvents. Cases in point are the use of a strong solvent in adsorption chromatography and the use of complex ions, such as alkyl sulfates, alkyl sulfonates, or tetraalkyl ammoniums, in ion-pair *Author t o whom correspondence University of Tennessee.
should
be addressed, a t the
chromatography. Under these experimental conditions, the injection of a sample may result in more peaks appearing on the chromatogram than there are components in the injected mixture. The additional peaks are called system peaks. They result from the perturbation which the injection of the sample mixture causes to the equilibrium of the mobile phase additive(s) between the two phases of the chromatographic system. The sample components compete with the mobile phase components for access to the stationary phase. The addition of the sample components to the mobile phase causes a change in the equilibrium concentration of the additives in the stationary phase, because of this competition. Accordingly, the sample injection generates two sets of zones that will migrate along the column. The first set of zones are those corresponding to each of the sample components. The second set of zones correspond to the perturbations of the composition of the various mobile phase components (except for the “weak” solvent, which is assumed not to be retained). System peaks in liquid chromatography have been reported long ago (I).Vacancy chromatography, which is a special case of system peaks where the sample contains only the weak solvent, was described much earlier, in gas chromatography (2).The occurrence of system peaks has been explained by
0003-2700/90/0362-0923$02.50/00 1990 American Chemical Society
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ANALYTICAL CHEMISTRY, VOL. 62, NO. 9, MAY 1, 1990
solvent displacement, solvophobic interactions in the mobile phase, preferential evaporation of one component of the mixed mobile phase during storage ( 3 ) ,preferential solvation ( 4 ) , intramolecular hydrogen bonding, or hydrophobic interactions ( 5 ) . The origin and properties of system peaks have been reviewed recently by Levin and Grushka (6, 7). Helfferich et al. have published a detailed analysis of the phenomena which take place in a chromatographic system following the sample injection (8). Knox and Kaliszan have considered the theory of solvent disturbance peaks in their investigation of void volume determinations (9). Crommen et al. have discussed the response model for nonionic systems (10) and ionic systems (11). Recently, Stahlberg and Almgren have reviewed the theory of system peaks in ion-pair chromatography (12). The use of system peaks in ion-pair chromatography has been discussed by Sokolowski (13)and by Arvidsson et al. (14). One of their most common applications is in the UV detection of compounds that do not have the necessary chromophore (15, 16). A compound with a strong UV-absorption band is added to the mobile phase and the components of the mixture that do not absorb on the monitored wavelength are detected by their corresponding additive perturbation peak (Le., system peak). In previous papers, we have discussed on a theoretical basis the properties of system peaks at high sample concentration in the case of pure compounds (17) and of binary mixtures (18). We have calculated the profiles of large concentration bands under various sets of experimental conditions and predicted the occurrence of extremely unusual band shapes. In companion papers we have described sets of experimental conditions under which the various profile shapes predicted for either pure components (19) or binary mixtures (20) could be observed. The purpose of the present paper is an in-depth, quantitative discussion of the system peaks for nonionic systems, under experimental conditions such that the phenomenon appears to be linear. Although a number of important papers have been published in this area (3-14), there is still no theoretical model permitting a quantitative prediction of the system peaks. I t seems important to emphasize at this stage that system peaks cannot be dealt with properly inside the framework of linear chromatography. Strictly speaking, they should be discussed as a linear perturbation in nonlinear chromatography, unless the concentration of the additive is very small, which is exceptional. Admittedly, the injection of a small sample in a chromatographic system using a mixed mobile phase creates linear perturbations and Gaussian band profiles. The position of the solute band and of the system peak, however, depends on the additive concentration. Although the solute zones move at the velocity predicted by linear chromatography, with a constant apparent column capacity factor proportional to the slope of their isotherm a t the concentration origin, the mobile phase bands move at a velocity that is determined by the slope of the isotherm at the additive concentration in the mobile phase. Chromatography may be linear as far as the sample components are concerned, because their concentration is small. It is not linear as far as the additives are concerned. In fact, the whole purpose of using additives is to manipulate the retention times of the analytes as needed for their rapid and easy separation (17). By definition, this is no longer linear chromatography.
THEORY In order to discuss a multicomponent problem on a theoretical basis, we must write a mass-balance equation for each component of the system. Those include all the sample components and all the additives and solvent modifiers or so-called “strong” solvents. The only exception is the main component of the mobile phase, the lesser retained one, which
by convention is assumed not to interact with the stationary phase and serves as reference to define the equilibrium composition. In the following, we assume that the sample contains N components, numbered i, and that the mobile phase contains P additives, numbered j, and one “weak”, bulk solvent. I. Mass Balance Equations, The mass-balance equations are written as follows for the components of a multicomponent sample ( N components)
ac, + F-aq, = D.-a2ci u-aci + az at at az* i = 1, 2, ..., N and for the additives
ac, + ac, + F aqi -=
U-
at
az
at
j = 1, 2,
a2cj
Dj-
a22
..., P
In these equations, t and z are the time and column length, respectively, Ci and qi are the concentrations of the component i in the mobile phase and the stationary phase, respectively, Cj and q j are the concentrations of the additives in the mobile and the stationary phase, respectively, u is the mobile phase linear velocity, D is the axial dispersion coefficient, and F = u,/u, = (1 - e)/, (where 6 is the column packing porosity) is the column phase ratio. 11. Competitive Isotherm Equations. In order to be able to solve this system of partial differential equations, we need a series of relationships between the two sets of functions, Ci and qi, initial and boundary conditions. We shall assume here that the column efficiency is high and, thus, that the compositions of the two phases are always such that they are near thermodynamic equilibrium. In practice, Giddings has shown that the slight deviation from equilibrium acts as a contribution to the axial dispersion. We may assume that the composition of the stationary phase is given by the set of equilibrium isotherms of the compounds involved in our experiment, while the apparent axial dispersion coefficients account for the actual column efficiency. In the following, although any equilibrium isotherm could be used, we have chosen the Langmuir competitive isotherms. An essential feature of the isotherms which are used is that they must account for the competitive behavior of the compounds involved. The Langmuir isotherms have this property, while remaining simple and being easy to derive from measurements made successively on each of the components of the system. The Langmuir competitive isotherms of the sample components are written
aiCi qi = 1
+ C j b j C j + CibiCi
The equilibrium isotherm of the additives is written ajCj
” = 1 + C j b j C j + CibiCi
(2b)
The third term of the denominator, xibiCi, which could be neglected for the sample components, cannot be forgotten in the case of the additives. Its presence will explain the system peaks of additives associated to each sample component. 111. Boundary and Initial Conditions. The initial conditions describe the state of the column at the beginning of the simulated experiment. Usually, the column is filled with mobile phase, in equilibrium with the stationary phase, and contains no sample component. Hence the initial conditions are C,(z,O) = 0
(34
ANALYTICAL CHEMISTRY, VOL. 62, NO. 9, MAY 1, 1990
925
Table I. Matrix of the Differentials of the Isotherm Equations“ 1
...
2
1
aq1/ac,
0
2
0
aq2jac2
i
0
0
N
0
N+1
aqN+l/aCl
N +J
%N+j/aCl
N
1
... 0 ... o
... ... ... ...
N+l
N
N
e..
...
0 0
0 0
0
0
... ...
.-.
+j
N
0 0
... ...
0 0
0
...
0
+P
0
... ...
0
aqN/acN
0
...
0
...
0
aqN+l/ac2
***
?qN+l/aci
‘**
aqN+l/acN
aqN+l/aCN+l
***
aqN+l/aCN+j
“‘
aqN+lIaCN+P
aqN+j/aC2
**‘
aqN+jIaci
‘**
?qN+jIacN
aqN+j/aCN+l
:’’
aqN+j/acN+j
“*
aqN+j/acN+P
bqN+p/aci
. . . a.q N + P / a c N
’%N+P/acN+l
’**
aqN+P/aCN+j
***
aqN+P/aCN+P
+ P aqN+PIaclaqN+plac2...
aqi/aci
* * e
“Differentialsof the stationary phase concentration of the kth compound (row), by respect to the mobile phase concentration of the kth compound (column). which expresses that the column contains no sample component, and Cj(2,O)
=
cjo
(3b)
while q,(r,O) is given by the corresponding eq 2b. The latter equation states that the column is equilibrated with a mixed mobile phase of composition Cj”. The boundary condition describes the perturbation made a t the column origin. If we inject a sample pulse of a mixture containing N components, with a certain composition, the boundary condition for the ith sample component is
Ci(0,t) =
c:
0 I t I t,
(44
with
t 1 t,
Ci(0,t) = 0
(4b)
where C: is the concentration of the component i in the sample and t, is the duration of the rectangular injection pulse. There are, of course, N conditions such as eqs 4a and 4b. Since the mobile phase pumped in the column contains a constant concentration, Cj”, of the j t h additive, we also have either Cj(0,t) =
cjo
(4c)
if the sample also contains the additive at the same concentration as the mobile phase, otherwise Cj(0,t) = 0
0 I t I t,
to the retention mechanism. We assume also that the perturbation caused by the injection is small. Under constant temperature and pressure, the stationary phase concentration perturbation for the ith component of the system is given by the following expansion:
k = 1, 2, .... N
+P
(5)
There are N + P such equations (one for each of the N components of the sample and the P additives). For small perturbations such as those we are concerned with in linear chromatography, the derivatives can be evaluated at the initial, unperturbed, steady state (zero concentration for the sample components and initial concentration, C y , for the additive). The differential element is replaced by the perturbation. Then, eq 5 becomes ‘qk =
(3) ac, cj,c,o *c1+(%) acz
Ac,+ ...
cj=cjo
k = 1, 2, .... N
+P
(6)
It is convenient to summarize the whole set of equations under matrix form (23)
(7)
A$ =
(44
and
if the sample contains no additives. Conditions 3 and 4 are the most common sets of initial and boundary conditions. Other conditions can be written for specific problems, e.g., for vacancy chromatography. The system of eqs 1 to 4 cannot be solved analytically, in closed form, but it is possible to write a computer program based on the finite difference methods, which calculate the solution corresponding to any complete set of experimental conditions. I t is especially simple to account exactly for the finite column efficiency, by a proper choice of the integration increments of space (62 = HETP) and time (at = 2Ht,/L). The calculation errors are equivalent to the diffusion term corresponding to a finite column efficiency (21, 22). IV. Properties of System Peaks. In all the calculations discussed here, we have assumed that we have an N + P + 1 component system, the sample containing N components and the mobile phase being a mixture containing P additives dissolved in or “strong” solvents mixed with a main, “weak” (unretained) solvent. These “strong” solvents or additives are retained and compete with the sample components for access
In eq 7 , Atj and AC are ( N + P) X 1 column vectors, representing the perturbations caused by the injection to the compositions of the stationary and the mobile phase, respectively, i.e., the changes-in concentrations of the N + P components of the system. 0is an ( N P) x_(N+ P) square matrix, whose element is aqk/aC1. Finally, b is a 1 X ( N + P) row vector whose elements are the N + P partial derivatives of the stationary phase concentration qk of the kth component of the system (components and additives). All the matrix elements are derived by differentiation of eqs 2 and evaluated for the initial, unperturbed composition of the stationary phase. The matrix can be divided in four parts (see Table I): square N x N and square P X P matrices and two rectangular matrices, N X P and P X N matrices. The square N x N matrix contains the differentials dqi/dCc of the concentration of a sample component in the stationary phase with respect to the concentration of another sample component in the mobile phase. Since all the sample components are present a t infinite dilution in the case of an analytical problem, eq 2a shows that all the cross-partial differentials are zero and only the diagonal elements of the matrix are different from 0. These nonzero elements are equal
+
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ANALYTICAL CHEMISTRY, VOL. 62, NO. 9, MAY 1, 1990
to the apparent column capacity factors for each solute (uI/(l + &blC,)). The square P X P matrix contains the differentials dql/aCl! of the concentration of an additive in the stationary phase with respect to the concentration of another additive in the mobile phase. As shown by eq 2b, all the elements, diagonal or not, are different from zero. The elements in the rectangular N x P submatrix are the differentials d q , / X , of the concentration of a sample component in the stationary phase with respect to the concentration of an additive in the mobile phase. As shown by eq 2a, these differentials (-utb,Cl/(l + C,blC,)2) are zero because the initial concentrations, C,, are 0 for sample components. Finally, the elements in the rectangular P x N matrix are the differentials dq,/aC, = -a,b,Cl/(l+ C,blCl)2of the concentration of an additive in the stationary phase with respect to the concentration of a sample component in the mobile phase. As shown by eq 2b, these elements are all different from zero. Accordingly, the perturbations of the concentrations of the sample components are decoupled
l S i l N
l S k l N + P
(9)
where Ckois the initial value of the concentration of the kth compound (e.g., 0 for the sample components). For each of these N sample components, only one single peak is observed on the chromatogram. This peak appears a t the time corresponding to the column capacity factor of this compound a t infinite dilution ( k : = k',,i/(l CjbjC,), where Cj is the concentration of the j t h additive). For the additives, on the contrary, all the cross-partial differentials are different from zero and there is no decoupling, unless their initial concentration is negligibly small. The perturbation is such that
+
kIN+P,
l l j l P
(10)
+
Thus, for a mobile phase containing N P constituents ( N sample components and P additives) and even if the perturbation is extremely small, there will be N + P bands for each additive. These bands are called the system peaks. The system peaks corresponding to different additives and a given component are not resolved from each other. V. Chromatographic System Containing a Single Additive. Suppose that we inject a sample containing N components in a binary mobile phase. The single additive is identified with the subscript s. In this case, eq 9 applies, unchanged. Accordingly, to each sample component corresponds a single peak, for which the column capacity factor is given by
As expected in linear chromatography, each sample component gives a peak that elutes a t the same time as a pulse of the pure component. For a Langmuir isotherm we have
Since we have only one additive, eq 10 becomes
Equation 13 is valid at any point in the column and particularly at the exit. A perturbation, q8, of the stationary phase concentration of the additive takes place every time a perturbation of the mobile phase concentration of a sample component, i , takes place. These perturbations move along the column at the constant component velocity (u/(l + kb,,)). When they reach the end of the column, they are accompanied by the corresponding solutes which coelutes with them, as a system peak. An additional perturbation q, accompanies the perturbation of the mobile phase concentration of the additive, C,. This perturbation, which we call the additive system peak, cannot be treated within the framework of linear chromatography. It moves at the velocity u / ( l + k',), where k', is the apparent column capacity factor of the additional peak. It is proportional to the slope of the isotherm a t C, = C,O and is given by k f S= F(
$)
c,=c,o
The equilibrium isotherm of the additive is not linear, unless its concentration is very low, and the apparent capacity factor of the additive is different from the capacity factor a t infinite dilution. For a Langmuir isotherm, it is given by (see eq 2b)
Thus, the additive gives N + 1 peaks. In many cases, the literature refers to the additive peak only as the system peak. The N + 1 additive peaks, however, result all from the perturbation caused by the injection of the sample to the equilibrium of the additive between the two phases of the chromatographic system and they should all be called, collectively, system peaks. These system peaks are not detected when a detector selective for the sample components is used, which is the general case in analytical applications of chromatography and explains why they are generally forgotten. If, however, a nonspecific detector is used, we see only one extra peak, the one that is eluted a t the time characteristic of the additive. The N other peaks are eluted at the same time as each of the sample component peaks, interfere with them, and, in general, cannot be distinguished from them. In the case when a detector selective of the single additive is used, as in the case when the sample components have no chromophore and a strongly UV absorbing compound is used as additive (15),the N + 1 additive peaks are observed, which is the basis of this detection scheme. The position, sign, and relative importance of the system peaks associated to the sample components can be obtained from eq 13 if we assume the equilibrium isotherms of all the compounds involved to be Langmuirian (see eqs 2a and 2b). Since the competitive Langmuir model assumes that the column saturation capacity is the same for all compounds involved (q8 = ai/bi = a s / b 8 )we , can write each partial differential for the additive as follows:
(
~ ) C ~ = C
-ai -bia,C, =--- b,C* ; = (1 + b,C,)2 1 + b,C, 1 + b,C,
where 4, is the fraction of the surface that is covered by the additive. If the column saturation capacities are different for some compounds, a constant proportionality factor has to be included in the result. Combination of eqs 13 and 16 gives
ANALYTICAL CHEMISTRY, VOL. 62, NO. 9,MAY 1, 1990
927
+ ‘0 9
T*
9
11
‘1. 0
E?
W N
L; 0
0
2 9
v
I
2’
a
6
4
10
12
8
6
4
Time ( s e d Flgure 1. Calculation of system peaks. Case of a weakly sorbed additive: 1, compound injected; 2, additive. Column length, L = 25 cm; phase ratio, F = 0.215; flow rate, 1 mL/min; dead time,t o = 205.2 s; column efficiency, N o = 5000 theoretical plates; Langmuir comCb,C, b,C,), with a = 12, a, petitive isotherms, 9, = a,C,/(l = 0.5s 1, qw = 2, b, = a , / 9 , . Concentration of additive in the mobile phase, C, = 0.01; injected amount of solute, 0.167 pmol, dissolved in the mobile phase. Peak 1, addtive system peak; peak 2, sample and component peaks.
+
+
On the other hand, we know that for the system peak which coelutes with the ith sample component, we have A48
F-
= kb,iaptand ACk = 0 except for k = i (18)
ACS
Thus, for the system peaks that coelute with the sample components, we obtain
which can be rewritten as
AC, -
ACi
-
4& b,i.pt k’s - kb,i.pt
10
12
Time ( s e d
(20)
This equation has been derived previously by Crommen et al. (IO). According to this equation, the system peak associated with a sample component corresponds to a positive fluctuation of the additive concentration if the component considered is eluted before the additive system peak (i.e., if k’,> kb,ia,). Conversely, if the peak of a component is eluted after the additive system peak (k’, < kb, ), ita associated system peak will be negative. Equation 20 s o w s that the area of a component system peak is proportional to the perturbation, i.e, to the amount of each component that has been injected. This equation gives also the response factor in the case of an additive used for detection purposes. From this equation we
Flgure 2. Calculation of system peaks. Case of a weakly sorbed additive. Same as Figure 1, except the solute is dissolved in the pure weak solvent. There is no additive in the injected sample.
derive the relationship between the ratio of the concentrations, C, and C,, of two components in the sample and the ratio of the peaks areas, A, and A,, observed for their associated system peaks, when the detector does not respond to these two components -A, =-=-
A,,
AC,,
kb,l klg - kb,,! C, k b,,, k lg - k b,, Clt
(21)
The sign and size of the additive system peak can be obtained by a mass balance of the additive. The amount of additive injected (zero if the same is dissolved in the mobile phase, negative if the sample is dissolved in an additive-free solvent) is equal to the sum of the amounts corresponding to all the component system peaks (derived from eq 20) and to the additive system peak.
RESULTS AND DISCUSSION We discuss first the results obtained by calculations carried out under a variety of experimental conditions corresponding to the cases discussed above. Then we compare our theoretical conclusions to a number of experimental results that have been reported in the recent scientific literature. Finally, we discuss certain characteristic properties of vacancy chromatography. I. Results of the Numerical Calculations. The results of the theoretical analysis reported above are valid for any equilibrium isotherm, i.e., for any functional dependence of the amount of a compound found at equilibrium in the stationary phase on the composition of the solution used as mobile phase. Therefore, these results are valid for all modes of chromatography. Their numerical applications, however, require that a proper representation of the distribution function is available. We have chosen, for the sake of simplicity, the competitive Langmuir isotherm model, in spite of theoretical and practical
928
ANALYTICAL CHEMISTRY, VOL. 62, NO. 9, MAY 1, 1990 t
‘0 9
T*
9
0 N
h
i
L. 0 E:
Y
v
0
0
9
0
o I
a
IO
12
14
,
16
9
N
18
:
Time ( s e d Figure 3. Calculation of system peaks. Case of a low concentration of a strongly sorbed additive. Same conditions as for Figure 1, except a , = 2a,.
limits to its validity. When experimental results are simulated, the best set of equilibrium isotherms available to account for the phase equilibrium of the compounds investigated is easily introduced in the general equations. Figure 1shows the chromatogram calculated in the simplest and most general case, when a binary solution (additive s dissolved in nonadsorbed weak solvent) is used as mobile phase, the sample is a pure compound (i) and the additive is less strongly retained than the sample (a, < ai).Under such conditions, we have
Therefore, the additive system peak appears first, followed by the sample peak accompanied by its associated system peak. The latter system peak is negative, as predicted by eq 20. The former system peak is positive and equal in size if, as assumed for Figure 1, the sample is dissolved in the mobile phase (zero amount of additive injected). On the contrary, if the sample is pure or dissolved in a solvent containing no additive, the additive system peak is also negative, as shown in Figure 2. Intermediate cases are possible, of course, including the one where a small amount of additive is injected and the first peak disappears entirely. On these two figures, as well as on the following ones, the positions of the sample component and the additive system peaks are the same as those predicted by the theory. The sample component has a column capacity factor given by k bapp = k b / ( l + b,C,)). For the additive system peak, k4 = Fa,/(l + b,C,)*). The sign of the system peaks that coelute with the sample component agrees with eq 20. Figure 3 shows a chromatogram calculated for conditions identical with those of Figure 1,with the only difference that this time the additive is more strongly adsorbed than the pure
a
10
12
14
16
18
Time ( s e d
Flgwe 4. Calculation of system peaks. Case of a low concentration of a strongly sorbed additive. Same as figure 3, except the solute is dissolved in the pure weak solvent. There is no additive in the injected sample.
sample (a, > aL). In this case, we have selected an additive concentration, C,, which is small enough and we have
(23) The additive system peak appears after the component peak and its associated system peak. The first (i.e., the component) system peak is positive, as predicted by eq 20. The second (Le., the additive) system peak is negative and equal in size to the first one, since we assume that sample to be dissolved in the mobile phase. In Figure 4, identical assumptions have been made, except that the sample is now dissolved in a pure solvent, with no additive. The component peak and its associated system peak are identical with those shown in Figure 3. The additive system peak, which is negative, is larger in Figure 4 than in Figure 3, since a vacancy of additive has been injected with the sample. Figure 5 shows a chromatogram calculated for the same conditions as those used for Figure 3, except that the concentration of the additive is increased so that, although the additive is more strongly adsorbed than the pure sample (a, > a J , we have
Now, the additive system peak is eluted before the sample peak, since it is the slope of the additive isotherm at C, = Ca,O which determines its apparent column capacity factor, not the slope of the additive isotherm a t the origin (i.e., k , is equal to F a , / ( l + bsCs)P,not to Fa,). The system peak associated with the sample is eluted later, with the sample peak, and, according to eq 20, is negative. With the assumption that the sample is dissolved in the mobile phase, the additive system
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Time ( s e d Figure 5. Calculation of system peaks. Case of a high concentration of a strongly sorbed additive. Same conditions as for Figure 3, except c, = 0.2 M.
peak is positive and equal in size to the sample system peak. 11. Comparison w i t h t h e Experimental Results Published in t h e Literature. The fact that N + 1additive peaks are observed when an N component sample is injected in a mobile phase containing an additive is the basis for a wellknown scheme for the UV detection of compounds having no chromophores (15,16,24-26). An additive that has a small or negligible effect on the relative retention and resolution of the sample components but has a strong extinction coefficient on an appropriate wavelength is added to the mobile phase. Upon injection of a sample, a chromatogram is obtained, with N system peaks corresponding to the N mixture components and an extra system peak for the additive. The areas of each peak associated to a component of the mixture are proportional to the concentration of this component in the sample. This technique is widely used in ion-pair chromatography (24), in ion-exchange chromatography (25),using UV absorbing ionic additives, and in reversed-phase chromatography, using neutral additives (26). Figure 6 shows a comparison between an experimental chromatogram (Figure 1,ref 26) and the result of a simulation of a chromatogram for a multicomponent sample and an additive in the mobile phase. The calculation was carried out by using Langmuir competitive isotherms (see eqs 2a and 2b) and numerical values of the parameters which permit the best approximation of the experimental results. In this experiment, a sample containing acetonitrile, N,N-dimethylformamide, ethyl formate, 2-butanol, ethyl acetate, and 1-pentanol is eluted on a C-18 chemically bonded silica phase (Nucleosil C18), using a water-methanol solution containing 0.000 25 M salicylamide, the UV-absorbing additive. The agreement is excellent and demonstrates the validity of our theoretical approach. The system peaks associated with the first five components, which are eluted before the additive system peak
Figure 6. Calculation of system peaks. Simulation of the experimental results of Heren et al. (26). Column length, L = 10 cm; dead time, t o = 160 s; phase ratio, F = 0.25; column efficiency, N o = 5000 theoretical plates, competitive Langmuir isotherms, with 9sal= 2, b , = a,/q,, for aH compounds. Column capacity factor for the different compounds: (1) acetonitrile, ko,, = 1; (2) N,Ndimethylformami, ko,2 = 2; (3) ethyl formate, ko,3 = 3; (4) 2-butanol, k o , , = 4; (5) ethyl acetate, ko,5= 6; (6) salicylamide (additive), ko,6= 11; (7) 1-propanol, ko,7= 15. Concentration of additive (peak no. 6) in the mobile phase, C, = 0.000 25 M. Amount of compounds injected, 0.835 pmol of compounds 1 to 5 and 0.417 pmol of component 7. Compare to Figure 1 of ref 26. Since the solutes have no chromophores, the chromatogram reported is the additive concentration profile at column outlet.
(no. 6), are positive, as predicted by eq 20. The system peak associated with the last component (no. 7) is negative. The direction (sign) of the additive system peak is determined by the mass balance of this compound. In Figure 6 it is negative. On the contrary, Figure 7 has been obtained by using exactly the same experimental conditions as for Figure 6, except that the amount of 1-pentanol injected has been doubled. The additive system peak is now positive. Equation 21 predicts the relative response factor of each component. The mass balance of the additive permits the prediction of the sign and size of the area of its system peak. Our results permit a quantitative explanation of the system peak phenomenon and an accurate prediction of the chromatograms, provided the competitive isotherms are available. The relative response factors, which are important for quantitative analysis, can be deterined directly from the chromatograms, however. Their determination does not require any prior information regarding the equilibrium isotherms. 111. Vacancy Chromatography. In this particular implementation of chromatography (2,27),a dilute solution of the sample in an appropriate mobile phase is pumped through the column. A steady-state equilibrium between the mobile and the stationary phases is reached. When an analysis is required, a sample of pure mobile phase is injected in the column. This vacancy pulse leads to a series of negative peaks.
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ANALYTICAL CHEMISTRY, VOL. 62, NO. 9,MAY 1, 1990
:
u)
'0
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0
_1
'1-
E"
Y C
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0 0 N
i
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C
t
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Time ( s e d Flgure 7. Calculation of system peaks. Siulation of the experimental resuits of Heren et at. (26). Same conditions as for Figure 6, except amount of compounds injected, 0.835 pmol for all compounds 1 to 5 and 7.
The problem is nearly identical with the one discussed previously. The only change is in the boundary (eqs 25 and 26) and initial (eq 27) conditions which become here Ci(0,t) = co,i t > t , (25)
Ci(0,t) = 0 Ci(X,O) =
0 It It ,
co,i
x
>0
As when a small pulse of sample is injected in a mobile phase that does not contain any of its components, the elution band remains Gaussian if the sample of pure mobile phase is small enough. However, the system remains uncoupled only if the concentrations of the sample components in the mobile phase are so small that the equilibrium isotherms remain truly linear. Very small deviations of these isotherms from linear behavior are sufficient to trigger the apparition of nonlinear behavior (21,28). When the system is uncoupled, the velocity of each concentration perturbation is equal to the velocity of the band in linear chromatography and the position of the peak observed for each component of the mixture gives its limit column capacity factor at infinite dilution. If the concentration of the sample components, or of some of them, is too large, the equilibrium behavior is no longer linear, each band moves a t a velocity that depends on the mobile phase composition, even if the vacancy injected is very small and the signal observed remains Gaussian or nearly so. Figure 8 shows the chromatogram calculated for the injection of a small vacancy in a chromatographic system when the mobile phase contains two additives at very low concentrations (linear isotherms). The two peaks are decoupled, each elutes with an apparent column capacity factor that is equal to the column capaciy factor a t infinite dilution. The chromatogram is identical with the one observed for a small injection of the binary mixture in the same chromatographic
5
10
15
Time ( m i d Flgure 8. Vacancy chromatography. Calculation of the chromatogram obtained with two compounds dissolved in a solvent (moblle phase). Case when the additive concentratlons are low. Same column as for Figure 1. Competitive Langmuir isotherms, 9, = ap,/(l 4- cb,C,), with a , = 12, a = 1.5, 9set = 2, b, = a,/9,. Constant concentrations of the two compounds: C, = C2= 0.0001 M. Injected amount: 4.9 pmol of pure solvent.
system, but with no sample dissolved in the mobile phase, except that the peaks are negative. In Figure 9, we show the vacancy chromatogram obtained for a single compound, when the isotherm is not linear. Although the vacancy pulse is very small and the peak recorded is Gaussian, its retention time depends on the concentration of the component in the mobile phase. The apparent column capacity factor is proportional to the slope of the isotherm at the actual concentration in the mobile phase ( k ' = F dq/dC, not k' = Aq/AC). Thus, changing the additive concentration and measuring the value of k ' at increasing concentrations gives dq/dC and permits the determination of the equilibrium isotherm which is given by q = S 0c k ' d C This is the principle of the method of elution on a plateau or step and pulse method (27, 29). When the number of components in the mixture studied increases, the situation becomes complicated since the system is now coupled and the number of peaks observed increases rapidly. If we assume that we have a chromatographic system where the mobile phase contains N components (weak solvent included) and that we inject a vacancy pulse of only a single one of these components, we observe the apparition of N 1 peaks. The velocities of these peaks are given by the eigenvalues of the matrix of the differentials aqi/aCj of the competitive isotherms corresponding to the experimental conditions (composition of the mobile phase). If we inject simultaneously vacancy pulses of several components (up to N - l),the negative peaks coelute with each other a t the N - 1 positions we have just determined and they interfere.
ANALYTICAL CHEMISTRY, VOL.
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Time (mid
Time (mid Figwe 0. Vacancy chromatography. Calculation of the chromatogram obtained with one compound dissolved in a solvent (mobile phase). Case when the additive concentration is important. Same column as for Figures 1 and 8. Langmuir Isotherm, q = alCl/(l -IblC,) wlth a =, 12,qsclC = 2, 6 = a ,/qrat = 6. Constant concentration of the additive, C, = 0.01;injected amount, 4.9 pmol of pure solvent.
62,NO. 9,MAY 1, 1990 931
Figure 10. Vacancy chromatography. Calculation of the chromatogram obtained with two compounds dissolved in a solvent (mobile phase). Case when the additive concentrations are Important. Same conditions as for Figure 8, except C1= C, = 0.01 M.
In Figure 10 we show the chromatogram calculated for the same components, under the same conditions as used for Figure 8, except that the concentrations of the additives have been increased from 0.0001 to 0.01 (Co,l = Co,z= 0.01 M). In this case, the matrix j3 (eq 7) is a 4 X 4 matrix and the mobile phase concentration is a 2 X 1 vector. As a result of the vacancy pulse injection, we have two perturbations, one for each component. Each perturbation contains two peaks, but a chromatograph would detect only two peaks, not four. Each of these two peaks is the result of an interference between the signals (positive or negative) of the two components which take place at that time. The velocities, hence the retention times or volumes of these pulses, depend on the competitive isotherms of the two components and on the mobile-phase composition. They are given by the eigenvalues of the matrix
r
1
2
Thus, the retention times of the two signals observed,' which have a Gaussian or nearly Gaussian profile in most cases if the vacancy pulse is small enough, are not simply related to the equilibrium isotherms of each component. Furthermore, the relative size of the two signals eluted a t each retention time change markedly with the composition of the mobile phase and the coefficients of the isotherms. As an example, Figure 11 shows the calculated chromatographic signals of the two components used for Figures 8 and 10, but their concentrations in the mobile phase are C,J = Co3 = 0.20 M. The retention times of the two pulses observed on
1
2
3
4
5
6
Time (mid Figure 11. Vacancy chromatography. Calculation of the chromatogram obtained wlth two compounds dissolved in a solvent (mobile phase). Case when the additive concentratlons are large. Same conditions as for Figure 8, except C, = C2 = 0.2 M.
932
Anal. Chem. 1990. 62, 932-936
Figures 8, 10, and 11are different, as well as their relative size. Under certain sets of experimental conditions the second signal may disappear, by cancellation of the responses of the detector for the two concentration perturbations. We have discussed above the properties of solutions of one or two solutes dissolved in a pure solvent used as mobile phase. We have shown that the velocity of each pulse can be related to the derivatives of the isotherm. It is easy to generalize our results to the case of a N component solution. Because of the coupling effect between the components of the mobile phase, the velocity eigenvalues are related to the slopes of the tangents to the multidimensional surface in the N - 1composition path directions. These slopes can be calculated when the isotherm surface is known. Conversely, the systematic measurement of the retention times of very small vacancy pulses for various compositions of the mobile phase may permit the determination of competitive equilibrium isotherms, provided a suitable set of isotherm equations is available. Least-squares fitting of the set of slope data on the isotherm equations allow the calculation of the isotherm parameters. Finally, the method discussed here for the prediction of system peaks and of vacancy chromatograms can be used as well for the calculation of the column response to more complex injections combining positive pulses of certain compounds and negative pulses of others in a mobile phase containing some of these compounds. Only the initial and boundary conditions have to be changed.
ACKNOWLEDGMENT We gratefully acknowledge support of our computational effort by the University of Tennessee Computing Center. LITERATURE CITED (1) Solms, D. J.; Smuts, T. W.; Pretorius, V. J. Chromatogr. Sci. 1971, 9 , 600.
(2) Zhukhovitskli, A. A.; Turkel'taub. N. M. Wl.Akad. Nauk 1962, 743, 646. (3) McCormick, R. M.; Karger, B. L. J . Chromatogr. 1980, 199, 259. (4) Berek, D.; Bleha, T.; Pevana, Z. J . Chromatogr. Scl. 1978, 14, 560. (5) Perlman, S.; Kirschbaum, J. J. J . Chromatogr. 1988, 357, 39. (6) Levin, S.; Grushka, E. Anal. Chem. 1988, 58, 1602. (7) Levin, S.; Grushka, E. Anal. Cbem. 1987, 59, 1157. (8) Hemerich, F.; Klein, G. Multicomponent Chromatography. A Theory of Interferences; M. Dekker: New York, 1970. (9) Knox, J. H.; Kaliszan, R. J. Chromatogr. 1985, 349, 211. (10) Crommen, J.; Schiil, G.; Herne, P. Chromatographie 1988, 25, 397. (11) Crommen, J.; Schill. G.; Westerlund, D.; Hackzell. L. Chromatographla 1987, 24, 252. (12) Stahlberg, J.; Aimgren, M. Anal. Ch8m. 1989, 67. 1109. (13) Sokolowski, A. Chromatographla 1986, 22, 177. (14) Arvidsson, E.; Crommen, J.; Schill, G.; Westerlund, D. J. Chromatogr. ism. 461.429. . - .. (15) Bilingmeyer, 8. A.: Deming, S. N.; Price Jr., W. P.; Sachok, 9.; Petrusek. M. J. Chromatoor. 1979. 786. 419. (16) Denkert, M.; HackzelcL.; Schill, G.; Sjogren, E. J . Chromatogr. 1981, 218, 31. (17) Golshan-Shirazi, S.; Guiochon, G. J . Chromatogr. 1989, 461, 1. (18) Golshan-Shirazi, S.; Guiochon, G. Anal. Chem. 1989, 6 7 , 2373. (19) Golshan-Shirazi, S.; Guiochon, G. J . Chromatogr. 1989, 467, 19. (20) Golshan-Shirazl, S.; Guiochon, G. Anal. Chem. 1989, 67,2360. (21) Guiochon, G.; Golshan-Shirazi, S.; Jaulmes, A. Anal. Chem. 1988, 6 0 ,
----.
1856.
(22) (23) (24) (25) (26) (27)
Lin, 9.; Guiochon. G. S e p . Sci. Techno/. 1989, 24, 31. Glover, C. J.; Lau, W. R. AIChf J. 1983. 29, 73. Gnanasambandan, T.; Freiser, H. Anal. Chem. 1982, 5 4 , 1262. Small, H.; Miller, T. E., Jr. Anal. Chem. 1982, 54, 462. Heren. P.; Renson, M.; Crommen, J. Chromatographie 1984. 79, 274. Reilley, C. N.; Hildebrand, G. P.; Ashley, J. W., Jr. Anal. Chem. 1982, 34, 1198. (28) Jauimes, A.; VidaCMadjar, C.; Ladurelli, A,; Guiochon, G. J Phys. Chem. 1984, 88, 5379. (29) Helfferich, F.; Peterson, D. L. Science 1963, 742, 661.
RECEIVED for review November 29,1989. Accepted January 24, 1990. This work has been supported in part by Grant CHE-8901382 of the National Science Foundation and by the cooperative agreement between the University of Tennessee and the Oak Ridge National Laboratory.
Flow System for Direct Determination of Enzyme Substrate in Undiluted Whole Blood T. Buch-Rasmussen Radiometer Medical AIS, Emdrupvej 72, DK-2400 Copenhagen, Denmark
Enzyme sensors requiring reagent and contrdled pH to detect substrate at non-steady-state conditions are described. Hematocrit dependence in sample transportatlon, in sample and reagent mixing, and In sample dialysis Is mlnknired in the system by the use of segmented sample injection, membrane deposited reagent, and membranes of low permeablllty. The clinical features of the flow system are illustrated by a B-0glucose determination. Other aspects of the flow system are illustrated by L-lactate and creatinine detennlnatkns. All three assays end up with a NADH detection at a chemically modified electrode (CME).
INTRODUCTION A time-consuming step in blood analysis is the often necessary separation of blood cells and plasma. This separation
is done to avoid the blood cells, which influence many sample operations in flow systems. Harrow et al. (1,2),when determining pH and COPcontent in whole blood, were the first to point out the difficulties associated with the varying hematocrit level in blood samples, observing that the measurement was unreliable unless minimum dispersion in a flow system was accomplished. However, at minimum dispersion the power to do chemistry is limited. The errors are related to a dilution effect and a hydrodynamic effect of the red blood cells on the dispersion. Petersson et al. (3)demonstrated in a urea determination that it is possible to perform chemistry on whole blood samples without interference from the blood cells, by use of time/space kinetic discrimination in a flow injection analysis (FIA) system with minimal dispersion. Another interference, the blood cell volume effect, is seen in dialysis of whole blood. The cells reduce the active sample diffusion volume in front of the dialysis membrane. Risinger
0003-2700/90/0362-0932$02.50/00 1990 American Chemical Society
