J. Phys. Chem. 1996, 100, 17305-17309
17305
Dependence on Composition of GC Retention on a Self-Associating Solvent/Hydrocarbon Mixture: Its Interpretation by the Computation of Available Sites for Interaction with Solutes† S. Costiner Department of Mathematical Sciences, Carnegie-Mellon UniVersity, Pittsburgh, PennsylVania 15213
T. Kowalska* The Institute of Chemistry, Silesian UniVersity, 9 Szkolna Street, 40-006 Katowice, Poland
E. Gil-Av‡ Department of Organic Chemistry, The Weizmann Institute of Science, RehoVot, Israel ReceiVed: June 20, 1996X
Diamides, e.g., N-lauroyl-L-valine-tert-butylamide, self-associate in appropriate media through hydrogen bonding. Such selectors form with apolar solvents binary phases, for which the contribution of the polar constituent to the overall retention is the product of the retention on the pure diamide, its weight fraction (x), and a concentration dependent factor (µx). In this paper a model is presented to interpret the effect of the weight fraction on µx. The diamide in the apolar solvent forms a mixture of structures of association degree 1 to n, between which n - 1 equilibria can be written. Every associate has at its terminals a site capable of hydrogen bonding to appropriate solutes (e.g., R-amino acid derivatives). It is proposed that µx ) Nx/N1, where Nx is the number of available sites for hydrogen bonding at weight fraction x and N1 is that of the pure diamide (x ) 1). On the assumption that the above equilibria have closely similar constants, equations were developed to compute the number of the available sites by both numerical and analytical procedures. A good fit between the computed and experimental µx values was found. The equations developed also permit the estimation of the minimal value of the equilibria constants as well as various other features of the diamide association-dissociation system.
x Vgmix ) µxVgp(x) + Vga(1 - x)
Introduction publications1,2
In two recent the correlation of the gas chromatographic retention volume of hydrogen-bonding solutes with the weight fraction of a self-associating polar solvent, diluted by an apolar component, was discussed. The binary stationary phase consisted of the diamide N-lauroyl-L-valinetert-butylamide mixed with either squalane or tetracosane3,4 and the solutes separated were N-trifluoroacetyl(TFA)-R-amino acid isopropyl esters. Purnell and co-workers5 have found that in many systems GC retention on a binary phase can be expressed by eq 1, even if weak interactions, the extent of which should x Vgmix ) Vgp(x) + Vga(1-x)
(1)
change with composition, can be expected in the binary phase. These authors proposed that in the systems studied by them mixing does not lead to “microscopic partition”5 of the two components, but each of the solvents continues to behave like a pure component. In the case of the diamide/squalane system, however, eq 1 does not hold and a concentration dependent correction factor (µx) has to be introduced to account for the variation of Vgmix with x, as shown in eq 2:
where Vgxmix, Vgp, and Vga are the specific retention volumes of a solute on, respectively, the binary mixture, the pure chiral components, and the pure achiral components and x is the weight fraction of the chiral component in the binary system. The diamide strongly self-associates, thereby partially blocking hydrogen bonding to solutes. On dilution with an apolar solvent, the degree of association is reduced, and more sites become available for interaction with solutes; the factor µx expresses the resulting higher affinity of the solutes (Scheme 1). One of us (T.K.) had previously introduced a model (model A) to correlate retention in LC with the weight fraction of a self-associating solvent in a multicomponent mobile phase.6-8 Applying this same approach to GC in the diamide system of concern, it was proposed1,2 that µx and x are correlated by the simple expression, eq 3:
µx ) x1/x
Dedicated to the memory of Professor Emanuel Gil-Av. * To whom correspondence should be addressed. ‡ The model and theory presented in this article were developed in large part by Professor Emanuel Gil-Av. Tragically, he passed away shortly after finishing this work. X Abstract published in AdVance ACS Abstracts, October 1, 1996.
S0022-3654(96)01849-7 CCC: $12.00
(3)
Strictly speaking, the difference in the density of the binary phase components has to be taken into account in this equation, which leads to eq 4.
µx ) x(1.196 - 0.196x)/x
†
(2)
(4)
where 1.196 is the ratio of the density of the diamide investigated over that of squalane at 95 °C (the experimental temperature). The data reported in Table 1 (taken from ref 2) demonstrate good agreement between eq 4 and the mean values © 1996 American Chemical Society
17306 J. Phys. Chem., Vol. 100, No. 43, 1996
Costiner et al.
SCHEME 1
TABLE 1: Correction Factors (µx) for the Binary System N-Lauroyl-L-valine-tert-butylamide/Squalane and N-Trifluoroacetyl-r-amino Acid Isopropyl Esters as Solutesa x µexp of N-TFA amino acid isopropyl esters of
x µcalc
x
L-Ala
D-Ala
L-Abub
D-Abub
L-Nvac
D-Nvac
L-Val
D-Val
L-Leu
D-Leu
mean valued
with eq 3
with eq 4
1.000 0.675 0.350 0.250
1.0 1.2 1.8 1.9
1.0 1.2 1.8 1.8
1.0 1.3 2.1 2.3
1.0 1.3 2.1 2.3
1.0 1.3 2.1 2.3
1.0 1.3 2.1 2.3
1.0 1.2 1.9 2.0
1.0 1.2 1.9 1.9
1.0 1.3 1.9 2.2
1.0 1.3 1.9 2.1
1.00 1.26 ((0.016) 1.96 ((0.06) 2.11 ((0.06)
1.00 1.22 1.69 2.00
1.00 1.26 1.80 2.00
x x Comparison of the experimental data (µexp ) and the values calculated (µcalc ) according to eqs 3 and 4. b R-Aminobutyric acid. c R-Amino-nd valeric acid. The mean error on all observations is given in parentheses. a
of the directly measured experimental µx data. Thus, model A seems quite satisfactory for the prediction of µx values as well as for that of retention volumes and partition coefficients.2 Model A is based on a formal analogy2 between the electric conductance of electrolytes in dilute solution and chromatographic retention on a self-associating solvent component in a binary phase. Indeed, the association-dissociation occurring in solution is an essential, initiating step of both phenomena. However, this analogy cannot be taken very far. Whereas the equilibrium of the dissociating salt in solution can be represented by an equation of the type AB h A + B, the self-associating solvent, admixed with apolar hydrocarbons (see below), distributes between many species of degree of association 1 to n and which are related by n - 1 equilibrium equations. To simplify this complex situation, it was assumed that the dissociation taking place may be represented approximately by the following equation, involving a hypothetical species of average degree of association n′, which dissociates to the hypothetical (n′ - 1)-mer: K′
n′-mer y\z (n′ - 1)-mer + monomer
(5)
where K′ is the hypothetical equilibrium constant and [(n′ 1)-mer]/[n′-mer + (n′ - 1)-mer] is considered the equivalent of the degree of dissociation of an electrolyte in dilute solution. Furthermore, in the case of electrolytes, the undissociated salt does not contribute to conductance. On the other hand, undissociated structures of the self-associating solvent, say the n′-mer, still contain at their terminal sites (Scheme 1) functional groups capable of interacting with solutes. These and other difficulties2 in the derivation of eq 3 led to an attempt to establish the empirically confirmed relationship between µx and x by another approach, to be discussed in this paper. In the diamide/N-TFA-R-amino acid ester system, the solventsolvent and solute-solvent interactions are figured to occur as illustrated in Scheme 1. Diamides, derived from R-amino acids,
are known to associate by hydrogen bonding leading to pleated sheet β-structures,9-12 as shown in Scheme 1. In solution, an equilibrium of associates, growing continuously from one to n units of diamide, will form in proportions depending on the weight fraction (x) of the diamide. These various species are related to each other by mass equilibrium equations as given below:
Wx2/2M
Kn-1 K1 Wxn/nM ) ) , ..., (6) x x x 2 V Vx (W1/M) (Wn-1/(n - 1)M)(W1/M) x where Wx1, Wx2, ..., Wxn are the weights of the various associated molecular species and M, 2M, ..., nM are their respective molecular weights (M ) 354 for the particular diamide studied). At each of the terminals of these linear associates, there is a site that is free to interact by hydrogen bonding with the N-TFAR-amino acid esters. Retention should increase with the number of such sites per unit weight of diamide. Table 1 shows that at a given weight fraction of diamide the spread of the experimental µx data for the different derivatives corresponds at most to an error of (3% on the mean of all observations. Such a relationship would be expected for a binary phase with a composition dependent proportion of sites capable of interaction with solutes. It is proposed that the relevant sites are the terminal ends of the diamide associates, free to hydrogen bond with solutes (Scheme 1) and, furthermore, that the magnitude of µx should be determined by the number of moles of such terminal sites per standard mass (see below) of diamide at weight fraction x (Nx). More precisely, it is proposed that according to this approach (model B)
µx ) Nx/N1 ) µBx
(7)
where Nx is the number of available sites for x, and N1 is that for x ) 1 (i.e., for the pure diamide). Counting the terminal sites of all diamide species for a given weight of the diamide
Composition Dependence of GC Retention
J. Phys. Chem., Vol. 100, No. 43, 1996 17307
gives eq 8.
(
Nx ) 2
)
Wxn Wx1 Wx2 + + ... + M 2M nM
TABLE 2: Comparison of the Experimental Correction x ) with Those Computed by Model B (µxB)a Factors (µexp
(8)
(As to differences in affinity of the two sites, see below.) In addition to eqs 6 and 8, knowledge of the mass balance is required to determine N. Arbitrarily, the mass unit of the solvent, used here throughout, was fixed as E ) 905g, i.e., the weight of 1 L of diamide at 95 °C, the temperature at which the gas chromatographic measurements were carried out.4 Thus, the mass balance is expressed by eq 9.
E ) Wx1 + Wx2 + ... + Wxn
(9)
To facilitate computation, a number of simplifications were made on the mode of association of the diamide, justified a posteriori, by the results. Initially, for the development of model B, linear self-association was chosen to occur in the parallel pleated sheet (Scheme 1) and not the antiparallel pleated sheet structure. Furthermore, most importantly for derivation of eqs 10-16, the assumption was made that the constants K1, K2, ..., Kn (eq 4) all have closely the same value (K). A justification of this latter postulation is that at each consecutive association step (in the parallel β-structure) the same two NH‚‚‚OC bonds are formed. Similarly, the interaction of an N-TFA-R-amino acid ester with the free sites at the ends of each diamide associate involves the same types of hydrogen bonds, so that the corresponding affinity for the solute (∆∆G) should be largely the same. It has, however, to be mentioned that the two terminal sites available for binding, in the diamide species will be divided between C5 (HN-C-C-O) and C7 (HN-C-C-C-O) moieties, which can interact with the solutes, forming somewhat different hydrogen bonded solute-solvent structures. This means that in eq 8 the number 2 should be multiplied by a certain correction coefficient to account for the non-equivalence of the sites. However, since the purpose of the computation is to determine the ratio Nx/N1, this coefficient cancels out in the calculation of µxB. The primary purpose of this paper is to check the applicability of model B. To do this, eq 7 must be solved for the weight fractions appearing in Tables 1 and 2 (x ) 0.076-1.0) and the computed µx values compared with the experimental ones.
x x µexp
0.675
0.35
1.26 ( 0.016b
0.25
1.96 ( 0.06b
2.11 ( 0.06b
0.076 3.46c
K (mol/L)
µBx
µBx
µBx
µBx
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0
1.17 1.19 1.20 1.21 1.22 1.22 1.22 1.23 1.23 1.23 1.24 1.24 1.24 1.25 1.25 1.25 1.25 1.25 1.25
1.43 1.52 1.56 1.59 1.61 1.62 1.64 1.65 1.65 1.66 1.70 1.72 1.73 1.73 1.74 1.74 1.75 1.75 1.75
1.56 1.69 1.76 1.80 1.83 1.86 1.88 1.89 1.90 1.92 1.98 2.01 2.02 2.04 2.05 2.05 2.06 2.06 2.07
1.90 2.23 2.43 2.57 2.68 2.76 2.83 2.88 2.93 2.97 3.22 3.34 3.41 3.46 3.50 3.53 3.56 3.58 3.60
a x µB ) Nx/N1 (eq 7). b See Table 1. c µx value fitting best the experimental partition coefficients (Table 3).
by calculating zx from eq 12 numerically, then using zx to calculate Nx from eq 11, and then µxB from eq 7. Equation 12 has a unique positive solution zx for any given E, K, Vx, and M. The distribution of Wxi versus i was determined by the numerical procedure using computed values of zx and eq 10. In Figure 1, corresponding plots are shown for K ) 50 mol/L and various x values. It is seen that for higher degrees of association, the plots tend asymptotically to the x coordinate without ever reaching it, indicating the presence of small quantities of diamide species for large n, e.g., for n g 70, and thus justifying the assumption of large n values in the system indicated. The analytical procedure of computing µxB is based on the following reasoning: Examination of eq 12 shows that for any given values of E, Vx, M, and K, for sufficiently large n (e.g., for n > 5, for our E, Vx, M, and K values), the positive solution of eq 12, zx, is less than 1. Hence, znx ≈ 0 when n becomes sufficiently large, e.g., for n g 70. If znx ≈ 0 one gets the simplified relations (eqs 13-15)
Numerical and Analytic Procedures of Computing This section presents a numeric and an analytic procedure for computing µxB for given E, K, Vx, and M. The numeric procedure holds and is very accurate for any n, while the analytic one holds and is accurate only for sufficiently large n values. Setting zx ) Wx1K/VxM, the terms in eq 6 can be expressed by eq 10:
( )
K Wix ) i(Wx1)i VxM
i-1
Vx ) i Mzix, i ) 2, ..., n K
(10)
By combining this equation with, respectively, eqs 8 and 9, eqs 11 and 12 result:
Vx Nx ) 2 zx(1 - znx )/(1 - zx) K E)
Vx Mz (nzn+1 - (n + 1)znx + 1)/(zx - 1)2 K x x
(11) (12)
By the numerical procedure of computing µxB, µxB is obtained
E ) MVzx/[(K(1 - zx)2)]
(13)
Nx ) 2E(1 - zx)/M
(14)
µBx ) (1 - zx)/(1 - z1)
(15)
From eq 13, zx is obtained analytically. Denoting δ ) M/(KE) and γ ) 1 - zx1 ) (-Vx1δ + (4Vx1δ + Vx21δ2)1/2)/2 the analytic expression of µxB as function of Vx results:
µBx ) (-Vxδ + (Vxδ + V2x δ2)1/2)/(2γ)
(16)
The values of Vx for the different experiments (Table 1) were calculated by Vx ) 1 + 1.196(1 - x)/x. By combining the latter equation with eq 16, the analytical expression of µxB as a function of x (and not Vx) is obtained. The analytical eqs 13 and 14 are simpler than the corresponding numerical formulae 11 and 12 and lead to expressions of µxB which can be more readily used in further analysis. For lower n values, the analytical results may not be close to the
17308 J. Phys. Chem., Vol. 100, No. 43, 1996
Figure 1. Distribution of the species (Wi) vs the degree of association in E(905g) of the diamide. K ) 50 mol/L; x ) 0.076 (1), 0.25 (2), 0.35 (3), and 0.675 (4).
Costiner et al.
Figure 3. Plot of µBx as a function of the equilibrium constant (K) for x ) 0.675 (1), 0.35 (2), 0.25 (3), and 0.076 (4).
TABLE 3: Comparison of the Experimental Separation Factors (rexp)on a Diamide/Squalane Phase of x ) 0.076 and the Calculated Values (rcalc), Taking µx ) 3.46a N-TFA O-isopropyl ester of
Rexpb
Rcalcb
% error of Rcalcd
Ala Val Leu Abu Nva
1.12 1.07 1.16 1.10 1.13
1.13 1.10 1.18 1.13 1.15
0.9 2.8 1.7 2.7 1.8
a Value used as µx b exp in Table 2, last column. Corrected retention volume of the L-isomer over that of the D-isomer. c The partition x coefficients were calculated by determining Vgmix for each of the two x enantiomers using eq 2 and varying µx until the ratio of Vgmix for the d L- and the D-isomers did fit experiment (see also ref 2). % error ) Rcalc - Rexp/Rexp × 100.
Figure 2. Convergence of µBx computed by the numerical (1) and the analytical (2) procedures as a function of n. K ) 50 mol/L; x ) 0.25.
numerical ones, which are accurate for any n. The two procedures of computing µxB as a function of Vx or x give very close results for Vx and K in a wide range if n is large enough (see Figure 2). Broad domains, which depend on n, Vx, K, and M, where the analytic expressions are accurate, can be found in a robust way using the numeric procedure. A report on additional derivations and details on the numeric and analytic procedures, and a program which implements these procedures can be obtained by anonymous ftp to iris.wisdom. weizmann.ac.il, in the directory pub/sorin, in the report NumericAnalytic-Models.ps and the program chromatographic.f, respectively. Results and Discussion When the results of the computations are presented, it should be remembered that for given values of E and M, the variables to be considered are K, Vx (or x), and n. However, it has already been mentioned above that in the system considered, n g 70, so that µxB can be obtained accurately by the analytical procedure and is then a function of K and Vx only. Plots showing the dependence of µxB on K for different values of x are given in Figure 3. As K increases, µxB tends rapidly to a finite limit, the magnitude of which fits the corresponding experimental data. Thus, when K is large enough, model B and eq 7 are seen to be substantiated. The effect of K on µxB can be better scrutinized by examining Table 2, in which the calculated and experimental data in the range of K ) 1.0-
100 mol/L are listed. For x ) 0.076, 0.25, and 0.675, the deviation for the two sets of data, above K ) 50 mol/L, reaches at most 4%, whereas for x ) 0.35 it does not surpass 10%. It should be mentioned that for x ) 0.076, no directly measured µx values were available but only partition coefficients. However, it is possible to derive from these partition data (see Table 3, footnote a) the corresponding µxB, which was found to be 3.46. Thus, it can be stated that where the computed µxB values approach the finite limit of the plot, there is agreement with experiment. The correction factor µxB is seen to be little sensitive to changes in K (Figure 3), and this supports the assumption made above that K1 ) K2 ) ... ) Kn-1 ) constant ) K. This insensitivity of µxB to K makes it difficult to estimate the magnitude of the equilibrium constant. However, it is possible to estimate the minimum value of K (Kmin) required to get a good fit of model B with experiment. If a deviation of 5% from µxB is tolerated and the corresponding K values on the plots (Figure 3) for the different weight fractions of the diamide are read, it is seen that Kmin tends to shift to higher values as x increases. As the basic assumption is that all equilibria in eq 6 have the same constant, only the highest Kmin determined will have a magnitude such as to lead to µxB values close to the experimental ones for all plots of Figure 3. Given the present data, the estimate for the corresponding Kmin is ∼50 mol/L (see Figure 3 for x ) 0.076). The true magnitude of the equilibrium constant is to be found in a wide range of values over and above the Kmin. An equilibrium constant higher than the above Kmin does not seem unlikely, if one recalls that at every association
Composition Dependence of GC Retention
Figure 4. Plot of Nx, computed by the analytical procedure vs V at high dilution. K ) 50 mol/L (1), 100 mol/L (2). Limit ) 5.112.
Figure 5. Plots of µBx , computed by the analytical procedure vs V at high dilution. K ) 50 mol/L (1), 100 mol/L (2).
step two hydrogen bonds of the type NH‚‚‚OC are formed, which could very well correspond to a ∆∆G of 3 kcal or more. At 95 °C, 3 kcal would mean an equilibrium constant of 59 mol/L. Scheme 1, on which model B is based, took into consideration a parallel pleated sheet structure of the diamide associates. As is in the present case, the system is nonbiological, the anti-parallel structure involving C5‚‚‚C5 and C7‚‚‚C7 rings is, however, equally possible. Minimal energy computation (M. Eisenstein, unpublished data) indicates that the formation of C7‚‚‚C5, C7‚‚‚C7, and C5‚‚‚C5 rings involves much of the same interaction energy. Furthermore, there is only a negligible difference in that energy between two C7‚‚‚C5 rings, as compared with one C7‚‚‚C7 plus one C5‚‚‚C5 ring; such proportions of hydrogen-bonded forms are indeed the rule in the parallel and antiparallel associates, respectively. From here it can be concluded that the K values are also the same in the two types of structures and that model B, as presented for the parallel association, and the equations derived from it are true and have identical form also for the antiparallel association. The fit of model B with experiment, although based initially on a parallel pleated sheet β-structure, sustains the validity of the above conclusions. This actually means that in the latter general case, the weight fractions in eqs 6, 8, 10, etc. represent associates of the same degree i without differentiating between parallel and anti-parallel structures. It should be mentioned that model B takes into account only hydrogen bonding, which, no doubt, essentially determines the stereoselectivity. For more rigorous treatment, possible contributions to selectivity involving non-
J. Phys. Chem., Vol. 100, No. 43, 1996 17309 bonded interactions should be considered; such effects should vary with the substituents on the solute considered. Information on certain structural features of the diamide in squalane solution can be derived from model B, as has already been demonstrated by the estimation of the lower limit of the equilibrium constant of eq 6, and the plot of the weight distribution of the various species of associates under different conditions (Figure 1). With relevance to the discussion in the preceding paragraph, it should, however, be recalled that all such data refer not to clearly defined structures, as shown in Scheme 1, but to mixtures of varying proportions of both the parallel and the antiparallel pleated sheet β-structure, having in common the same degree of association i. In Figure 1, such important parameters as the weight fraction of the monomer (Wx1) and the degree of association (i) of the associate of highest abundance at a certain x are readily discerned. The average molecular weight (2E/Nx) and the average degree of association (2E/NxM) are readily calculated by eqs 13 and 14. It is further of interest to compute the maximum possible value of Nx and µxB, which should be reached at high dilution, where the diamide must be completely dissociated. Equation 2 cannot give an answer to this problem, since it goes to infinity when x tends to zero. On the other hand, eqs 14 and 16 derived from model B permit computation of Nx and µxB as a function of Vx for a given K and yield a finite answer for high dilution. Figure 4 shows Nx for very high dilution with K being, respectively, 50 and 100 mol/L. It is seen that in both cases, Nx has the limiting value of 5.112. Indeed, this is the number of free hydrogen bonding sites (2E/M) of the monomeric diamide contained in E and represents complete dissociation. This state will be reached for any K, provided that dilution is high enough. Figure 5 shows plots of µxB versus Vx. At high dilution, Nx has the constant value of 5.112 (see above). But N1 decreases as K increases, and since µxB ) Nx/N1, the asymptotic values for µxB will increase with K. Compare plot 2 for K ) 100 mol/L (N1 ) 0.31) with plot 1 for K ) 50 mol/L (N1 ) 0.43; see Figure 5, plot 1). It is of interest to note that for K ) 100 mol/L, µxB at high dilution is more than 16 times higher than that of the undiluted diamide. Further theoretical and experimental work is indicated to provide additional support for the present conclusions as to the associative structure of the diamide in the binary system and to eventually refine the retention model in accordance. References and Notes (1) Kowalska, T.; Hobo, T.; Watabe, K.; Gil-Av, E. Paper presented at the IUPAC ’91 ICAS, International Meeting on Analytical Chemistry, Chiba, Japan, 1991. (2) Kowalska, T.; Hobo, T.; Watabe, K.; Gil-Av, E. Chromatographia 1995, 41, 221. (3) Hobo, T.; Watabe, K.; Gil-Av, E. Anal. Chem. 1985, 57, 362. (4) Watabe, K.; Gil-Av, E.; Hobo, T.; Suzuki, S. Anal. Chem. 1989, 61, 126. (5) Laub, R. J.; Purnell, J. H. J. Am. Chem. Soc. 1976, 98, 35 and references therein. (6) Kowalska, T. Chromatogr. 1989, 27, 628. (7) Kowalska, T. J. Planar Chromatogr. 1989, 2, 44. (8) Kowalska, T. J. High Resolut. Chromatogr. Commun. 1989, 12, 474. (9) Pauling, L. The Nature of the Chemical Bond, 3rd ed.; Cornell University Press: Ithaca, NY, 1960; pp 473. (10) Mizushima, S.; Shimanouchi, T.; Tsuboi, M.; Sugita, Y.; Kato, E.; Kondo, E. J. Am. Chem. Soc. 1953, 73, 1330. (11) Ichikawa, T.; Itaka, Y. Acta Crystallogr. 1969, B25, 1824. (12) Feibush, B.; Balan, A.; Altman, B.; Gil-Av, E. J. Chem. Soc., Perkin Trans. 2, 1979, 1230.
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