a = s e p a r a t i o n factor; e q u a l to r a t i o of k' values f o r t w o solutes in same
LSC s y s t e m
6 2 = secondary s o l v e n t p a r a m e t e r s ( E q u a t i o n 4), v a r y i n g w i t h s o l v e n t a n d solute, f o r solvents 1 a n d 2 a n d same solute Ao = a s o l u t e p a r a m e t e r m e a s u r i n g t h e e x t e n t of s o l u t e localization; E q u a t i o n 5 to = solvent s t r e n g t h parameter t l , t2, e,, €b, Cab = values o f € 0 f o r solvents 1 , 2 , A, B, or b i n a r y solv e n t A-B, respectively
AI,
ACKNOWLEDGMENT I am much indebted for the critical evaluation of the present manuscript by D. L. Saunders of the Union Oil Co. of California, J. J. Kirkland and J. J. DeStefano of DuPont, and E. Soczewinski. RECEIVEDfor review January 21, 1974. Accepted May 7 , 1974.
Theoretical Comparison of Chromatography and Spectrometry as Methods of Measuring Weak CompIexation Constants Claude Eon and Georges Guiochon Laboratoire de Chimie Analytique Physique, €cole Polytechnique, 7 7, rue Descartes, Paris 5', France
This paper gives a theoretical approach to the rneasurement of complexation constants by gas-liquid chrornatography and establishes some similarities with the spectrometric methods. It is shown that the difference usually found between the results given by these different methods can be explained by a misinterpretation of chromatographic measurements. When corrections are made, we can expect the two kinds of methods to give reliable results in good agreement, provided that the solvent used has the same refractive index as the main additive and that none of the species studied is strongly polar. I n all cases, however, complexation constants do depend on the nature of the inert solvent used and there is no direct access to the absolute constants.
During the past few years, chemistry of charge transfer complexes has been very intensively investigated by means of various spectrometric methods (UV, NMR, IR, etc.). Many data have been reported (1, 2 ) and the large discrepancy found between results obtained by different authors and methods is striking. One can be perplexed about the reliability of these numerical results, especially when dealing with weak molecular complexes. Have we not seen even negative equilibrium constants reported ( 3 )! More recently, a chromatographic method that allows one to measure complexation constants has been described ( 4 , 5 ) . We have shown previously that this method could lead to large errors when the non-ideality of the solution is neglected and we have given a new set of equations to correct for this effect (6-8). At that time, it ( 1j R . Foster, "Organic Charge Transfer Complexes," Academic Press, New York, N . Y . , 1969. ( 2 ) J. Rose, "Molecular Complexes,'' Pergamon Press, New York, N . Y 1967 ( 3 ) M. W Hanna and A. L Ashbaugh, J , Phys. Chem.. 68,811 (1964). ( 4 ) B W Bradford, D Harvey, and D. E. Chalkley, J . Inst. Petroi., London, 41, 80 (1955). ( 5 ) J H Purnell, "Gas Chromatography 1966," A. B Littlewood, Ed , The Institute of Petroleum, London (G B ) , 1967, p 3. (6) C Eon. C Pommier, and G Guiochon, Chromatographia, 4 , 235 (1971) (7) C Eon. C. Pommier, and G Guiochon, Chrornafographia, 4, 241 (1971) ( 8 ) C. Eon, C Pommier and G. Guiochon. J . Phys. Chem., 75, 2632 (19711.
seemed that the chromatographic method would be more useful than the spectrometric ones, not only because measurements are faster and can be carried out accurately a t very low concentrations, but mainly because this method appears to derive from a direct thermodynamic approach. However, there are cases where the chromatographic method cannot be used, particularly when none of the solutes is volatile. Thus, it is necessary to investigate the extent of agreement between data derived by both kinds of methods. In one of the rare comparative studies of the same chemical reaction by chromatography and UV spectrometry, a good agreement (20%) between the results of the two methods was found (9), but a much larger discrepancy was recently observed for other chemical reactions by Purnell et al. (IO).On the basis of these last results, Purnell et al. concluded that chromatography was much better than all the spectrometric methods. As no theoretical study of this problem has been made so far, the basic question remains: can chromatographic and spectrometric methods lead to the same value of the complexation constants as they basically measure quite different physical properties? In what conditions and within which range can we expect their results to agree? There are many reasons that could explain the disagreement between the results. Among them, however. the most important is probably that activity coefficients are most often neglected and, moreover, they vary with the composition of the solution. This is especially important as sets of solutions of largely different composition are used in all methods of measurements of complexation constants. In other words, all physical interactions are neglected and it is assumed that all the deviation from ideal solution behavior comes from the chemical reaction. Obviously, this is certainly not valid when dealing with weak molecular complexes. Consequently, the role of the solvent which makes up the activity coefficients is not really taken into account and the measured complexation constants are often meaningless. This approximation, which is bad in any case, can hardly be understood when applied to chromatographic methods which precisely measure the complexation constants from activity coefficients. The aim of this paper is to clarify those points and to (9) C. Eon, C. Pommier, and G. Guiochon, C.R.Acad. Sci., 168, 1553 (1969) (10) J. H . Purnell and 0. P. Srivastava.Ana/. Chem., 4 5 , 1111 (19731
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show that both chromatography and spectrometry should lead to the same reliable results when due corrections are made and when the solvent is properly chosen. CHROMATOGRAPHIC METHOD Relation between Partition Coefficient and Association Constant. The determination of the association constant corresponding to the reaction A + B e AB, is carried out by systematic measurements of the retention times of the volatile solute B on chromatographic columns prepared with solutions of the nonvolatile additive A in an inert solvent S as stationary phases, using different concentrations of A. Whether any chemical reaction takes place or not between A and B, the chromatographic partition coefficient of B is given by (11 ):
K,
=
RT
fc Y B * m V A , c
(l
-
XhB
We shall now consider the limiting slope of the curve us. XA when X A becomes 0; in such a case 'yLmis unity and YB is no longer different from y ~ (Cf. * Equation 3). Equation 5 then becomes: KR V A , s o
(1)
In this equation, f ~ is ' the fugacity of B in the gas phase, yB*" its macroscopic activity coefficient in the solution, and V A , s o the molar volume of the solution A,S used as stationary phase. The macroscopic activity coefficient, which measures the deviation from ideal behavior of the total solubility of B in the solution, is defined in such a way that its value is unity for the pure substance. When chemical association occurs, the activity coefficient YB* is different from the microscopic one, Y B , which takes into account only the physical interactions between the solute and its surroundings. As shown by Langer et al. (12), the two quantities are related by the equation: YB* =
from chromatographic measurements, as we have no way of knowing the microscopic activity coefficients of all the species involved. For this reason, another complexation constant, OK is defined and used. This constant refers to the other definition of the activity coefficients 'y8, which is such that O y L becomes unity when X,becomes 0. Notice that OK varies from solvent to solvent as the solute-solvent interactions are now incorporated into "K. Obviously, this leads to the relationship:
(2)
where x is the fraction of the solute molecules which are engaged in complex formation in the solution. The solute B is supposed to react with A and not with S,and Equation 2 can be written:
(3)
where K is the familiar thermodynamic equilibrium constant of the reaction defined as a function of the microscopic mole fraction X,and the activity coefficients yi by:
xq-
0
The LHS term of Equation 7 accounts for both physical and chemical interactions of the B molecules in the (A,S) solution, whereas the two terms of the RHS of this equation account separately for these physical and chemical interactions, respectively. X A is always much larger than X, in the range of concentrations that is required to make measurements and thus when the solvent is chemically inactive, X.4 is equivalent to the macroscopic mole fraction of A ( X A * ) .Then it is clear from Equation 1 that the left term of Equation 7 can be derived from a plot of VA,sOKR us. X,A*. As y ~ * "can be derived from measurements of KRo with the help of Equation 1, we see that the only unknown quantity that prevents us from calculating "Kis the variation of l / y B m with X A*. This brings a major problem that has been overlooked in the past: chromatographic measurements do not allow the determination of the association constant from the variation of the activity coefficient with the additive concentration. On the contrary, the classical procedure ( 5 ) gives the sum:
(4) Notice that K does not depend on the nature of the solvent in which the reaction takes place as the activity coefficients correct for the solute-solvent interaction. From Equations 1 and 3, we can write the variation of the product of the two experimental quantities, KR and V&', as a function of the composition of the stationary phase. Taking XA as the variable, this leads to:
Equation 5 does not allow the direct determination of K (11) S. Dal Nogare and R . S. Juvet, "Gas Liquid Chromatography." Interscience, New York, N.Y , 1962, Chap 1 7 . (12) S. H Langer, C Zahn, and G. Pantazopios, J , Chromatogr.. 3, 154
(1960).
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Obviously, the weaker the eomplex, the greater the error. At this stage, we have to emphasize that the best we can expect from chromatography is not a measurement of complexation constants but rather an estimation. This is due to the fact that we have to use a molecular model to estimate the variation of the microscopic activity coefficient of B with the concentration of A. This is certainly not a simple matter as most of the molecular theories of solution are still too qualitative to be accurate enough, as shown by the limited results obtained in the prediction of activity coefficients and retention volumes (13, 14). At the present level of theory, we feel that it may be better to try to minimize the correcting term by choosing a proper solvent. As we shall see, this goal can be achieved only when weakly polar compounds are studied, so that we shall consider first this type of solution. Solution Thermodynamics and the Correction for the (13) R Kelier, B Karger, and L. Snyder, "Gas Chromatography 1970," R . Stock, Ed., The Institute of Petroleum, London (G E . ) , 1971, p 125. (14) L. Snyder, E. Karger, and C Eon, J . Chromaloyr. Sci.. submitted.
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Microscopic Activity Coefficient. Case of Non-Polar or Slightly Polar Compounds. According to Equation 7 we have to investigate the factors that contribute to the activity coefficient Y B in order to calculate its variation with XA and to make certain the conditions in which this variation is negligible. There are four main contributions to Y B , those of the dispersion, the orientation, and the induction forces and of the structural effects. For slightly polar molecules (p up to 0.5 D) we can neglect the effect of orientation and induction forces and split YB into only two parts, corresponding to the thermal and athermal contributions respectively:
isLE
where A T I E and are the excess molar enthalpy and entropy of solute i. With the assumptions made above, tyl accounts for the effects of the dispersion forces whereas ayLaccounts for the configurational effects. E f f e c t s of the Dispersion Forces. This calculation can certainly be made, although it is only one of the possibilities, through the new generalized solubility parameter theory as developed by Keller et al. (13) and by Snyder, Karger, and Eon (14). They defined a dispersion solubility parameter (DS,) that is related to the enthalpy of vaporization (ADHL)that the solute would have if the dispersion forces only were responsible for its cohesion:
(15)who found an important variation of 'y= with the concentration in additive for various uncomplexed solutes eluted on mixtures of 1,3,5-trinitrobenzene and di-n-nonylphthalate. However, this may illustrate as well the fact that trinitrobenzene and di-n-nonylphthalate can give complexes together, so that the solvent is not chemically inert (see discussion of this case latter). This demonstrates clearly that the solvent has to be chosen with special care, to avoid the necessity of other correction factors than those given in the present work. Configurational Effects. We have just seen that t y m remains almost constant when the refractive indices of A and S are nearly the same. The other source of variation of yia with the concentration of A is the size difference between the molecules of A and S. that affects the athermal activity coefficient. The importance of this effect is not always understood but can be illustrated by the following example. It is well known that there is a linear relationship between the entropy and enthalpy of reaction of a group of analog compounds B with a given complexing additive A. Direct measurements of the thermodynamic functions of the reaction between the pure species is difficult however. When the partial molar properties are measured from the macroscopic activity coefficients through the well-known equations (16): -
AGBE = RT In yH*"
(9: From this parameter, it is possible to estimate the thermal activity coefficient of B (14):
In this last equation, & stands for the volume fraction of species i in the solution. The interest of the dispersion solubility parameter lies in the possibility to predict it. It is beyond the scope of this paper to review the work in references (13, 1 4 ) but it was found that the dispersion energy should, in first approximation, be proportional to the electronic polarizability per unit of volume, a,\', which is related to the refractive index by: =
3 - 1 m (n 2 m )
where N is the Avogadro number. Then, the dispersion solubility parameter should be approximatively proportional to the refractive index Lorentz-Lorenz function X = (n2 - l)/(fl + 1).In fact, the plot of the dispersion solubility parameter of 96 hydrocarbons us. X reveals a slight deviation from linearity (13), the data being best approximated by the following expression:
D6, = -2.24
+ S X ,- 58X,* + 22X,J
(12)
The most important thing to underline, however, is the very strong correlation between D6 and X so that we can say that compounds t h a t have the same refractive index have almost the same dispersion solubility parameter. Consequently, it is advisable to choose a solvent S that has nearly the same refractive index as the additive A, so that the RHS of Equation 10 does not change greatly in the whole range of concentration. When this condition is not fulfilled, tym may change widely and this probably explains the results of Castells
ASH)
aH,' - LGBF
= ______
2'
these functions can be considered as the sum of two contributions accounting for the chemical and the physical effects, respectively. Then it is not surprising that the plot us. 1 T H E for a group of solutes that can react of with A is not always linear. This is confirtned by earlier results (17 ) regarding the complexation reaction between volatile solutes and di-n-propyltetrachlorophthalatein the absence of any solvent (17), and it is illustrated by Figure 1.
As a first approximation we can assume that the contribution to the reaction enthalpy due to physical effects is nearly constant when the composition of the solution varies, but this is certainly not so for the contribution of these effects to the entropy term as the solutes studied have quite different molar volumes. A configurational entropy correction should be made. This can be done simply, using the Flory and Huggins equation (16)
In that equation, @ A stands for the volume fraction of the stationary phase, A. The limit of Is,,,,;,,when XH becomes zero is then subtracted from DEand the difference is plotted as a function of 1 in Figure 2. Comparison of Figures 1 and 2 which are drawn on the same scale shows that the linearity of the plot is much improved by the correction. The regression coefficient which was 0.74 in Figure 1 becomes 0.90. This clearly (15) R . C Castells, Chromatographia. 6. 57 (1973) (16) E. Guggenheim. "Therrnodynarnlcs.' North-Holland, Amsterdam (The Netherlands), 1957, Chapter 5 (17) S H Langer and J H Purnell. J Phys Chem.. 70, 904 (1966)
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0.5
015
-
c -
0 L.
-2.5
~
--L ---L-------L-*
0
0.5 AH:
1
Jcoi.mde-'1
L 05
Figure 1. Plot of the molar partial excess entropy vs. the molar partial excess enthalpy for compounds dissolved in di-n-propyltetrachlorophthalate (after ref. 17) 1 methyl cyclohexane; 2. benzene: 3. toluene; 4. ethylbenzene; 5 propylbenzene: 6. o-xylene; 7 . m-xylene: 8 p-xylene; 9 . 1-methyl-2-ethyl benzene: 10. 1-methyl-4-ethyi benzene: 11. 1,2,3-trimethyl benzene; 12 1.2.4-trimethyl benzene; 13 1,3,5-trimethyI benzene; 14. isopropyl benzene; 15. tert-butylbenzene; 16. styrene; 17. anisole; 18. fluorobenzene; 19. chlorobenzene; 20. bromobenzene; 21. iodobenzene; 22. chlorotoluene; 23. o-dichlorobenzene; 24. rn-dichlorobenzene; 25. p-dichlorobenzene: 26. rn-chlorobromobenzene
demonstrates the importance of the configurational entropy term. As shown by us previously ( I @ , we can write for most solutions of weakly polar compounds:
-
I
C
L _ i -
05
E$
.mole'
K ~ O
Figure 2. Plot of the molar entropy vs the molar enthalpy of the chemical reaction of complexation for various compounds with di-n-propyltetrachlorophthalate Compounds numbered as in Figure 1
S.Thus, from Equation 1, we should have y ~ * "V4,s" constant. Even the strongest defenders of Equation 2 have never explained why it should be so. At the very best their conclusions are based on the fact that the so-called activity coefficient " y ~ ~( "7 =~YB*" ~ VA,so) is more constant from solvent to solvent than is ye*". Even backed by some experimental results, this fact alone does not prove anything and certainly does not allow one to conclude that "yet" is a fundamental quantity that remains constant, as this fact can easily be explained by solution thermodynamics. It is easy to show, by considering Equation 14, that ifyDISp(B)mis constant:
Determination of t h e Association Constants. Combining Equations 7, 14, and 15-gives: 1 = -($ x4-0
where
YB*=
+
O K )
(16)
4 is:
Integrating Equation 16 and combining with Equation 1 gives the equation derived earlier (18), provided that X A is nearly equal to XA*:
This rather long derivation allows us to understand better the meaning of this equation, its range of validity (Sand A should have the same refraction index) and to compare it with its classical counterp.art:
Apart from the fact that the meaning of K, is questionable, as shown previously (28), Equation 19 would imply that the partition coefficient of an uncomplexed solute remains constant in the whole range of concentration of A in (18) C. Eon and B. Karger.J/. Chrornatogr. Sci., I O , 140 (1972)
1396
In most experimental studies made by gas chromatography, VBO is small and Vt,' is not very different from V s o , so that the exponential term is practically constant, and as a result "yBC" appears also to be almost constant, but this is accidental. For these reasons. Equation 19 is valid only in very special cases that are almost never met in actual practice. We have shown, for example, that the error that can result from its application can be up to 50% of the OK value of complexes which are not extremely weak (18) and this is a t the origin of the large discrepancy between the results reported in reference ( I O ) . This does not mean however that Equation 16 is exact in all cases. It is based on a model of solution and is valid only as far as this model is a good description of the solution in which measurements are made. From Equation 18, we see that there is no reason why a plot of KRVA,S"us. XA* should be linear. When deviation from linearity occurs, it becomes questionable to explain it by considering that a E-I reaction occurs (see later discussion), as it might result from a variation of the activity coefficients as well. The important result, however, is that
A N A L Y T I C A L CHEMISTRY, VOL. 46, NO. 1 1 , SEPTEMBER 1974
the slope of this plot a t the origin can be safely related to the thermodynamic constant OK. Case of Polar Compounds. When strong polar interactions occur, the situation becomes so complicated that no theory is a t the moment reliable enough to predict the variation of y m gwith the composition of the solution A,S. As neither the athermal nor the thermal parts of the activity coefficient can be calculated, a reference solute is often used to estimate a ( l / y B ) / d x A . Such a solute (R), often referred to as the counterpart solute, is supposed to give all kinds of interactions but chemical similar to those of the studied compound, so that:
The choice of this solute is obviously difficult. It is usually believed that its physical characteristics (i. e., refractive index, dipole moment, polarizability, etc.) and its molecular shape should be as similar as possible to those of B, apart from the fact that it must give no complex with A. Although such a procedure is certainly acceptable in some cases, its limitations and dangers should be pointed out. This can be illustrated by one of the simplest possible examples, the search for the counterpart of a monofunctional linear alkyl derivative. As R should behave as the original compound B with respect to the solvent, R and B certainly must have the same dispersive solubility parameter, which implies that their refractive indices must be about the same. The work of Weinier and Prausnitz (29) shows that they should also have the same polar solubility parameter ( p a ) which is related to the total solubility parameter (6) by (13): pa2
=
62
-
D62
(22)
For a series of monofunctional linear hydrocarbons, Snyder, Karger, and Eon (14) have shown that p6(,! can be written as a function of the dipole moment (pL)and the molar volume ( K ") of the solute i:
It turns out that C' is the same for all studied series but C is not. This prevents us from choosing a counterpart solute from the mere comparison of the dipole moments. Figure 3, taken from the work of Snyder et al. (14), illustrates this effect; it shows the variation of p6 us. pi for linear compounds with a common molar volume of 78 cm3/mole. Such a plot shows how much caution is necessary in the choice of a counterpart solute and that identical dipole moments is not a general criterion for this choice, although it does work in some cases. As the risk of error is certainly much larger with more complicated molecules, one has to be extremely cautious when trying to find a good counterpart and it seems that no safe rule can be given a t this time. We are never sure that a solute behaves in the same way as its counterpart and this is an inherent limitation to the accuracy of chromatographic determinations which has not been realized in earlier work. In fact, we are now a t the heart of the problem: whether we deal with nonpolar molecules or with polar ones, we cannot calculate the a ( l / y m B ) / d XA* term without using a model that will introduce some error in the determination of OK. Consequently, we cannot expect the results to be closer to the actual values of the complexation constants to better, than a t least a few per(19) R . Weimer and J . M . Prausnitz. Hydrocarbon Process., 44, 237 (1965).
cent, depending upon the degree of polarity of the molecules. As we shall see later, the spectroscopic method does not suffer from the drawback of having to make some critical assumptions regarding the value of Y-B or its variation but, unfortunately, it has some other specific disadvantages. Case of Non-Unique Reaction. Isomeric Complexes. The simplest case is the one of the isomeric complexes, first suggested by Orgel and Mulliken (20). Such complexes occur when the solute B may react in two (or more) different ways with the additive to give as many isomeric complexes 1-1: A + B S A B A f B S AB' . . . e
(24)
I . . . . . . . . . . . . . .
The total concentration in solution is:
(x.4~ of )the ~
complexed species
and there is an apparent complexation constant related to (XAB)T and given by:
('KAB)T
The preceding theoretical treatment remains valid if we substitute all the way XAB by ( X . A B ) T and 'KAB by ('KAB)T.So, whether we use the $ or the counterpart approach, the chromatographic method leads only to the determination of ('KAB)T and does not allow separation of the complexation constants corresponding to the formation of each isomer. Complexes 1-11 and 11-1. Unfortunately, analytical chromatography is useless to calculate the association constants of 11-I complexes ( i . e . , formation of B2A). The main reason is that the concentration of the solute B that is usually unknown, and varies largely from column inlet to outlet, appears in the theoretical expression of K R . On the other hand, analytical chromatography is quite successful in the case of 1-11 complexes and allows us to measure the complexation constants of the following reactions:
............................ A + A,B A,+,B Klhn+l"I
It is easily shown that Equation 18 becomes:
In that case $ is still given by Equation 17. Alternatively, the counterpart method can be used, when solvent-solute physical interactions are strong. I t should be emphasized a t this point that we must now assume that the ratio ~ A , , ~ , s ~ / ~ ~ . ~ R ,does . A , s not , change with the concentration in A if we want to calculate O K A B , ' K A ~ B. ,. . . This is not necessarily true in all cases, as (20) L. E. Orgel and R S. Mulliken, J. Amer. Chem. Soc.. 79, 4839 (1957).
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0
1
3
5
f;
Figure 3. Variation of the polar part of the solubility of functional derivatives with normal alkyl chains as of the dipole moment
parameter a function
pointed out before, but still in most instances Equation 28 can be safely used to get a good estimation of the complexation constants as illustrated by our previous results (8). Influence of the Chemical Reactivity of the Solvent. As pointed out above, the choice of the solvent is critical, the change of solvent can lead to very large variations in " K which reflect not only the variation of the activity coefficients but also the reactivity with the additive (19). The solvent should be as inert as possible toward the solutes studied, but total inertness is not always possible in chromatography. Usually, the solubility of very polar compounds or of strong electron donors or acceptors in hydrocarbons is very small and the great solubility that is observed in other solvents reflects the fact that chemical effects occur. When this is so, the microscopic mole fraction of A can become quite different from the macroscopic one and, as a result, a large systematic error on the value of " K occurs, if the calculations are carried out using the known value X A * instead of the unknown X,A in Equation 7 . The stronger the AS complex, the larger the error. It is clear from Equation 4 that the ratio between the true value of OK, that would be measured via the use of XA,to the apparent value of OK which is measured via the use of X A * , is equal to the ratio X A * / X , . Thus the error made can be calculated if we know the complexation constant of A with s. Assuming a 1-1 reaction, A S ~ r AS. ! and neglecting the activity coefficients it is obvious that:
+
XA'-
where cpB and cpAB are the shifts of the studied nucleus (H for example) in B and AB, respectively, and w and P A B , the proportion of this nucleus which is in the state B and AB ( i e . , 2 pL = 1). It is a simple matter to relate the chemical shift to the association constant K ( I ) : (31) n
with
A
= PB
-
YAB ~p
= cp~(1
A0
= (PB
and
- XEJ
-
(PABXAB
PAB:
Equation 31 is nothing more than the conventional Benesi-Hildebrand equation which is too often written with a complete and unwarranted neglect of the activity coefficients term. Importance of the Choice of the Solvent. K is constant in Equation 31 but the yL's depend on the solvent and change with the concentration in A as we have already shown. So it is not possible to derive K directly from Equation 31. To discuss this problem, it is convenient to transform Equation 31 to what is often called the "half-reciprocal equation" (23):
0
Results found b y Equation 7 should t h u s be multiplied b y "KAs+l to obtain the correct complexation constant of t h e reaction AB. As a consequence, results obtained with the chromatographic method, when the "inert" solvent in which the complexation reaction is studied presents some acidic or base function as it is often the case, should be considered very carefully and critically. SPECTROSCOPIC METHODS AND THE BENESI-HILDEBRAND EQUATION Background. Whatever spectroscopic method is selected, UV, IR, or NMR, the Benesi-Hildebrand equation is the basic relationship used for the determination of the complexation constants. The theory of these measurements has been developed with many details in the excellent text book by Foster ( 1 ) and there is no need to review it. 1398
Our purpose now is to discuss critically the validity of the methods derived from the Benesi-Hildebrand equation. This discussion will be valid for all spectrometric methods as it is basically thermodynamic in nature. However, we need an example and we have chosen Nuclear Magnetic Resonance because this technique is now more widely used than the other spectrometric ones. This is mainly because it avoids the classical pitfall of deviation from Beer's law (21) and that the only restrictive conditions for its use are the need for the donor or acceptor molecule to contain protons or other magnetic nuclei and the necessity to achieve a convenient resolution of the spectrum. Usually, the concentration of one of the solutes, B, is kept as small as possible and the variations with increasing concentration in A of the chemical shift of one of its nuclei is measured. In the case of a simple 1-1 complex that makes a rapidly exchanging system the measured chemical shift ( c p ) is a composite function of the mole fractions of B and AB (22)given by:
The usual procedure to derive the association constant from experimental data is to plot X..Z*/Aas a function of XA*. The linearity of the plots obtained is usually satisfactory. When the activity coefficients term and its variation with X A * are neglected, the slope is 1 / l $ B and the intercept is l / A # B ° K . Most measurements are based on this assumption which unfortunately is wrong and usually leads to large error on the determination of "K. In fact, it is easy by differentiation of Equation 32 to show that the slope of the plot is given by: (21) P H . Ernslie, R Foster, C A . Fyfe, and I . Horrnan, Tetrahedron, 21. 2843 (1965). (22) R. Foster and C. A. Fyfe, "Progress in Nuclear Magnetic Resonance Spectroscopy," Vol. 4, J . w. Ernsley, J. Feeney, and J. H. Soteliffe. Ed.. Pergarnon. Oxford ( G . B . ) , 1969, Chap. 1 (23) R . L . Scott, Reci. Trav. Chim. Pays-Bas B e @ , 75, 787 (1956).
ANALYTICAL CHEMISTRY, VOL. 46, NO. 1 1 , SEPTEMBER 1974
we can write: 1 p
(33)
Although the concentration of B is always greater in NMR than in chromatography and is usually larger than 0.05 mole fraction, we have assumed in writing Equation 33 that Y B = y B m . If we compare Equation 32 with Equation 5 , it is clear that whenever experimental data gathered on a given A, B, S system lead to a linear plot of KRVA,s0us. KA* in chromatography, the plot of XA*/A us. XA* in NMR should be linear, as in both cases this can be obtained only if d (-yA / r A B m ) / d x A * is nearly zero. T h e practical consequence of this result is that the concentration range in which linear plots are obtained is the same in chromatographic and spectrometric methods. In this range of linearity, the variation of Y A B /yA ~ with X.4 * can usually be neglected in Equation 33 but not the variation of l/yBm . This leads then to:
In complete analogy with chromatography, we have to The retake into account the variation of with xA*. quirements to make this correction as small as possible are the following: 1) The solutes should be slightly polar or nonpolar. 2) The inert solvent and the additive should have nearly the same refractive index. 3) The molar volumes of A and S should be nearly the same, in order to cancel the variation of l / y B m due to the variation of the configurational entropy, otherwise we should introduce the I+L function. As the difference between the molar volumes of the solvent and the main additive is usually much smaller than in chromatography, the use of the I+L function is much less critical. The origin of the intercept (Equation 32) is always given by: 1 1 1 = - -1 yAPyBm Aom OK
K-
TAB
Knowing the slope of the plot, i t is now possible to measure an association constant, OK, that has a thermodynamical meaning as discussed in the previous section. This constant however refers to the reaction between A and B a t infinite dilution in S. It is a function of the solvent s. Only when this solvent is the same as the one used in chromatography can we expect a good agreement between'the results of the two methods. Formation of Isomeric Complexes. Let us consider again what happens when the solute B reacts with the additive A to give two or more isomeric complexes as shown by Equation 24. Equation 30 becomes: (36) P = PBVB + PAB'FAB + PAB'VAB' being the chemical shift of a given nucleus of the molecule B, the electronic environment of which is perturbed by the complex formation. We have:
p
XAB
= PAB(xB
+
XAB
+
x.4B')
(351
and
X A B=~ P A B ~ X+B XAB+ XAB')
In this equation, neither OyARm nor 'yAR#=+ appear as they should be unity. Then we can write:
(39) Combining Equation 39 and the similar equation we can write for P A R , with Equation 36, we obtain:
v
= PBVB
+ (PAB)TPT
with (PAB)T
and
=
PAB
+
PAB'
(40)
1
Equation 40 is quite like Equation 30 and shows that association constants determined by NMR in the case of isomeric complexes are the total association constants ('KAB)T-i.e., the sums 'KAB f ' K ~ R. This expresses the same result as Equation 26 and, consequently, neither chromatographic nor spectrometric methods distinguish isomeric complexes. On the contrary, they both measure ('KAB1 T .
CONCLUSIONS Although large disagreements between different methods of studying complexation constants are reported, it appears that experimental results should be in much closer agreement if users were more aware of the requirements that should be fulfilled when chosing the inert solvent. This paper shows that in spite of superficial differences due to the change of the properties studied, chromatographic and spectrometric methods of determination of association constants lead to very similar equations. In both cases, we can expect to get meaningful results when the reactants have a low polarity and if the solvent used has nearly the same refractive index as the additive. In addition, the molar volumes of the solvent and main additive must be nearly equal in NMR. This is not necessary in the case of the chromatographic method but then a correction has to be applied, using for example the function $. Then chromatographic and spectrometric methods should give association constants in close agreement. These constants, however, do depend on the inert solvent used to make the measurements. The problem of the derivation of the absolute constants can be resolved only for groups of similar compounds (18). Although these conditions are not too difficult to achieve in practice, they are unfortunately not fulfilled in many of the works published up to now dealing with the determination of complexation constants. Thus, most of the data reported should be reconsidered. Unfortunately, the case of very polar solutes is not resolved a t the moment and further investigation in this area is needed.
RECEIVED for review June 8, 1973. Accepted February 12, 1974.
A N A L Y T I C A L C H E M I S T R Y , V O L . 46, NO. 11, S E P T E M B E R 1974
1399
