more ready attack on the carbonyl in some conformations of the alkyl chain than the equatorial alkyl group, so a significant axial population would tend to increase the overall rate by this mechanism. The degree of correlation with solution chemistry is less than quantitative, in any case, for the n-alkylcyclohexanones, though solution chemistry, where it exists, provides a useful qualitative guide to reactivity. Third, we note a factor of approximately two in the rates of the isomeric decalones 4 and 5. The trans isomer 5 reacts more rapidly. This is to be expected from examination of the compounds, which exist in the conformations shown below, 4 being able to convert between several stable forms and 5 being constrained to exist in one stable form. In the cis form 4, the hydrogen a t the 8-position
H
v
\
\H 5
4
tends to shield the carbonyl oxygen from attack, as shown, in any stable conformation; thus the rate of this compound should be slower. We again see, therefore, that this sort of ion-molecule reaction parallels effects of stereochemical parameters, and that isomers react a t predictably different rates, as we observed once before (7). Experiments with non-hindered isomers of the cyclohexanones are under way to confirm the assignment of a steric effect.
These conclusions form the basis of more general remarks. The steric environment of a functional group in a large molecule clearly influences its rate of reaction for a specific ion-molecule process when the reacting species are large enough and competing channels for products do not drain off enough of the collision complex to obscure the reaction of interest. This should be significant for chemical analysis when the mass spectra of isomers are very similar or when the mass spectra differ in ways which cannot be easily predicted. It now becomes important to study the interactions of more complex acylating reagents to determine if there is a correlation of ionic size with power to discriminate between functional groups of different steric environment. Perhaps more effective gaseous ionic reagents may be found by this route. It also becomes important to study the effect of ionizing energy on relative rates of reaction of isomeric neutrals. It has been shown that there is a dependence of relative rate constants on ionizing energy (26). Optimized conditions should be sought for analysis, perhaps the ideal case where one isomer reacts to produce a detectable amount of product on some set ICR time scale but the other does not. Studies designed to test these suggestions are in progress. RECEIVEDfor review February 19, 1974. Accepted June 12, 1974. This work was supported by the National Institute of General Medical Sciences (GM15994) and the Alfred P. Sloan Foundation. The instrument was purchased through funds from Hercules, Inc., the Shell Companies Foundation, the North Carolina Board of Science and Technology (159), and the National Science Foundation (GU 2059). (26)
M.L. Gross and J. Norbeck, J. Chem. Phys., 54, 3651 (1971)
Determination and Comparison of Association Constants for Weak Organic Complexes by Thermodynamic, Resonance, and Optical Methods Daniel E. Martire Department of Chemistry, Georgetown University, Washington, D. C. 20007
The usual gas-liquid chromatographic (GLC), Le., thermodynamic, methods of studying “acceptor/donor” systems yield the sum K1 a l , where K1 is the 1:l AD association constant and ai is a measure of the effect of noncomplexing A-D interactions. With weak complexes (small K1) the latter term may be an important contribution to the sum. a1 may be interpreted either in terms of the solution nonideality of uncomplexed solute (A or D, as the case may be) or in terms of contact pairing between A and D. The techniques of nuclear magnetic resonance (NMR) spectrometry and ultraviolet/visible ( U V / V ) spectrophotometry yield K , separately. However, with small K1 values, mixed solvency effects could produce serious error in the NMR and U V / V experiments. Equations are derived and supporting examples are given which illustrate important considerations relative to the analysis, interpretation, and comparison of measurements from GLC, NMR, and UV/V.
+
1712
This study was prompted by reports ( I , 2 ) that, for a given “acceptor/donor” pair, quite different values for “association constants” are obtained by thermodynamic (gasliquid chromatography (GLC)) and spectral (nuclear magnetic resonance (NMR) spectrometry and ultraviolet/visible (UV/V) spectrophotometry) means. In one such investigation ( I ) , GLC produced seemingly reasonable values for all systems studied, while NMR and UV/V gave much smaller (and sometimes negative) values. In another study ( 2 ) , the GLC results were uniformly larger than the NMR results. The equations derived in the present paper provide a basis for explaining these apparent discrepancies and should serve as a guide to potential pitfalls in determining and comparing “equilibrium constants” from the three methods. (1) J. H. Purnell and 0. P Srivastava, Ana/. Chem., 45, 1111 (1973). (2) D. E. Martire. J. P. Sheridan, J. W. King, and S. E. O’Donnell, J. Amer. Chem. SOC..submitted for publication.
A N A L Y T I C A L C H E M I S T R Y , VOL. 46, NO. 12, OCTOBER 1974
We shall consider liquid mixtures containing two neutral organic species, an “acceptor” (A) and a “donor” (D), dissolved in an inert solvent (I), where one of the conditions C D >> C A or CA >> C D obtains and where Ci refers to the concentration of uncomplexed i. As will become evident, the thermodynamic properties of such systems and their spectrometric behavior are governed not only by the extent of complex formation, but also by noncomplexing interactions between A and D. Complex formation is an association of definite stoichiometry between A and D. When we say that a donor and acceptor form a 1:l complex in an inert solvent, we mean that the number of adjacent DA pairs is in excess of the number to be expected on the basis of random encounters. This complex has a lifetime which i5 long compared to the duration of a molecular collision. In addition to these longer-lived pairs (which usually require a special mutual orientation of A and D for formation), there also exists a range of loose contacts between A and D, whose lifetimes are of the order of the duration of a molecular collision. (The former are sometimes classified as “chemical” interactions and the latter as “physical” ones.) Orgel and Mulliken (3) termed these loose interactions “contact pairs.” In their simple model, all AD pairs are arbitrarily divided into two classes, namely, 1:1 complexes and contact pairs, where the former reduces the effective number of free A and free D in solution, while the latter does not. Their model will be applied to the NMR and UV/V procedures for obtaining association constants and will be one of two applied to the GLC approach. I t will be shown that, if the number of contact pairs is small compared to the number of complexes, then, under these conditions (Le.,for “strong” complexes), valid equilibrium constants ( K )can be obtained directly by any of the three experimental methods. However, with “weak” complexes, the effect of contact pairing must be taken into account, particularly in the GLC experiment. The concentration of weak complexes in solution cannot be determined from thermodynamic measurements alone without the aid of a detailed model. The GLC method provides no exception. In the other model applied to the GLC method, solution theory is used to describe the effect of loose collisions in terms of the solution nonideality of uncomplexed solute (D or A, as the case may be). Both models lead to the result that the usual GLC approaches never yield K separately, but always a sum K + ci, a difference which is unimportant only for strong complexes ( K >> ( 1 ) . With the Orgel-Mulliken ( 3 ) model, CY is regarded as a quasi-K for “contact pair formation” (see later), while in the other (and more explicit) model. LY is related to combinatorial, energetic, and structural effects in solution. For the spectral methods, it will be shown that the presence of contact pairs affects only the spectroscopic properties of the “complex,” and, hence, that K can, in principle at least, be obtained separately. This is in agreement with other workers (3, 4 ) . However, wih NMR and UV/V, a potential pitfall in determining K for weak complexes is related to the solution nonideality of D in I (taking C D >> C A ) .
D be the electron donor additive, I be the inert solvennnt, and M be the mixed solvent (D I) a t concentration Cu. The usual condition C D >> C A applies where Ci is the concentration of uncomplexed i. Recently, it was shown (8) that the generalized equation relating the apparent parti, the partition coeffition coefficient of A in M, ( K R ) M to ~ , the 1:l AD ascient of uncomplexed A in pure I, ( K R ) and sociation constant (K,) is given by
+
where y ~ *and y ~ are * the infinite dilution activity coefficients of uncomplexed A in, respectively, 1 and M (based on the mole fraction convention that y L A 1 as x~ -* l), is the activity coefficient of D in M a t C D (based on the con1 as CD 0), and 1‘1 and c h are ~ the vention that y~ molar volumes of I and M, respectively. Also (8, 9 ) ,
-
+
+
where xtAis the interaction parameter of uncomplexed A with i ( i e . , the sum of the structural and energetic contribution to In y t A ( 9 ) ) , and In (y!“)“ is the combinatorial or molecular size contribution to 1nyLA.Previously ( 8 ) , we showed that
(3 Assuming that u I 1 U D , and letting XM* = xn“41, + xrAdl( v * / v n ) x ~ ~ @( I ~O @ ) , where ~ CbL IS the volume fraction of component i in M, one obtains (klA -
4
kM
=
(IDXI*
- /‘DxD*
D
-
)CD
-
( 1 4 1D \ I D ) ~ D ’
(4)
where xnA is the interaction parameter of uncomplexed A with D, XI^ is the interaction parameter of I with D, and the u ’ s in Equations 3 and 4 represent molar volumes. Thus, Equations 2-4 combined give
‘D\DA
lAkID)ICD
-
1
-
lA(D\I
D
’
=
@,CD -
(5)
where ~1 and nq, the coefficients of C D and C112, respectively, are independent of CU. Note that for the case CI) I C I , X M =~ XD*@D + XI*QI - ( U A / U I ) X I ~ Q D @( I O ) , and Equation 5 becomes
ASSOCIATION CONSTANTS FROM GASLIQUID CHROMATOGRAPHY Let us consider the GLL method utilized by Purnell ct al. ( I , 5-7j for obtaining “association constants” of organic complex formation. Let A be the electron acceptor solute, (3) L E. Orgel and R S Mulliken, J. Amer Chem. Soc , 79, 4839 (1957). (4)R . L. Scott, J. f h y s . Chem., 75, 3843 (1971). (5) J. H. Purnell, “Gas Chromatography,” A. B. Littlewood. Ed., Elsevier, Amsterdam. 1966,p 3. (6)D. F. Cadogan and J. H.Purnell. J Chem. SOC ( A ) . 2133 (1968). (7) D F. Cadogan ana J H Purnell. J Phy; Chem, 73,3489 (1969)
(8)
H. L. Liao, D. E. Martire, and J. P. Sheridan, Anal. Chern
45, 2087
(1973). ( 9 ) D. E. Martire, Anal. Chem., 46, 626 (1974). (10) H.Tompa, “Polymer Solutions,” Butterworths, London, 1956, p 182. cq 7.22.
A N A L Y T I C A L C H E M I S T R Y , VOL. 4 6 . N O . 12. OCTOBER 1974
1713
+
ception being that the sum (cu12/2) cy2 in Equation 8 is replaced by cu12 in Equation 14. At sufficiently low C D , ( K R ) Mshould be a nearly linear function of CD, enabling . a1 may be inone to measure the sum K1 ( ~ 1Moreover, terpreted either in terms of the nonideal behavior of uncomplexed A in M, or in terms of an “equilibrium quotient” for contact pairing. Note that the former interpretation leads to an explicit relation for a1 and to the result that a1 depends not only on A and D, but also on the nature of the “inert” solvent (Equation 5 or 5’). In Equations 8 or 14, the C D term ~ can be neglected only a t low C D and/or when the coefficient of the Co2 term is sufficiently small. Also, 131, which is a measure of the nonideal mixing of D and I, can be either negative or positive (see Appendix), whereas K1, K2, and cq are positive, while c u p (Equation 8) can be negative or positive ( ~ can 1 be~ negative or positive-see Appendix). Therefore, the CD* term could be positive, negative, or (with fortuitous cancellation) zero. In any event, if a large enough C D range is studied, the coefficient of the CD* term may be determined empiri- l ] / c D us. cally from the slope of a plot of [((KR)M/(KR)IJ CD, which also yields K 1 a1 directly as the intercept. Thus, in general, GLC studies yield the sum K1 cy1, and not K 1 separately. With strong complexes ( K 1 >> a l ) , this need not be worrisome. Similarly, if no long-lived complexes exist (K1 = O), GLC would yield only the “equilibrium quotient” 011. With weak complexes (small K l ) , however, GLC indiscriminately measures the sum of equilibrium constants for all isomeric long-lived 1:l AD complexes present (see Equation 9) p l u s short-lived A-D contact interaction pairing. The latter may not be negligible, a situation which can be advantageous or disadvantageous, depending on the molecular information one hopes to infer from the investigation (see Discussion). It should be noted that workers in the field often attempt to account for the 01 term (or to justify its neglect) by studying an arbitrary noncomplexing (N) solute such as a cycloparaffin or paraffin (6, 8). The usual argument that a zero value for c y I of N implies a zero value for a1 of A is not justifiable, in general, on theoretical grounds (see Appendix). Under certain circumstances it may be possible to select and study a proper nonpolar analog (11, 12) of molecule A in order to estimate
+
where
*/D
will be taken as the following polynomial in CD: 7~ = 1
+
$ ~ C D+
@2cD2
+
m . .
(7)
With Equations 6 and 7 and an additional term to allow for the possibility of AD2 complexes, we have, up through terms in CD’,
(KR)M=
(I(E)I
[
+
(Ki
+
+
+
where K S is the equilibrium constant for the reaction D AD + AD2 ( 5 ) . I t can readily be shown (2, 5 ) that Equation 8 also applies when two isomeric 1:l complexes, type a and type b, are present. In that event.
+
and K2 would be a more complicated function which need not concern us here. In the derivation of Equation 8, account was taken of the difference between the partition coefficient of uncomplexed A in M and that in I. This was implemented through an activity coefficient correction (the ratio ~ 1 % I/ y ~ in Equation b ~ l), and led to the a terms in Equation 8. Alternatively, one could account for this difference by utilizing the Orgen-Mulliken model ( 3 ) , which, in effect, would attribute this difference to noncomplexing A-D interactions. or contact pairing, in M. It arbitrarily divides A-type molecules into three classes: (a) unperturbed and uncomplexed A (as in pure I), with its concentration denoted by CAC,(b) uncomplexed A undergoing loose interactions (contact pairing) with D, and (c) A in complexes with D, the concentration being given by c.4~. The concentration of contact pairs (CA-D)is assumed to be proportional (proportionality constant of c q ) to the product CACD, where C, is the concentration of uncomplexed i ( 3 ) .Hence, a1 may be regarded, roughly, as a quasi-K for short-lived contact pairing. Furthermore, the total concentration of uncomplexed A is the sum C A * CA-D, and we have the following equilibrium expressions:
+
+
a1.
If one neglects the CD* term, Equations 8 (or 14) and 9 give:
(ER) I
A*(gas)
(10)
A*(liq)
Among the possible combinations that a GLC measurement of “K1” for a presumed donor/acceptor system might be reflecting are: (i) K1,a K1,b 011, (ii) Kl,a cy1, (iii) K I , +~ Kl,b, (iv) K I , ~or, (v) q.Spectroscopic and/or dielectric studies can often be used to help distinguish among these various combinations (see later). In addition, the GLC experiment alone might provide circumstantial evidence as to whether one equilibrium constant or the sum of two or more equilibrium constants is being determined. For example, it can be shown ( 3 )that with (iii):
+
AD
+
IC?’
D ZZ AD,
(13)
where the prime superscripts denote stoichiometric (concentration units) equilibrium quotients. Correcting the K’s for possible nonideality of D in I, and following the derivation set o u t by Purnell (5) for class A(ii) equilibria, with terms higher order than CD’ neglected, one obtains from Equations 7 and 10-13: (ICRIbI
= (fcR)T[l
(IC1
(f> CA, and CA is the concentration of uncomplexed acceptor. T o be completely general, assume that there are four distinct proton frequencies: completely free proton ( OA*), proton perturbed through collisional contact pairing or loose A-D interaction ( U A - D ) , proton perturbed by 1:l complex formation of type a (VAD,), and proton perturbed by 1:l complex formation of type b (u*nh). Because of the assumption of contact pairing, VA-D should be regarded as an averaged value over all orientations ( 3 ) .Again letting CA-D = a l C ~ C uone , obtains for the time-averaged or observed proton frequency:
Thus far, only the Purnell approach to determining “association constants” by GLC has been discussed; it is the most general (8). The method devised by Martire and Hiedl ( 1 3 ) ,which is applicable under certain weil-defined conditions, employs the following expression for the GLC determination of K 1:
where, as before, C A = CA+ CA-D. Rearranging terms, Equation 19 becomes
In the above, D refers to the pure electron donor solvent, I to the pure inert reference solvent (chosen to have approximately the same molar volume, structure, a c d polarizability a s D), V,” is the specific retention volume, A refers to the electron acceptor solute, N to an arbitrarily chosen alkane solute, and y L Jto the infinite dilution solute (j) activity coefficients (based on the mole fraction convention that y l J ---*1 as xJ 1). The other symbols have already been defined. Note that -,D* represents the value for uncomplexed A in pure D. In the past (13-15), it has been argued that the solute activity coefficient ratio on the r.h.s. of Equation 17 should be approximately unity. I t is now found that this is not justifiable, in general, on theoretical ground. The value for this ratio turns out to be 1 tulC~,+ ( a l 2 / 2 ) C n 2where , a1 is given by Equation 5 with U D i= V I (see Appendix). Also, Y D is close to unity (see Appendix). Therefore, with y~ = 1 + @ 1 c D , Equation 17 leads to a form similar to Equation 8 and identical to Equation 14 through C nL:
+
where
w he r e A*Da = u A D a
-
+
AADb
=
AA-D
vA-D
CD
1
- -~ ~
Kl’A,
A
+
-
-
v** vA*
T o simplify matters, the possibility of termolecular AD2 complexes has been ignored in the above derivations. For an excellent discussion and treatment of the effect of termolecular complexes the reader is referred elsewhere (16, 1 7 ) . Equation 20 is in the form used by Foster and Fyfe ( 1 8 ) for plotting their NMR results. I t is equivalent to that suggested by Scatchard (19) for treating UV/V data. Deranleau ( 1 6 ) demonstrates the clear advantage of this type of plotting equation over the Benesi-Hildebrand or doublereciprocal equation ( 2 0 ) and the Scott or half-reciprocal equation (21 ). The latter is given by ~
The approach based on the above equation has the advantage of‘ requiring on157 relative retention measurements on two columns, but has the potential disadvantage that the Co2 term with pure D may not be negligible (8). In any event, with the CIj2 term assumed negligible, both the less general approach and the Purnell approach give the sum K 1 cy1 as the “apparent association constant.”
Vmb
VA*
~
-
cC DT
A,
(25)
To convert ths equilibrium quotient K 1’ to an ecluilibrium constant K 1, we assume that A and AD are sufficiently dilute in I, $0 that y * ~ / y ~ 1 (based on the convention that y L 1 as C, 0 ) .Thus, with Equation 7:
-
+
-+
h’l’
h’l)D
fDr)*CDlsI (5a 1 where ( 7 ~)"," the infinite dilution mole fraction activity coefficient, is found from Equation l a to be
From Equations la, 5a, and 6a:
When x
I
>> x D, Equation 7a may be approximated by
+
Thus, for PlCD
