of pulsed source time resolved phosphorimetry are apparent. Morphine, codeine, and ethylmorphine have only slightly different phosphorescence lifetimes, but morphine can still be temporally-resolved from either or both of the other two. Because ethyl morphine and codeine are metabolites of morphine, in vivo, this measurement illustrates the possibility of analysis of drugs and metabolites that would be difficult to separate by physical means. Time resolution of a short- and long-lived species is illustrated by the analysis of morphine and quinine. This particular combination also has a potential practical application because quinine is used as a diluent for heroin. Morphine, being the immediate metabolic product of heroin if adequately separated from the urine mixture, could be determined in the presence of quinine, assuming the quinine is also extracted along with the morphine. Spectral overlap of amobarbital and phenobarbital prevent spectral isolation; time resolution can be applied to the determination of this mixture. However, spectral and temporal overlap ‘of amphetamine and methamphetamine do not allow either/or both spectral and/or time resolution of this mixture. Cocaine and phenobarbital similarly do not show a sufficient difference in phosphorescence lifetimes (see Table 11) to allow temporal resolution of a mixture of these species. No results are given here for real street samples. However, time resolved phosphorimetry should save considerable time for a number of specific analyses, such as some of the ones mentioned above. Most street drug analyses involve simply a qualitative analysis via TLC and/or colorimetric methods. Quantitative analyses of drugs of abuse in biological fluids and in street samples generally involve extensive “clean-up” procedures (14-18), as well (14) D . Sohn, J Chromatogr. S c i . . 10, 294 (1972) (15) A. Romano, J . Chromatogr S c i . . 10, 342 (1932)
as one or more separation steps (14-18), and finally some sort of measurement step including gas chromatography (16-18), fluorimetry (16, 18), and absorption spectrophotometry (16, 18). The greater selectivity of time resolved phosphorimetry could certainly enable the analyst to omit or shorten some of the steps prior to the final measurement. Qualitative Analysis. Because excitation and emission spectra and phosphorescence lifetimes of a molecule in a given environment are characteristic of that molecule, and because these parameters may change to a different degree for different molecules with a change in environmental conditions, time resolved phosphorimetry could be a useful technique for molecular identification. From an examination of the data in Table I and Figure 2, it is obvious that, of these drugs, any single (phosphorescing) component sample could be identified qualitatively uia phosphorescence excitation and emission spectra and lifetime, except for the binary mixtures of meperidine and any of the amphetamines and ethyl morphine and codeine. (It could well be, however, that with additional measurements of lifetime of these species in other solvents, these combinations would be resolvable and therefore identifiable.) Therefore, qualitative analysis could be accomplished on multicomponent samples as long as the species are spectrally- and/or temporally-resolvable. Received for review November 21, 1973. Accepted March 19, 1974. Research was carried out as a part of a study on the phosphorimetric analysis of drugs in blood and urine, supported by U.S. Public Health Service Grant No. GM11373-11. (16) H . E. Sine, M . J. McKenna, M . R . Law, and M. H . Murray, J Chrornatogr. S c i . 10, 297 (1972) (17) L. Kazak and E. C. Knoblock, Anal. Chem.. 35, 1448 (1963). (18) “ A Bibliography of References on the Analysis of Drugs of Abuse,’ J. Chromatog S C I . 10, 352 (1972)
Kinetics of Chromatographic Processes with Application of the Steepest Descent Approximation Method Kuang-Pang Li,’ David L. Duewer, and Richard S. Juvet, Jr.* Department of Chemistry, Arizona State University, Tempe, Ariz. 8528 1
This paper extends earlier fundamental studies of the mechanism of chromatographic processes to include, in addition to simple partitioning, the interaction of the solute with the stationary phase through either physical or chemical reaction. The steepest descent approximation method is applied in overcoming the inherent mathematical difficulties introduced by these additional considerations. It is evident from computer simulation and solution of the appropriate differential equations that, not only is retention a function of the stability of the association product formed between the solute and the liquid phase, but also the elution peak profile is a function of the rates of formation and dissociation of the association product formed on the column. This work id:,
’ Present address, Department of Chemistry, University of FlorGainesville, Fla. 32601.
- T o whom correspondence concerning this contribution should he addressed.
suggests the direct evaluation of the rate of solvolysis or the rate of solute complex formation from peak profile studies. The often observed shift in base line following peak elution may also be explained solely on rate considerations and need not involve adsorption or irreversible chemical reactions on the column.
The theory of chromatography is well established and numerous authors have contributed to the description of on-column phenomena. Early theoretical studies, such as the pioneering work of Lapidus and Amundson ( I ) , van Deemter e t al. (2), and Giddings and Eyring ( 3 ) ,empha(1) L. Lapidus and N. R . Amundson. J. Pbvs. Chem.. 56.984 11952) (2) J J van Deemter, F J Zuiderweg, and A Klinkenberg, Chem Eng S o , 5, 271 (1956) (3) J C Giddings and H Eyring, J Phys Chem 59. 416 (1955)
A N A L Y T I C A L C H E M I S T R Y , VOL. 4 6 , NO. 9 , AUGUST 1 9 7 4
1209
Mobile phase
have an average velocity a over the wfiole column. However, in addition to pure partitioning, it is assumed in this model that the solute molecules, A, can interact, not necessarily reversibly nor in equilibrium with the stationary phase, Sp, by either physical or chemical reaction. The complex or reaction product, A-Sp, in this treatment is assumed to have negligible vapor pressure and is not eluted. Rate Theory. The process can be described with the following rate equations:
-i
A kl Stationary phase
1
4 I(
ki
A + % ,
ki
, A.Sp
-x
Flgure 1. Model of solute association and definition of kinetic terms
a2ci -
Dsized the development of mathematical models. This early work was later extended by Giddings (4-7) to include more complex mass transfer phenomena. The generalized approach thus developed rendered an important service in being immediately applicable to a large number of real chromatographic problems; however, the theory is based on the assumption that on-column reactions proceed under near-equilibrium conditions. This assertion is made to permit mathematical simplifications where otherwise extreme difficulty would be encountered. Recent research considers numerical analysis of elution curves (8-10). Different curve fitting techniques have been employed and statistical moments up to 6th order have been studied. Although high precision and accuracy have been achieved, a fundamental understanding of the mechanistic characteristics of chromatographic processes has not been emphasized in these studies. The inherent mathematical difficulties encountered in earlier theories lie mainly in the inverse of the Laplace transform obtained from the rate equations which are chosen to describe the chromatographic processes. In this paper, an attempt is made to overcome these obstacles so that more complicated on-column processes can be treated quantitatively. In this context, the steepest descent approximation method (11-14) appears to be the most promising of the currently available methods. The method offers both versatility and ease of computer implementation.
-d2hi -_dxz
h s3 U ' d h i - _1 D dx D
~~
+
(10) (11) (12) (13) (14)
S2(k
c, c,
at
(1)
when x = 0, t 2 0
= = CZ =
c,
when x
= 0
>
0, t = 0
Equations 1-3 can be solved by the method of the Laplace transformation. By definition C1, Cz, and C3 can be expressed as follows:
e [ c , ( x ,t ) ] =
e - S t C i ( x t, ) d t = h i ( x , s ) = hi (4)
Jm
0
where i = 1, 2, 3, and s is a parameter for transformation. Elimination of hz and hs gives
+
J. C. Giddings, J. Cbem. Pbys., 31, 1462 (1959). J. C. Giddings, J. Cbromatogr., 3, 443 (1960). J. C. Giddings, J. Cbromatogr., 5, 46 (1961). J. C. Giddings, J. Pbys. Cbem., 68, 184 (1964). 0. Grubner, Anal. Chem., 43, 1934 (1971). S. Wicar, J. Novak, and N. Ruseva-Rakshieva, Anal. Cbem., 43, 1945 (1971). S. N. Chesier and S. P. Cram, Anal. Cbem., 43, 1922 (1971). M.R.HoareandT. W.Ruijgrak, J. Cbem. Pbys.,52, 113(1970). M. R. Hoare, J. Cbem. Pbys., 52, 5695 (1970). K. H. Lau and S. H. Lin, J. Pbys. Cbem., 75, 2458 (1971). C. Kittel, "Elementary Statistical Physics," Wiley, New York, N.Y., 1958.
1210
ac,
at
~
model to represent the system and then to develop a theory which is rigorously applicable to the model. Discrepancies of the theory and experiment will indicate the weakness of the model and thus be of help in finding more realistic models for the system. A uniformly packed column may be envisaged ideally as a capillary column of unit cross-section, having its inner wallcoated with a homogeneous film of a stationary liquid (see Figure 1);Laminar flow of the carrier gas is assumed to (4) (5) (6) (7) (8) (9)
+ac2 +
where C1, Cp and C3 are, respectively, concentrations of the solute a t time t in the mobile phase, in the stationary phase, and in the complex form at a position x cm from the inlet of the column; k and k' are the sorption and desorption rate constants of the unassociated solute molecules across the gas-liquid phase boundary; and k1 and k-1 are the rates of association and dissociation, respectively, of the solute with the stationary phase. Instrumental void volume is not included in these equations and is assumed negligibly small. Moreover, since solute diffusivity in the statimes smaller than that tionary phase is generally ca. in the mobile phase, it is sufficiently small to be neglected. The following boundary condition and initial values were chosen for the above partial differential equations:
such as a chromatographic column, it is rather difficult, sometimes impossible, to consider all parameters a t the same time. An easier approach is to choose an idealized
-
ax
at
and
THEORETICAL CONSIDERATIONS Idealized Model. To deal with a highly complex system
~
,. -aci
aCi
a x2
k'
sz
+ k l + k - 1 ) + s ( k k - 1 + k'k-1 + k k i ) + S(k' + k - 1 + k i ) + k ' k - i
= 0 (5)
This equation must be solved subject to the conditions,
-
hi = [Co(t)l = h&) hi = finite as x 05 By making the change of variable, y = hl exp(-ax/2D), Equation 5 is reduced to the convenient form
where
A N A L Y T I C A L CHEMISTRY, VOL. 46, N O . 9 , AUGUST 1 9 7 4
g ( s ) = i[(s D k , = k'
+
+
9)+
k,
+
k-1
uz
A = k + - - - k2 40
2
A
+
encouraged to consult other references (11-14) for a more detailed discussion. F l ( t ) may be written in terms of Bromwich’s integral formula (25): The solution for y is obvious,
(7)
y = ho(s)e-xm)
where
Let F l ( t )be the inverse Laplace transform of the function
e-[-
+A
$Js
“1
+ B~2 s -+E2
+
That is,
Bs
+
The integration in Equation 14 is to be performed along a line s = y in the complex plane where s = s1 isz. The real number y is so chosen that s = y lies to the right of all the singularities but is otherwise arbitrary. Hence, we can rewrite Equation 14 as a real integral,
“)] (8)
s2 - E 2
fi(t) =
~11
+
-
e f ( s ) d2s
(15)
Then from the rule for translation of result function (15) c-i
[ ( ’ - X l Z )
] = e-k2t’2Fl(t)
(9 )
Accordingly, the inverse transform of y is 2‘’ [y] = e - ‘ [ h , ( s ) e -X m ]
s,i
=
C,(t - T)e- ( k 2 / 2 ) T F i ( r ) d (10) T
The last expression is obtained from the convolution theorem (15). Since y = hl exp ( - i i x / 2 D ) , therefore
c1
-
2 - 1[hi] =
-
erx /2D
c-1 [ e ; X l 2 D y
] =
ebx/2D2-1
C,(t - r)e- ( k 2 / 2 ) ~(-r)dT ~ ,
[Y 1
c,
it
C o e r x ’ 2 D e‘ ( k z / 2 ) TFi(T)dT
(12)
A distorted S-shaped curve is formed. The expression foi C1 in Case (ii) is
c,
t
= c O e r x I 2 DL
s22
+
. ..
(16)
where the point (so, 0) is the root of f’(s) = 0. For first order approximation, the higher terms in Equation 16 can be neglected. Combination of Equations 15 and 16 gives,
Substituting Equation 17 into Equations 1 2 and 13, we obtain,
and
t > T
where T is the duration of the injection pulse. The former is encountered in frontal analysis while the latter is generally applied for elution studies. In Case (i), Equation 10 becomes E
2
O s t s T
C,(t) = 0
C,
f(s) = f(s0) --f”(s0)
(11)
Equation 10 is applicable for any form of feeding function, C o ( t ) .However, two special forms are of major interest in chromatography: (i) C o ( t ) = Co, a constant a t all time t > 0; and (ii) Co(t) is a pulse function. That is,
Co(t)=
In this problem, it is always possible to find one minimum ( f ” ( s ) > 0) along the real axis. The position of the minimum is denoted by the point s = ( S O , 0). The value of the integrand falls off sharply a t points away from the real axis a t (SO, 0). Now if f ( s ) is expanded about the point (so, 0) in the imaginary direction a t which the function has a minimum, Equation 16 is obtained.
T
e - ( k : , / 2 ) TI(. F
)dT
(13)
Steepest Descent Approximation. In both cases, the evaluation of C1 requires the knowledge of the function F l ( t ) . Although there is a unique relation between a Laplace transform and its inverse, to find the inverse transform is by no means an easy task. However, the evaluation of F l ( t ) is achieved rather conveniently and with sufficient accuracy by means of the approximation method of steepest descent. Readers who are interested in this method are
(19) for the frontal analysis and elution studies, respectively. Method of Computation. The major computational difficulty encountered in the application of the steepest descent method occurs in the evaluation of the root (so, 0) from the equation f’(s) = 0. A desirable root must be a positive real number which lies to the right of all singularities of the function f ( s ) . Furthermore, the second derivative of f ( s ) at ( S O , 0) must be positive as required by the theory. In the given system, the evaluation of S O involves the solution of a 9th order polynomial:
f ( s ) = s9
+ Ais8 + A2s7 + , . .. . +
= 0
(20)
with coefficients: Ai = A - L2/4Dt2 A2 = B - 4E2 A3 = C - 4AE2 (2E2 B)L2/2Dt2 A4 = 6E4 - 3BE2 -k C L 2 / D t 2 A5 = 6AE4 - 3CE2 - (6E4 2BE2 B2)L2/4Dt2
+
(15) G. A. Korn and T. M. Korn, “Mathematical Handbook for Scientists and Engineers,” 2nd ed., McGraw-Hill. New York, N.Y., 1961.
+ A,
A,s
+
+
+
A N A L Y T I C A L C H E M I S T R Y , VOL. 46, N O . 9 , A U G U S T 1 9 7 4
1211
L = 2 5 cm ii= 3 cm sec-’ D=0.1 cmzsec-’
t
L 25 cm ii- 3 cm sec’ D-0.1 cm2sec-’ k-100 s e d 8 sec-’
al ln
.,
100
150
Figure 2. Variation in simulated chromatographic peaks as a function of the effective rate of desorption, k3, for large values of the complex dissociation constant, k-’, and typical values of column length for complex formation studies, L = 25 cm; linear gas velocity, (5 = 3 cm sec-‘; and solute diffusivity in the gas phase, D = 0.1 cm2 sec-’. Rate of sorption maintained constant at k = 1 0 0 sec-’
+
RESULTS AND DISCUSSION The family of curves shown in Figure 2 is representative of that obtained from Equation 19 with a feeding pulse of very short duration, T , and large k-1. Since dissociation of complex ASp is fast, chemical equilibrium may be assumed to be maintained instantaneously. Solute molecules are eluted with an “effective rate of desorption,” (16) D. D. McCracken and W. S. Dorn, “Numerical Methods and FORTRAN Programming with Applications in Engineering and Science,” Wiley, New York. N.Y., 1968, pp 133-5.
1212
300
350
Time, sec
Figure 3. Variation in simulated chromatographic peaks as a function of the complex dissociation constant, k-’. Equilibrium constant, kjlk-1 = 1; true rate of desorption, k’ = 8 sec-’; effective rate of desorption, k3 = 4 sec-’; and rate of sorption, k = 100 sec-’. Column characteristics as in Figure 2
+
A6 = 3BE4 - 4E6 - (2CE2 BC)L2/Dt2 A7 = 3CE4 - 4AE6 + (2E6 - BE4 - 2C2 - B2E2)L2/2Dt2 As = E’ - BE6 (CE4- BCE2)L2/Dt2 Ag = AEs - CE6 - (E8- 2BE6 + B2E4)L2/4Dt2 where A, B, C, and E are defined as functions of rate constants in the previous section. L is the column length. Since there must be a definite time elapsed, to > 0, before one starts to observe a positive signal a t the outlet of the column, coefficients A1 . . . A9 are real. All possible roots of Equation 20 (for the initial time to) can be evaluated with Bairstow’s method of quadradic factors (15). These roots are then flagged as to their acceptability as the “prime” root according to the constraints established by the steepest descent method. Roots that pass the criteria are evaluated by running a trial function evaluation and, if necessary, a trial peak evaluation. This process continues until all roots are checked or a root is proved to be the “prime” root. Once the “prime” root is found for the initial time, the function is evaluated for following times with Newton’s method (16).The “prime” root for the immediately preceding time is used as the first approximation to the current one. As Newton’s method can find only one root a t a time, if two or more roots to the polynomial are in the real domain and sufficiently close in magnitude, than it is possible that either a “wrong” root or no root at all will be found. This is, indeed, occasionally a problem near the maximum of a sharp elution peak. Bairstow’s method can be used to evaluate the function in this region whenever it proves necessary. Once the prime root has been established, the evaluation of C1 for the given time is a straightforward computation. The solution has been implemented on a PDP-8/E minicomputer in FOCAL, and on the GE 440, Univac 1110 and CDC 6400 computers in FORTRAN. Execution on the high-precision CDC 6400 is by far the most convenient.
250
200
Time, sec
k 3 -
k‘
1
+ k,/k_,
Solute molecules with greater effective rates of desorption are eluted rapidly and the peak widths of the elution curves are narrow, while those with smaller k3 values have longer retention times and much broader elution peaks. Broadening is ascribed to the increased effect of the longitudinal diffusion of the solute molecules in the gas phase as desorption becomes slower. If the chemical reaction is maintained at equilibrium at all times, but the partitioning processes need not be so, k3 is essentially equivalent to the desorption rate constant given in the Lapidus and Amundson equation (1) only its numerical value is less than the latter by a factor of (1 k l / k - l ) . Thus, the Lapidus and Amundson model (and also van Deemter’s) is actually a special case of the system described in this paper. Moreover, if all column parameters and the sorption-desorption rates are identical, a given solute molecule will have longer retention time in GLC than in GSC. For Gaussian-like peaks, as would be expected from symmetry, the retention time measured by the position of the peak maximum approaches that of the centroid (the center of gravity or the first statistical moment) defined as,
+
sw
tC,(t)dt
Pi =
0
k m C 1( t ) d t That is, t,,, = t , for a Gaussian-like peak. This relationship has been determined by others (17-19). We have established numerically that the elution peak centroid, located a t t,, is given by,
t, =
pi =
(1
+
k/k,)L/E
(22)
for all elution peaks generated from this model, irrespective of peak symmetry. Thus the centroid, rather than the observed retention time, would appear to be a more desirable method to express peak position if it could be readily evaluated. For asymmetric peaks, the observed peak maximum and the centroid may differ appreciably. If the chemical reaction is not instantaneously at equilibrium--i.e., dC,/dt # 0 a t all t > 0-the peak shape and (17) H. Yamazaki, J. Chromatogr.,27, 14 (1967).
(18)0.Grubner, Advan. Chromatogr.,6, 173 (1968). (19) E. Kucera, J. Chromatogr., 19, 237 (1965).
A N A L Y T I C A L CHEMISTRY, V O L . 46, NO. 9 , AUGUST 1 9 7 4
k ’ = 16 k,. 8
I
k ’ ~ 1 6
k’:16
k,=
k,:
4
2
4
\
k’=8
k’=8
k,= 4
kl- 2
(Y
?:4
-m
C (Y
200
400 600 T l m e . sec
800 $ 6
p
+
k’= 4 k,z 2
200
400
600
800
T I m e , see
Figure 4. Variation in simulated chromatographic peaks as a function of the true rate of desorption, k’, and the effective rate of desorption, k3. Rate of sorption, k = 100 sec-l. Column characteristics as in Figure 2
peak position will depend not only on the sorption rate constant, k , and the effective desorption rate constant, ks, but also on the magnitudes of the chemical reaction rates, k l and k-1. The family of elution curves of Figure 3, computed from Equation 19, demonstrates clearly the effect of the on-column reaction kinetics on the peak characteristics. In this figure, the ratio of the reaction rate constants, hllh-1, is maintained constant but the absolute values of k l and L1are varied. When h l and k-1 are small compared to h and k’, although the solute molecules are constantly partitioning at the gas-liquid surface, few become associated with the liquid phase. However, once a solute molecule is immobilized by association, it will lag behind the bulk of the zone until it is released in the dissociation reaction. The time which a molecule will stay in the complex form is a completely random parameter. But the statistical lifetime of the complex is closely related to the reciprocal of the dissociation rate constant, k-1. Because of the lag in elution in the presence of complex formation, the elution curve exhibits an extremely long tail. Thus the solvolysis or association rate may be sufficient in itself to explain the experimental observation that the base line occasionally appears to shift upward following peak elution and does not immediately return to its original value. Curves generated under the conditions given in Figure 3 for h-1 values of and lo-*,” sec-’ are examples of this. Such occurrences need not involve adsorption of the solute on the solid support nor chemical reaction of the solute with trace impurities in the liquid phase forming less volatile products eluted slowly from the column, the explanations most often given for this observation. As a larger fraction of solute molecules participates in the complexation reaction owing to an increase in the values of both k l and h-1, the elution peak becomes less spike-like and its tailing effect becomes more significant.
At intermediate values of h-1, the peak maximum, and thus the retention time, becomes more and more ill-defined. Tabletop-like curves are obtained under the condi~ sec-I. Such curves tionsgiven inFigure 3 when h - 1 10-1.5 have been observed experimentally by Juvet and coworkers (20) in the gas chromatographic elution of aluminum chloride from a sodium tetrachloroaluminate column. The elution curve eventually approaches Gaussian shape when h1 and h-1 become sufficiently large that every solute molecule is complexed with the solvent molecules at least once during its elution. The observed retention time measured a t the peak maximum is obviously different from that observed when chemical interaction is insignificant. Variations in the degree of solvation or association account for differences in retention time for a solute in different stationary phases although all other experimental parameters are maintained constant. Thus, the interaction between a solute and different liquid phases may be studied by the variation in retention time providing column parameters are identical and peak maximum is well defined (20). A family of curves with general features similar to those given in Figure 3 has been simulated by Villermaux (21) using a different numerical method and a less-general model. The families of curves given in Figure 4 demonstrate the separate and combined effect of h’ and k 3 on the peak shape and peak position of a given solute. The partitioning rate constant, h, is held constant in all cases. A decrease in the effective rate of desorption, k3, produces an increase in retention when k l and h-1 are large but has little, if any, effect on retention when hl and h-1 are small. On the other hand, a decrease in k’, the rate of desorption of the undis(20) R. S. Juvet, V. R. Shaw, and M. A . Khan, J. Arner. Chern. SOC.,91, 3788 (1969). (21) J . Villermaux, J. Chrornatagr.,83, 205 (1973).
A N A L Y T I C A L CHEMISTRY, VOL. 46, NO. 9, AUGUST 1974
1213
sociate solute molecule, produces an increase in retention when k l and k-1 are small but has little, if any, effect on retention when k l and k-1 are large. The above analysis is based on the assumption that all reaction rate constants are known so that the elution peak can be simulated by means of a computer. However, in practice it is the peak shape which is known and the reaction rate constants are the sought-for information. In principle, all on-column kinetics can be obtained from an analysis of the peak characteristics. However, this is not an easy task. Work is continuing along these lines in our laboratory.
ACKNOWLEDGMENT We wish to thank S. H. Lin for suggesting the use of the steepest descent approximation method in the solution of this problem and Joseph Fulton for programming the plotter package used in the preparation of Figures 2-4. Received for review January 7, 1974. Accepted April 19, 1974. Financial support from the National Science Foundation under grant GP25553 and from the National Defense Education Act for a Fellowship for one of us (D.L.D.) is also gratefully acknowledged.
Determination of Ionization Potentials by Gas-Liquid Chromatography Richard J. Laub and Robert L. Pecsok Department of Chemistry, University of Hawaii, Honolulu, Ha waii 96822
The possibility of determining vertical ionization potentials by GLC is explored. Charge transfer formation constants (K1)were determined for benzene, toluene, and the xylenes with 2,3-dichloro-5,6-dicyanobenzoquinone (DDQ) in di-nbutyl phthalate (DNBP). These were found to vary inversely with ionization potential. Flfteen substituted butadienes were then examined with 2,4,7-trinitrofluorenone (TNF) in DNBP. Several of these yielded abnormally low formation constants, resulting in high apparent ionization potentials. Steric hindrance (mutual approachability) is postulated as a limitation of the method. Other ionization potentials determined by this technique are shown to be in excellent agreement with predicted values.
Gas-liquid chromatography (GLC) has been used in conjunction with charge transfer complexes for some time (13). Purnell ( 4 ) has reviewed various classes of solute-solvent interactions. Gil-Av and Herling ( 5 ) ,and Muhs and Weiss ( 3 ) have proposed a method of calculating the charge-transfer formation constant, Kf. For D A s C,
+
and
where D, A, and C represent donor, acceptor, and complex species, respectively. KLO is the distribution coefficient for a donor species on a column containing only “inert” stationary phase. KL is the distribution coefficient on a column with acceptor concentration [A] dissolved in the liquid phase. Other methods of determining Kf by GLC have also been proposed (6). R. 0. C. Norman, Proc. Chem. Soc.,151 (1958). E. Gil-Av, J. Herling, and J. Shabtai, J. Chromatogr., 1, 508 (1958). 84, 4697 (1962). M. A. Muhs and D. T. Weiss, J. Amer. Chem. SOC., J. H. Purnell, in “Gas Chromatography-1966,’’ A. B. Littlewood, Ed., Adlard, London, 1966, pp 3-20. (5) E. Gil-Av and J. Herling, J. Phys. Chem., 66, 1208 (1962). (6) D. E. Martire and P. Riedl, J. Phys. Chem., 66, 1208 (1962).
(1) (2) (3) (4)
1214
Kf is related to the true thermodynamic equilibrium constant, Keq,by:
K , = Kf
YCa
(3 )
YBaYAa
where yimis the infinite dilution activity coefficient of the ith species. Eon and Karger (7) have recently calculated “corrected” and “relative absolute thermodynamic” formation constants from Purnell’s excellent data (8).These were for toluene, ethylbenzene, and the three xylenes, with T N F in a variety of alkyl phthalate ester solvents. If true charge transfer forces are operative, one would expect the measured formation constant to be a function of the ionization potential (Id)of the donor, and the electron affinity ( E a )of the acceptor. If solvent interactions are negligible, or can be taken into account, this is approximately true. When large solvent effects occur, no correlations between Kf, Id,or Ea are expected. This is apparent in spectroscopy where the Benesi-Hildebrand equation (9) (or various modifications) is used to determine the formation constant. Anomalies are amply illustrated by Foster and Fyfe (10); and Brown, Foster, and Fyfe ( 1 1 ) who employed proton NMR and 19FNMR spectroscopy. The latter technique particularly would be expected to overcome solution effects. They found, however, that the use of different solvents made it impossible to compare either Kf values, or the ratios of such values. When log Kf (1,3,5-trinitrobenzene and fluoranil) was plotted against Idfor 26 alkylbenzenes, the correlation was poor (12). The same lack of linearity was noted for -AHp us. I d (13).Dewar and Thompson (14) have severely criticized charge transfer theory, partic(7) C. Eon and B. L. Karger, J. Chromatogr. Sci., 10, 140 (1972). (8) D. L. Meen, F. Morris, and J. H. Purnell, J. Chromatogr. Sci. 9, 281 (1971). (9) H. A. Benesi and J. H. Hildebrand, J. Amer. Chem. SOC., 71, 2703 (1949). (10) R. Foster and C. A. Fyfe, Trans. Faraday SOC.,62, 1400 (1966). (11) N. M. D.Brown, R. Foster, and C. A. Fyfe, J. Chem. SOC.B, 406 (1967). (12) P. H. Emslie, R. Foster, I. Horrnan, J. W. Morris, and D. R. Twiselton, J. Chem. SOC.B, 1 16 1 (1969). (13) M. I. Foreman, R. Foster, and C. A. Fyfe, J. Chem. SOC.B, 528 (1970). (14) M. J. S. Dewar and C. C. Thompson, Tetrahedron SuppL, 7, 97 (1966).
A N A L Y T I C A L C H E M I S T R Y , VOL. 46, NO. 9, AUGUST 1974
