adsorber is being heated, GC No. 1 door is open and the chromatograph a t 25 OC for the next run, V2 is in mode 1 (closed), GC No. 2 is cooling to 25 O C . I t is apparent that samples are being simultaneously collected and analyzed in a consecutive repetitive automatic pattern. I t is important to note that the adsorber is purged with helium for 3 min prior t o being thermally eluted t o drive off any oxygen before heat is applied. Performance Experience. The monitor has been in continuous operation since July 1972. Between installation and the end of 1974, 35 000 data points have been generated. This monitor has played a major role in ensuring safe operating conditions in our plant. The automatic monitor described in this report could be readily adapted to selectively monitor almost any gaseous pollutant a t the ppb level.
LITERATURE CITED (1) "Occupational Safety and Health Standards: Part Ill", Department of Labor, Occupational Safety and Health Administration, Fed. Regist., 39, 3756-3797. Tuesday, January 29, 1974.
(2) J. L. Gargus, W. H. Reese, Jr., and H. A. Rutter, Toxic. Appi. Pharmacol.. 15, 92 (1969). (3) B. L. Van Duuren, A. Sivak, B. M. Goldschmidt, C. Katz, and S. Melchionne, J. Natl. Cancer inst., 43, 481 (1969). (4) S. Laskin, M. Kuschner, R. T. Drew, V. P. Cappiello, and N. Nelson, Arch. Environ. Health, 23, 135 (1971). (5) R . W. Nichols and R. F. Merritt, J. Natl. Cancer lnst., 50, 1373-1374 (1973). (6) J. C . Tou and G. J. Kallos, Anal. Chem., 46, 1866-1869 (1974). (7) L. Collier, Environ. Sci. Techno/., 6, 930 (1972). (8) R . A. Soloman and G. J. Kallos, Anal. Chem., 47, 955 (1975). (9) K. P. Evans, A. Mathias. N. Mellor, R. Silvester, and A. E. Williams, Anal. Chem., 47, 821 (1975). (10) L. A. Shadoff, G. J. Kallos, and J. S. Woods, Anal. Chem., 45, 2341 (1973). (11) S. B. Dave, J. Chromatogr. Sci., 7, 389 (1969).
ACKNOWLEDGMENT We wish to acknowledge the substantial contributions of L. J. Gibson, E. M. Sioma, L. Walecki, and K. Wallisch to the success of this project.
RECEIVEDfor review July 9, 1975. Accepted December 24, 1975. This analytical monitoring system is covered by United States Patent No. 3,807,217, April 30, 1974.
Rate Dependence of Statistical Moments of Chromatographic Profile on Solute-Solvent Interactions Kuang-Pang Li" and Yue-Yue H w a Li Department of Chemistry, University of Florida, Gainesville, Fla. 326 7 7
Statistical moments and related parameters, such as the skew, S, and the excess, E, of a chromatographic profile carry all macroscopic information about the kinetics of oncolumn interactions. To establish the rate dependence of these parameters, the steepest descent approximation method is employed. Simplified equations for these parameters are derived on the basis of a hypothetical model In which complexation of solute molecules with the stationary phase is included. Computation with these equations indicates that the statistical moments are always less involved than S and E. Accordingly, they may be more useful in the real-time determination of on-column reaction rates even though S and E may be more conveniently measured experimentally.
Chromatography has been recognized as a powerful tool for chemical separations and determinations. I t has been regarded as a device which most nearly approaches the chemist's dream of a magic tube into which a complex sample is placed and out of which emerges a complete analysis ( 1). This impressive achievement of chromatography has tended to overshadow its other potentialities to study reaction kinetics directly on the column. The use of a chromatographic column both as a reactor and as an analyzer has a number of advantages. First, since separation of reaction products is initiated instantaneously and effected continuously during the entire course of chromatography of the reactant, reverse reaction and side reactions are minimized. The forward reaction can be studied with little interferences. Second, retention data and relative concentrations (or rates of formation) of all eluting re-
action products are available. This information may throw important light on the mechanism of the reaction. Third, stop-flow technique can easily be employed (2, 3 ) . With this method, it is possible to follow in one experiment the rate of reaction from the first 0.1% of reaction to the stage where the reaction is 99.9% complete. In addition, the chromatographic reactor benefits from other features inherent in chromatography, i.e., small sample size and simple control of operating variables. Since chromatography is based upon rate processes, each physical-chemical reaction which the eluting molecules undergo, will have profound influences on the distribution of the molecules. In other words, the resulting chromatogram should carry all of the macroscopic information about these reactions. Earlier theories of chromatography have clearly demonstrated that pure partitioning, or simple sorptiondesorption processes, gives rise to a gaussian distribution in the limit (4-6). Thus, substantial deviations from gaussian distribution may be attributed to processes which contribute to the non-linearity or non-ideality of the column. These processes could be dissociation (7, 8 ) , rearrangement (91, etc., of the eluting molecules, or mutual interactions, such as solvation or complexation of the eluting molecules with the stationary phase ( I O ) . If the nature of the on-column process is known, then an appropriate profile analysis would reveal the kinetics of the reaction. There are several ways to characterize a peak profile. The use of statistical moments and related parameters has been the most widely adapted in chromatography. Several authors (11-15) have discussed the importance and theoretical evaluation of these parameters. Some of them have demonstrated their usefulness in the discernment of overlapping chromatographic peaks ( 1 6 ) and in the diagnosis of ANALYTICAL CHEMISTRY, VOL. 48, NO. 4, APRIL 1976
737
peak contamination ( 1 7 ) . However, the rate dependence of these parameters has been overlooked, and the possibility of using these parameters for on-column reaction kinetic studies has been ignored. Although Kucera (12) has derived expressions for the first five central moments in terms of rate constants for linear non-equilibrium elution chromatography, his equations are too complicated to be of any practical value in real analysis. In the present report, we employed the steepest descent approximation method to evaluate the moments. By this method, the expression become highly simplified and the rate dependence of these parameters can easily be evaluated. We hope that this simplification will provide a practical way for the estimation of on-column kinetics in real time. Moreover, the method reported here can be extended to more complicated reactions, such as combination, rearrangement, decomposition, and catalytic reactions. The normalized nth moment of a profile is defined by
-u
k ’
Stationary
Figure 1. Idealized model employed to represent tographic system
a
elution chroma-
S t n C ( t )d t
n = O , l , 2,. . . S C ( t ) dt where C ( t ) is the concentration distribution of the eluting molecules a t the end of the column a t time t . In other words, C ( t ) is the peak profile of the resulting chromatogram. By this definition, the zeroth moment, mo, always has a value of unity. The first moment, rnl,defines the center of gravity of the peak. If there is no chemical complexation, the peak is symmetrical, and ml coincides with the peak maximum. For moments higher than the first moment, it is more convenient to calculate them around ml, i.e., m, =
Mobile Phase
1
-4
b ,
-3
-2
-1
0
Figure 2. Dependence of various moments on the complex exchange rates Numerical values used for computation are: L = 25 cm, ti = 3 cm s-’, 0.1 cm2 s-’, k = 100 s-’, k‘ = 8 s-’, and k l / k - , = 1
The second central moment, m 2 , is seen to be the difference of m2 and m1*. I t is the variance of the distribution and is, therefore, a measure of the peak width; m3 and all other odd moments furnish information on the asymmetry of the profile, m 4 and higher even moments measure peak flattening as well as peak width, and m 3 and m4 are usually expressed in terms of the skew, S , and the excess, E , which are defined, respectively, by
(3)
I
1
Log k
D=
Laplace transform of the function
where s is the Laplace coordinate and y is any contour of integration. A, B, C, and E are rate dependent constants,
(4)
These quantities measure directly the deviations from a gaussian peak. I t is seen from Equations 1 and 2, that in order to estimate the rate dependence of the moments, we need to know the functional form of C ( t ) , which is very much dependent upon the nature of the on-column reaction. In the present report, only the solute-solvent interaction will be discussed.
B = -kk’ C = -1 (kk’kl
+ kk’ * - kk’k-1)
2
k2 = k’
+ k l + k-1
Based on an idealized model (IO), Figure 1, in which the eluting molecules, A , are assumed to interact with the stationary phase, S p , to form a non-eluting complex, A - S p , the distribution of A following a pulse input of amplitude COand width T can be represented by
and k and k’ are the sorption and desorption rate constants of the unassociated solute molecules across the mobile-stationary phase boundary; k l and k-1 are the rate constants of association and dissociation, respectively, of the solute with the stationary phase. By making the change of variable, t = 1 - (2/k2)s, and carrying out the integration, Equation 5 gives
where the average velocity of the mobile phase, a, and the longitudinal diffusion coefficient, D, are assumed to be constant and L is the column length. F l ( t ) is the inverse
where @ ( z ) is obtained from $(s) by substituting s with kZ(l - 2)/2. Substitution of C ( t ) into Equation 1 gives
THEORY
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ANALYTICAL CHEMISTRY, VOL. 48, NO. 4, APRIL 1976
shape of the eluting profile in a very complicated manner. Li, Duewer, and Juvet (10) have simulated a series of chromatographic peaks in terms of the rate parameters. For ease of comparison, they assumed a constant rate ratio for the forward and backward reactions but varied the magnitude of these rates. In other words, the rate of exchange between A - S p and A in the stationary phase is varied but the effective desorption rate defined as
where
k3 =
In the integration of C ( t ) with respect to t , it is seen that the condition 0 < Re z < 1,must be fulfilled to make m, finite. Within this region, i t can be shown that the function z P 2 exp(-@) is decreasing while exp(kzTz/2) - 1is increasing monotonically. If the latter is greater than exp(-@) a t the point Re z = 1,then the product
will have a saddle point a t z = z o along the real axis. If we now choose the contour of integration to be a circle about z = 0 with the radius 2 0 , the contour will pass through z o in the imaginary direction. Thus along the contour, the integrand, G&), will have an extremely sharp maximum a t the point z = 20. If elsewhere along the contour, there is no maximum comparable in height to this one, the contribution to the integral comes solely from the neighborhood of 2 0 . We can now expand the function f(z) about z o in the imaginary direction and neglect all terms higher than 2. Go(z) can be integrated to give
where f ” ( z 0 ) is the second derivative of f(z)a t the point z = zo. The same approximation can be applied to the function Gn(z),
where z 1 is the saddle point of the function
Substitution of Equations 9 and 10 into Equation 8 gives,
2
mn
=
n
(G)
n!ef(z+n
In z1/v‘277(f”(21)
+ n/z12)
e f ( z o ) l v ‘ 2 ~ f (”2 0 )
n = 1 , 2 , . . . (11) If z 1 is not significantly different from zo, we can assume that 2 1 = z o = z*, then Equation 11 can be simplified further to give Equations 12 and 13,
k’
1
+ hl/k-I
is kept constant. When the exchange is very slow, the profile is a sharp spike with a long tail. On the other hand, when the exchange is very fast the peak becomes nearly symmetrical but with a different centroid. In the intermediate region of exchange rates, the profile varies from distorted peaks to table top-like curves. Substitution of the same numerical values for the rate constants into Equations 11-13 and 3-4 gives the indicated values of the first five moments and their related quantities (Table I). I t is seen that Equations 1 2 and 13 give comparable results for the moments when the exchange rate is fast. For slow exchange, the results are very erroneous. This is because z 1 and z o are not close enough to be considered as equivalent in these cases. The estimation of G o ( z ) is very far off because of the steep descent when z is away from the saddle point 2 0 . All m, values, particularly the higher moments, seem much too high. This renders the S and E very insensitive to the rates. The rate dependence of the moments calculated with Equation 11 is given in Figure 2. For simplicity, only ml, m 2 , S, and E are plotted. In the presence of chemical complexation, the sample molecules seem to be eluting as two different species, the free molecule A , and the bonded A in the complex A-Sp. When the exchange rate is slow, most of the molecules elute as free A. Only a very small fraction exists as the bound A . The center of gravity of the resulting profile, ml, is coincident with the profile maximum. When the exchange rate becomes faster, ml increases rapidly, though a maximum can still be seen a t the retention time of A . This shift of ml is expected because more A is immobilized by complexation, heavier weights should be placed on the bound species. After passing a maximum, ml starts to decrease gradually to a constant value and approaches once again the maximum of the peak. ml is neither the retention time of the free A nor that of the bound A but is a weight average of the two. The free and bound species exchange so rapidly that they are kinetically indistinguishable. They seem to be eluting as a single species a t the average zone velocity. The variation of f i 2 with the exchange rate follows the same trend as ml but those of S and E.are much more complicated. If we express the elution profile in terms of the GramCharlier series ( 1 8 ) ,Le.,
n = 1 , 2 , . . . (12) = n!
2 , (G) ,when z *
n
* )
