Anal. Chem. lQ86, 58,1617-1625
interactions to w ’ - 4 5 ’ when w is in the range 40-120’ and then it rotates back to w‘ -180’ in the range w = 140-340’. The carboxylate also rotates along the Cl-C2 axis, and the ammonium slides from one face of the planar carboxylate to the other face. The motion of the isobutyl group in 9 is even more complex. The third and final point is that the calculations employed an effective dielectric constant o f t = 1.50. This is reasonable for ionic CSP’sin hydrocarbon solvent, but the results become less tenable for more polar solvents. Additionally, even though solvent molecules were not included in these calculations, explicitly including nonpolar organic solvents would not change the qualitative trends presented above. Our results are consistent with chemical intuition and highlight the fact that these ionic CSP’s are even more conformationally flexible that one would a priori predict. Both syn and anti conformations exist, and the barrier to interconversion is small. Furthermore the pendant ammonium group plays a key role by coordinating with both the carboxylate and with the amide C=O. The conformational flexibility of the phenyl group in the phenylglycine CSP and the isobutyl group in the lycine CSP is great enough to allow for a wide range of orientations for templating purposes; perhaps it is this inherent flexibility that serves to make these CSP’sso effective. We are currently investigating the conformational and dynamic behavior of the covalently bonded analogues and will report this at a later date. Registry No. 1, 102208-23-1; 2, 102208-24-2; 3, 74927-72-3; 4, 102073-88-1; 5, 98243-65-3; 6, 102073-89-2; 7, 98243-66-4; 8, 102073-90-5;9, 102110-09-8.
LITERATURE CITED (1) Asymmetric Synfhesis; Morrison, J. D., Ed.; Academic Press: New York, 1983.
1817
(2) March, J. Advanced Organic Chemistry, 3rd ed.; Wiley: New York, 1981; Chapter 2. (3) Willstatter. R. Ber. Msch. Chem. Ges. 1904, 3 7 , 3758. (4) Prelog, V.; Wieland, P. He&. Chlm. Acta 1944, 2 7 , 1227. (5) GICAv. E.; Feibush, 6.; Charles-Sigler, R. Tetrahedron Lett. 1966, 1009. (6) Blaschke, G. Angew. Cbem., Inf. Ed. Engl. 1980, t 9 , 13. (7) Schuria V. Angew. Chem., Int. Ed. Engl. 1984, 2 3 , 747. (8) Armstrong, D. W. J . Liq. Chromafog. 1084, 7(S-2), 353. (9) Plrkle, W. H.; Finn, J. M. In Asymmetric Synthesis; Morrison, J. D., Ed.; Academic Press: New York, 1983; pp 87-124. (IO) Alienmark, S. J . Biochem. Biophys. Methods 1984, 9 , 1-2$. (11) Davankov. V. A.; Kurganov. A. A.; Bochkov, A. S. Adv. Chromatogr. 1084, 2 7 , 71-116. (12) Pirkle, W. H.: Finn, .I. M.; Hamper, B. C. J . Am. Chem. Soc. 1981, 703, 3964. (13) Pirkle, W. H.; Finn, J. M. J . Org. Chem. 1981, 46, 2935. (14) Pirkle, W. H.; Schrelner, J. L. J . Org. Chem. 1081, 46, 4988. (15) Pirkia, W. H.; Finn, J. M. J . Org. Chem. i982, 4 7 , 4037. (16) Pirkle, W. H.; Hyun, M. H. J . OIg. Chem. 1984, 4 9 , 3043. (17) Plrkie, W. H.; Hyun, M. H.; Bank, B. J . Chromatogr. 1884, 376, 585. (18) Pirkle, W.H.; Hyun, M. H.; Tsipouras, A.; Hamper, B. C.; Banks, 6. J . Pharm. Biomed. Anal. 1084, 2 , 173. (19) Wainer, I . W.; Doyle, T. D. LC Mag. 1084, 2 , 68. (20) Boyd, D. 8.; Lipkowltz, K. B. J . Chem. Educ. 1982, 59. 269. (21) Osawa, E.; Musso, H. Angew. Chem., Int. Ed. Engl. 1983, 2 2 , 1. (22) Burkert, U.;Aillnger, N. L. Molecular Mechanics; American Chemical Society: Washington, DC, 1982; ACS Monograph 177. (23) Clark, T. A Handbook of Computetional Chemlstty: A Practical OUMe to Chemical Structure and Energy Calcu&tions ; Wiley-Interscience: New York. 1985. (24) Dewar, M. J. S.; Thiei, W.J . Am. Chem. Soc.1977, 99, 4899. (25) Stewart, J. P. QCpEBuH. 1083, 3(2), 455. (26) Allinger, N. L.; Yuh, Y. H. Ouantum Chem. hogram Exch. 1081, 73, 395. (27) Meyer, A. Y. In The Chemistry of Functional Groups, Supplement D ; Patai, S., Rappoport, Z., E&.; Wiley: New York, 1983; Chapter 1.
RECEIVED for review December 20,1985. Accepted March 4, 1986.
Measurement of Reaction Rate Constants in the Liquid Chromatographic Reactor: Mass Transfer Effects Alexander H. T. Chu and Stanley H. Langer* Department of Chemical Engineering, University of Wisconsin-Madison, Madison, Wisconsin 53706 The adaptation of llquid chromatographk c o h n s as chemkal reactors for r e a c t h klnetk studies Is examlned. The overall for pyrldlne and 4-plcollne catalyzed rate constants, k-, esteriflcatlon reactlons of tetrachloroterephthaloyl chlorlde were determined for a llquld chromatographk reactor (LCR). I t Is shown that wHh a rate constant for reactbn In the moMle phase, a valid rate constant for thls system can be obtalned for reactbn In the statlonary phase. Thus conversbns In each phase cuukl then be evaluated. PosSiMlttks for complkatlons from mass transfer effects in the LCR system were examlned, and reglmes where they should be consldered are determined. Longitudinal dtffuslon, micropore diffusion, and interfacial sorptlon and desorption were specifically considered. However, analysls shows that none of these processes slgnlllcantly affected the klnetk measurements carried out in this study where chemlcal reactlan rates are slow. Experimental llmltatlons on rate constant measurements for flrst-order and pseudo-flrst-order reactlons are lndlcated and dlscussed.
The extension of chromatographic reaction rate measurements to reactions occurring in liquid chromatographic col0003-2700/86/0358-1617$01.50/0
umns not only expands the application of these systems but also has promise of providing special information about the stationary phase to complement information obtained from physical measurements and other chemical methods (I, 2). Such information initially would come from the application of chromatographic theory with the use of appropriate kinetic systems obeying simple rate laws. However, this application is predicated on the column operating as an ideal chromatographic reactor, a behavior which could be more limited in liquid chromatography than with gas chromatography because of mass transport processes. Therefore, the effects of these processes are examined here. Earlier we showed that with a reversed-phase column and a reactive solute solvolysis reactions in the stationary phase had to be considered and that an octadecylsilyl stationaryphase composition could be investigated with rate constant measurements (I, 2). As such, on-column chemical rate measurements can be subject to mass transfer effects from transport (physical) rate processes in the packed chromatographic bed; these effects must be examined systematically so that they can be eliminated or minimized and the limits of reaction rate measurements in liquid chromatographic columns identified. The criteria for “ideal chromatographic 0 1986 American Chemical Society
1618
ANALYTICAL CHEMISTRY, VOL. 58, NO. 8, JULY 1986
reactor” (ICR) operation proposed earlier (3, 4 ) can be extended to the chemically bonded liquid chromatographic system to provide guidelines for reaction rate measurements. With the column dimensions and operating conditions of high-performance liquid chromatography (HPLC), the potential effects of a number of physical rate processes including longitudinal diffusion, micropore diffusion, and sorptiondesorption kinetics across the mobile-phmtationary-phase boundary can be examined for a first-order or pseudo-firstorder reaction occurring in both phases. This makes it possible to assess the applicability of the liquid chromatographic reactor (LCR) for kinetic studies, defining suitable chromatographic operating conditions, and utilizing it for other purposes. Earlier studies dealing with mass transfer dynamics in HPLC have mainly focused on their effects on column efficiency and band broadening in the absence of reaction (5-9).
THEORY During the development of chromatographic reactor principles, models of various degrees of complexity have been used to describe the dynamic behavior resulting from the interaction of reaction processes with concerted separation in a packed bed. In general, two basic types of models are found in the literature: (i) the pulse-elution model (3, 4 ) and (ii) the plate or mixing cell model (10, 11). For a liquid chromatographic reactor in which solute reactant or reaction mixture is introduced as a pulse onto the column, partial differential equations can be derived based on a continuous elution model for evaluating rate parameters for the fmt-order reactions occurring in both the mobile and stationary phases. Nonideal transport processes can then be considered to determine regimes where rate constant measurements could be affected. For an isothermal, homogeneous chromatographic reactor column, a complete material balance on a differential section of the liquid chromatographic column at position x along the column and time t can be represented by (1-4)
where C, f , a,and r are reactant concentration, volume fraction, diffusion coefficient, and reaction rate for the designated phase (m = mobile; s = stationary); u(x) is the linear velocity of the mobile phase. The terms on the right-hand side include longitudinal diffusion, convection, and reaction rate effecta. If sorption (k,) and desorption (kd) rate processes across the two-phase boundary are both first order, a second material balance on the reactant concentration in the stationary phase becomes (12)
ac,/at
= k,c, - kdc,-
assuming negligible diffusion in the axial direction. The phase ratio (4 = f,/f,) is Significant in modern reversed-phase liquid chromatographic operation. Therefore, volume fractions of each phase, f, and f,, are employed here instead of the porosity or the void volume fraction of the column, e, used elsewhere. A generalized “sorption” treatment can accommodate either a dynamic partition or an adsorption process operating across the interface boundary; combinations also may be involved in the column retention mechanism. Characteristics of an ideal chromatographic reactor can then be considered in detail to highlight considerations and potential oversimplification in treating the liquid chromatographic system. These are as follows.
(1) The liquid chromatographic column is homogeneous. Thus, (i) no significant long-range inhomogeneity should exist in the packing and (ii) no significant expansion of the mobile phase should occur along the column so that the mobile-phase velocity is essentially constant without a pressure correction complication. Linear flow velocity in a porous medium, uo, is u s d y given by Darcy’s law: UO
= - K / V dP/dx
(3)
where dP/dx is the pressure gradient along the column (negative), the mobile-phase viscosity, and K the column permeability. Therefore, when variations of viscosity and liquid compressibility with pressure are not significant as with relatively slow flow rates in a homogeneous column, the superficial velocity is linear with a constant pressure drop throughout the column. (2) Peak spreading and axial diffusion are not significant. The order of magnitude for the solute diffusion coefficient calculated by the empirical Wilke and Chang cm2 s-l for reequation is usually between lo4 and versed-phase liquid chromatography. This value is of the order of 10-4 times smaller than that encountered in a gaseous mobile phase (13). Thus, axial diffusion effects in the liquid chromatographic reactor are not significant at common flow rates, a major factor in the high efficiency of modern liquid chromatography (14). However peak spreading from eddy diffusion and finite rate of mass transfer can still result in a 10-fold or greater dilution of reactant as has been pointed out by Karger et al. (15). Such general effects make the real chromatographic reactor most suitable initially for studies with first-order reactions, which are least sensitive to dilution and peak spreading. (3) The distribution coefficients of reacting solutes are linear and concentration independent. Thus, low solute coverage (low initial reactant concentrations) is desired so that competitive sorption on the surface of stationary phases is not a complicating factor. Then, the distribution isotherm can be treated as linear. For the octadecyldimethylsilane (ODs)-bonded stationary phase used here hydrocarbon chains are presumably fully solvated in the methanolietetrahydrofuran mobile-phase solvent (16),participating as a thin bulk liquid layer accessible to small-size reacting molecules. This feature is different from an adsorption-type liquid-solid chromatographic reactor in which a nonlinear isotherm may be fostered around sites so that a number of separate experiments are required to determine the adsorption isotherm (17-19). For an analytical scale chromatographic column with a low concentration (10-4-10-3 M in this study) of reactant in the mobile phase, a linear distribution isotherm can exist between the stationary phase of high chain length and the solvent mobile phase (20). (4) The reaction is chemically controlled; mass
transfer, adsorption, and desorption rates are relatively fast. This criterion is of more concern with a liquid chromatographic reactor than with a gas chromatographic reactor unless the pertinent reaction rate is slow and mass transfer resistance in the monomolecular layer is not significant, a reasonable condition for a silica surface treated with monochlorodimethylsilanes (21). Where the distribution equilibrium is established instantaneously between two phases, with a linear isotherm, the distribution constant, K , can be expressed as
K = CJC,
= k,/kd
(4)
( a ) Negligible “Film” Resistance of the Mobile Phase at the Particle Boundary. As a rough estimate of the order of magnitude for the convection mass transfer coefficient in
ANALYTICAL CHEMISTRY, VOL. 58, NO. 8, JULY 1988
1619
First-Order Reactions with and without Equilibrium Distribution between Two Phases. For a reversed-phase LCR with negligible axial diffusion and constant flow eq 1 simplifies to
with initial and boundary conditions
at t = 0
Cm(x,0 ) = C&X,0 ) = 0 Cm(O,t ) = &) Figure 1. Schematic diagram indicating rate processes for consideration in a chemically bonded liquid chromatographic reactor: k, and k,, sorption and desorption rate constants between two phases; k,, mass transfer coefficient at the particle boundary; k, and k,, reaction rate constants in mobile and stationary phases; K,, partition coefflclent for species i.
packed beds, the free surface model developed by Pfeffer and Happel (22)can be used to predict the fluid-to-particlesurface rate coefficient. The estimated rate of the transport process across a "stagnantn mobile-phase film is relatively fast (10 8 ) compared to the reaction rate range between lo4 and 10" s-l of this study. Thus the bulk mobile phase with its hydrodynamic boundary layer (thickness N 100 8)can be treated as a continuum with homogeneous composition. ( b ) Zntraparticular Diffusion Resistance. Some recently reported results address the internal mass transfer problem. Horvath and Lin (6) claim that such resistance is not important for ODS-bonded microparticle (5 pm) columns compared with sorption-desorption kinetics; internal mass transfer effects are virtually eliminated because the pools of the stagnant mobile phase are shallow (14). On the other hand, Knox and Scott (23) estimated the C values in the van Deemter equation by measuring the band broadening in 50-pm and 540-pm ODS-bonded silica columns and suggested that equilibration is limited only by the solute diffusion rate in the stationary zone (mainly the pores of the particles). Herbut and Kowalczyk (24) claimed that mass transfer inside the micropores is a significant contributor to the band broadening for silica gel packed columns (adsorption chromatography without surface derivatization). Here, the mobile-phase solvent and the chemically bonded stationary phase are considered as two "homogeneous continua" in which micropore phenomena, such as internal diffusion and interaction with the silica substrate, are not significant. Variations of solute concentrations inside the pores in both phases initially are neglected but are considered subsequently (vide infra).
(c)Mass Transfer Resistance due to Sorption-Desorption. Kinetic resistance resulting from reversible eluite binding has been reported to be the most significant contributor to band broadening in columns packed with uniform spherical 5-pm Spherisorb-ODS particles (6). The plate height is controlled by slow binding kinetics at the stationary-phase surface, especially for strongly retained species. Therefore, for kinetic studies in an LCR with a hydrocarbon chemically bonded phase, the major mass transfer terms as a first approximation are considered to be the first-order "partitioning" of reacting solutes through transfer in and out of the nonpolar hydrocarbonaceous stationary layer, k,C, and kdc,. A schematic representation of the rate process considerations of this dynamic model is shown in Figure 1. Two mass transfer resistance extremes can be considered: (i) equilibrium distribution between two phases is established rapidly, Le., K = c,/C, = k,/kd, and (ii) equilibrium distribution is not established, and both transfer rates across the boundary are finite.
at x = 0
(6)
(7)
where d t ) is the reactant input function at the column inlet. Assuming then fiit-order kinetics in both the mobile and the stationary phases,
Case i. With equilibrium distribution C, = K C,, then following earlier treatments Wout/Win
= exp(-kmtm - k,tJ
(9)
With addition of an inert reference material to the reaction mixture, and the standard method for extracting the rate constants from the experimental data ( 3 , 4 ) , then
where AR and AI are the peak area of reactant and inert standard, WR,inand WI are the weight of inlet reactant and inert standard, SRand SIare the detector sensitivity factor for reactant, and inert standard (Si = Wi/Ai), respectively, and kappis the apparent column rate constant, which is
With the logarithm of the area ratio, ARIA*,plotted against the reactant retention time in the system, kappis obtained from the slope. Since the ratio t,/ t, is constant for a particular chromatographic column, k , and k, cannot be decoupled by simply varying the flow rate. Case ii. Where reactant sorption and desorption rates are commensurate with chemical reaction rates, a general solution for the concentration distribution, C(x, t ) ,with consideration of these transport processes can be obtained where micropore phenomena can be neglected as justified earlier. Then from eq 8 and a mass balance on reactant for the stationary phase
a c , / a t = kaCm- (kd
+ k,)C,
(12)
By Laplace transform from eq 8 in the time domain
(13) while eq 12 becomes
s or
C, = kaCm- (kd + k J C ,
(14)
1620
ANALYTICAL CHEMISTRY, VOL. 58, NO. 8, JULY 1986
Substituting eq 15 into eq 13 gives
k,' = f & A / f m
=
(26)
ta/tm
Therefore, considering the total retention time of the reactant, tR
or where hap/
with the input function, +(t). The solution for this first-order differential equation is
Analytically solving for C,(n, t) with inverse Laplace transforms can be involved under such circumstances. However, the statistical moment method can be utilized to obtain the conversion, X A , and the retention time of reactant, t ~at , the column outlet instead where the nth moment is defined as
For the zeroth moment, the amount of reactant, F~ = 1- X A , thus
and the first absolute moment, which is the reactant time (see the Appendix),
Equations 20 and 21 represent an initial attempt to incorporate mass transfer rates, k, and k d , into the continuous flow model of the liquid chromatographic reactor. If kd >> k,, these two equations simplify to the familiar expressions, respectively
X A = 1 - exp(-k,t,
- kat,)
(22)
and
The reaction rate constant in the stationary phase, k,, plays a more significant role relative to that in the mobile phase, k,, when a chemical reaction is combined with other physical transport processes. Using the inert reference method and with mass transfer a significant factor, then
In
(2) ( = In
WR,in/SR
WIISI
)
- kappltm
(24)
from eq 20 where
(25) To express the equations above in terms of experimentally accessible quantities, they can be rearranged and use can be made of the capacity factor relationship
= I,(
t) (
+ kka/kd s
)(k)
(28)
Thus, it appears then that a finite kd value would cause underestimation of the observed k, value, since
With equilibrium distribution between two phases established instantaneously, the observed ks,obsdapproaches the intrinsic reaction rate constant in the stationary phase. EXPERIMENTAL SECTION Liquid Chromatographic Reactor System. The reactor system incorporating two 25-cm Altex Ultrasphere-ODS HPLC columns in series connected with a 1/2-in.length of 1/16-in.tubing (0.01 in. i.d.) has been described earlier (1). Reagents and Solvents. The source and preparation of all reagents used in this work were also reported earlier. The mobile phases consisted of a relatively dilute base catalyst (pyridineand 4-picoline) in a methanol-tetrahydrofuran solution that was 0.25 M in THF. Two concentrations were used for each catalyst at two temperatures (25 and 35 "C). The injections were 20 FL of a methanol solution 0.0003 M in tetrachloroterephthaloyl chloride (TCTPC12)reactant, 0.01 M in inert standard, and approximately 0.25 M in THF. Either UV-sensitive n-phenylheptane or n-phenyloctane (Aldrich Chemical) was utilized as an internal standard for the pyridine or 4-picoline catalyzed reaction, respectively. RESULTS AND DISCUSSION Estimation of t h e Apparent Rate Constants from the Pulse-Elution Liquid Chromatographic Reactor. The p h l u t i o n technique (1-4) was employed to study reaction kinetics for the first step of the base catalyzed esterification of TCTPClz in methanol; the structure of the intermediate product has been identified as quaternary N44-chlorocarbonyl-2,3,5,6-tetrachlorobenzoyl)pyridinium(M) or picolinium chloride (M') salts with 13C FT-NMR and other techniques (25) (eq 30a and 30b). The rate constants of reactions 30a and 30b in bulk methanol solvent, k,, were measured by sampling the reaction mixture from an independent batch reactor followed by analysis in the HPLC system at different reaction times; the results expressed in terms of Arrhenius parameters are reported elsewhere (25). A typical series of reaction chromatograms at various flow rates at 25 "C for the 0.0075 M pyridine catalyzed reaction in the liquid chromatographic reactor are illustrated in Figure 2. Because the reactant peak (R) was only partially resolved from the product peak (M) due to the continuous (first-order) formation of M or M' from R during chromatography, the areas for R and M or M' were measured and allocated as described earlier (26,27).To correct for overlapping product, a midpoint line is drawn between a horizontal and extrapolated product line. Error analysis indicates less than a 1% error in the rate estimate. The correction is discussed in detail by Yurchak (27). By use of eq 10 or 27, the apparent rate constants, Itapp, were estimated from the slope of the plots of logarithm of ARIAIvs. reactant retention time, tR,as illustrated for the TCTPClz reaction in Figure 3. The eluent flow
ANALYTICAL CHEMISTRY, VOL. 58, NO. 8, JULY 1986 CI \
/
'F
CI
1621
RII
"VC' CI-
CI \
CI
/
.=.*
so00
RETENTION TIME W c l
Flgure 2. Series of liquid chromatograms for the TCTPC12 esterificatlon reactlon catalyzed by 0.0075 M pyridine in methanol at 25 OC: R, reactant (TCTPCI,); I, inert (l-phenylheptane); M, Intermediate product (N-(4-chlwowbonybtrachb~oyl)pflnlum chloride); H,half ester impurity (methyl-CI-TCTP); and C, catalyst vacancy peak (pyrldine). Conditions are as follows: (a) flow rate = 0.32 mL/min, AP = 900 psi; (b) flow rate = 0.21 mL/mln, AP = 700 psi; (c) flow rate = 0.1 1 mL/min, AP = 500 psi.
CI
CI
3000
0 '1000
CI
r
(M')
rate was randomly varied from 0.05 to 0.90 mL/min to give a range of reactant residence times. Linear behavior (correlation coefficients = 0.969-0.997) was observed for each set of experimental runs with data points ranging in number from 30 to 80. The lower linear coefficients for the set at 0.005 M pyridine catalyst concentration (see Figure 3) are due to the relatively small conversions (0.3-0.5) resulting from the slow reaction rates. Where any external mass transfer effects are dominating conversion should be proportional to the reciprocal of the one-third power of the flow rate (28). The apparent rate constants for TCTPClz reactions at various catalyst concentrations were obtained from eq 10 by plotting In (ARIAI) vs. tR. The experimental reaction kinetics in the stationary phase, ks,owwere estimated by using known values of k , in the expression
I
i
I
I
I
I
I
4000
assuming a homogeneous methanolic mobile phase where the mrption-demrption rate is not limiting (Le., k, = k w because kd >> k,, vide infra). The reactant retention time in the mobile phase, t,, was determined by a homologous series method using C5-Cl0 n-alkylbenzenes together with nonlinear regression analysis (29,30). The retention time in the stationary phase, t,, was obtained by subtracting t, from the total retention time, tR. The mobile phase volume, urn, determined by the n-alkylbenzene series (4.6 mL for two Altex Ultrasphere-ODS columns in series) agreed with that determined with uracil in methanol. With a 9.1-mL retention volume for TCTPC1, (at 25 "C), the times that the reactant spent in the mobile and stationary phases are quite comparable (t,/t, = ( VR- Vm)/ V , = 4.514.6); thus, the calculation of k , using eq 31 should be reasonably accurate. Some kappvalues together with calculated rate constants for the initial reaction of TCTPClz in both phases at 25 and 35 OC are shown in Table
I. Since finite sorption-desorption kinetics would result in underestimating the apparent k, values as indicated in eq 29, the k , measurements here are the intrinsic ones free of any mass transfer complications. However, for related chroma-
1
8000
RETENTION TIME (sec)
,,
Flgure 3. Firstsrder plot of area ratio, A ,/A vs. reactant retention time on a sermilogarithm scale for TCTPCI, reaction at 25 OC (0)and 35 O C p):moblle phase, 0.005 M pyridine, 0.245 M THF in methanol; stationary phase, bonded Ultrasphere-ODS (5-l.tm) packing. Table I. Experimental Rate Constants for TCTPClz Reactions in Both the Mobile and Stationary Phases at Indicated Catalyst Concentrations at 35 and 25 "C
T, apparent k,, catalyst/concn, M OC 4-picoline/0.00615 4-picoline/0.0082 pyridine/0.0050 pyridine/0.0075 pyridine/0.0050 pyridine/0.0075
35 35 35 35 25
25
lo4
4.99 f 0.07 6.69 f 0.32 1.79 f 0.10 2.73 f 0.06 0.95 f 0.06 1.33 f 0.06
batch k,,
exptl k,,"
10-48-1
1048-1
7.26 f 0.06 9.68 f 0.08 2.55 k 0.10 3.80 k 0.15 1.45 f 0.06 2.15 f 0.09
2.41 f 0.27 3.62 f 0.40 0.77 f 0.12 1.56 f 0.28 0.37 f 0.07 0.56 f 0.10
aCalculated from eq 31. tographic reactor applications where kinetic measurements might be of interest, it is pertinent to consider where mass transfer limitations might be operative. This can be done by
1622
ANALYTICAL CHEMISTRY, VOL. 58, NO. 8, JULY 1986
considering each physical rate process separately to establish initial guidelines for future physicochemical work. Criteria for Possible Mass Transfer Effects on Rate Constant Measurements i n t h e LCR. Types of potential mass transfer complications in the two-phase liquid chromatographic reactor analogous to those encountered with heterogeneous catalytic flow reactor systems (see Theory section) include (starting from the bulk fluid) the following: (1)longitudinal or interparticulate diffusion, (2) film resistance at the external boundary layer of the packed bed, (3) intraparticulate diffusion into micropores, and (4)sorption and desorption rate processes across the two-phase boundary. In this study, steps 1, 2, and 3 have been neglected up to this point in the treatment of the chromatographic reactor. However, they can be considered for the purpose of establishing guidelines for larger scale or less ideal (nonequilibrium) chromatographic reactor systems. Operating criteria also can be identified for reaction kinetic studies in conjunction with developing other mathematical models. Since mass transfer through the hydrodynamic film a t the particle boundary normally is not a major concern in chromatography ( 4 ) ,attention can be directed to longitudinal dispersion, micropore diffusion, and interfacial mass transfer where potential complications can arise from these physical rate processes. An outline of the levels a t which to address mass transfer problems in various models is presented in Table 11. Mathematical Modeling of the Linear, Nonideal LCR. Assuming a linear isotherm, constant mobile-phase velocity and first-order reactions in both phases, a more general mathematical model can be considered to take into account radial micropore diffusion axial diffusion (coefficient am), (coefficient a),) with sorption and desorption kinetics ( k , and kd). Then,
for kinetic measurements. With the Laplace transform treatment, the total amount of reactant upon injection is
For the total amount of reactant injected as a pulse emerging at the reactor outlet, x = L
w,,~,,= 1
5 Cm(x = L, s) = iim $(s) S-4
Dividing eq 40 by eq 39 and introducing the inert reference peak area, AI, as before give
where AR and AI are proportional to the weight of reactant and inert, respectively. (1)Longitudinal Dispersion in the Packed Column. Under operating condition such that reaction rates are slow relative to diffusion in the mobile phase, then
(2) (z)t=o(
Using a binomial expansion of the Taylor series in eq 41 gives
In For a spherical particle of packing in which reaction occurs
a2C,
2
ac,
at
(33)
Here a), is constant at low solute concentrations. The interfacial mass transfer in a dilute system is
aC, _ - k,Cm - (k,j + k,)C, at
(34)
where the average concentration in the particle is
c, =
L R C S ( r ,x , t ) r 2dr
(35)
JRr2 d r
Csis introduced to eliminate any radial (r) dependence of the stationary-phase concentration in eq 32 (31, 32). The initial and boundary conditions for the chromatographic bed are then a t x = 0,
cm(x,t ) = q!4t)
at x = cm(m, t ) = 0 a t t = 0, c&, t ) = Cs(x, t ) = 0 03,
(36) (37) (38)
The above equations can be solved in the Laplace domain but not explicitly in the original (x, t , r ) domain. Statistical moments can be employed to extract useful rate expressions
= In
- kmtm
+
+
1 ksts k,/kd
)
(43)
equivalent to eq 27 derived earlier. Thus, the micropore diffusion coefficient, a,, does not appear in the expressions for first-order reactions, although the original equations involving the mobile-phase concentration, C,(L, s), in the Laplace domain contain functions of as(32). An equation similar to eq 41 has also been developed by Kocirik (33) allowing for both longitudinal diffusion and slow sorption and desorption kinetics. An important criterion derived from eq 42 is that the inverse Peclet number (Bm/uL)should be much greater than the reaction terms, C k i t i (i = m or s). For reactions in the chromatographic system studied in our laboratory, (4.6 mm, i.d.), linear velocities, uo,were between 0.3 and 5.4 cm/min with nominal flow rate between 0.05 and 0.9 mL/min methanol. Using numerical values in eq 42 gives
(0.06-0.90)
