Znd. Eng. Chem. Res. 1991,30,1489-1499
1489
Rate Process Analysis in the Liquid Chromatographic Reactor: An Application of the First Statistical Moment Chawn-Ying Jeng and Stanley H.Langer* Chemical Engineering Department, University of Wisconsin, Madison, Wisconsin 53706
A statistical moment method based on the overall elution profile is developed for a (pseude)firstrorder chemical reaction occurring in a chromatographic reactor. This approach overcomes difficulties in resolving and analyzing reactant and product overlap in elution traces and involves conversion of chromatographic reactor data to the first absolute moment for in situ kinetic parameter analysis and evaluation. This conversion is facilitated by detector adjustments. Application of this method is illustrated for a hydroquinone oxidation catalyzed by an iron-modified chromatographic silica in an organic environment. The effects of some mass-transfer processes on intrinsic kinetics for reactions occurring in a liquid chromatographic column are also examined. A simple reaction model and a model incorporating finite adsorption and desorption rates together with reaction are both developed. A comparison is then made of the fit of kinetic data in these two models. The results for the hydroquinone oxidation indicate that adsorption processes can significantly influence reaction rates and affect kinetic measurements. Introduction Gas chromatographic reactors have been shown to have a number of advantages and to possess the potential for providing a rich variety of information about reactions. Several reviews summarize much of this information and provide a guide to other relevant literature and earlier reviews (Laub and Pecsok, 1978;Conder and Young, 1979; Coca and Langer, 1983). Early applications of liquid chromatographic columns as chemical reactors also have shown the possibility of achieving reaction conversions exceeding equilibrium values and have indicated the potential for special applications and kinetic studies (Wetherold et al., 1974;Schweich et al., 1980;Cho et al., 1980). With the further development of high-performance liquid chromatography (HPLC), possibilities for using liquid chromatographic columns for carrying out reactions and investigating features of the liquid chromatographic reactor (LCR) have begun to receive more recognition (Bolme and Langer, 1983;Jacobson et al., 1984;Benedek et al., 1984;Chu and Langer, 1985;Moriyasu et al., 1985; Hanai et al., 1986). In contrast to batch reactors, LCR operation can be directed toward providing in situ measurements under continuous flow conditions and information on reactor bed characteristics as well as selective conversions of special materials. Other advantages include the simplicity of operation and control of HPLC arrays, the capacity for working with small sample sizes or dilute solutions, and the elimination of interference from impurities and undesirable side reactions. There is also a versatility that can be extended to both homogeneous and heterogeneous reaction studies. Nevertheless, LCR applications have been limited, to date, compared to the many uses of gas chromatographic reactors. A number of sophisticated mathematical treatmenta for chromatographic reactors have been developed, but only a few have been applied in LCR experiments to evaluate kinetic parameters. This is a consequence of the leas than ideal liquid environment in the LCR and a limitation from the slower physical and chemical kinetic processes that occur in the liquid phase at moderate temperatures. The higher purity and inert character of many stationary phases used in HPLC columns also limit potential catalytic reactions. An additional problem arising in kinetic studies is the difficulty in interpreting reaction
* To whom correrpondence should be addressed. 0888-5886/91/2630-1489$02.50/0
chromatograms where overlapping concentration profiles of reactants and products mask information about individual species and where small mobile-phase composition changes result in the shifting and masking of "inert standard" peaks (Langer and Patton, 1972). Some of these problems can occur in analyses of gas-phase-reactor chromatographic data as well. Figure 1 illustrates a typical elution profile for a reactor chromatogram and the overlap between reactant and product that can occur to make it difficult to interpret (cf. Figure 1 caption). Statistical moment approaches frequently have been applied in chromatographic reactor studies, and most models have dealt with the statistical moments of reactants or producta separately (Suzuki and Smith, 1975;Yamaoka and Nakagawa, 1976). Applying these solutions to overlapping concentration profiles directly, however, is difficult since determination of individual moments is complicated by distortion and shifting of peak shapes and locations in reaction chromatograms. This is particularly evident upon examining the product wave shown in Figure 1. For a reaction chromatogram involving only two species, peak deconvolution procedures have indeed been used to obtain data on the reactant or the product (Langer et al., 1969;Langer and Patton, 1972;Moriyasu et al., 19851,but these can be time-consuming and susceptible to error at high conversions as can be seen in Figure 1. With numerical deconvolution techniques available nowadays, coeluting peaks of Gaussian shape can be resolved with less time and higher accuracy. However, models of greater complexities are usually needed for describing asymmetric profiles such as tailing peaks and the continuous elution of product waves which overlap Gaussian reactant peaks. Other methods such as "stopped flow" (Muller and Carr, 1984;Powell et al., 1987)and control of on-column reactions through mobile-phase composition variation (Benedek et al., 1984)have been invoked for specific systems to distinguish products from reactants. Consequently, a more general approach would be desirable for kinetic parameter estimation and interpretation particularly with liquid chromatographic reactors. Thus, an objective of this work has been to develop a practical approach for LCR data treatment and to illustrate its application. In particular, a statistical moment approach based on the overall profiles of reactant with product is developed here for rate studies. Earlier, we showed how the hydroquinone oxidation reaction could be used as a probe reaction for detecting catalytic activity that presumably originated from tran0 1991 American Chemical Society
1490 Ind. Eng. Chem. Res., Vol. 30, No. 7, 1991 2
Figure 1. Illustrative reaction chromatogram for kinetic measurementa using inert standard (I) technique; P = product. Area of reactant (R)after passage through column (12351) approximated as 123641. The shaded areas 145 and 536 should be approximately equal (Langer and Patton, 1972).
sition-metal ions associating with silica (Jeng and Langer, 1989). Since the transport and kinetic properties of the species involved in this reaction seemed suitable for LCR kinetic investigations, this oxidation catalyzed by iron (Fes+)-modified silica packing in an organic solvent environment is chosen here for exploring and illustrating the application of a general statistical moment approach to liquid chromatographic reactors. Although approaches similar to those of this study have been used to treat isomerization kinetics in HPLC columns (Jacobson et al., 1984; Hanai et al., 1986), mass-transfer effects were not investigated. Many mass-transfer processes associated with separation and reactions in liquid Chromatographic reactors can be neglected especially where liquid-phase reactions are slow and rate determining (Hanai et al., 1986; Chu and Langer, 1986). On the other hand, special care must be taken with faster reactions (e.g., surface-catalyzed ones), since some mass-transfer rates then can be commensurate with reaction rates (HorvMh and Lin, 1978; Muller and Carr, 1986; Hage et al., 1986; Marshall et al., 1986; Arnold and Blanch, 1986; Lin and Ma, 1989). Among those that might be important, strong adsorption and slow desorption have been reported most often. This is a common consequence of the use of silica-based packings, since they can interact strongly with the polar functional groups of some solutes. Another goal of this work, then, is to extend a previous study of masstransfer effects in LCR reactions (Chu and Langer, 1986) to consider the potential influence of adsorption processes. Possible mass-transfer effects related to catalyzed hydroquinone oxidation kinetics are examined first. Chromatographic reactor models then are developed with and without consideration of finite adsorption and desorption rates. The significance of adsorption processes is established through a comparison of the "goodness" of kinetic parameter fit for the two models using experimental data from the liquid chromatographic reactor system.
Theory Many mathematical models describing concerted reaction and separation in chromatographic columns have been developed with the use of varying assumptions about isotherm linearity and ideality, aa well as reaction type and order in the system. Methods used to solve the equations and expressions also have varied. However, few models derived in prior studies are applicable directly to reaction chromatograms in which elution concentration profiles of reactants and products significantly overlap as indicated earlier. To address this problem, a statistical moment approach based on the overall profile of the elution curve (reactant and product) is developed here for rate evaluation with a (pseudo-)first-order irreversible reaction in a linear, liquid-solid, chromatographic reactor. The ideal chromatographic reactor model free of complication from mass-transfer processes is developed initially, followed by a second model where adsorption and desorption rates commensurate with reaction rates are considered. Results from these models can be extended with modifications to
the bonded-phase type of liquid chromatographic reactor as well as situations where reversible reactions are involved. A material balance on a solute species participating in a chemical reaction at time t and position x in the axial direction in a differential section of the liquid chromatographic column (where uniformity in the radial direction is assumed) can be written as
ac,
ac, a2Cm a + r, (1) at at ax2 ax where C, is the mobile-phase concentration and C, is the average concentration in the particle defined previously (Babcock et al., 1966; Ruthven, 1984; Chu and Langer, 1986) to eliminate any radial dependence of the stationary-phase concentration; D, is the axial dispersion coefficient and u(x) is the linear velocity of the mobile phase at x , both based on the interstitial space; rmand r, are the reaction rates in the mobile and stationary phases, respectively (the values are negative for reactant and positive for product). For the stationary phase with nonequilibrium distribution, the material balance is - + - + D, -+ -[[u(x)C,] = rm
where k , and kd are adsorption and desorption rate constants. The stationary-phase concentration C,in eqs 1and 2 is based on the column void volume; it is related to the surface concentration C,' commonly used in adsorption studies by C, = (p,/t)C,', where C,' is based on the total packing mass. Here ps is the packing bulk density and e is the void volume fraction; the ratio p l / e in adsorption chromatography has a significance analogous to the phase ratio of bonded-phase liquid chromatography. The use of C, instead of C,' here (as well as K instead of K' later) facilitates elimination of e from our equations to obtain general expressions for LCR models that can be extended to the bonded-phase situation with only minor modifications. External mass transfer resistance across the particle boundary and pore diffusion inside the particle are not considered in the equations above. These processes are neglected as well as axial dispersion later on the bases of an earlier analysis of their influences on reaction rate measurements (Chu and Langer, 1986). Criteria for considering possible mass-transfer effects from longitudinal dispersion, intraparticle diffusion, and interfacial sorption/desorption were developed where the zeroth and fmt moments from reaction chromatograms were used to evaluate reaction rates. When these were applied in studies of base-catalyzed solvolyses of tetrachloroterephthaloyl chloride in a reversed-phase system, insignificant effects were found on the rate measurements. This is because the intrinsic reaction rates (on the order of lo-' s-l) were limiting relative to typical mass-transfer rates. A similar analysis is used here to examine the iron-catalyzed hydroquinone oxidation during passage through silica columns. The reactor system parameters and hydroquinone reactant properties are shown in Table I. In Appendix I, criteria for evaluating the significance in rate measurements of axial dispersion, external mass transfer, intraparticle diffusion, and adsorption effects are presented separately together with means of obtaining meaningful numerical values for important constants. The arguments that, with the exception of adsorption proceaaes, mass-transfer processes generally have negligible effects on reaction rates in liquid chromatographic columns are developed in Appendix I. We also conclude that it is only
Ind. Eng. Chem. Res., Vol. 30, No. 7, 1991 1491 Table I. Representative LCR Parameters for the Hydroquinone Oxidation Study chromatographic column characteristics column diameter, de 0.3 cm 10 cm column length, L cm particle diameter, d, 0.72 column porosity, c mobile-phase properties' 0.670 g/cm3 density, p 0.00403 g/(cm s) viscosity, 1 linear velocity, u 0.246 cm/s 0.698 min mobile-phaae residence time, t, hydroquinone properties' 2.086 min stationary-phase residence time, t, diatribution coefficient, K 2.99 mobile-phase rate constant, k, 0.0003 min-' stationary-phase rate constant, k, 0.25 m i d (maximum value) 3.6 X lod cmz/s molecular diffusivity,* Dm 'Experiment: CYJ4-34#17, at 25 "C and 0.75 mL/min. bFrom the Wilke-Chang equation.
the reactant that may be subject to limiting adsorption and desorption rates for the oxidation under study. The assumption in the treatment that the axial dispersion effect can be neglected does not imply an absence of contribution to peak profile broadening, as can be seen in the chromatograms here. However, this axial dispersion essentially results in equal spreading relative to the peak mean and, thus, does not influence evaluation of the first absolute moments (Suzuki and Smith, 1975). Although only the iron-catalyzed hydroquinone oxidation is addressed, Appendix I illustrates how similar analyses of mass-transfer effects can be applied to other reactions. Conclusions analogous to those of Appendix I have been reached for solvolyses (Chu and Langer, 1986) and interconversions (Hanai et al., 1986). Comparison of a model that treats reaction alone with one that also incorporates consideration of finite adsorption and desorption rates is developed below. Because only a limited range of reaction rates is viable for LCR studies, the results should be applicable to many other on-column reaction systems. Modeling of First-OrderReactions with and without Adsorption Equilibria in the Liquid Chromatographic Reactor. With negligible longitudinal dispersion (justified in Appendix I) and a constant linear velocity u throughout the column (since the fluid is essentially incompressible), eq 1 simplifies to
ac, + ac, + u ac, = r , + r , at at ax
(3)
Considering a first-order or pseudo-first-order reaction occurring in both mobile and stationary phases of the type R + other reactant(s) P + other product(s) such that only R and P are detectable (This can be accomplished through choice of an appropriate absorbance wavelength in an ultraviolet detector or by using a specific detector, e.g., electrochemical detection.), the material balances for the reactant R and the product P become, respectively
-
dc,P acmP + -ac,p +u at at ax
k,C,R
+ k,CnR
(5)
where the superscripts in the concentration terms represent the designated species (R for reactant and P for product); k, and k, are the (pseudo-)firsborder reaction rate constants in the mobile and stationary phases, respectively.
Initial and boundary conditions are C,R(x,O) = Cmp(x,o)= 0 at t = 0 (6) C,R(o,t) = &), CmP(O,t)= 0 at x = 0 (7) d t ) is the reactant input function a t the column inlet. Since pulse injections are used in this work, 4(t)can be approximated as a delta function 6(t). Because the nature of axial dispersion is such as to have a negligible effect on reaction rate evaluation, a simple boundary condition can be used here instead of the commonly used Danckwerts type which includes dispersion at the column inlet. Specific assumptions about reactant and product concentration distributions between the mobile and stationary phases now permit the development of two models: I. Adsorption equilibrium is established for both reactant and product, and chemical reaction is the only important rate process. 11. Adsorption and desorption rates for the reactant are important and must be considered together with chemical reaction rates. The general strategy in treating both models is conversion of C(x,t)to ita Laplace transform C(x,s); Le., the system can be treated with a set of ordinary differential equations. Analytical solutions in the Laplace domain are then accessible for each model. van der Laan's theorem is utilized to convert the Laplace transform solutions for reactant and product concentrationsto their corresponding nth-order statistical moments:
[""
m, = JmC(x,t)tndt = (-l)nlim -(C(x,s)) M dsn -
1
(8)
Model I: Reaction with Equilibrium Distribution between Phases for both Reactant and Product. Equilibrium distributions are established instantaneously between mobile and stationary phases, and linear isotherms are assumed for both reactant and product since low reactant concentrations are utilized in these kinetic studies. (For example, the distribution constant variation for either hydroquinone or benzoquinone is less than 2% within the concentration ranges used here.) Thus, equilibrium constants are concentration independent and can be expressed as
Here, K is the distribution coefficient for the designated species and t, is the corresponding total retention time; t, is the mobile-phase residence time. Again, the distribution coefficient K is based on C,, the stationary-phase concentration in units of moles per void volume defined earlier. This expression is equivalent to the capacity factor and represents the ratio of residence times in stationary to mobile phases. The relationship between K and the adsorption isotherm equilibrium constant K' based on C,' is given by K = (p,/t)K'. With linear equilibrium distributions between the two phases, eqs 9 and 10 can be substituted into eqs 4 and 5. Then using Laplace transformation eqs 4 and 5 become dCmR (1 + KR)sCmR+ u -= -krpp,ICmR (11) dx
and
1492 Ind. Eng. Chem. Res., Vol. 30, No. 7, 1991 Table 11. Comparison of Relationships for the Two Statistical Moment Models model I1 (reaction and adsorption) model I (reaction only) equilib distribn Yes not established for reactant aC,R concn relationship between KR = C>/CmR (9) = k,CmR- (kd k,)C> mobile and stationary At phases (10) K p = C,P/CmP
-
+
“I
(16)
where kspp,Iis the apparent rate constant for Model I, a function of the reaction rates in the mobile and stationary phases, and k,, is the corresponding stationary-phase rate constant. The assumptions involved in this model are basically those of the “ideal chromatographic reactor” system (Langer et al., 1969; Langer and Patton, 1972). The reactant concentration profile in the time domain can be obtained by solving eq 11 analytically and applying the inverse Laplace transformation. The Laplace expression for the product profile, however, is difficult to invert analytically without use of the statistical moment approach. The application of the Laplace transformation with statistical momenta to solve the equations above is explained in Appendix 11. The final expressions for the zeroth and first momenta of the reactant and product are given in Table 11. The expression for the first absolute moment of the overall chromatographic elution curve containing both reactant and product, pltot, then can be obtained by combining eqs A2, A3, A6, and A7 (See Table 11). mlR + mlP ’ltotl = moR +
-
Here, t,P is the product retention time and A ~ L C is R the shift of the first moment resulting from reaction in the liquid chromatographic column. The meaning of pltot is illustrated in the hypothetical reaction chromahgram shown in Figure 2. It is essentially
( A ) ICPUT PULSE
(B)REACTION CHROMATOGRAM
Figure 2. Illustration of first momenta in a hypothetical reaction chromatogram; R = reactant and P = product profile. plbt = first moment of elution profile with central axis at ab. See text for explanations.
the “average”retention time of the overall elution profile, Le., the center of mass (at line ab) of the distribution of P + R. It is expressed in eq 14b as the s u m of the product retention time and an LCR reaction-dependent term, AtLcR, which includes contributions from the remaining reactant peak and the broadening occurring from continuous product formation during the passage through the column. Where the reactant is retained longer than the product as shown in Figure 2 (and in this study), the A t u ~ term from eq 16 is positive so that the center of mass plht for the elution curve is located to the right of t,P, and vice versa. Casting eq 16 in the form
two limiting cases can be used to illustrate the physical meaning of plht and A~LcR. Thus, with no chemical reaction in the column or with an infinitely high flow (such that kapptmapproaches zero), A ~ L C in R eq 17 becomes t? - t,P from L’Hbpital’s rule. Then, plht becomes the first absolute moment of the reactant plR with no product profile complication, i.e., the retention time for reactant t,R (Figure 2). At the other extreme, for very fast reaction (an infinite k value), A t z becomes ~ zero from eq 17 and plht becomes“P‘ t,b, i.e., the reaction is complete immediately
Ind. Eng. Chem. Res., Vol. 30, No. 7, 1991 1493 upon reactant entrance into the column so that elution gives essentially a product peak. Model 11: Reaction without Equilibrium Distribution of the Reactant between Two Phases. When reactant adsorption and desorption rates are comparable to chemical reaction rates and with other mass-transfer processes not significant as explained earlier, eq 2 becomes appropriate in place of eq 9 to describe the reactant distribution in the stationary phase. Application of the Laplace transformation to eq 2 gives
Substitution of eqs 18 and 10 (with a linear equilibrium distribution for the product) into eqs 4 and 5 together with Laplace transformation gives
(20) The derivation for model I1 is similar to model I but is more involved as shown in Appendix 11. Solutions for model I1 are also shown in Table 11. The expression for the overall first absolute moment for model I1 is mlR + mlp = t,(l + Kp) + = moR +
where KadR
- up
and now (23) where k d and k , are the apparent and stationary-phase rate constants of model I1 and KadRis the adsorption equilibrium constant for the reactant (or ka/kd). When there is an instantaneous reactant equilibrium distribution between the two phases, KadRbecomes the distribution coefficient KR of model I. An important advantage of utilizing plW in the LCR rate evaluation, of course, is that it solves the problem resulting from reactant and product overlap. T h e plm value can be obtained directly from the elution curve of a reaction chromatogram as a function of the mobile-phase residence time, t,. By varying mobilaphase flow r a m , data for the first momenta at different t, values can be generated. Kinetic parameters in eqs 14 and 21 for models I and I1 can then be estimated by using measured quantities together with a nonlinear regression routine. Another advantage of the first absolute moment method is that it allows the use of simplified models in which consideration
of broadening effects from axial dispersion and other mass-transfer processes can be ignored. Such simplified models do not provide exact descriptions of the real concentration profiles of eluting species since spreading phenomena due to these processes are not considered. On the other hand, these models can be applied in reaction and adsorption rate evaluation because the first absolute moment is little affected by the broadening from these processes (Appendix I; also Suzuki and Smith, 1975; Hanai, et al., 1986). Therefore, relatively simple analytical solutions become feasible and kinetic parameter estimations less formidable. More exact response curves can be constructed by utilizing models incorporating dispersion factors and simulated by using numerical techniques such as fast Fourier transformations (Hsu and Ernst, 1990). However, evaluation of actual kinetic and dispersion parameters then requires more sophisticated methods and highly precise data.
Experimental Section The high-performance liquid chromatographic array used for this work incorporated a Waters Model 590 solvent delivery pump, a Rheodyne Model 7125 six-port syringe injector with a 20-pL sample loop, and a LDC/ Milton Roy spectroMonitor D variablewavelength detector in a standard confiiation. A Perkin-Elmer LC-235 diode array detector was used to determine the monitoring wavelength which gave equal sensitivity for reactant and product, from scanning ultraviolet spectra of the quinhydrone standard (1:l hydroquinone:benzoquinone molecular complex). The diode array detector was also used to confirm the identity of the product wave by comparing spectra with standard reference material. All fittings and high-pressure tubing (1/16 in. by 0.01 in. id.) between the injector and the detector were of PEEK material (polyether ether ketone; Upchurch) to minimize any metal leaching into the column of the type that might originate from conventional stainless steel material (Jeng and Langer, 1989). A Hewlett-Packard 3396A integrator equipped with a 9114A Disc Drive was used to store reactor chromatagram signals at the acquisition rate of 20 pointa/s for subsequent data analysis. A program for conversion of the signals to the first absolute moment was written and executed by use of the BASIC language installed in the integrator system. Microbore 10-cm glass columns (3 mm i.d.; Omnifit) designed to withstand pressures up to 1200 psi were used to observe color changes in the column from iron treatments. The packing material (Spherisorb Slow 10." silica gel) was dried at 150 OC overnight before slurry packing into a glass column using isopropyl alcohol at ca. 800 psi with an in-house slurry packing apparatus. Only moderate column efficiency was attained under the lowpressure conditions; however, observation of successful metal doping with the transparent columns was more important than high efficiency in this study. Toluene was used to measure column void volume, and all retention volumes were corrected for extra-column volume of 0.10 mL. The use of glass columns and titanium frits (2-pm porosity; Upchurch) in the Tefzel column endfittings as well as in a precolumn filter minimized active metal contact with the reactor bed. In situ modification of the silica surface by ferric ions was carried out by first equilibrating the column with 100 column volumes of buffered isopropanol solution (0.04 M potassium acetate + 1% (v/v) acetic acid). This activated the silica surface to facilitate subsequent ferric ion immobilization (Eiceman and Janecka, 1983). Before metal doping, 50 column volumes of neat isopropyl alcohol were
1494 Ind. Eng. Chem. Res., Vol. 30,No. 7, 1991
used to remove residual salts. Then, 10-12pulses of 1 mL of FeCl+opropyl alcohol (2.72mM) were injected into the column; effluent was monitored at 450 nm to estimate total uptake of ferric ions (ca. 10 "01). The column packing gradually turned light yellow as it approached saturation. The development of an uniform color throughout the column and uniformity of resulting reactor chromatograms were good indicators that the metal ion treatment of the packing was homogeneous. During the experiments described here, the catalytic activity of the treated column was constant with no observed loas of ferric ions, indicative of a stable modified silica surface under operating conditions. A constant temperature bath with circulator (Lauda MT3) was used to achieve isothermal conditions in the column which was encased in a water jacket (fO.l "C). Twenty feet of tubing (1/16by 0.02in. i.d.) immersed in the bath after the pump and 10 ft of tubing (1/16by 0.01 in.i.d.) encased in another water jacket before the column were employed to preheat the mobile phase and reduce any temperature gradient at the column entrance. Kinetic experiments were performed at temperatures between 15 and 45 "C. At each temperature, three to four flow rates were utilized with four replicate experiments for each flow rate. The column was equilibrated at each temperature for at least 2 h before kinetic experiments to allow thorough heat transfer. All solvents used in this study were HPLC grade. The mobile phase of 8% tert-butyl alcohol (Adrich)-hexane (Burdick & Jackson) was ultrasonicated for at least 1/2 h before use. All chemicals in this study were purchased from Aldrich Chemical Co. and used without further purification.
Results and Discussion Catalyzed Hydroquinone Oxidation in the Liquid Chromatographic Reactor. The redox reactions of hydroquinone and benzoquinone have been studied in high-performance liquid chromatographic systems previously both in aqueous modified solvents (Huang et al., 1984; Gattrell and Kirk, 1987) and organic environments (Jeng and Langer, 1989): H
O
e HO
O
H + 02
-
0
0
0 + H20
(H202)
(24)
BO
This oxidation also has been found to be catalyzed by supported metal ions (Jempty et al., 1981;Radel et al., 1982). In the present LCR kinetic study, the catalyst bed was obtained by doping active ferric ions onto the silica packing. Application of the models discussed here to eq 24 requires irreversible pseudo-first-order kinetics. The negative standard value of the redox potential for the reverse benzoquinone reduction (-0.53 V) indicates that it is unfavorable even at low oxygen concentrations. More convincingly, injections of benzoquinone samples onto the iron-treated silica columns gave only a single benzoquinone peak eluent with no indication of hydroquinone formation. As for the assumption of pseudo-first-order kinetics, the oxidation rate is known to be first order in hydroquinone and catalyst concentrations and either independent of or proportional to the oxygen concentration based on the rate-determining step (LuValle and Weissberger, 1947). Oxygen solubility in the organic mobile phase used here is ca. 10 mM; this guarantees oxygen excess well over the maximum hydroquinone pulse concentrations (0.72mM). Thus, a pseudo-first-order expression can describe the
v
-TME(My) m
Figure 3. Seriea of liquid reaction c h r o m a t o g " for hydrquinone oxidation catalyzed by iron-modified silica at 25 O C . R = hydroquinone reactant; P = benzoquinone product; S = toluene standard; mobile phase, 8% tert-butyl alcohol (v/v) in hexane; aample, 20 p L of 0.72 mM hydroquinone; wavelength, 267.5 nm. (a) Flow rate (0 = 0.3 cm3/min; (b) F = 0.5 cms/min; (c) F = 0.75 cm3/min.
on-column hydroquinone oxidation, with catalyst concentration incorporated into the stationary-phase rate constant. A representative series of liquid reador chromatograms for the iron-catalyzed hydroquinone oxidation under various flow rates at 25 "C is shown in Figure 3. With detector operation at 267.5 nm, absorptivities of hydroquinone and benzoquinone were equal (1.00 f 0.01).This permits direct use of response without sensitivity corrections as discussed later. Product P in the chromatograms was confirmed to be benzoquinone through spectra comparison from the diode array detector. First Moment of the Elution Profile. The first absolute moment of the overall elution profile can be obtained from a reaction chromatogram where detector signals are acquired in small time increments (At = 0.05 s in this study) using the following equations: mlbt* Plbt*= mgtot* and t2
mnbt* =
c A(t)t"At tat,
(26)
where pl"* is the measured overall first absolute moment based on detector response; Q"* and ml"* are measured zeroth and first moments of the elution curve; A ( t ) is the detector signal of effluent at elution time t (in absorbance units for spectrophotometers). The summation covers the time range of the entire elution profile enveloping reactant and product, (tl,t2),and is calculated through the BASIC program stored in the integrator. For the experimental moment data to be applied in eqs 14a and 21a of models I and I1 which are based on concentrations, the following conversion is necessary: mlR* + mlp* - SRmIR + SPmlP ""* mOR*+ mop* SRmoR + SPmoP (27)
-
where m,* and m, are the nth-order statistical momenta based on the detector signal and concentration respectively (mn* = S X m,) while S is the detector sensitivity factor. With the UV detector absorbance wavelength at 267.5 nm in this study where absorptivities for hydroquinone (SR) and benzoquinone (s9 are equal, cclW is equivalent to cclW and can be applied to eqs 14 and 21 directly. When the detector cannot be operated with equal semitiviti- for the two species (e.g., a fixed-wavelength detector), eq 27 can be used and the measured fmt moment will depend on the relative absorptivity (Sp/SR).In either situation, experas a function of flow rates are reimental data for pcltot* quired. Values for t, and t: at each flow rate as well as K p and KR(the last value is needed only in model I) can be obtained from chromatograms of standard compounds;
Ind. Eng. Chem. Res., Vol. 30, No. 7, 1991 1496 then, AtwR is readily calculated from eq 14b or 21b. Subsequently, eqs 16 and 22 can be utilized for kinetic parameter determination for each model using nonlinear regression as illustrated later. Kinetic Parameter Estimation. Equations 16 and 22 permit an apparent rate constant estimation for each model. This constant represents a result from all reaction paths operative in both phases. A distinction between reaction rates in each phase frequently is important. A simple approach is to calculate the stationary-phase rate constant with eq 13 or 23 when the mobile-phase (homogeneous) rate constant is known. The homogeneous rate constant, of course, can be obtained from batch kinetic studies (Bolme and Langer, 1983; Chu and Langer, 1985). For surface-catalyzed reactions such as the one studied here, the mobile-phase reaction rate is usually insignificant compared to the stationary-phase rate and can be neglected. Batch reaction rate constants for hydroquinone oxidation were measured in the organic environment of the column mobile phase. Both the uncatalyzed reaction and the homogeneously catalyzed reaction with dissolved Fe3+ ions (FeClJ were studied at 25 O C . The uncatalyzed reaction reached equilibrium in about 1 week (k = 0.0003 mi&) while the homogeneously catalyzed one was close to complete after 3 days (k = O.OOO6 min-'). With reaction rates quite slow relative to on-column rates (Table I), the assumption of negligible mobilephase rate constants seems valid. These measurements also support the conclusion of Huang et al. (1984) that the hydroquinone oxidation was accelerated by the immobilized catalyst on silica. With surface-catalyzed reaction rates dominating the overall conversion, equations for the two models (eqs 16 and 22) reduce to the following nonlinear expressions: model I (reaction only)
model I1 (reaction and adsorption)
A modified Arrhenius relation can be used with all rate constants in eqs 28 and 29 so that In ki = di,' - 0i,2(1/RT- l/RTB) (30)
where Bi,l = In Ai - E ~ / R T B = In ki(TB)
ei,2 =
(31)
(32) Here Bi,l and Bi,* are unknown kinetic parameters in the rate constant expression, while R is the gas constant and TBis a baae temperature (1303.15K)i n t " d to reduce correlation between the preexponential factor Ai and the activation energy Ei (Hunter and Atkinson, 1966). F b " t i o n of eq 29 shows that independent estimation of adsorption and desorption rate constants in model I1 is not possible because of coupling between these two quantities in the expression. Therefore, a modified expression with two rate parameters is applied instead:
f l
OMod.1
I
1OOO/T(K) Figure 4. Arrhenius plots of stationary-phase reaction rate constants for on-column hydroquinone oxidation. The k, values are obtained by using experimental data with eqs 28 and 33 for models I and 11. Table 111. Kinetic Parameters for Hydroquinone Oxidation in the Two Models4 In k.(Td E., cal/mol model I (reaction only) -1.208 4100 model I1 (reaction/adsorption) -2.343 13040 k,(TB),min-': the stationary-phase reaction rate constant at 303.15 K (base temperature). E,, cal/mol: the activation energy for stationary-phase reaction.
Data analyses using this expression were conducted with a VAX 11/785 computer and the GREG (General Regression) package developed by Stewart (1986). The program involves a FORTRAN subroutine utilizing a least-squares algorithm for parameter estimation of nonlinear equations. The first attempt incorporated data at all temperatures to fit kinetic parameters e,, and 6 for k, in model I from eq 28 and k, and k, in model Iyfrom eq 33. This met with little success especially for model I1 since kWpa is a composite rate constant (eq 23) with a nonlinear combination of activation energies and preexponential terms. Thereafter, a separate estimate was made of the rate constantsto fit eqs 28 and 33 at each reaction temperature. The logarithms of the estimated k, values from models I and I1 are plotted against temperature in Figure 4. The values of ,e, and ,8 for the stationary-phase rate constant in each model could then be calculated from the intercept and slope in Figure 4 using eq 30. These values are listed in Table 111. Estimates of k, from model I deviate from those for model I1 significantly at low temperatures, but closer values are obtained at higher temperatures as can be seen in Figure 4. A greater contribution to AtLCRfrom the reactant is expected at low temperatures with low conversion where hydroquinone adsorption is important. At higher temperatures with benzoquinone product dominating the elution profile, the two models with the same assumptions on product behavior tend to give similar results. The exceptionally low activation energy value for reaction with model I (Table 111)also suggests that it may be inadequate to describe the LCR system under study. However, overestimation of the stationary-phase rate constant by model I relative to model I1 shown in Figure 4 is contrary to the previous conclusion that a smaller observed k, value should be obtained from the simple model (Chu and Langer, 1986). This is because the IP values used in eq 28 were obtained with the bare silica column and these are somewhat different from the KdR values for the iron-modified silica surface. However, an accurate KR value is often difficult to obtain directly from the reaction chromatogram because the locus of the center of the reactant peak (cf. Figure 1)on the chromatographic response can be seriously distorted by a contribution from
1496 Ind. Eng, Chem. Res., Vol. 30, No. 7, 1991 2.01
\T=25*C
i\ f
1.51
'
'
0.4 ROW
'
0.6
'
0.8
'
WE (arr'ldn)
1.0
'
Figure 5. Simulation curve comparison for two models using the LCR experimental data for hydroquinone oxidation: 0,25 O C ; +, 35 O C ; 0 , 4 5 O C . (a) Model I, reaction only; (b) model 11, reaction and sorption processes.
the overlapping product wave. Comparing Models. Alternatively,the adequacy of the two models for description of the LCR system can be examined through comparison of predicted responses with the experimental data. Simulation curves are calculated from eqs 16 and 22 on the basis of the estimated parameters as functions of flow rates at various temperatures, as shown in Figure 5. The ordinate of plots, [l - exp(-ha ptm)]/h,pp, is a corrected AtLCR term respresenting sole& the contribution from on-column kinetics, free of the dependence on equilibrium constant variations due to temperature changes. In Figure 5a, predicted values based on model I significantly deviate from experimental data in low-temperature regions with high flow rates; this is where contributions from the hydroquinone reactant dominate the system since there is small conversion. However, model I1 yields a good fit under all operating conditions (Figure 5b). This difference reflects the importance of the reactant adsorption process in low conversion situations. At higher temperatures and lower flow rates where reactant conversion is high, the reactant adsorption contribution is minimized so that model I also gives an adequate fit to experimental data. The advantage of using model I1 in preference to model I then becomes less significant. Conclusion A statistical moment approach based on the overall elution curve is developed here that can be utilized for kinetic parameter studies of first-order-type reactions occurring in liquid chromatographic reactors. The application is illustrated with a study of hydroquinone oxidation catalyzed by iron-modified chromatographic silica. A reactor model incorporating finite reactant adsorption and desorption considerationsfits experimental data better than a simple model where chemical reaction rate alone is considered. The former is especially advantageous for the low conversions occurring at low temperature and high flow rate. The main goal of this work has been to bridge a gap between a realistic mathematical model and experimental reador chromatograms. As noted earlier, many theoretical treatments of chromatographic reactor models have been successfully developed, but few are readily applicable to kinetic parameter estimation from chromatograms. Hopefully, this development can provide a convenient
means for managing overlapping elution profdea for kinetic studies as well as some insights into associated masstransfer processes. The use of the total elution curve can eliminate a need for internal or external standards. This is advantagous since such standards can complicate chromatograms in a number of situations (e.g., during solvent variation when elution orders are shifted). Mass-transfer processes in chromatographic columns have been studied successfully by using the plate theory in the past (Horviith and Lin, 1978; Muller and Carr, 1986). However, when chemical reactions confound chromatographic processes, plate-height evaluation can be difficult. During in situ chromatographic reactor studies, it also can be difficult to separate adsorption and desorption effects from chemical reactions. The present approach to demonstrating significant contributions from finite rates of adsorption in the liquid chromatographic reactor has potential for extension to biochemical-based reacting systems, such as protein denaturation in the column, where strong adsorption and slow desorption may play important roles (Benedek et al., 1984; Hanai et al., 1986). Acknowledgment We appreciate support from the University of Wisconsin and the US.Army Research Office. We thank Professor Warren Stewart for the use of parameter estimation programs m well m Ms. Vicki Injeski and Mr. Paul Christoffel for their help with a number of calculations. Nomenclature A ( t ) = detector response signal, AU (absorbance unit) C, = concentration in the mobile phase, mol/cms C, = average concentration in the stationary phase, mol/cms C,' = stationary-phase concentration based on the packing mass, mol/g C ( x , s ) = Laplace transform of C(x,t) d, = column diameter, cm d, = particle diameter, cm DAB = molecular diffusivity, cmz/s D, = axial dispersion coefficient, cm2/s D, = intraparticle diffusion coefficient, cm2/s E = activation energy, cal/mol F = flow rate, cm3/s k , = adsorption rate constant, l / s kd = desorption rate constant, l / s k , = external mass transfer coefficient, cm/s k , = firet-order mobile-phase rate constant in the column, l / s (l/min) k, = first-order stationary-phase rate constant in the column, l / s (l/min) k, = apparent rate constant in the column, l/s (l/min) K = distribution equilibrium constant = C,/C, K' = distribution equilibrium constant = C,'/C,, cms/g K,d = adsorption equilibrium constant = k,/kd L = length of the column, cm m, = nth-order statistical moment based on concentration m,* = nth-order statistical moment based on detector signal NPe= PBclet number for axial dispersion = d u/Dm NRe= Reynolds number on the basis of superhcial velocity =i
d#ur/p
Nsc = Schmidt number = p / p D m Nsh = Sherwood number = k , d , / D , r, = reaction rate in the mobile phase, mol/(cm3 8) r, = reaction rate in the stationary phase, mol/(cms 8) R = gas constant = 1.987 cal/(mol K) S = detector sensitivity factor, AU/ (mol/cms) t = time, s (min) t, = column residence time in the mobile phase = Llu,s (min) t , = column residence time in the stationary phase, s (min) t, = total retention time in the column = t, + t,, s (min)
Ind. Eng. Chem. Res., Vol. 30, No. 7, 1991 1497
T = absolute temperature, K u = linear velocity of the mobile phase, cm/s x = axial position in the column Greek Letters c = void volume fraction, i.e., total column porosity p = viscosity of the mobile phase, g/(cm s)
pl
= first absolute moment based on concentration, s (min)
pl* = fmt absolute moment based on detector signal, s (mid p = density of the mobile phase, g/cm3 p, = bulk density of the packing g/cm3
4(t) = input function of the reactant at column inlet 3(s) = Laplace transform of 4(t) 7 = tortuosity factor
Appendix I. Examination of Mass-Transfer Processes on Intrinsic Kinetics in the LCR. Case Study of On-Column Hydroquinone Oxidation Catalyzed by Iron-Modified Silica Mass-transfer effects involving hydroquinone reactant are evaluated first using parameters from Table I, followed by a discussion for benzoquinone product. (a) Axial Dispersion. Earlier, reaction rate constant measurements were shown to be unaffected by longitudinal dispersion (Chu and Langer, 1986) where the following relationship applies: (kmtm + kst,/(l + k,/k&)
