I n d . E n g . Chem. Res. 1987, 26, 2148-2151
2148
to the drum, a mirror rotation rate of 217.4 rev sT1 and a linear scanning rate of 2.3 X lo-* cm s-l (translation frequency of 8696 Hz), led to 9.5 X W delivered to step 10 of the sensitivity guide, the last solid step held after heating in vacuo for 30 min at 40 "C. This solid step was calculated as a sensitivity of 2 x J cm-2, essentially equivalent to the sensitivity found by projection imaging with white light. Thus, there was no reciprocity failure. Large plates (16-in. X 221/2-in. X 0.009-in.) were used with a Laser Pagefax plate system (Image Information Inc., Danbury, CT). This system uses a He-Ne laser to scan the copy, the information being transferred to a 4-W argon laser, which was used multiline (488.0/514.5 nm) to expose the plate. It was found that with 40 mW of power on the plate, optimum printing conditions were realized when the fifth to sixth steps on a Stauffer Sensitivity guide was held, a sensitivity of 6.8 X J Under these conditions, a full range, 5-909'0, of 65-line halftone dots could be held (at lower exposures, the 5% dots were lost during printing), and a full-size newspaper plate could be imaged in 60 s. A press run on a commercial offset press (Dual-Lith, Davidson Corp., Brooklyn, NY) was halted after 25 000 copies with no evidence of plate wear.
Acknowledgment We are indebted to L. Kangas for development of the hydroperoxide analysis and to B. H. Gingrich, M. J. Centuolo, and J. J. Dolan for technical assistance. We also thank the Hercules Printing Plant, the Gyrex Corporation for assistance in developing a coating system for the plates, Image Information Inc. for the use of their laser system,
Kingsport Press for the use of their Opti-Copy projector and facilities, and Photocolor Inc., Newark, DE, for use of their camera. Registry No. 5B,68892-29-5; diethyl 4,5-dimethyl-trans-4cyclohexene-1,2-dicarboxylate, 68658-27-5; 3,4-dimethyl-3-cyclohexenecarbonyl chloride, 69815-57-2; acrylyl chloride, 814-68-6; 2,3-dimethylbutadiene, 513-81-5; bisphenol A, 80-05-7; zinc tetraphenylporphyrin, 14074-80-7;vanadyl acetylacetonate, 315326-2; pentaerythritol triacrylate, 3524-68-3; pentaerythritol tetraacrylate, 4986-89-4; oxygen, 7782-44-7.
Literature Cited Adams, D. R.; Wilkinson, F. J. Chem. SOC.,Faraday Trans. 1 1972, 11, 586. Breslow, D.S.; Simpson, D. A. U.S. Patents 4 271 259 and 4 272 610, 1981. Bourdon, J.; Schnuriger, B. Photochem. Photobiol. 1966,5, 507. Dubosc, J. P.; Mercier, C.; Bourdon, J. Bull. SOC.Chim. Fr. 1971, 3286. Dzhagarov, B. M. Opt. Spectrosc. 1970,28,33. Foote, C. S. Acc. Chem. Res. 1968,1, 104. Foote, C. S.; Wexler, S.; Ando, W.; Higgins, R. J . A m . Chem. SOC. 1968,90, 975. Gollnick, K. Adv. Photochem. 1968, 6, 1. Kopecky, K. R.; Reich, H. J. Can. J. Chem. 1965,43, 2265. Meyer, G. Bull. SOC.Chim. Fr. 1970,702. I. M.: Guronovich. G. P. ODt. Petsold. 0.M.;. Bvteva. . Spectrosc. _ 1973,'34, 343. Walker, P.; Webus, Z. J.; Thommes, G. A. J . Photogr. SEI. 1970,I8, 150.
Received for review January 16, 1987 Revised manuscript received June 23, 1987 Accepted July 11, 1987
Some Interesting Kinetic Observations on the Aqueous Permanganate Solutions Zhi-Xin Lin D e p a r t m e n t of Chemistry, W u h a n University, W u h a n , China
Thomas T.-S. Huang* Department of Chemistry, E a s t Tennessee S t a t e University, Johnson City, Tennessee 37614
Photochemical decomposition products in the UV+sible region and reaction products with hydrogen peroxide of aqueous permanganate solutions have been studied under neutral, basic, and acidic conditions. Although the product distributions of the reactions are different at different pH levels, the two types of reactions, i.e., the photochemical reactions and the reduction reaction by hydrogen peroxide, basically follow the same reaction path. This suggests that both types of reactions start with the same elementary step (probably a charge-transfer step) and then proceed to give different products according to the environmental condition of the solutions surrounding the permanganate ions. The industrial use of permanganate solutions will continue to increase as long as new processes are developed, utilizing its high oxidation potential in the field of water and air pollution control (Obuchi et al., 1974), as in the sterilization (Hamilton, 1974) and deodorization (Prosselt and Reidies, 1965) of water and air. The decomposition and oxidation-reduction reactions of permanganate solutions are thus an important subject of research studies. Photochemical decomposition of aqueous permanganate ion was studied (Zimmerman, 1955) under a variety of conditions in 1955. This study revealed that (a) the quantum yield, +, is small and decreases with increasing wavelength, (b) is independent of the pH and the concentrations of reactants and products, (c) at longer
+
wavelengths there is a small positive temperature coefficient of 4, decreasing with decreasing temperature, and (d) all of the atoms of the photochemically produced O2 come from Mn0,- ions. It concluded that the photochemical decomposition was a simple dissociation of Mn04into two fragments and that no appreciable electron exchange between the dissociated fragments destined to become O2and either H,O or OH- occurred. The mechanism of the oxidation-reduction reaction of permanganate ion reported in the literature was, however, much more complicated. Different paths were proposed for different situations (Wilberg and Gear, 1966; Symons, 1953). Generally speaking, these dealt with the kinetic explanation of product distribution by proposing various
0888-588518712626-2148$01.50/0 1987 American Chemical Society
Ind. Eng. Chem. Res., Vol. 26, No. 10,1987 2149 Table I. Other Experimental Data" neutral dissolved 02,ppm PH color change
"ChlnO4-= 2.00
X
before 5.81 4.12 purple
basic after
7.70 9.38 yellowish brown
before 5.80 12.80 purple
acidic after
7.80 12.70 yellowish green
before 5.92 0.65 purple
after 9.00 0.60 reddish yellow
M. Irradiating time = 1 h. T = 25.0 f 0.1 O C .
200
400 Xhn)
200
600
Figure 1. Repeated spectra of Mn04- in neutral solution at fixed time intervals of 30 s: CMlno4-= 1.00 X lo4 M.
subsequent reactions following the initial decomposition step. The decomposition of permanganate solutions in sulfuric acid was investigated quite thoroughly (Shafiiovich et al., 1978; Shafirovich and Shilov, 1978) for the purpose of determining the role of manganese ions in the liberation of oxygen in photosynthesis. It was established, using labeled compounds, that after acidification, permanganate exchanged oxygen with H,'80 and that the rate of exchange increased with increasing acid concentration (McDonald, 1961). In summary, the literature reports of the reaction of aqueous permanganate solutions are very diversified in their mechanistic interpretations; furthermore, there seems to be no connection among them. The purpose of this investigation is to provide a consistent viewpoint for understanding the stability of Mn04- in solution.
Results and Discussion Complete spectra (absorbance, A , versus wavelength) of the KMn04 solutions under constant irradiation at time intervals of 30 s are given in Figures 1,2, and 3 for neutral,
600
800
Figure 2. Repeated spectra of MnOc in basic solution at fixed time = 2.00 X intervals of 30 s: CM,,O~M, CKoH = 0.50 N. 0.61i
I
A
05
0
200
400 X(nm1
600
Figure 3. Repeated spectra of MnO, in acidic solution at fixed time = 2.00 X lo4 M, C H ~ S = O 0.50 ~ N. intervals of 30 s:
P
0.4
:-?
ot
0
Experimental Section Reagents. All chemicals used were of analytical reagent grade and were used without further purification. Permanganate solutions were reduced to four-fifths of their original volume by boiling, filtered through a fine sintered glass funnel, standardized by using the classical Na2C204 method, and stored in brown bottles. The solutions were found to follow Beers' law with a molar absorptivity for Mn04- of 250 m2 mol-' at 546 nm. Experiment. Spectra were obtained by using an HP-8451 spectrophotometer; a thermostated cell holder maintained at 25 f 0.1 "C was utilized. Complete spectra (A from 190 to 820 nm) can be scanned in 0.1 s. A computer program was written to obtain the dynamic plots of our solutions (absorbance, A , at fixed wavelength versus time, t ) ,using repeated 0.1-s scans at fixed time intervals. When strong irradiation was needed, an MGR-100 photochemical reactor was used with tubes of different wavelength ranges. Otherwise, the spectrometer lamp itself was used as the irradiation source, since for integration times of 0.9 s or less, the shutter of the spectrometer remains open between measurement frames, i.e., the sample is continuously irradiated by the spectrometer lamp. Dissolved oxygen in the solutions was monitored with a YSI-57 oxygen meter having a sensitivity 0.02 ppm. All pH values were measured by using a Fisher pH meter with a precision of 0.02 unit.
400 Xhm)
I
5 1
I50
t (sac)
300
Figure 4. Absorbance of MnOc as a function of time at constant wavelengths in neutral solution: CMno4-= 1.00 X lo4 M, at (1)312, (2) 350, (3) 526, (4) 546, and (5) 606 nm. (See Table I1 for assignments.)
0 0
300
600
t ( sed
Figure 5. Absorbance of MnO; as a function of time at constant = 2.00 wavelengths in 0.5 N basic solution: CKOH= 0.50 N, CM~O,X M; at (1)312, (2) 350, (3) 546, and (4) 606 nm. (See Table I1 for assignments.)
0
-
O
t bed
Figure 6. Absorbance of MnO; as a function of time at constant O 0.50 ~ N, C M ~ O=~2.00 wavelengths in 0.5 N acidic solution: C H ~ S = X lo-* M; at (1)508, (2) 526, (3) 546, and (4) 566 nm. (See Table I1 for assignments.)
basic, and acidic solutions, respectively. The pH values remained practically unchanged for the basic and acidic solutions. For neutral solutions, however, there was a large increase in pH value. These pH values, along with the
2150 Ind. Eng. Chem. Res., Vol. 26, No. 10, 1987
10
0
0
300
600
X (nml
t (sed
Figure 7. Absorbance of MnOL as a function of time at constant wavelengths in 0.5 N acidic solution (addenda to Figure 6): Chlno4 = 2.00 X lo4 M, CH2SOr = 0.50 N; at (1) 220, (2) 350, (3) 430, and (4) 546 nm. (See Table I1 for assignments.) Table 11. Some Characteristic Peaks of Mn in Solutions possible A, nm possible species A, nm species 486 MnO(0H) 210-230 Mn2+ MnOL MII(OH)~-M~O(OH), 508-546 312-350 606 Mn0,2430-440 MnO, Table 111. Zero-Order Rate Constants under Different M/min) Conditions (106k, neutral run H2O D,O acidic at pH 0.60-0.65 I 3.68 3.38 14.7 2 3.79 3.43 15.8 3 3.25 3.40 15.4
dissolved oxygen content and color change before and after a typical reaction, are collected in Table I. The dynamic plots (absorbance versus time) of these scans at different wavelengths are shown in Figures 4-7. Characteristic peaks were assigned and are shown in Table 11. In neutral solutions, the reaction seems to be simple. Three distinct and clean isosbestic points were observed at X = 204,518, and 577 nm (Figure 1);the reaction is zero order (Figure 4 and Table 111). While Mn04- (representative peaks at X = 526 and 546 nm) is decreasing, soluble Mn(1V) species (representative peaks a t X = 312 and 350 nm) are increasing and a negligibly small amount of Mn042-(represented by X = 606 nm peak) is forming. In basic solutions, the situation is more complicated. While the isosbestic points at 204 and 577 nm are reasonably clear (Figure 21, there no longer exists an isosbestic point at X = 500 nm. Figure 5 demonstrates the wellknown first-order decrease of Mn04-species and a corresponding increase in Mn0,2-; we also see that after an equilibrium between Mn04- and Mn02- is reached, the formation of Mn(1V) species becomes zero order once again. In acidic solutions, the reaction is even more complicated. Initially peaks representing Mn04- exhibit a zeroorder decrease (rate constants are collected in Table 1111, in a manner similar to that observed in neutral solutions but with Mn2+ increasing correspondingly as shown in Figures 6 and 7. Initially, two distinct isosbestic points are located at X = 216 and 300 nm. However, as the reaction proceeds, the isosbestic point at X = 300 nm is completely lost, and most strikingly, peaks representing Mn(1V) and Mn(VI1) species increase while that of Mn2+ starts to decrease accordingly. In addition, we have obtained two more pieces of relevant information about the decomposition of MnO,-. Firstly, the reaction shows no solvent isotope effects; the zero-order rate constants of three repeated runs in H 2 0 and in D 2 0 are nearly the same in either medium (see Table 111). This means that the participation of water in the initial step of this reaction is minimal. Secondly, when Mn04- is reduced by using stepwise additions of small increments of Hz02,the reaction rate is rather fast but the product pattern after each increment of H202exhibits
0
200
400 X (nm)
600
Figure 8. Mn0,- spectra in neutral solution showing the effects of M. (1)irradiation and (2) addition of H,02: CMnO4-= 2.00 X
exactly the same features as when repeated scans at constant time intervals were made to illustrate photochemical decompositions in neutral, basic, and acidic solutions (Figures 1, 2, and 3). Figure 8 demonstrates this point; for example, in neutral solution, the spectra shown before and after the photochemical reaction (1)and those showing the reduction by H2O2(2) are practically identical. This suggests that the reactions follow the same mechanism. In the rate-determining step, energy is supplied by radiation in the photochemical process, while the activation energy is provided by the reaction of Hz02with Mn04- in the reduction reaction. In the case of the photochemical reaction, Mn04- dissociates presumably into two fragments destined to become O2 and either H 2 0or OH- as described by Zimmerman (1955). It is, however, more difficult for us to suggest how H20zreacts with Mn04- to result in the same fragmentation pattern. The product distribution is governed by the environment surrounding the Mn0,- ion, which in turn depends on whether the medium is neutral, basic, or acidic. It is likely that the initial step is a charge transfer induced by radiation or interaction with H202,which causes the reduction of Mn(VI1) and the breakage of Mn-0 bonds. To explain what we observed followed this initial break-up step, the following mechanism is suggested. We must emphasize that it consists of overall reactions, which could be described by elementary processes only if further experimental data were available: Mn04- + 3H20 Mn04- + 2H20
-
-
+ 0 + OH- + OH MnO(OH), + 0 + OH- + OH Mn(OH),
--+
In neutral solutions, the reaction then simply proceeds by 20H
H20
20
1/z02
0 2
which is consistent with our observations. In basic solutions, complications occur because of an equilibrium reaction: MnO,-
+ OH-
-
MnOd2-+ OH
This causes the observed first-order decrease in Mn04until equilibrium is reached and then proceeds via the slower photochemical reaction as in the neutral solutions. In acidic solutions, other reactions occur simultaneously and play an important role on product distribution. Possible reactions are 2Mn04- + lOOH + 6H+ 5 0 2 + 2Mn2++ 8H20
-
and
Mn04-
+ 2Mn2++ 2Hz0
-
Znd. Eng. Chem. Res. 1987, 26, 2151-2157
2MnO(OH)z + Mn3+
+ + 2Mn3+
Mn2+ Mn4+
MKI(OH)~ MnO(OH)z
HzO
MnOz + 2Hz0
This yields an initial zero-order decrease of Mn04- and a concomitant formation of Mn2+. As the reaction proceeds in acidic solution, the formation of solid MnOz starts slowly, initially at the expense of MnO(OH)z,and accelerates later on. We believe that this sol (solid MnOJ formation is the reason why there is an apparent increase in the MnO, peak, as well as in all other peaks in the range X = 200-570 nm for the acidic solutions. The scattered radiation due to Tyndall effects gives the appearance of increasing concentrations for all absorbing species. The above description of the Mn04- reaction is a rather general one. However, it does explain the rather complicated assortment of reaction products in neutral, basic, and acidic solutions. Furthermore, it is consistent with many previously published results concerning the reaction
2151
of Mn04- ion and provides a starting point for the interpretation of experimental results within a broad spectrum of starting conditions. Registry No. Mn04-, 14333-13-2; H202,7722-84-1.
Literature Cited Hamilton, J. J. J . Am. Water Works Assoc. 1974, 66, 734. McDonald, H. 0. Diss Abstr. 1961, 21, 454. Obuchi, A.; Okuwaki, A.; Okabe, T. Nippon Kagaku Kaishi 1974, 1425. Prosselt, H. S.; Reidies, A. H. Znd. Eng. Chen. Prod. Res. Deu. 1965, 4 , 48. Shafirovich, V. Ya; Shilov, A. E. Kinet. Katal. 1978,19(4),877-883. Shafirovich, V. Ya; Khannamov, N. K.; Shilov, A. E. Xinet. Katal. 1978,19(6) 1498-1501. Symons, M. C . R. J. Chem. SOC.1953,3956. Wiberg, K. B.; Gear, R. D. J. Am. Chem. Soc. 1966, 88(24), 5827. Zimmerman, G. J. Chem. Phys. 1955,23(5),825. Received f o r review February 10, 1986 Revised manuscript received June 29, 1987 Accepted July 27, 1987
Lumping Strategy. 2. A System Theoretic Approach Pamela G. Coxson+ Department of Mathematics, The Ohio S t a t e University, Columbus, Ohio 43210
Kenneth B. Bischoff* Department of Chemical Engineering, University of Delaware, Newark, Delaware 19716
Lumping analysis is placed in the context of linear systems theory, providing simpler and new ways of looking a t the problems, as well as justification to the strategies used in part 1. In particular, the relationships between lumpability of a kinetic scheme and the concepts of observability, controllability, and minimum realization of a system are developed. This also leads to the use of a generalized inverse to find the new lumped rate coefficient matrix from the original full matrix for those applications where this is known and a reduced order representation is desired. Finally, the theorems provide useful guides in identifying appropriate lumping schemes. 1. Introduction It is common practice to view similar chemical species as a single lump or pseudospecies for the purpose of formulating a relatively simple model of a complex reaction network. This approach has worked well in industry (Weekman, 1979) and is supported, in the case of monomolecular reactions, by the theoretical works of Wei and Kuo (1969), Kuo and Wei (1969), Hutchinson and Luss (19701, and Ozawa (1973) and reviewed by Bischoff and Coxson (1987). In part 1 (Coxson and Bischoff, 1987), we developed a systematic method for determining appropriate lumping schemes from experimental data. In this paper, we place lumping analysis in the context of linear systems theory. This approach provides considerable simplification of many known results and fresh insight into some old problems, and it permits a variety of diverse results to be housed under the same roof. Finally, systems theory provides motivation and justification for the lumping strategies presented in part 1. A summary of results from linear systems theory is given in section 2 and those of lumping analysis in section 3. In section 4, the two are combined. The results which emerge are applied to the practical problems of lumping in the remainder of the paper. 'Current address: Member of Technical Staff at the Aerospace Corporation, Los Angeles, CA 90009-2957. 0888-5885/87/2626-2151$01.50/0
2. Linear Systems A brief introduction to the relevant results of linear systems theory is given below. For further details, there are numerous sources varying in level and emphasis (see, for example, Luenberger (1979), Brockett (1970), or Ray (1981)). The reader familiar with the concepts of controllability, observability, and realization might omit this section. A linear dynamic system is given by a system of linear difference (for a discrete system-DS) or differential (for a continuous system-CS) equations forced by inputs u and observed via the measured values y = Mx: x(h + 1) = Gx(h) + cu(h) x(hJ = xo y(h) = Mx(h) k(t) = Kx(t) + Lu(t) x(to) =
(DS) XO
y(t) = Mx(t) (CS) where the value of x at a particular time lies in the state space X ( = Y P ) ,u takes values in an input space U (=Yl'), and y lies in the space of measured values M X . If (DS) is a discretized (sample) version of (CS) with x ( h ) = x(to h7), then G and K are related by G = eK7.The systems (DS) and (CS) aboye are completely determined by the ordered triples ^(G,L,M)and (K,L,M),respectively. A system (G,L,M) or (K,L,M)is n-dimensional if X i s an n-dimensional vector space and G (respectively, K), L,
+
0 1987 American Chemical Society
