edited by
JAMESP. BIRK
computer series, 150
Arizona State University Temp, AZ 85281
Temperature as an Index of Chemical Reaction Progress Temperature-Time Curves and Exponential Curve Peeling Kenneth Kustin Brandeis University, Box 9110, Waltham, MA02254-9110 Edward W. Ross Department of Mathematical Sciences, Worcester Polytechnic Institute, 100 Institute Road, Worcester, MA 01609-2280
In this paper we recommend two chemical reactions for the production of temperaturetime curves in adiabatic reactors, and we present a method for determining the kinetics parameters of these curves, based on the systematic application of statistical "best-fit" criteria. Evaluating Reactions
A well-known reaction. sodium thiosulfate with hvdrogen peroxide in liquid water, is shown not to be optimal for use as a kinetics standard due to its comdex stoichiometw and multistep rate law. lodide ion catalisis of this reaction in chloroacetate buffer at pH 3, however, has unambi~mous stoichiometry and straightfo&ard kinetics; it is r&ommended as a thermochemical kinetics standard. By varying the concentration of tI1, a convenient range of halflives is accessible. A second recommended reaction is the heterogeneous iron-promoted reaction between magnesium and water in saline solution. It has the advantages of simple stoichiometry, convenient first-order rate constant, and large reaction enthalpy. Variation of the concentration of NaCl confers control over the reaction half-life. Dieitized ex~erimentaltemoerature-time curves for a strongly exothermic reaction system collected under nearadiabatic conditions in a calorimeter are essentiallv noisefree, but possibly contaminated by a weaker thermk leakage, especially i n the later stages of the reaction. Estimation of meaningful reaction-rate constants and enthalpy values despite this contamination required systematic nonlinear data fitting. The weak (thermal leakage) effect was modeled, along with the strong (chemical)process. Parameters were calculated by sequential, exponential peeling, using statistical methods to ensure that the parameter estimates were meaningful. The importance to chemical technolom of monitor in^ temperature in adiabatic reactors containing exothermi;: reactions can hardly be overstated. Because the rate constant of even a simple, single-step reaction increases exponentially with increasing temperature, the reaction rate increases autocatalytically,and chemical instabilities such as multiple steady states and oscillations can, and do, occur in such reactors (I).To test theoretical predictions of chemical reactor behavior, an exothermic reaction of known stoichiometry, exhibiting single-step kinetics, with accurately determined rate law, was sought as an experimental model of such reactor systems (2).
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454
Journal of Chemical Education
The exothermic reaction between sodium thiosulfate and hydrogen peroxide would seem to be such a reaction (3).It has been selected as a paradigm for studying chemical instabilities in adiabatic or near-adiabatic chemical reactors (I, 4) due to its supposedly single-step rate law and simple stoichiometry. This reaction appeared to be so well-understwod that it had been selected to serve as the basis for a ~edaeoeicallvinterestine nndereraduate laboratorv in ~ h rethk chLm1;al engineering c ~ r r i c u l u k(5).~ l t h o u this action is hiehlv interestine and useful in reactor desien. - . it has neither a simple rate law nor a sim le stoichiometry (6). The oxidized products are both so4' and S40s2-.and their ratio depends on the pH and the ratioofinitial hydrogen peroxide and thiosulfate ion concentrations, [HzOzlo and [SzO8~Io, respectively. Selection of a better calorimetric kinetics standard by modifying or replacing this reaction should be possible.
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Treating Coupled Nonlinear Effects
From an instrumentation viewpoint, temperature is an excellent choice as a monitor of reaction progress. Through the use of thermistors and analogldigital conversion hardware, temperaturetime curves consisting of several hundred accurate, virtually noise-free pairs of data are easily obtained with relatively inexpensive calorimeters (Parr Instrument, Solution Calorimeter, Model 1455). The data, however, are difficult to interpret due to coupling of two or more ~ossiblvnonlinear effects. a situation similar to the dynamics of drug delivery to multicompartment biological systems 171, radioactive decay r81,and chemical inntahility (91.
Statistical Methods Statistical methods developed to analyze such curves have been incorporated into software such as Sigmaplot and Statgraphics as resident nonlinear curvefitting options. Large data sets, typically requiring the estimation of four or five parameters, are treated rapidly by such programs with excellent "fits" between the observed data points and the calculated "curve." Parameters obtained by a curvefitting routine to produce a "best fit" curve may not be meaningful, however, because neither physical and chemical constraints nor objective statistical criteria have been applied during their implementation. Rate constants, for example, must be positive quantities and, for cyclic reaction networks near equilibrium, must conform to microscopic reversibility; concentrations must conform to mass
balance; temperature changes must conform to energy balance. Therefore, a student or investigator should have the ability to experiment with the fit, and to discriminate among comoetinn sets of "best fit" oarameters to bring the results into conflormity with phy.&cal and chemicaleonstraints.
Graphical Methods Traditionally,graphical methods have evolved for treating coupled nonlinear effects (10).One of the most effective is exponential curve peeling. The method is based on plotting the logarithm of the signal against time. If the plot is linear, a single, fust-order effect is present. If only firstorder effects are present, curvature indicates multiple effects: these are treated by drawing a straight line through the end ol' the curve (the slowest effect,. and obtainine an amplitude and time constant for this of the c&e. The swnal due to this effect is subtracted.. or oeeled nwav. from tge full set of coupled effects. The process is repeate"d until all the effects have been evaluated. Though the method is approximate, it can still serve the useful purpose of providing initial "guess" values for more exact methods of analysis carried out on a computer. Such an exact method should also confer control of data evaluation upon the investigator. It should also permit application of constraints and obiective statistical criteria to the fitting of temperature-tike curves. This paper therefore addresses two main issues: &
recommending chemical reactions in which temperature is a reliable index of readion progress providing systematic methods of data treatment for these temperature-time curves to obtain their kinetics parameters From a practical viewpoint, it would be desirable if the systematic method of data evaluation could take advantage of regression programs in commercially available spreadsheets, such as Lotus 1-2-3, Minitab, Quattro, and Excel, to cam, out and evaluate the analysis of coupled exponential processes with regard both to accuracya n d significance of the curve's parameters. Experimental Temperature-Time Cuwes The calorimetric measurement of enthalpy, whether for reaction, solvation, or phase transition, is a basic chemical technique, well-described in textbooks (11,12). Nonideal effects in a calorimeter, such as gain or loss of heat due to thermal leakage and gain of heat due to stirrer work, are measured, and their effects are separated from the process under study. In these measurements, absorption or evolution of heat bv chemical reaction. for examole. is discernible from noiideal effects because the chekihal effect is more rauid and. over the time of observation. more energetic. In calorimetric measurement of kinetics, however, the temporal evolution of heat is being studied. The shape of the temoerature-time curve is the fundamental measurement ofinterest, and the temperature change due to reaction is often comuarable to the temoerature chanee due to thermal leakage'and stirring work, 'as the chemica?conversion approaches completion. To separate the two effects, it is assumed that the dependence of temperature on time for nonideal effects is either constant (stirring) or first-order (Newton's law of cooling). For this assumption to be valid, the temuerature channe in the calorimeter due to chemical reaction should not exceed 1-2 'C.
tor, and stirred with a magnetic stir-bar. Temperature can be recorded from a thermocou~le.ulatinum resistance thermometer, or thermistor. &s setup would allow the student to monitor simultanmuslv a second variable (e.e.. . -, pH). Superior research-quality data can be obtained with more sophisticated instrumentation of modest price, especially if a personal computer is available. In a Pam calorimeter, analog temperature is converted to a digitized voltaee a t oreset time intervals and is cautured on a Dersonal-computer with, for example, Lotus ~ e a s u r eThean. alon - outout - of a thermocou~le.olatinum resistance thermometer, or thermistor may &'digitized with an analog/ digital conversion board (Metrabyte DASH 8) and processed for storage and analysis by Labtech Notebook. This instrumentation has the advantage of providing time in seconds to the base 10, rather than in some other format to the base 60, which must be converted to a base 10 time in seconds. Pseudo-First-OrderIi&-N;~S203
The reaction between hydrogen peroxide and thiosulfate. whose kinetics were fust svstematicallv investigated by E. Abel more than 85 years &o (131,has variable and often complex stoichiometries in different pH ranges. To simplify the stoichiometry in neutral, unbuffered aqueous solution, and to try to make this system suitable for kinetics measurements, the reaction was run with a large stoichiometric excess of H202over thiosulfate. The temoerature-time curve should then be independent of reaction stoichiometry. A representative kinetics curve (Fig. 1)shows that even the pseudo-first-order reaction exhibits complexity around pH 7. The rate law is not simple; multistep reactions of competitive time constants are present, as visual inspection of the curve in the range 200-600 s shows. Evaluating the data between 50 and 150 s with exponential curve peeling indicates the presence of two effects with half-lives of approximately 40 and 75 s. The half-life of the single-step reaction calculated h m the second-order rate constant a t this temperature is 57.5 s (4). We therefore do not recommend the use of this reaction in an undergraduate instructional laboratory. An auuarentlv simule rate law (second-order reeime) would 6; achieved wit'h the penalty i f having an uncertain stoichiometry, or a simple stoichiometry (pseudo-firstorder regime) would be achieved with the penalty of having a complicated multistep rate law. When run together
Time
Experimental
Data can be collected in a simple, inexpensive setup consisting of a beaker placed in styrofoam, or similar insula-
(s)
F~gure1. TemperatLretlme cuwe forthe reactlon between Na2S2& 0 01~ 1 ~ . and Hz& n water wlth H 2 0 2 in stochlometncexcess l ~ ~ & M IH202],, 0 685 M, nmal pH, 6 61, so utlon volume. 100 mL Volume 70 Number 6 June 1993
455
I
23.4 0
Time (s) F~gure2. Temperature-time curve for the Nai-catalyzed react'on be0.1 M: [H2O2lo. 0.05 tween Na2S20, and H,02 in water: (S203270. M; Ihailn.0 1 M. total DJfferconcentratlon.0.5 M lmonochioroacetic acid);inzial pH, 2.99; solution volume, 100 mL.
with continuous pH measurement, for which several effects are clearly seen (data not shown), the experiment affords a glimpse into the complexity of aqueous sulfur chemistry. Iodide Ion Catalyzed Hz@-NazSz03 The studies wnducted by Abel on the kinetics of the hydrogen peroxide thiosulfate reaction were carried out for over two decades starting from 1907. In the course of these studies. Abel investi~atedthe thiosulfatehvdroeen oeroxide rea&ion's stoichiometry and kinetics over dikeknt pH ranges. He discovered and explained several interesting phenomena, such as the inactivation of copper(I1)catalysts of this reaction at elevated tem~eratures(14). Later. more modern studies refined his resuits but didnot significantly revise them (15). At pH 2-5, Ahel found that the only oxidized product is tetrathionate, and the stoichiometry is shown below (13). The reaction is slow, and it is suitable as a calibrating reaction when catalyzed by iodide ion. By varying the wncentration of P it is possible to obtain a reaction of known stoichiometry with a convenient time constant (see Fig. 2). The reaction rate constant determined in this experiment, in which reaction progress is followed by temperature change, is different from that given by Abel. He followed reaction progress by sampling and titrating unreacted thiosulfate with iodine, or unreacted peroxide by addition of base followed by titration with HOI. The temperature-evolving step in the presence of catalyst and the rate-determining step for the overall reaction are not the same.
500
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2000
1500
2500
3000
Time (s) F~gure3. Temperature-1 me curve forthe iron-promoted,chloride ion catalyzed reaction Detween magnesium and water; 0.0443 g Mg with 5 moi% Fe: [haCI]2.00 M: sotunon volume, 100 m ~ .
action. Thus, the time required for wmplete reaction can be controlled by varying the concentration of added electrolyte. In the presence of 2 M NaC1, the reaction lifetime is approximately 17 min (Fig. 3) (18). The iron-promoted, chloride ion catalyzed Mg-HzO reaction is the basis for an inexpensive, effective, and safe flameless heater, used to provide heat when fire and service equipment are inappropriate or unavailable (e.g., heating foods while camping; warming or thawing sealed pouches under water). The mixture is easily prepared from readily available chemicals, and it has been tested and suggested as a classroom experiment (19). The reaction stoichiometry and energetics are
where Q is the (extensive) amount of heat produced, and the enthalpy of reaction is AHn = -352.96 kJ (mol Mg)-'. Kinetics of Heat Evolution from the MagnesiumWater Reaction The rate law expression in molls for this heterogeneous reaction (20)is
where n is moles of Mg (n = n(t)); nw is moles of HzO; Vis reaction volume of HzO (mL); and ?& -.is a heterogeneous rate constant.' Whenx = dt). where x is moles ofMe reacted. the stoichiometries for and HzO are
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Iron-PromotedMagnesiumWater Reaction in Saline Solution The reaction between solid magnesium metal and liquid water is very slow, despite a strong thermodynamic driving force, because a thin oxide layer prevents direct contact between magnesium and water (16, 17). However, two changes in reaction mixture composition speed up the reaction. Most importantly, 3-5 mol% iron mixed into and firmly contacting magnesium disrupts the oxide barrier. Upon wntact with water the reaction readily initiates. Additionally, electrolytes, especially those containing simple anions such as chloride ion or sulfate ion, catalyze the re456
Journal of Chemical Education
where no and nwoare moles of Mg and HzO, respectively, present at to. Substituting eqs 2-4 into eq 1yields 'The equation for reaction rate per square centimeter of surface area for a heterogeneous reaction requires the use of two different concentration units. Assuming the surfacearea remains wnstantexcept at the very end of the reaction, when the change in area may be neglected, allows cancellation of area on both sides of the rate equation, and leads to eq l (mol s-l) for this heterogeneous reaction.
Water is the reagent in excess, so nwo >> 22. Redefining the rate constant as follows
we obtain
sisting of calorimeter contents and jacket will be in a steady state (SS). Then we get (21) ATT= ATQsT Let the temperature T ('0, in the calorimeter over the timeranget=to=Otot=tbe T=To+ATR+ATT (12) Substitution of eqs 10 and 11into eq 12 yields the complete expression for T. T = T~+ A T ~ ~-(&'RI'))+
The heat evolved, Q = Q(t), is proportional to the moles of Mg reacted, where the proportionality constant is the reciprocal of the heat of reaction, A& (kJ (mol MgT1), according to
~ ~ ~e(-k+)) ~ ( 1 (13) -
Data Analysis by Consistent Application
of Exponential Peeling
The general model expected to fit the data is of the form T=F(t)+g(t) (14) F(~)=A,+A@-~"~'w)
gtt) = 0 Upon substituting eqs 6 and 7 into eq 5, we obtain the heat evolution rate equation.
g(t)=A4
1
(15) fort < tss
+&(l-e-(Agt'm))
(16)
fort 2 ~ S S (11)
where A is another constant arising from the fitting procedure.
This can be integrated using the integrating factor to yield Q = e(-*'
((n&&
'k
" + C)
n
Att=to=O,Q=O,andC=-n&IR Thus,
AT, = A5 To =Al AT, = A2 where w is a scaling factor, used to scale the times obtained in the experiment. The value w = 1000 s is convenient for the magnesium-water reaction, but other values could be used. Here F(t) is the primary or "strong" effect, which predicts the emss overall behavior of the data. The function e(t) is the Gcondary or "weak" effect, which is a small correction that usuallv shows U D durine the latter Dart of the data. In fact, g(t)may be absent altogether ( i . e . ; ~=A5 ~ = 01, or it may be present but difficult to detect. The procedures described below furnish a method for finding both F(t) and g(t). In this report we use the statistical program M i t a b , hut the method is readily adapted for use with Lotus 1-2-3 or similar spreadsheet. The fitting is done by first ignoring the secondary effect and removing later portions of the data. A nonlinear Gauss-Newton procedure is used to estimate the parameters Al, A2, and A3 and their statistical significance. This procedure is done a number of times, removing increasing amounts of data from the end and keeping track of the estimated mean-square errors (s2)andthe Student t value (t3) of the estimate for As, which is the parameter most sensitive to changes in the data. As the data list becomes shorter, accuracies in the estimates ofs2andA3improve. In other words, s decreases and t3 increases, at first, but eventually become poorer. The best estimates ofAl,AZ,andA3 are obtained from the data range at which these accuracies reach their optimum values. The portion of the data most affected by the weak effect has been removed at this stage; estimates for A1-A3 obtained by this procedure should be "pure" F(t).
~... ~
Because Q = 0 at t = 0, the increase in temperature in the calorimeter due to reaction initiated at t = 0 (ATRo)is zero. The defition of heat capacity integrated between the limits Q = 0, AT, = 0, and Q = Q, ATn = ATn gives
where pv is the density (g mL-'1 of the solution whose volume is V (mL); C, is the specific heat capacity (J g-'"C-'1. Substitution of eq 9 into eq 8 gives, upon rearrangement,
AT^ = A T ~ ~-(e(-hnt)) I
(10)
where
which represents the maximum expected temperature increase C C ) due to the exothermic magnesium-water reaction. Through a similar development, the change in temperature due to thermal leakage and stirring work is given by where kTis a composite "rate constant" proportional to the thermal leakage modulus (21); ATTis the temperature increase or decrease due to nonreactive heating or cooling; and ATT ATE>AT=), and the system con-
Finding Errors and Deviations These values are then used to fmd the errors or deviations in the fit at all points in the data. These deviations will be very small in the early part of the data because the parameter estimates are based on minimizing the deviations fmm the observed data in that part of the temperature-time curve. The deviations in the later part of the Volume 70 Number 6 June 1993
457
Results of Trial Fittings for Strong Effect tss/(s)
2900
2000
1200
1000
800
600
500
AiICC)
23.541
23.518
23.500
23.492
23.484
23.479
23.477
tl
3160
4490
6140
7510
10540
12060
1 1 090
A21Cc)
1.273
1.268
1.257
1.247
1.232
1.216
1.207
178
256
353
416
517
409
267
2.685
2.903
3.121
3.248
3.403
3.536
3S92
h
88.5
114
132
142
165
140
103
S
0.0222
0.0148
0.0079
0.0053
0.0043
t2
1,000x n3'W1)
0.0100
data may or may not be much larger than in the early part. If the deviations in the later part of the data are much larger than in the early part, we conclude that the weak effect is present. To learn as much about it as we can, we fit these deviations by the model of eq 17.
This is essentially the same model as eq 15 for F(t) but with different parameters and possibly a different scaling factor v . This procedure may be demonstrated by application to the data of Figure 3, which consists of 186 measurements of temperature ('C) as a function of time (s).The first step in the procedure, the determination by a Minitab version of the nonlinear Gauss-Newton Method, led to the results shown in the table. Each column presents the estimates for a calculation using data in the range 0 5 t 5 tss in seconds. The first column, for tss = 2900 s, uses all the data, and other columns show results for various subsets of the data, down to that with tss = 500 s. The choice of tss is seen to be slightly ambiguous: The largest absolute values of ta and tz occur when tss = 800 s, while the least value of a, the estimated standard-error, is at tss = 600 s. After some consideration, tss = 800 s was selected, giving the following as the main model parameters. To =Al = 23.484CC) ATRM=A2 = 1.2317CC)
The deviations from this model are plotted against
500
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2500
Time (s) Figure 4.Errors for Figure 3 data after fittingof strong effect. 458
Journal of Chemical Education
Calculating the Secondary Effect
The secondary effect is calmlated by fitting the deviations d t ) to the model (eq 18)in theregion 800 < t 2 2900 s. Using the same Gauss-Newton procedure in the estimation of the primaw effect. the followine estimat& were bbtained.
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It is clear that this secondary model is highly significant in a statistical sense (all absolute t values greatly exceed 2) and fits the data well. The deviation of this fit is displayed in Figure 5; its standard error is appmximately one-fourth that of the primary-effect fit in the range 0 2 t 2 800 s. Expressing these results in terms of eqs 14-17 leads to T(t)=F(t)= 23.484+ 1.2317(1 -e4'0"M3S') CC) fort < 800 s
T(t)=F(t)-0.07963+ 0.31668(1- e-0.wau4m')
CC) for t 5 800 s
The absolute overall error in this model is less than 0.02 'C. There is a discontinuity in the model a t t = 800 s. However, its magnitude, appmximately 0.013 'C,is comparable to the errors of the fitting in the range 0 S t 5 800 s, so attempts to remove it are not useful. Discussion
t11000 for all t values (Fig. 4). As expected, the deviations
0
0.0044
are small (less i n magnitude than 0.02) and somewhat random for 0 5 t I 8 0 0 s, but clearly become much larger fort > 800 s. This result im~liesstronrrlv that a weak or secondary effecttakes place, starting at about t = 800 s.
The requirement for a n exothermic chemical reaction with temperatur+time curves suited for dynamics studies can be met by implementingeither of two reactions:
3000
Time (s) Figure 5. Errors in the range tss < t for Figure 3 data after fitting of weak effect.
icdide-catalyzed oxidation of thiosulfate to tetrathionate by hydrogen peroxide iron-promoted, chloride-catalyzed reduction of water to dihydrogen by magnesium
where
These two reactions have several desirable characteristics:
The exponential is then approximated quadratically,
simole rate laws o v e k l l reaction rate constants that can be adjusted to provide convenient reaction times simple, reproducible stoichiometlies Running these reactions in a calorimeter, and monitoring the increase in temperature as a function of time, can be an excellent index of chemical reaction progress. The basic mathematical ~roblemin this studv was to find a useful way to fit experimental data to &e model given by ea 13. Numerous a t t e m ~ tto s carrv out this fittine k t h existing nonlinear software' proved m k l y unxuccessful. It appeared that the trouble arose from the different levels ofaccuracy that were required to fit the data a t early and later times. Perhaps a weighted, nonlinear leastsquares approach would have worked, but the exponentialpeeling method seemed to fit the oeculiarities of the situation more naturally. As mentioned above, the exponential-peeling method has usually been invoked a s a way to find simple, first approximations for use in a more elaborate. com~letelvnonlinear curve-fitting. Its present form appdars &go siightly beyond the customary procedure in that nonlinear fitting is done at each stage, and explicit use is made of the statistical information (Student t values) furnished by stronp-effeet calculations to choose a near-optimal value for tss. The success or failure of the weak-effect computation is also judged by the Student t values. While the method worked fairly well in the example shown above, it is still subject to all the uncertainties that commonly afflict nonFailure can he caused by excessive linear curve-fitting (10). noise, poor initial estimates of parameters, or an inauspicious model having variables that are toomrrelated. In addition, of course. it will not do well if the weak effect t w strongly overlaps the main effect. The nonlinear Gauss-Newton Method that was used here within the framework of Minitab consists of three small programs. The f r s t estimates the sum of squared errors for the initial parameter values. The second, which can he repeated any desired number of t i e s , produces a new set of parameter values from the preceding one. This program is iterated until the parameter values no longer change. Then the third program estimates the final sum of squared errors, the variance-covariance matrix. the standard errors in each parameter, and the corresponding StuDromams are available from dent t values. List~ngs - of the . either author.
.,
Linear Regression
In some cases, the fitting of the weak effect can be accomplished by linear regression, as follows. First, make a scale change in the independent variable hy defining 'T=-
t - t
