James E. Finholt
Carleton College Northfield, Minnesota 55057
The Temperature-Jump Method for the Study of Fast Reactions
Before the last 10 or 15 years chemical reactions could be divided into two groups, those whose rates could be measured and those whose rates were too fast to be measured. Too fast meant any reaction with a half-life of less than low3sec. The major problem in attempting to study fast reactions was that it was not possible to mix two reagents in less than sec. The mixing problem has not been solved, but several end run techniques have been developed to avoid the need for mixing. I n one approach an analysis of the shape of nmr or esr absorption bands is used to get kinetic information. From the Heisenberg uncertainty principle the uncertainty in the energy of a state, AE, is related to the average lifetime of a state, At, by the following equation: AEAt cz h
(1)
where his Planck's constant. From this basic relationship it is possible to derive expressions relating rate law parameters to the shape of the observed absorption bands. A second general approach is to establish a competitive system resulting in the establishment of a steady state system. A third general approach is to rapidly perturb an equilibrium system and follow the return to equilibrium by some fast analytical method. Procedures utilizing this third technique are called relaxation methods. Reactions with half-lives as short as 10-lo sec have been studied using these new methods. A reaction with a half-life of 10-'0 sec is the fastest possible reaction in solution because at this rate the reaction is controlled by the rate a t which the reacting species can diffuse together through the solvent. Thus, the entire range of solution reactions is now open to kinetic investigation. The purpose of this paper is to provide an introduction to the study of fast reactions. Rather than attempting to discuss all of the methods being used in this field, attention will be focused on one representative technique, the temperature-jump method. This method is a relaxation technique and hence an understanding of the temperature-jump method will aid in understanding other relaxation methods. The temperature-jump method has been used to study a wide variety of chemical systems. Fortunately, it is also relatively simple and easy to understand. Experimental
As the name implies, the temperature-jump method uses a very rapid temperature change to perturb the system being studied from an equilibrium state. After the temperature change there is a net chemical reaction as the system moves to an equilibrium state at the new temperature. The resulting concentration changes are 394
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lournal of Chemical Education
observed as a function of time by measuring some property related to the concentration of one of the species in the reaction. Temperature changes of about 10' can be produced in lo-= sec by discharging a capacitor through the sample cell. This sounds like a simple task, but building the required apparatus is a formidable job. Concentration changes during the experiment are usually determined spectrophotometrically. Monochromatic light is transmitted from a monochromater through the sample cell and into a photocell detector. The wavelength is chosen to maximize the absorbance change during the experiment. After suie able amplification, the output of the detector is fed to the vertical plates of an oscilloscope. A simple linear time sweep is applied to the horizontal plates of the oscilloscope at some known sweep rate. The system is set so that the time sweep is triggered by the capacitor discharge. The net result is an oscilloscope trace in which a signal proportional to absorbance is plotted versus time. From such a trace, together with a knowledge of the initial concentration of all species, the parameters of a rate law can be extracted after performing some mathematical juggling. Mathematical Theory
Example 1
I n order to understand how rate law parameters can be determined from the raw data of a temperature-jump experiment it is useful to examine a simple reaction scheme such as the following:
Our goal is to relate kr and k, to the experimentally observable information. As a first step it should be noted that the oscilloscope deflection is proportional to the difference between the concentration of a species at a given time and the equilibrium concentration of that species. It turns out that the mathematical analysis is simpler if this fact is explicitly acknowledged by expressing the concentration in terms of an equilibrium concentration and a concentration difference. This is done in the following equations where [XI and [El represent equilibrium concentrations.
- [a [B] - [El
&[A] = [A]
(4)
A[B] =
(5)
A conservation equation can be written from consideration of eqn. (2). A[Al
+ A[B] = 0
(6)
[B]is obtained by con-
A relationship between [A] and sidering that a t equilibrium
dAIA1 dt - kdA[Al
+ l ~ l ) ( ~ I B+l @I)
-
k.(A[CI
Equation (8)is obtained by the substitution of eqns. (4) and (5) into eqn. (2).
+ [el)
A[A]A[B] G 0
Equation (8) is simplified by substituting from eqns. (6) and (7).
(20)
Equation (20) can be simplified by making substitutions from eqns. (17)-(19). Further simplification can be obtained by making the approximation (21)
This is a valid approximation because the experiment involves very small concentration changes. Equation (20) after simplification gives
Integration of eqn. (20) gives an expression relating the rate law parameters to the relaxation time.
Equation (10) is obtained by integrating eqn. (9).
It is convenient a t this point to define a quantity r , the relaxation time, to be the time required for A[A] to change from A[A]o to 1/eA[AIo. I n terms of r eqn. (10) becomes
The relaxation time can be obtained directly from the oscilloscope trace since it is the time required for the displacement to drop from its maximum value to l/e times its maximum value. Thus for this example the sum, k, k., can be obtained by using eqn. (11) and an experimental value of the relaxation time. If the equilibrium constant for the reaction can be obtained from a separate experiment, separate values for k, and k, can be calculated. It is interesting to note that it is not necessary to know the actual concentration change taking place during the experiment.
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Example 2
I n general, in order to determine rate law parameters a mechanism must be assumed. From the assumed mechanism a relationship between the rate law parameters and relaxation times must be derived, and this relationship tested to see if it fits the observed data. This process is complicated by the fact that the relationship between rate law parameters and relaxation t i e s can get very complicated for mechanisms only slightly more complex than the examples discussed here. Fortunately, some general methods for handling complex mechanisms have been developed, but it is beyond the scope of this paper to discuss them here.' A Typical Temperature-Jump Experiment
A recent study of the kinetics of the reaction Fe(H2O)a"
+ Fe(CN)2- = Fe(H20)sFe(CN)s+ HnO
(24)
provides an excellent example of the use of the temperature-jump method.= The mechanism of the reaction was postulated to be
The reaction system described in eqns. (12) and (13) is an example of a slightly more complicated situation.
- -d[A1 =
dt
kr[A][B]
- k,
[C]
(13)
The same procedure used to analyze the preceding example can be used in this case. The first step is to express the concentration of each species in terms of an equilibrium concentration and a concentration difference.
- 1x1 [B] - [B] [CI - [Cl
A[A] = [A]
(14)
A[B] =
(15)
A[Cl =
The species Fe(H20)6Fe(CN)B was assumed to be an ion pair in rapid equilibrium with the initial reactants with K = [Fe(HzO)6Fe(CN)61/[Fe(H~O)6w] [Fe(CN)6s-]. The problem was to design a series of temperature-jump experiments which would provide evidence to support the proposed mechanism and allow the rate constants in the mechanism to be evaluated. If the reaction in eqn. (25) is very much faster than the reaction in eqn. (26) the following expression can be d e r i ~ e d . ~(The derivation is presented in Appendix A.)
(16)
Next all applicable conservation equations are written A[A]
- A[B]
A[A]
+ A[C] = 0
(18)
kr[x] [B] = k,[e]
(19)
=
0
(17)
At equilibrium The rate law in terms of concentration differences and equilibrium concentrations is
'HAMMES,G. G., AND SCHIMMEL, P. R., J. Phy8. C h a . , 70, 2319 (1966). D. L.,AND SWINEAART, J. H.,I n ~ r g .Chem., 6 , SINGLETON, 1536 (1967). he changes in concentration caused by the temperature change are very small, so final equilibrium concentrations are about the same as the initial concentrations, i.e., for speci!s S, [S] c [S]. In eqn. (27) and all following equations the [S] notation will be abandoned. Volume 45, Number 6, June 1968
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395
To test the proposed mechanism a series of temperaturejump experiments were run at d i e r e n t ( [ F e ( H 2 0 ) P ] [Fe(CN)6a-]) values. A plot was then made of l / r versus ([Fe(H20)P] [Fe(CN)P]). According to eqn. (27) the resulting curve should be linear when K([Fe(H20)6a+]t [Fe(CN)6a-])> 1. The experimental results were in accord with these predictions. At 29S0, kl,[HzO] was evaluated from the intercept of the plot to be 15.0 e 1.0 sec-', and from the initial slope, k2,K was found to be 1750 a 2 5 0 M-'sec-'. No unusual experimental conditions were required in these experiments. The reaction was conveniently followed spectrophotometrically a t 500 mp since a t this wavelength the product is the only species which absorbs. Afost runs were carried out in 0.5 M HClOn with ([Fe(H20)6a+] [Fe(CN)e3-1) ranging from 0.001 to 0.2 M . The resulting T values varied from 0.2 to 0.02 sec. A photograph of a typical oscilloscope trace is presented in the report of this investigation.
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Limitations of the Temperature-Jump Method
Although the spectrophotometric analytical method is not the only analytical technique that can be used in a temperaturejump experiment, it is so convenient that it almost always is used. This fact raises the problem of studying a reaction in which none of the species absorbs in the visible or ultraviolet region. A common solution to this problem is illustrated in a study of the rate of deprotonation of some substituted malonic acids.' The reaction is
The reaction in eqn. (28) was coupled to a color producing reaction by adding an acid-base indicator to the reaction system and following the change in absorbance a t the wavelength at which the indicator absorbs. Cresol red or phenolphthalein was used as an indicator and the indicator reaction was
The rate of the indicator reaction is very much faster than the rate of reaction under investigation so, KI, = [HIn-] [OH-]/[InZ-] relates the quantities in eqn. (29). The derivation of the relationship between the relaxation time and the rate law parameters is very similar to that of the Fe(H20)63+ Fe(CN)63- reaction. The result is
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I n the case of the malonic acid study the indicator term, [HIn-]/(KI. -!- [OH-]), was small enough so that it could be neglected. Another limitation required by the temperaturejump method is that AH for $he reaction being studied must be large enough to produce a detectable change in concentration with the 2-10" temperature jump pro-
'
MILES,M. H., EYEING, E. M., EPSTEIN, W. w., and OSTUND, R. E,, J. Phys. Chem., 69, 467 (1965).
396 / Journal of Chemical Education
duced in the experiment. If this condition is not met, a second reaction which does have a large enough AH can sometimes be coupled with the reaction of interest in much the same way that an indicator reaction was used to provide a color change. Brief mention should be made of a number of additional l i t a t i o n s of the temperaturejump method. I n order to use the capacitor discharge heating method the reaction system must have an appreciable concentration of ionic species present to conduct the discharge current. For this reason most temperaturejump experiments have been conducted in aqueous solution with an inert salt a t 0.1 to 1.0 M concentration. Relaxation times of less than sec cannot he observed with the temperaturejump technique. It is often quite difficult to establish the relationship between the parameters of a proposed mechanism and the observed relaxation times. This problem is present in all fast reaction studies regardless of the method. And, as stated previously, it must be admitted that the construction of a temperature-jump apparatus is a difficult and expensive undertaking. Applications
Appendix B lists a few recent papers utilizing the temperaturejump technique. The list is by no means complete, it is presented only to indicate that the temperaturejump method is an important tool in the investigation of many areas of chemistry. Appendix A , The D e r i v d o n of Equation (27) Let A = Fe(HnOhJ+,B = Fe(CN)s'; AB = F ~ ( H I O ) ~ Fe(CN)e, and C = Fe(HzO)sFe(CN)a. The proposed mechanism for the reaction is
Equations (3146) relate concentrations to equilibrium wncentrations and concentration differences.
The conservation equations are AIAI - A[B] = 0
(7) (8) It is assumed that the reaction in eqn. ( I ) is much faster than the resetion in eqn. (2), therefore an equilibrium expression is d i d for reaction (1). AIA]
+ AIAB] + AIC] = 0
The rate law for the mechanism of eqns. ( I ) and (2) is
Sub&ution fr? eqns. (3) and (5) into eqn. (10) and notinglhrtt karIAB1 = ks, [C I [H20I gives
Suhstilution from eqns. (a), (4), and (5) into eqn. (9) gives
K(A[AI
+ I ~ ) ( A I B+I [BI)
=
(A[ABl
+ [ml
(12)
Amuming that all concentration changes are small A[A]A[B]
"
(13)
0
Rearranging eqn. (12) and substituting from eqns. (7), (a), (9), and (13) gives A[AB] = - A
A Selected Annotafed Bibliography
1
LC]
SWINEHART, J. H., Electron Transfer in the Flavin Mononucleotide System, J. Am. Chem. Soc., 88, 1056 (1966). A,, AND SH~AO, YAPEL, A,, BAN, M., LUMRY,R., ROSENBERG, D. F., Studies of the Chyrnotrypainogen Family. V. The Effect of Small Molecule Contaminants on the Kinetic BeJ. Am. Chem. Soc., 88,2573 (1966). havior of ~~Chymotlypsin,
1
(14)
CAWIN,E. F., "Fast Reactions in Solution," John Wiley & Sons, Inc., New York, 1964, chap. 4. An excellent introduction to all methods for studying fast reactions. For advanced Substituting from eqn. (14) into eqn. (11) gives undergraduates and beginning graduate students. G., AND EIGEN,M., 2. Elektrochem., 63, 652 (1959). CZERLINSKI, A detailed description of the temperature-jump apparatus. EIGEN,M., AND DEMAEYER,L., "Techniques of Organic Chem, L.,LEWIS,E. S., AND WBIGSBEROER, ist$' ( E d t k 8 : F ~ m s s8. A,) Interscience Publishers (division of John Wiley & Sons, Finally, upon integration the desired result is obtained. Ino.), New York, 1963, Vol. VIII, Part 11, chap. XVIII. An important review of relaxation methods including the temperature-jump. The development is probably too sophisticated for an introduction to the field. Appendix B, Some Recent Investigations using the EYRING,H., AND EYEING,E. M., "Modern Chemical f(inetic8," Reinhold Publishing Corp., New Yark, 1963, chap. 6. A brief Temperature-Jump Method introduction. CATKOU,R. E., AND HAM ME^, G. G., Relaxation Spectra of G. G.. AND FASELIA. P.. J. Am. Chem. Soc.. 84. 4644 HAMMES. Ribonuclease. 111. Further Investigation of the Interaction (1962). 1nel"des a description of a temperature-iumb a p of Rihonuolease and Cytidine 3'-Phosphate, J. Am. Chem. Soc., paratus in English. 87. 4674 (196.5). G. G., AND SCHIMMEL, P. R., J. Phy8. Chem., 70, 2319 HAMMES, (1966). A development of a general method far relating relaxation times to rate law parameters. KING,E. L., "How Chemical Reactions Occur," W. A. Benjamin, 87, l(1965). Ine., New York, 1964, chap. 9. An excellent example of H u n m z , P., AND KUSTIN,K., Kinetics of Fast Electron-Transhow the topic of fast reactions can be introduced to beginning fer Reactions, Ifiorg. Chem., 3, 823 (1964). KOWALAK, A,, KUSTIN,K., PASTERNACK, R. F., AND PETRUCCI, students. G. C., HAMORI, E., DAVENPORT, G., AND SCEERAGA, KRE~EECK, S., Steric Effects in Fast Metal Complex Substitution ReacH. A,, J. Am. Chem. Soc., 88, 246 (1966). Includes a. detions. 11,J. Am. Chem. Soc., 89, 3126 (1967). scription of a temperature-jump apparatus. G. C., HAMORI, E., DAVENPORT, G., AND SHERAGA, KRESHECK, SW!N~HARI., J. II., J. CHEM. Enrc., 44, 524 (1967). D~seribes H. A,, Determination of the Dissociation Rate of Dodecyln s t u d w ~ kiwticq rxp~ritnrnt illu+rmtinp. the relaxation pyridinium Iodide Micellesby a Temperaturdump Technique, J. Am. Chem. Soe., 88, 246 (1966). ntethud but not requiriw any special equipment. +
K(IK1
+ lB1)
Volume 45, Number 6, June 1968
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