I CORRESPONDENCE I
I
Direct Measurement of Activation Energies: An Alternative Formulation of the Kinetics Problem Sir: The usual method for obtaining activation energies for chemical reactions is to measure the rate constant at a number of temperatures followed by plotting the logarithm of the rate constant 12 vs. the reciprocal of the temperature, the activation energy being obtained from the slope of the least squares straight line (1). Over wide ranges of temperature, observation of curvature in the data is not uncommon, but it is usually not possible to determine whether the curvature is real or due only to a fortuitous combination of statistical errors, or possibly a long-term drift in a critical calibration since rate constants at different temperatures are usually measured hours, and often days, apart. Perhaps more importantly, all theoretical considerations predict that the Arrhenius plot should not be linear (1);i.e., the preexponential factor should be a function of temperature, k ( T ) = A(T)e-E*/RT
(1)
From the definition of the activation energy which is
43) where E* is the energy barrier for the reaction and the second term is the temperature dependence causing the curvature in the Arrhenius plot. Further curvature may be due to changes in the Boltzmann distribution of reactants among rotational, vibrational, and electronic states with temperature, and the reaction may proceed on more than one potential energy surface with different energy barriers. That is, the observed rate constant is actually a sum of microscopic rate constants, k(T) = Z k i ( T ) = ZAi(T)e-Ei*'RT i
(4)
i
so that the exact temperature dependence is quite complex and varies from reaction to reaction. Conventional methods of reaction rate measurements are neither precise nor accurate enough to determine the functional form of the temperature dependence for a particular rate constant. Conventional Measurement of the Reaction Rate Constant and Inference of the Activation Energy. Consider the reaction
A + B + Products
(6)
for which the rate constant is defined by
The solution of this differential equation is well known and is given by
(7) 1074
ANALYTICAL CHEMISTRY, VOL. 49, NO. 7, JUNE 1977
Whenever possible, kineticists simplify the measurement of a second-order reaction rate constant by making one species in large excess over the other so that pseudo-firsborderkinetics applies. If B is in large excess over A, then Equation 7 simplifies to
Thus, if the reaction is made pseudo-first order, only the species in large excess need be measured absolutely. A relative measurement of [A] at two reaction times is all that is additionally required to measure the reaction rate constant. In practice, relative measurements of [A] at a number of reaction times are made and the least squares slope of a plot of In S A vs. t is taken to be 4 [ B ] where S A is the detector signal proportional to [A]. This is usually done for a range of values of [B] to demonstrate that k is independent of [B]. Considerable experimental ingenuity has been applied to the measurement of fast gas phase reactions. Stopped flow, flash photolysis, and discharge flow are particularly useful techniques (2). These methods have been used in conjunction with a variety of detection methods. Of these, atomic and molecular fluorescence and absorption and mass spectrometry have proved to be the most generally useful (2). Figure 1is a schematic diagram of a conventional fast flow apparatus for the measurement of gas phase reaction rate constants. Of the three techniques mentioned, this one will be discussed in greater detail since, as we shall find in the following section, if suitably modified it should lend itself particularly well to the direct measurement of activation energies. In the fast flow technique, reactants A and B are diluted in an inert carrier gas. Reactant B directly enters the main body of the flow tube and reactant A enters through a sliding injector. The flow tube is usually made of glass, and typical dimensions are 2.5 cm in diameter by 1m in length. A vacuum pump maintains a total pressure in the flow tube of between 0.5 and 5.0 Torr so that flow is essentially plug (radially flat velocity profile) with a horizontal velocity of typically 5 to 50 m/s. Reaction begins when A and B mix at the inlet jet near the end of the injector; either A or B is made to be in large excess and the minor species is detected downstream. Absorption is indicated in the diagram but, as previously indicated, many methods of analysis may be used. The reactanta A and B may be atoms, free radicals, or stable molecules and either A or B may be used in large excess. In the case of atoms and free radicals, a microwave discharge is used to dissociate a diatomic gas, and these atoms may be converted to appropriate free radicals by selective reactions either within or prior to entering the flow apparatus. The reaction time may be varied by translating the sliding injector, and the reaction time is calculated from the formula
ffP1 t=FR T
(9)
B
1
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I I
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A
A-
f
d
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-
T
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To Pump
Light Source
0
I Figure 1.
-
Conventional fast flow apparatus for the measurement of reaction rate constants
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P " d
"
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/L
"
sy,
*_ _ _ _ _ _ _ _ _ _ I
0-
I I I Rotating
Mirror
I
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Fast flow apparatus for the direct measurement of activation energies B is split between two flow tubes held at two temperatures, where (Y is the cross-sectional area of the flow tube, P is the T , and Tb. Reactant gas A enters the two chambers by way pressure in the flow tube, I is the distance from the inlet jet of two sliding injectors. The gas streams converge at a point to the detector, F is the total flow rate of gases into the flow beyond the detection system which leads to a single pump. tube, R is the gas constant, and T is the temperature. To A null point detection system is used with a single light source measure a reaction rate constant, the detector response to A and a single detector. Again, pseudo-first-order kinetics is (if B is the species in large excess) is obtained for a number to be used with either A or B in large excess. To directly of positions of the sliding injector. Also, there must be some measure the activation energy, the detection system is first absolute method of knowing the concentration of B. brought to null electronically with no gases entering the flow Only two positions of the sliding injector are required in system. With the two injectors held at the same position (1, order to obtain the rate constant by Equation 8. From a = Ib) and with A flowing but not B, the detector is brought measurement of the rate constant at a second temperature, to null by adjustment of the needle valves on the two injectors the activation energy may be evaluated. Combination of (so that [AIO,,= [A],,b). Then with both A and B flowing, any Equations 2, 8, and 9 yields for E, two positions of the two injectors are found that will once again Ea = Ta Tb bring the detector to null. The activation energy is given by the measurement of two lengths, 1, and lb, and two tem(Tb - T a ) peratures, T, and T b , and is given by Figure 2.
Ta Tb
E, = (Tb
where the subscripts a and b represent the two different temperature measurements. Note the application of the logarithm function three times. If absorption is used for detection,
so that the logarithm function is applied seven times. Note that a systematic error in the calibration of P, F, 1, or [B] will cause the rate constant to be in error, but not the activation energy. Any drift in the calibration factor for these variables between measurements a t the two temperatures will result in an error in the measurement of E,. Furthermore, any statistical error in the measurement of T , P, F, I , [B], or In ([Al1/[AI2) does not cancel since these measurements are independent. Direct Measurement of the Activation Energy. Direct measurement of the activation energy implies directly measuring the temperature derivative of the reaction rate constant: hence two temperatures are required. Figure 2 is a schematic diagram of a "double-barreled" fast flow system designed for this purpose. In this apparatus, the reactant gas
-
Ta
Tb 21a
)
In (-)
Ta21b
when B is chosen to be in large excess. Length and temperature are two physical measurements that can be made to a very high degree of accuracy. In arriving at Equation 12, it must be recognized that the two flow tubes are at the same pressure but not the same temperature, so that according to the ideal gas law
Furthermore, since the pressure gradient in each flow tube is the same, the flow of B is split equally between the two tubes and, since the pumping speed is the same for both tubes, we also have
That is, the difference in concentration of B in the two tubes is due to a difference in residence time, Le.,
The difference in residence times (linear velocities) also applies to the minor species A. The higher concentration of B and longer reaction time in the flow tube at the lower temperature explains the appearance of the T b 2 / T," term in the logarithm function of Equation 12. ANALYTICAL CHEMISTRY, VOL. 49, NO. 7, JUNE 1977
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We see that by directly measuring the activation energy using optical absorbance as just described, the logarithm function is applied to the experimental data only once, not seven times. Also no absolute measurement of either [A] or [B] is required. No absolute flow rate or pressure measurements are required. Since A and B enter the two flow tubes from the same two sources, any drift or fluctuation in the flow rates will affect both tubes equally and effectively cancel. Similarly, any fluctuation in the pumping speed affects both tubes equally and also does not affect the detector null. Also, any fluctuation in the lamp intensity or detector response does not affect the null point. Clearly, the major sources of error in the measurement of activation energies can be eliminated by a direct measurement similar to the one just described. Other null point detection systems can be applied, observation at 90” being the only modification required for fluorescence. The application of mass spectrometry would be possible, but would require some ingenuity. By beginning at either the high or low end of a temperature range of interest and measuring the activation energy over many small temperature intervals, the temperatures of the two flow tubes being alternately changed (“leapfrogging”), it would be possible to obtain the activation energy as a function of temperature. One could then arrive a t an appropriate temperature function for expressing a given rate constant. The
apparatus just described can, of course, measure the rate constant as well, the additional information required being [B], F , and P, so that measurements of the rate constant a t one or more temperatures could be fit to this functional form. Such a procedure should considerably improve the accuracy and precision with which a reaction rate can be predicted a t any given temperature. Such improvements are necessary in order to accurately model important chemical problems in which the temperature is not a constant. Examples include the modeling of flames (7’ = 300-3000 K) and the modeling of the ozone chemistry of the stratosphere (2’ = 200-300 K). LITERATURE CITED (1) H. S. Johnston, “Gas Phase Reaction Rate Theory”, Ronald Press, New York, 1966. (2) R. T. Watson, “Rate Constants of Cl0,of Atmospheric Interest”, J. phys. Chem. Ref. Data Ser., in press (1976).
John W.Birks School of Chemical Sciences University of Illinois a t Urbana-Champaign Urbana, Illinois 61801 RECEIVED for review November 22,1976. Accepted February 25, 1977. Acknowledgement is make to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for support of this research.
Pulsed vs. Continuous Wave Atomic Fluorescence Spectrometry Sir: It is well known that the ideal experimental setup for atomic fluorescence spectroscopy has to combine the use of a high intensity source of excitation and a low background, highly efficient atomization reservoir. In recent years, several authors have attempted to assemble a fluorescence setup consisting of a pulsed excitation source and a gated detection system ( I , 2). Understandably, the aim of this research was twofold: (i) high source radiances could be achieved within a narrow time interval while still maintaining the discharge conditions at a practical level of operation, and (ii) background noise due to the atomizer and detector could be greatly diminished because of the gated operation of the detector. Experimental results (1-8) were obtained using both continuum as well as line sources, such as xenon arcs and hollow cathode lamps. Moreover, tunable dye lasers, operated with a low duty cycle, have also been investigated because of their unique properties of high peak power, wavelength tunability and narrow spectral bandwidth. Unfortunately results achieved with conventional sources operated in the pulsed mode are comparable to those obtained for similar sowces operated in the continuous wave (cw)mode, and so the predicted advantage o f signal-to-noise ratio has never been fully realized. Particularly disappointing are the results obtained for pulsed laser excitation where the experimental detection limits for flame atomizers are far less attractive than those predicted. The aim of this note is to emphasize some of the drawbacks inherent in pulsed fluorescence work, which have been overlooked from the beginning both because of the lack of experimental information on the source characteristics when operated in a pulsed regime and, in the case of laser excitation, because of the fundamental relationship between the rate of photon absorption from the source (and photon fluorescence rate) and its spectral irradiance. The experimental and theoretical background upon which the foregoing consider1076
ANALYTICAL CHEMISTRY, VOL. 49, NO. 7, JUNE 1977
Table I. Summary of Reasons Why Atomic Fluorescence Signal-to-Noise Ratio in Pulsed Luminescence Spectrometry Does Not Necessarily Increase with Peak Source Power Source type Reason Comment Pulsed hollow Self-reversal Experimental cathode lamps results conclusive Pulsed Probably self-reversal No experimental electrodeless results yet discharge lamps Pulsed xenon Shift in spectral Experimental lamp distribution; inefresults ficient conversion conclusive of input power to light output Tunable dye Saturationa Experimental laser approached in results luminescence conclusive (signal reaches plateau); scatter increases linearly with source power for resonance fluorescence; molecular fluorescence increases linearly with source power a Molecular fluorescence with dye lasers is not as prone to saturation and thus the signal level should increase nearly linearly with source flux even with lasers. ations are based is largely available in the fluorescence literature. Nevertheless, by collecting all essential points together, we hope to help the reader and/or the potential user of the pulsed fluorescence technique to gain a more realistic
