THE TEMPERATURE PATTERN METHOD IN THE STUDY OF FAST CHEMICAL REACTIONS BY DAVID GARVIN," VINCENT P. GUINN,** AND G . B. KISTIAKOWSKY Gibbs Memorial Laboratory, Department of Chemistry, Harvard University, Cambridge, Massachusetts, U.S.A. Received 18th January, 1954 Polanyi's technique of spherically symmetric " highly dilute flames " has been modified for the study of those fast gaseous bimolecular reactions which are not accompanied by useful optical phenomena. The new method involves the measurement of the temperature pattern within and without the reaction zone under conditions of very small thermal gradients and preponderantly diffusional mass and heat transport. An idealized theory of such measurements is developed and it is shown that the chosen experimental arrange- ment satisfies reasonably well the theoretical requirements.Some of the results obtained with reactions between boron trifluoride and a series of amines are discussed and the limitations of the method are emphasized. Nonetheless, it should be of a rather wide usefulness in the studies of very fast gaseous reactions. The development af the technique of " highly dilute flames " by Polanyi and his coworkers 1 was a major step forward in the field of chemical kinetics. It made possible a direct measurement of the rates of very fast gaseous reactions in terms of the frequency of molecular collisions. The method was applied success- fully to a variety of reactions of metal vapours with halogens and halogen com- pounds. Except for a few investigations in which the extent of the reaction zone was followed by the measurements of the spread of the solid deposits of the reaction products on the walls of the reaction vessel, the measuring techniques revolved around the resonance absorption and fluorescence of metallic vapours.Either gives, in effect, the total volume of the reaction zone of two inter-diffusing reactants. It seemed to us desirable to extend the principle of the method to chemical systems in which the reaction zone cannot be determined by optical density measure- ments and the condensation on the walls of the reaction products with an accom- modation coefficient unity is at least in question. Such extension is indeed possible with the aid of temperature pattern measurements within the reaction zone. In the following we shall present the idealized theory of the measurements, its prac- tical adaptation, a few of the results obtained and their critique.Consider a constant finite source of reactant Y in an infinite atmosphere of a uniformly distributed reactant Z . Let the two concentrations be denoted by y, z, respectively. If Y and Z undergo a bimolecular reaction with a rate given by kyz and the diffusion coefficient of Y in the atmosphere is given by D,, a steady state may be reached which is described by the Poisson equation, (1 1 .kz v 2 y - - - y = 0, D Y Making the assumption that the diffusion coefficient and the rate constant are invariant throughout the system, the equation yields for a spherically symmetric case with r being the distance from the source, 1 k c2 = - y = - (A1 exp(cr) + A2 exp(- cr} ; Dy '* * Department of Chemistry, Princeton University, Princeton, New Jersey.** Shell Development Company, Emeryville, California. 32DAVID GARVIN, VINCENT P . GUINN A N D G . B . KISTIAKOWSKY 33 Introducing the boundary conditions that y = 0 at r = co and that the flow from the source is b = 4nkz yr2dr, J: the solution is reduced to b exp(- cr) 4nDY r ’ y -= (3) (4) This is the form used by Polanyi, except that the steepness of the exponential permitted him to neglect the l / r term in estimating the value of c from the approxi- mate radius of the reaction zone. When the reaction is accompanied by a heat change Q, the thermal steady state is given by the equation, Qkz b exp(- cr) = 0, 02T-t ___ ___ __- K 4nDy r where K is the thermal conduction coefficient.Taking it also to be independent of r, and assuming that the source reactant is entering the system at the tempera- ture of the atmosphere, the solution is A3 Qb exp(- cr) T = - - - 4- A49 r 47r~ r One boundary condition is obtained from Laplace’s equation for heat conduc- tion at such large distances from the source that reaction is insignificant, Qb 4r2Kn V T + - - - z o o . (7) The other results when the temperature at a variable distance r is compared with the temperature at some fixed distance r l . Then The temperature difference AT is seen to be determined by a proportionality factor Qbl47rtc and a form factor c, which latter does not involve the thermal conduction coefficient. A particularly effective way to apply this equation to the experimental data, for which we are indebted to Mr.Paul C . Mangclsdorf, Jr., is to plot the quantity r n T in arbitrary units against r, as is shown in fig. 1 . The region outside the reaction zone is then represented by a rising straight line; the vertical distance between any point on the descending branch of the curve within the reaction zone and the extension of the straight line is Kexp(- cr). By taking such distances at two or more points, the proportionality factor K is eliminated and the ratio between the rate constant and the diffusion coefficient is obtained. The assumption that D,, k and K are independent of r, made in integrating eqn. (1) and (9, imposes an experimental requirement that the reactants be diluted with a large excess of an inert carrier gas, to maintain a nearly constant mean free path and to keep The much more dubious assumption that z is constant throughout the reaction zone, which we shall consider at the end of this article in more detail, certainly presupposes that the total quantity of 2 in the reaction system be much larger than that of Y .The bimolecular reactions to be studied must be fast, since otherwise the experiments would have to be carried out at such high pressures that convection could not be avoided. Tsmall. B34 TEMPERATURE PATTERN METHOD EXPERIMENTAL It is a Pyrex cylinder of 10 cm int. diani. and ca. 40 cm length, The " atmosphere ", a mixture of reactant 2 and a carrier gas at 0.05 to I mm pressure enters at B, flows through the reactor at linear velocities of the order of I to 10 cm/sec and, together with reaction pro- ducts, is eliminated a t D through a regulating gate valve, a cold trap, and a high-capacity mechanical oil pump.The " source " reactant Y, with added carrier gas, enters the reactor through the nozzle A at linear velocities of the order 103-104 cm/sec. The ratio of the flow rates of the two reactants in each experiment is such that a substantial excess of the atmospheric reactant is flowing through the reactor. The pressure in the reactor is measured by a McLeod gauge connected through C. The temperature pattern in the diffusion region surrounding the nozzle is measured by a movable thermocouple T, whose leads T' are brought out of the reactor through a thin-walled 3/32-in. stainless steel tubing joined to a stouter brass tubing F which can be slid and rotated in a double, packed, bushing G, the inner space E of which is evacuated by the pump.Above the cover plate which mounts the bushing are located suitable gears and scales to read the vertical and horizontal position of the tip of the thermocouple. The measurements are carried out in a reactor shown schematically in fig. 2. rnT I , cm FIG. 1 .-The theoretical radial temperature pattern and a plot of experimentally determined rAT (in cm microvolts) against distance (in cm) from the nozzle. A FIG. 2.-A schem- atic drawing of the apparatus. These scales are calibrated with the aid of a cathetometer in reference to the nozzle tip each time a new thermocouple is mounted. The " hot " junction of the thermocouple is a 0-5 x 0.5 x 0.1 mm silver foil to which are soldered in a V the 0.05 mm iron and constantan wires.The junctions to the copper lead-in wires are at the base of the stainless steel tubing. The thermocouple is held rigid relative to the tubing by a thin glass fibre cemented to the wire some distance from the tip. The reactor is immersed to near the cover plate in a liquid thermostat ; to insure a steady temperature at the " cold " thermocouple junction, its supporting tube is passed through an auxiliary bushing H cemented to the wall of the reactor. It is to be noted that in order to apply eqn. (8) the distance from the cold junction to the nozzle need not be known, provided the temperature of the former either follows the local gas temperature or (as in the present arrangement) stays constant.In the earlier experiments,% 3 the temperature differences were measured to 0.1 pV, i.e. 0402", by a Leeds and Northrup Wenner-type potentiometer and a mirror galvano- meter. In the more recent still unpublished work of Mr. Richard Williams, use was made of a Perkin-Elmer DC amplifier, gaining thereby nearly an order of magnitude in the sensitivity of the thermal measurements.DAVID GARVIN, VINCENT P . GUINN AND G . B . KISTIAKOWSKY 35 To date, all experiments with this system have been concerned with the reaction of boron trifluoride and an assortment of amines. These reactions were shown by othcrs 4 * 5 ~ 6 to result in the formation, through a co-ordinate bond, of bimolecular adducts, F3B-NR3 with very little, if any, side reactions.Their rates were found by us to be fast cnough for the application of the described technique. The reactants, mixed with suitable pro- portions of the carrier gas, are admitted to the reactor through fine capillaries from large supply flasks, the pressure in which does not vary significantly during each run. The flow through the capillaries is calibrated by the various gas mixtures used and is adjusted to the desired values by varying the pressure in the supply flasks. particular experiment the ratio v/Dy was 16 cm-1. Lower nozzle velocities (down to v/Dy == 5 cm-1) bring the centre of the radial gradient pattern closer to the nozzle tip but leave it spherically symmetric to the accuracy of these measurements. At still lower -' 336 TEMPERATURE PATTERN METHOD Assuming a spherically symmetric mass flow from the nozzle at an essentially constant pressure, the time required by a gas volume element to move a distance r from the centre of the nozzle tip is t == 4r3/3vr$, (9) where v is the linear gas velocity through the nozzle tip and ro is its radius.With the larger nozzle used (ro = 0.046 cm) and the data of a typical run (v = lo4 cm/sec), the time required for a displacement to a distance of 1 cm from the centre is then 0.05 sec. On the other hand, the Einstein diffusion equation gives 0.0005 sec as the time for an average displacement of 1 cm due to molecular diffusion in the atmosphere (P = 0.1 mm) surrounding the nozzle. The ratio of these calculated times decreases as one chooses smaller distances from the nozzle but it is evident that only within a few nozzle radii is the hydrodynamic flow fast enough to interfere materially with the molecular diffusion.Indeed, the replacement of the above nozzle by one with ro = 0.028 cm produced no measurable changes in the tem- perature pattern, although the hydrodynamic flow was reduced by a factor of three FIG. 4.-A plot of the observed rate constants in the system BF3 + NMe3 against the ratio of the linear nozzle gas velocity to the diffusion coefficient. because less carrier gas was used in the nozzle gas mixture. Perhaps a more sensi- tive test of the effects of hydrodynamic flow is an analysis of a large number of runs, to reduce the effects of accidental errors of rneasuremcnt. In fig. 4 are plotted rate constants for the chemical system BF3 + Me3N against v/D, char- acterizing each experiment. One does observe a slight downward trend in the k’s with increasing v/D,, as expected since the hydrodynamic flow tends to keep the atmospheric reactant Z from penetrating close to the nozzle tip.The effect, however, is slight, being just noticeable over the random scatter of experimental points and an extiapolation to vJDy = 0 would not change significantly any of the conclusions reached. Within the v/Dy range of 5 to 16 cm-1 the temperature patterns were found to fit eqn. (8) very well with several reactant pairs used interchangeably as the nozzle and the atmospheric reactants. Thus the circles in fig. 1 are taken from a run with BF3 as the nozzle reactant and Me2HN in the atmosphere, which was carried out by Mr.R. Williams. In some measure this exccllent fit, typical of a great majority of runs madc under properly chosen conditions, is surprising because a rough calculation of the heat transfer by the gas to the receiver plate of the hot junction of the thermo-D A V I D GARVIN, VINCENT P . G U I N N A N D G . B . KISTIAKOSWKY 37 couple and of heat conduction along its wires does indicate that the hot junction, notwithstanding its substantial area, cannot have the temperature of the adjoining gas layers to the accuracy of these measurements. A concern with this situation has led Mr. Williams in his current work to use thermocouple wires of only 0.017 mrn diameter and to arrange the thermocouple so that the wires are tangential to the isothermal surfaces, rather than nearly normal to them, as was the case for the V-shaped couples.The absolute temperature differences measured with the new couple may be slightly different from those previously obtained. The absolute values, however, are not involved in the calculation of the rate constants and the form of the temperature pattern has remained the same. One must conclude therefore that although significant heat losses may occur along the wires of the couple, their effect on the thermocouple junction is linear in AT and is therefore without influence on the interpretation of the data. The theory assumes no disturbance of the temperature pattern by the measuring instrument and this may not be true if the thermocouple acts as a catalyst for the reaction.In the course of runs with each thermocouple, it became lightly coated with non-volatile reaction products but this had no effect on the thermal measure- ments, In one series of runs the couple was coated with perfluorinated hydro- carbon, again without noticeable effect on the results. One is thus justified in concluding that surface reaction is insignificant which is anyway highly probable in view of the nature of the reactions studied. A condensation of the reaction products on the couple might drastically alter the predicted thermal gradients. With one reaction mixture tried, boron trifluoride and ammonia, the couple became rapidly coated with reaction products and the observed temperature patterns could not be interpreted by eqn.(8). With all the other reaction mixtures thc bimolecular adducts apparently did not condense on the couple because the slight deposits formed (in amounts smaller by orders of magnitude than in the ammonia reaction) were entirely non-volatile, in contrast to the known vapour pressures of the bimolecular adducts.4-6 An assumption is implicit in the theory of these experiments that the reaction energy, which is contained initially in thc adduct molecules, is converted so rapidly into the Boltzmann distribution of all the adjacent molecules that the only observ- able transport of energy is by ordinary thermal conduction. But it is well known 8 * 9 that the energy exchange between vibrational and translational degrees of freedom is quite slow, in some instances requiring as many as 105 collisions for the transfer of a vibrational energy quantum.In contrast to this number, if it is tentatively assumed that the thermocouple averages the temperature over a region of 1 mm3, the product molecules in a typical experiment suffer only a few hundred collisions before migrating out of such a volume. An attempt was made therefore to change the unknown rate of energy equilibration by substituting helium for nitrogen as the carrier gas. The rate constants were found to be unchanged 3 but this re- assuring finding might still not be due to a very rapid rate of energy equilibration. Let us make the extreme assumption that the product molecules do not equilibrate their energy with other gas molecules at all but do transfer it to the surface of the thermocouple.The product molecules are subject to diffusion which is described by an equation equivalent to that of heat conduction (6), except for the propor- tionality factor. The local rate of production of product molecules is proportional to the rate of local heat generation used in eqn. (6). Thus, on the basis of this extreme assumption, one should still obtain an equation for the apparent tempera- ture pattern which is equivalent to eqn. (8) except for the proportionality factor. Undoubtedly the reality lies somewhere between this extreme assumption and that which forms the basis of eqn. (6), because the experimental results do show that the product molecules lose some of their energy on collisions with other gas mole- cules or inevitably dissociate.A mathematical treatment of this intermediate proposition appears to be rather complex and one can only hope that its result is analogous to those following from the two extreme assumptions.38 TEMPERATURE PATTERN METHOD The experimental measurements lead to the quantity c = (kz/Dy)* and thus give in principle the ratio between the rate constant and the collision frequency. Unfortunately the numerical relation between the collision frequency and the diffusion coefficient 10 is not accurately known for multicomponent gaseous mixtures of different molecular weights and sizes. This uncertainty, however, is slight compared to other sources of error in the present method and therefore the simple kinetic theory equation 3 2 4 (RT)3/2 My + Ma 6 Dy = 4 (> NP(0y + MyMa ) ' was used for the computation of Dy.The symbols have their usual kinetic meaning, the subscript y referring to the nozzle reactant and subscript a to the atmosphere. The molecular cross-section and weight of the latter were calculated as mole fraction averages of the corresponding properties of the atmospheric reactant Z and the carrier gas. The results of the measurements of the reactions of boron trifluoride with mono- methyl, dimethyl and trimethyl amines have been already described.3 The scatter of the rate constants observed was rather large-nearly a factor of two in repeated experiments-which is not surprising considering the indirect relation between the thermal patterns and the rate constants. In essence the data show that the reactions proceed through a two-step mechanism, kr k- I Y + Z z Y Z * ; Y Z * 4- MTYZ + M, which leads to the observed pressure dependence of the rate constants, The following table gives the numerical data obtained.TABLE 1 .-REACTIONS OF BF3 WITH THREE AMINES amine NHzCH3 NH(CHd2 N(CH3h kl 8 x 1011 3 x 1013 4 x 1012 cm3/mole sec k2ik- 1 2 x 10s I x 107 1 x 109 cm3/mole Thus the rate constants kl for the formation of the energy-rich intermediates approach the collision number (ZO, ca. 1014) between the reactant molecules. This suggests that the activation energies involved are insignificant, an interesting con- clusion in view of the evidence 6 for strains which exist in the adduct molecules and the change of the BF3 configuration from a coplanar into a tetrahedral one. There appears to be no regular progression of the parameters of eqn.(11) in the series of the three amines, dimethyl amine having the highest extrapolated rate at high pressures but the shortest lifetime of the intermediate YZ*, as shown by the smallness of the ratio k21k-1. The large uncertainty of individual rate constants and the small number of runs made with dimethyl amine are apparently responsible for this conclusion. More recent experiments of Mr. Williams, shortly to be published, place dimethyl amine closer to the other two by modifying its parameters in eqn. (11). In line with this observation Mr. Williams finds that the rate con- stants of arnines more complex than trimethyl amine have no measurable pressure dependence. Evidently the lifetimes of these complex energy-rich intermediates are too long to be detected in the pressure range accessible by the present technique.These results are so consistent with the general theories of chemical kinetics that one is tempted to discontinue further critical discussion of the experimental method, but there does exist onc assumption in its idealized mathematical theory which may be responsible for a substantial inaccuracy of the absolute values of the rate constants reported. It is the assumption that the atmospheric reactantDAVID GARVIN, VINCENT P . GUINN AND G . B . KISTIAKOWSKY 39 Z is uniformly distributed right to the centre of the nozzle orifice. As noted above, the mass flow from the nozzle would by itself keep the atmospheric reactant only out of the immediate vicinity of the nozzle, a volume too small to affect substantially the calculated rate constants. But another source of concentration gradients of the atmospheric reactant-its depletion due to reaction near the nozzle-may be far more significant.The exact theory calls for the integration of partial differential equations (1) for both reactants, followed by the integration of the resultant heat flow equation. It has not been possible to accomplish this analytically. Recently Cvetanovic and LeRoy 11 have carried out an approximate integration of the diffusion equations, applicable to the boundary of the reaction zone and have shown that significant corrections must be applied in the calculation of rate constants from the radii of highly dilute flames because of the depletion of the atmospheric reactant.It may be anticipated that the corrections applicable to the present method are larger because it relies on the distribution of reactants within the reaction zone. The available experimental data tend also to demonstrate the existence of this effect. An inspection of the results obtained with boron trifluoride and trimethyl amine 3 shows that when the smaller-and hence faster diffusing-BF3 was chosen as the atmospheric reactant, the calculated rate constants were slightly higher than with the reversed arrangement. In Mr. Williams’ experi- ments, with still larger amine molecules, the ratio of rate constants observed when the reactants were interchanged rose to almost a factor of two. Some of this discrepancy may be due to the incorrect calculations of the diffusion coefficients,lo owing to uncertainties in molecular cross-sections, etc., but the reality of the dis- crepancy can hardly be doubted.While the difficulties of an analytical integration of the system of equations involved appear rather formidable, work on this problem is now in progress in this Laboratory and it is expected that the solution will be available in the near future. With the availability of this solution, the method of temperature patterns should become one of rather wide applicability and of quite acceptable accuracy, compared with other methods applicable to very fast reactions. One of its attractive features is that the concentration of the nozzle reagent need not be known, provided it is low enough not to exhaust the atmospheric reagent. Thus the method should be rather suitable for reactions of atoms and free radicals introduced through the nozzle, provided, of course, that the overall reaction mechanism does not involve several reaction steps with comparable rate constants. 1 Polanyi, Atomic Reactions (Williams and Norgate, London, 1932). 2 Vincent P. Guinn, Thesis (Harvard University, 1949). 3 Garvin and Kistiakowsky, J. Chem. Physics, 1952, 20, 105. 4 Laubengayer et al., J. Amer. Chem. SOC., 1948,70,2274; 1945,67, 164; 1943, 65, 5 Burg and Green, J. Amer. Chem. SOC., 1943, 65, 1838. 6 Brown et a!., J. Amer. Chem. SOC., 1948, 70, 2793, 2878 ; 1947, 69, 1137 ; 1945, 7 Heller, Trans. Faraday SOC., 1937,33, 1556. 8 Kneser, J. Acous. SOC. Amer., 1939, 5, 122. 9 Kantrovitz, J. Chern. Physics, 1946, 14, 150. 10 Kennard, Kinetic Theory of Gases (McGraw-Hill, New York, 1938). 11 Cvetanovic and LeRoy, Can. J. Chem., 1951,29, 597. 884. 67, 1765, etc.