J. Phys. Chem. 1983, 87, 1000-1004
1800
Approximate Expressions for the Yields of Unlmolecular Reactions with Chemical and Photochemical Activation J. Troe Institut fur Physikalische Chemie, UniversiBt @ttingen, D 3400 Giirtingen, West Germany (Received September 2 1, 1982)
Approximate expressions for the quantum yields, the effective rate constants, and time-dependent product formation are derived for chemical and photochemical activation systems with competition between collisional and unimolecular processes. The results are compared with accurate numerical solutions of the appropriate master equations.
Introduction Unimolecular reactions with thermal activation at low pressures are governed by collisional energy transfer. The rate coefficient includes a total collison frequency for energy transfer, which is approximated by the Lennard-Jones collision frequency Zu, and a “collision efficiency” p,. It is possible to express 0,in a simple way by the average energy transferred per collision ( AE)(including all up and down transitions) via the approximate relationship’ -(M) Pc E (1) 1 - pc1/2 FEkT where FE is given by JE,”f(E) dE/f(E,)kT with the Boltzmann distribution f ( E )and the threshold energy E,,. (For large molecules at high temperatures, at FEk 4, eq 1 becomes invalid and p, approaches a nearly constant valuea2 Equation 1 is practically exact for exponential collision probabilities; for other cases minor deviations O C C U when ~ ~ ~ ~p, exceeds about 0.2. Equation 1shows the relation between average energy transfer and collision efficiencies in a very clear way for thermal unimolecular reactions. Obviously, the collision efficiencies required for unimolecular reactions with other types of activation will differ from eq 1. We shall try to derive in the present work similarly simple expressions for collision efficiencies yc in unimolecular systems with competition between unimolecular and collisional processes such as chemical and photochemical activation systems. It is our aim to elucidate for these systems the relation between the collision efficiencies ye and the properties of energy transfer, such as (AE),and of the specific rate constants k(E) of the unimolecular process. We derive our simple expressions from well-known stochastic models of the r e a ~ t i o n . ~A comparison with accurate numerical solutions of the appropriate master equations demonstrates the surprisingly good performance of the approximate results. Therefore, our expressions allow for a quick computation of collision efficiencies without explicit solution of a master equation. Definition of Yields and Collision Efficiencies Steady-state chemical and photochemical activation experiments are analyzed by dissociation/stabilization product ratios Y or quantum yields 4, i.e, by
Y =D/S 4 = D/(D + S)
(2) (3)
where D is the number of dissociated and S the number of stabilized reactant molecules per unit time. Assuming a monoenergetic initial preparation of reactive states at an initial energy Ea, and a “strong”, one-step, collisional stabilization with a collision frequency 2, eq 2 and 3 lead to Y = k(Eac)/Z[M] (4
4 = k(Eac)/{k(Eac) + Z[Mll (5) with the specific rate constants k(E) of the unimolecular reaction. In reality, there are no strong collisions such that suitable collision efficiencies yc have to introduced into eq 4 and 5. Taking into account the finite width of the initial distribution of energy states from the activation around the average energy (Eac),we define collision efficiencies by y = D / s = k((Eac))/(YJLJ[Ml) (6)
4 = D / ( D + S ) = k ( ( E a c ) ) / I k ( ( E a c ) ) + rJLJ[M]\ (7) These collision efficiencies yc in general will be complicated functions of energy transfer properties, of the specific rate constants, of the initial energy distribution, and of the bath gas concentration [MI. In our work, we try to express these dependences in a simple approximate way. In chemical activation experiments it is often useful to define an effective rate constant4 by eq 8, as an alternative to the ratio D I S . This quantity again is related to the collision efficiency yc by (8) k a = Y ~ L [JM I D / S = k ( (Ea,) 1 Simple Stepladder Model for Collision Efficiencies The simplest, “stochastic”, model for a collisional stabilization sequence by many steps estimates the stabilization yield S / ( D + S) by a product of probabilities for stabilization rather than dissociation. A t the energy Ei such a probability is given by (9) pi = ZLJ[Ml/(ZLJ[Ml + k(Ei)l The total probability for deactivation rather than dissociation in T steps then is given by4
(1)J. Troe, J. Chem. Phys., 66, 4745 (1977).
(2) R. G. Gilbert, K. Luther, and J. Troe, Ber. Bunsenges. Phys. Chem., 87, 169 (1983). (3) D. C. Tardy and B. S. Rabinovitch. J.Chem. Phvs.. 45.3720 (1966): 48,1282 (1968); &em. Reu., 77,369 (1977); M . Quack &d J.’Troe h “Gas Kinetics and Energy Transfer”, Val. 2, The Chemical Society, London, 1977. (4) G. H. Kohlmaier and B. S. Rabinovitch, J . Chem. Phys., 38, 1692, 1709 (1963); 39,490 (1963).
( 5 ) H. Hippler, J. Troe, and H. J. Wendelken, Chem. Phys. Lett., 84, 257 (1981); J. Chem. Phys., in press. (6) H. Hippler, J. Troe, V. Schubert, and H. J. Wendelken, Chem. Phys. Lett., 84, 253 (1981).
QQ22-3654/03l2Q07-1800$Q1.5Q/Q 0 1983 American Chemical Society
The Journal of Physical Chemistfy, Vol. 87,
Yields of Unimolecular Reactions
Equations 9 and 10 are f i t approximations to the solution of a master equation; for accurate solutions see below. We estimate the number of deactivation steps T in the following way. In the absence of dissociation, molecules initially at an energy E after one collision will have an energy distribution given by the collisional transition probability P(E’,E). The average energy of the molecules after this collision will be E + ( A E ) where
(AE)= J-(E’ 0
- E)P(E’,E) dE’
(11)
Contracting the energy distributions after each collision at the average energy in absencepf dissociation, the steps Ei of the deactivation sequence are located at (12) Ei = (Eac)+ (i - 1)(AE) Then, the minimum number of deactivation steps T required for stabilization follows from the condition ET+1 EO (13)
In general, Stern-Volmer plots 4-l as a function of [MI will not be linear. Similarly, the effective rate constant k, of chemical activation will not be constant but pressure dependent: We neglect these effects in the present treatment and only consider the high-pressure region of the stabilization yields. Here, with k(EJ )
Flgure 1. Specific rate constants k ( € ) for toluene dissociation: example for a log k(€) log (€ - €,) representation; see eq 18. 10
16
Figure 3. Stern-Volmer plots for system of Figure 1 and relatively strong collisions: circles = exact solutions of the master equation, lines = simplified results from eq 7 and 25. (a: exponential collision model, eq 23, with (Y = 1208 cm-’ corresponding to - ( A € ) = 1183 cm-’; b: Poisson collision model, eq 24, with - ( A € ) = 1472 cm-‘; see ref 7.)
/
/
2
/’
I
0
1
0
0
40
80
120
160 200 240 280 320 2 ~1 IM 1 / k
360
€0,)1
given small energy range around E at time t. The collisional transition probabilities obey detailed balancing exactly. We investigate two basically different models for P(E’,E), an exponential down model
a)
E-E’
for 0 IE’ IE (23)
and a Poisson type down model
P(E’,E) 0:
(ET)E’ -
e x p (E -- Ey ’) for 0 IE’ 5 E (24)
with energy-independent (Y parameters. For all details of our solution of the master equation the reader is referred to ref 7. We have performed calculations for CY values
1 1
1
1000 - I cm-’
Flgure 4. Dependence of the collision efficiency yc on - ( A € ) . (System of Figure 1, 0 = exact results for exponential collision model, 0 = exact results for Poisson collision model. - -: eq 25 with n = 1, label y c l ( l - yc); ---: eq 15, label yc; -: eq 25, label yJ(1 7):; X: detailed evaluation of eq 10.)
between 100 and 4000 cm-’, i.e., for weak and relatively strong colliders. We characterize the results by the respective ( A E ) values calculated with eq 11,23, and 24, and detailed balancing. We represent our results at first in the form of SternVolmer or yield curves as a function of a reduced bath gas concentration such as Z u [ M ] / k ( ( E , , ) ) .Figures 2 and 3 compare our accurate numerical results with the corresponding data from the approximation given in eq 19 with n = 2. In view of the simplicity of eq 19, the agreement is quite acceptable. Higher precision of yc would require exact solution of the master equation together with accu, Equation 19 reproduces rate knowledge of k ( E )and.2 in a transparent way the general dependence of the collision efficiency yc on the various parameters involved. E.g., the dependence of yc on ( A E ) is shown in Figure 4. The numerical results (linearized Stern-Volmer plots be(7) J. Troe, J. Chem. Phys., 77, 3485 (1982).
The Journal of Physical Chemistry, Vol. 87,No. 10, 1983
Yields of Unimolecular Reactions
1 .o
TABLE I: Collision Efficiencies rC for Linearized Stern-Volmer Plotsa
2 0.5 .
YJapproximate)
-(AE)/ cm-’
model
n= 1 n= 0.0188 0.0192 0.0170 0.0192 0.0190
(exact)
eq 10
-
\
eq 19 YC‘
1003
n= 2
40.7 expon 55.5 Poisson 0.0251 0.0231 0.0261 0.0255 0.0261 183 expon 0.0788 0.0740 0.0856 0.0794 0.0862 497 expon 0.202 0.190 0.222 0.190 0.234 1183 expon 0.398 0.391 0.446 0.358 0.557 1472 Poisson 0.477 0.465 0.512 0.410 0.693 2518 expon 0.621 0.661 0.664 0.543 1.186 5072 expon 0.800 0.901 0.812 0.705 2.389 a Equation 7 between @-l= 1 and 6-l = 5 ; exact: solution of the master equation 22.
tween 4-l = 1 and 5) for the two collision models of eq 23 and 24, and for weak and strong collisions, which were shown in Figures 2 and 3 are included in the figure. Whereas the agreement with the “derived” low yc part, eq 15, of the curve is excellent, for higher yc the exponent n of eq 17 has to be fitted from the calculations. We find that, independent of the collision model, the exponent 2 gives very good results. Therefore, the collision efficiencies of Figure 4 can be fitted well by the relation
c
I
8
0.2 .
1
E
e
0.1
’
1.0
0
2.0 k () t
/6,
Flgure 5. Timedependent quantum yields 4 ( t ) for system of Figure 1; exponentla1 Collision model with ( E )= -40.7 cm-’. (Full line: + ( t ) = 1 - exp(-k((E,,))t); exact calculations for ( 0 )4 - = 0.99, (0) 4= 0.79, (X) 4 - = 0.59, (A)4 - = 0.38, (V)4 - = 0.22, (8) 4- = 0.12.) 1.0
8
e 0.5 1.
-c
I
8 0.2
-
Figure 4 shows nicely the insensitivity of the yc ( A E ) relation with respect to the collisional energy transfer probability function P(E’,E). It is also interesting to compare eq 25 with the results from an explicit evaluation of the stochastic equation 10. The Stern-Volmer plots predicted by eq 10 are generally more curved than the results from the master equation. However, if yc is evaluated by linerization of the Stern-Volmer plot between 4-l = 1 and 5, the agreement between the results of eq 10, 25, and the accurate solution of the master equation is remarkable as shown in Figure 4 and in Table I. For stronger collisions the exponent n in eq 25 can slightly depend on the particular system. Hence, explicit evaluation of eq 10 for large -(a) can be used for a quick adjustment of n. For weak collisions, where the exponent n becomes unimportant, the approximate equation 25 gives an amazingly better agreement with the accurate solution of the master equation than the starting equation 10.
Time Dependence of Dissociation and Stabilization Chemical and photochemical activation experiments can be conducted under steady-state conditions with very slow rates of preparation of excited states. In this case the observables of primary interest are the reaction yields 4 or Y (or other combinations such as D / ( D + S) and k,) discussed so far. Modern pulse techniques create very large initial concentrations of excited states in very short times such that the time dependence of product formation or of the stabilization sequence can be monitored in real time (see, e.g., ref 5 and 6). For such applications, the results from our solutions of the master equation are compared in the following with simple approximate relations based on eq 19. We represent the rate of product formation by a timedependent “quantum yield” 4 ( t )given by the ratio of the product concentration and the initial reactant concentration at time t. To a first approximation we assume that the initial rate of product formation can be given by the specific rate constant k ( ( E a c ) )i.e., ; we neglect the weighted
-
e
0.1
0.05 0
1 .o
2 .o k l < E a c , ) t/+,
Flgure 6. Timedependent quantum yields 4 ( t )for system of Figure 1, exponential collision model with ( A € ) = -1 183 cm-’. (Full line: + ( t ) = 1 - exp(-k((E,,))t); exact calculations for (0)4 - = 0.92, (0) 4 - = 0.67, (X) 4 - = 0.47, (A)4 - = 0.30, (V)4 - = 0.17, (8) 4 - = 0.093.)
averaging over the product k ( E ) n(E,t=O). Furthermore, we assume that the competition between unimolecular dissociation and collisional stabilization results in a nearly exponential approach of the final quantum yield 4(t--..) discussed in the previous sections. At this level we therefore assume 4(t)to be of the form 4(t) 4(t+m)[l - exp(-k((Eac))t/4(t-m))I (26) The final value 4(t--..) in the previous sections we have expressed by eq 7 with the collision efficiency yc approximated by eq 25. The time-dependent 4 ( t ) we have again derived from our numerical solutions of the master equation and represented in the form suggested by eq 26, Le., by plotting the logarithm of [$(t--..) - $ ( t ) ] / @ ( t - m ) as a function of k((Ea,))t/q5(t--..). The results for an exponential collision model and weak (( AE) = -40.7 cm-’) and relatively strong (( AE) = -1183 cm-’) collisions are shown in Figures 5 and 6. Indeed, eq 26 provides a very good representation for the rate of product formation in the presence of stabilizing collisions. The reaction proceeds slightly faster than given by the full lines. This is due to the fact that the average specific rate constant for the thermal energy distribution at 300 K carried up to the high energies is slightly faster than given by k ( ( E a c ) ) .This disagreement could be easily removed by deriving a suitable average ( k ( E ) ) .Apart from this, for the whole range +(t--..)= 0.1 1,eq 26 provides excellent results at least for one l / e time of the reaction. Equation 26 indicates how collision-contaminated direct measurements of k ( E )
-
J. Phys. Chem. 1983, 87, 1804-1808
1804
should be evaluated: a logarithmic plot of [4(t--) 4(t)]/+(t+m) as a function of time leads to a slope given by ( k(E)) / C#I(t+-), or approximately k( ( E a c )/)$( t--..). With a suitable estimate of 4(t---), then h((Eac)) can be extracted. The usefulness of this approach for experiments like those in ref 6 is immediately evident. (Obviously, this approach is only applicable as long as the decay can be characterized by a single rate "constant" lz (E).)
lision efficiencies yc and energy transfer averages ( a E )has been made visible. The expressions derived appear to be most useful for a quick interpretation of experimental data. It should be emphasized that, whenever sufficiently detailed knowledge of the molecular parameters involved is available and when the measurements are of very high precision, a detailed master-equation analysis should follow the present analysis.
Conclusion In the present work, simple expressions for the yields of unimolecular reactions with nonthermal activations under time-dependent or time-independent observation have been derived. The relation between empirical col-
Acknowledgment. The author thanks R. G. Gilbert and K. Luther for their most valuable comments. Financial support of this work by the Deutsche Forschungsgemeinschaft (Sonderforschungsbereich "Photochemie mit Lasern") is gratefully acknowledged.
Reactions of Mg('S,), Mg(3P,), and Mg('P,) with Diatomic and Triatomic Oxidants W. H. Breckenrldge' and H. Umemoto Department of Chemistry, University of Utah, Salf Lake City, Utah 84 1 12 (Received: October 7, 1982)
Laser techniques have been utilized to study the reactions of ground-state Mg(3s3s1So)and excited Mg(3s3p3PJ) and Mg(3s3p'P1) atoms with the oxidants Oz, NO, NzO, COz, and SOz. Ground-state Mg('So) reacts only with N 2 0 at 520 K to produce MgO(XIB+),as expected thermochemically,but the reaction rate is very slow, consistent with theoretical predictions by Yarkony of an entrance-channel activation barrier. The reaction of Mg(lP1) with NzO produces excited MgO(B'Z+). In contrast, C 0 2 and SOz react with Mg('P,) to produce MgO(XIB+), with little or no production of MgO(B'Z+) or MgO(A'H). Reaction of Mg(3PJ)with O2produces MgO(XIZ+) but no MgO(A'H). The initial vibrational and rotational quantum state distribution of the MgO(X'Z+)resulting from the Mg(3PJ)/0zreaction, determined by a pump-probe laser technique, was very similar to that reported by Dagdigian from molecular beam experiments, and was much "colder" than predicted by statistical models.
Introduction The reactions of ground-state and electronically excited metal atoms with simple oxidant molecules in the gas phase have been studied by several Much of the original work was triggered by the possibility of electronic-transition chemical laser action, since many of the oxidation processes are sufficiently exothermic to produce several electronically excited states of the metal monoxide products. Research in this area has continued, for the most part because new experimental techniques for identification of state-resolved products have created the possibility of detailed characterization of processes with several possible exit channels. Theoretical advances in understanding the dynamics of reactions involving multiple potential energy surfaces have also contributed to the interest in these processes. We present here a study using laser techniques of the chemical reactions of ground-state Mg(3s3slS0)and excited Mg(3s3p 3PJ)and Mg(3s3p 'P1) with the simple oxidant molecules 02, NO, NzO, COz, and SOz. Experimental Section The experimental apparatus and laser system have been described previ~usly.~Briefly, magnesium vapor, pro~
~~
~~~~
(1) W. H. Breckenridge and H. Umemoto, Adu. Chem. Phys., 50, 325 (1982). (2) P. J. Dagdigian, J. Chem. Phys., 76, 5375 (1982), and references - therein. (3) J. W. Cox and P. J. Dagdigian, J . Phys. Chem., 86,3738 (1982), and references therein. 14) M. Menzinger, Adu. Chem. Phys.. 42, 1 (1980).
duced in a resistively heated alumina crucible, is entrained in a stream of helium gas. Oxidant gases are added to the magnesium/He vapor stream via a "shower-head'' stainless steel mixing device located just above the alumina crucible. The crucible and mixing device are located inside a stainless steel vessel with several ports for laser beam access and/or detection of fluorescence. Typical pressures for the experiments reported here were as follows: He, 17 torr; Mg vapor, 104-10-z torr; oxidant gases, 0.3-1.0 torr. The Mg(3s3p3PJ)and Mg(3s3p 'PJ states were excited in the gas stream by absorption of -6-11s pulses of dyelaser radiation. The third harmonic of a Molectron MY32 Nd:YAG laser was used to pump a Molectron DL-200 dye-laser head. For excitation of Mg('P1), the dye-laser output was frequency doubled by means of an angle-tuned ADP crystal to the 'So-'PI resonance transition at, 2852 A. For excitation of Mg(3Pl),the dye laser was tuned t o 4571 A, the highly forbidden 1S0-3P1transition. The temperature at the laser interaction region, measured with a thermocouple probe, was -425 K for the Mg('P,) studies. For the Mg(3PJ) studies, higher heater power was used to generate the maximum attainable concentration of Mg vapor, and the temperature for the Mg(3P,J)work was measured to be -520 K. Chemiluminescence from the reaction zone was dispersed with a Jobin-Yvon monochromator (resolution 60 A), and detected with an HTV R910 photomultiplier, the
-
-
-
( 5 ) W. H. Breckenridge and H. Umemoto, J . Chem. Phys., 77, 4464 (19821, and references therein.
0022-3654/83/2087-1804$01.50/00 1983 American Chemical Society
