Two-Body and Three-Body- Atomic ~ecombinationReactions Jose M. Alvariiio and E. Martinez Departamento de Quimica Fisica, Facultad de Ciencias, Universidad del Pais Vasco, Apartado 644, Biibao, Spain Recombination reactions of atoms are of great interest. Due to their simplicity they are amenable to theoretical treatments. Also, they produce as reaction products diatomic molecules, which are, without exception, the object of study of general physical chemistry or quantum chemistry courses. From a chemical kinetic point of view these reactions are universally invoked as elemental steps of complex gas phase reactions such a linear chain reactions (e.g., synthesis of hydrogen halides) or reactions in flames. Last, but not least,
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I
stellar soace. . . where other &.rent reactions involvine free electrons and ions are also relevant. For very recent reviews see the contributions of J. N. Barsley and H. van Regenmorter in (1). These are exoergicl reactions in an amount equal to the dissociation energy of the formed molecule. In the vast maioritv of whvsical chemistrv or even chemical kinetics books. ~~
presence of a third body which carries away the excess energy of the molecule, i.e., A
M + B -.AB* +AB
where A and B are atoms, AB* is an excited molecule, AB a stable molecule, and M is the third body. I t seems, however, that the lack of the absolute necessity of a third body for a successful association was first recognized by Boltzmann as early as 1898, as Steiner (2)points out. In this paper we point out how in some circumstances the two-body reaction (i.e., the association in absence of a third body) is the onlv oossihilitv of association and how the reaction wroceeds .. in this case. Generally speaking, however, one has to take into account the comoetition between two- and three-bodv association reactions: In this general situation we discuss the relative importance of two-bodv aminst three-bodv recombinations and give the conditions k d e r which the former can be competitive. ~lthou~ inha slightly different context, i.e., emphasizing chemiluminiscence rather than formation of a stable molecule. two-body recombinations are discussed in more specialized hooks: see for example the classical work by Herzherg (31,and references 4 . 5 . However. as alreadv stated thev are ienored " in undergraduate textbobks.2 Let us first discuss the nossibilities of enereetic stabilization of the molecule which iniiially bears an energy excess such as to render it unstable against its dissociation products (atoms). We first consider that the atoms A and B approach each other. In order to have a problem in the general framework of chemistry3 we assume A and B to be in their respective (electronic) ground states and to have relative kinetic energies E 0.2 eV o T 2000 K (the so-called "thermal" energies). At these velocities it is acceptable to separate the electronic and nuclear movements in the Schrodinger equation for the A-B system. I t is then possible to interpret the electronic en-
--
--
I
I-De.Oi
,A
r,.~
Figure 1. Typical potential energy curve tor a bound state ot a diatomic molecule AB. The coordinates origin is taken at (-D,, 0). ergy as providing a potential field V(r) for the nuclear movement. If, as supposed, the atoms A and B are ground state atoms, the relevant V(r) curve is the curve for the electronic ground state (XIof the AB molecule. A typical curve is shown in Figure 1, where D, is the spectroscopic dissociation energy. Consider an A-B collision with energy El = 1/21rasu2 just WB-' is the reduced mass of above 0, where NAB-' = FA-' ) u is the the system A (with mass m ~-)B (with mass m ~and relative velocity v = VA - v ~For . the sake of simplification we assume head-on collisions, i.e., collisions with impact parameter b = 0.4Otherwise one would take E l equal to the component of energy along the line of centers since the tangential component is tied up in angular momentum and unavailable for reaction (7). If the system had to "move" exclusively on that V(r) curve, to form later a stable AB molecule (a molecule well in the well) the system would have to radiate (emit a photon) before thk
+
.
' n gas p n x = or, sli 1 oetter. mler s ng e co s cn condmonsic g in m o e c ~ a, ueilnlst ewocrg c ,or endoer~ictp a,s the role of e m Iherm c lor endothermiq ..nocr macroscop c cnno lions In me alter case, energy interchanged in a chemical reaction is more or less rapidly degraded into heat (kinetic energy). This need not (and. indeed, it
probably cannot) be the case when the molecular concentration is low. One referee pointed out to us that explicit consideration is given to the possibility of two-body recombination in the recently published book "Physical Chemistry," by R. C. Berry, S. A. Rice, and J. Ross, John Wiley and Sons, New York, 1980, p. 1125. But see (6)regarding the viewpoint of excited atom chemistry. 'For b # 0 a stabilization mechanism appears which differs from those discussed here. This "orbiting resonance mechanism" is due to resonance between un-bound states of the two-body system before Collision and quasi-bound states of the molecule temporarily formed during the collision. In this way, a metastable collision campiex is formed which can be stabilized in a further collision with a convenient partner. It is wolfh noting, however, that this mechanism is not a simple two-body process, but consists rather of two two-body steps. See, eg., Toennies, J. P., Welz, W., and Waif, G., J. Chem. Phys., 71, 614 (1979).
Volume 60
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January 1983
53
collision ends, i.e., before the A and B atoms fly apart. The collision time is of the order (4) of 0.1 ps (10-l3 S) and turns out to he much shorter than the shortest typical vibrational radiative lifetime (',,ib E 10-3-10-1 s (8).Moreover, one has to consider here the highest vihrational levels for which r,ib is much loneer (9)and in addition manv successive iumns ditional difficulties arise from the gross selection rule which forbids electric dipole vihrational transitions for diatomic homonuclear molecules (10). In conclusion, two-body recombination in this context is practically impossible. The atoms A and B approach each other as near as r , , (classical turning point) recoil hack and fly apart after having exchanged some kinetic energy (elastic collision, see (111). I t is easy to see how a third-body M could help in the stabilization of an excited AB molecule: it just would have to collide with AB* within the ahove cited 0.1 ps and take energy enough from AB* so as to allow the latter t o become AB. This (inelastic the ahove ideas are in the heart of the unwarranted assumption of the necessitv of a third body in atom-atom recomhinations. No matter how inefficient in this case it may he, we have introduced in the last lines a stabilization mechanism which is alternative to the collisional one, i.e., the possihility of emission of electromagnetic radiation by the excited molecule. We shall see that this mechanism can enhance its effectiveness because more than one electronic state correlates with a given stage of the separated atoms, i.e., the system AB can "move" on more than just one potential curve. This fact, together with the possihility of curve-crossing and the subsequent light emission, quantitatively explain the possibility of two-body recombinations as we shall now show. Two- and three-body recombinations processes using chemiluminiscence studies have been reviewed hy Carrington and Polanyi (4).Two-body recombination of atoms on a single curve have been reported for the Hez(AIZ,+) emission coming from recombination of ground and excited He atoms (12). Two-body recombination with diahatic curve crossing have been well studied (13) for the NO(C211)emission, in a process such as:
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N(W + 0(3P)e NO(a411)s NO(CZn) NO(X22)+ hu Golde and Thrush (5a) showed that the nitrogen atoms recomhination leading to emission from ahove dissociation threshold is characteristically two-body in nature and can he expressed as follows:
k2kl
I=-lNI2
=hpKIN12
k-1
where K is a pseudo equilibrium constant and can he calculated by statistical mechanics, and values uT Kkz are compared with measured values of IIlN12. Thus: (Kkd cale.1 (IIIN2)exp.1 Icm~mol-1s c l Icm" mol-Is-1 A3n,,u = 13 2.8 X lo4 6 X 10' aln,, u = ifi,J > 13 1.8 x 105 3 x 10'3 Good agreement between the calculated and measured values of IIIN12 confirms this interpretation of the stahilization mechanism at low pressures. But at pressures higher that 1 Torr three-body recomhination becomes the dominant formation mechanism.
Internuclear d i r t a n c e l i
Figure 2. Potential energy curves of
N2 relevant to this study.
Let us compare the efficiency of the formation of a stable molecule via a two-body and a three-body mechanism. For a three-body mechanism where the A and B atoms form a excited molecule AB*, (11,which might he stabilized (AB) by emission, (Z), or collision with a third body M, (3) we shall have:
k%
AB* - L A B
where Nz* represents the emitting state and Nz a lower-lying electronic state. As the possible emitting states Nz(B3lI,)-(visible)-and Ns(a'IIp)-(vacuum UV)-do not correlate with ground state atoms. the process must be one of inverse predissociation via an initially formed state. This is consistent with the wellknown predissociations in Nz(allI,), u = 6, J > 14 and in N2(B"I,), u = 13 to 16, which are believed to involve collisionless curve crossing to N2(5Z,+) followed by dissociation (see Fig. 2). The intensity of two-body emission is given by:
For an allowed predissociation, k-1 approaches 10'3 s-' while the values of k? for NdalII,) and (B3II,) are 7 X lo3 s-I
54
Journal of Chemical Education
+ hv
(2)
So the production of AB shall he given by
Consider now a two-hody mechanism where a quenching step is included. M is the atom A or B.
The corresponding expression for the production of AB would he
where
Condition (6)
+
-
The condition k-l >> (ka khlMI) is not fulfilled. This is the case for forbidden predissociations (k-I lo8) and allowed radiative transitions (ka lo8) having in a good approximation (for low pressures of A or B) k2 >> kslMI. Then
-
k1
R2U =
(k,+kfi:Ml)
The third-order rate constant for combination of atoms into a bound (excited) state can be taken as (5c) kz Z2.r where Z is the collisionnumber Z = 02(811ksTlfi)'J2 and T , the duration of a two-body collision, is about 10-l3 sec, so k~ 9 X 1 0 - 3 % m h o l e ~ u l e - ~s-'. For a given situation the mechanism will be predominantly two-body if RZ0> RQO,and third hody if RzO < RQO. Let us suppose a pure two-body mechanism, and a threehody mechanism is in the presence of a third hody M a t 1 mTorr pressure. In such a case the magnitudes to he compared are
--
-
and Rqn= halMl
Under these conditions RaO= 3.2 X 10-lg c d molecule-' s-1. For a two-body mechanism we shall have two possible situations. Condition ( A )
If hLl >> (h,
+ kaM)
+
v
--
+
then
N(4S) OPP) s NO(a411)
-
This condition is always fulfilled for allowed predissociations (k-l 1 O 1 k 1 ) ,and for forbidden predissociations (k-I los s-I) when the transitions to the ground state are forbidden (k2 < loGs-I). In both cases, except a t high pressures (above 100 mTorr) of A or B, even for a very efficient quenching, deactivation by A or B becomes inefficient ksIMI < lo"-'. For instance, assuming inefficient quenching by A or B for N atom recombination N2(X1Zp+)+ h u N + N s Nz("Zpt) s Nz(o'&) ~ mm 9 ~ i e ~ x~ I ~x e105- S-'~ R~ = hhz = 3 x cm" r n ~ l e c u l e -s-' ~ R2 = 2.1 X
-
I n such a case (k-I = lo8 S-' and kz = lo8 s-I) and taking for the sake of comparison, a K value of 3 X 10-2%m3 molecule-' (as for N N), one would have RaO= 1.5 X 10-IT cm" molecule-' ssl, that compared with the value obtained in the case of a three-body mechanism for a third-body pressure of 1 mTorr, Ran = 3.2 X lo-'?' cm3 molecule-I s-', shows preponderance of two-body stabilization in this case. As can be readily shown, no two-body mechanism can be successful versus three-body stabilization for pressures of 2 1 mTorr with k 2 5 loGs-I. On the other hand, a t a fixed he, the chance to get a predominant two-body mechanism is increasingly larger as k - ~rises as the efficiency of this process becomes higher as kz increases. Optimum values for RaOare obtainedfor large kz magnitudes with k-I > kz. MaximumRaO values are obtained for the maximum possible magnitudes of kz(-lo%-') and k-'(lo1'-1013 S-') as those are insensitive to k-,. once k-7 >> ko. For stronelv forbidden transitions from ~ I I ~ 111~.vwirtd ~ a t tk . I N, 1.: hl 41 i-, .I . r I < , ( , X I U Wlie ~ l ~mid~ 11 twt~u1d ~ dhart 1.1I)< ~,msi~Imcd lt,r diirerc~i~ Y,,IIII.S of ks. Under any conditions in our case (K 3 X 1 0 - 2 % m h o l ecule-'), two-body mechanism will he predominant versus three-body, a t pressures of a third body of PM > 1Torr. Long-lived excited states, such as the B"II(O,+) state of C l p = 3 X lod4 s (14) that is predissociated by the repulsive III(1,) state, correlating with ground state atoms would not be efficient in a two-body mechanism. One of the best known cases of this type is
That means that even a t very low pressures of third-body, (1 mTorr), the three-body mechanism is two orders of magnitude more efficient than the two-body mechanism. Things would be different if we assume very efficient quenching cm"molecu1e-' s-I, deactivation by A or B, ks = 2 = 3 X a t atom concentrations above 100 mTorr, h5IMI = lo%-', where two-body mechanism begins to be predominant. In this case, in which k-I >> (k2 k51MI), the two-body mechanism will be more efficient the larger kz or ks(M1 are. It means that allowed transitions to the ground state and efficient deactivation mechanism by atoms, will favor the twobody recombination kinetics. I t is easily understood that for large pressures of a thirdbody, namely above 1 Torr, the three-body mechanism will he predominant.
+
-
e NO(C211) NO(X2Z)+ hu (6 bands)
This is a particularly favorable example for several reasons. The perturbing state, a411,is fairly weuknown; the radiative transition C X is strongly allowed; and the lifetime of the predissociating levels of the C-state is sufficiently short so that third body effects are minor even up to pressures of a few torrs. Callear and Pilling (13)summarized rate data for NO(C2n, u = 0)
-
According to previous expressions and assuming a K value (as for Nn) of 3 X 10-'km7 molecule-', we shall have:
Experimental values reported by Mandelman, Carrington, and Young (15) for 6 hands (lying between 190 and 300 nm) of NO from the C", u = 0 state were
This value is in perfect agreement with our calculated parameter. The rapid removal of NO(C") by radiation and predissociation precludes observation of quenching by N o r 0 atoms; therefore, it is very probable that I;, is rigorously proportional to 1 NI 101. Volume 60
Number 1
January 1983
55
In conclusion, by collecting several examples i n t h e literature, we hope t o have shown how unwarranted i t is t o assume t h e absolute necessity of a t h i r d hody in a t o m recomhination reactions. N o m a t t e r how efficient a t h i r d body m a y be, i t is just a help, n o t at all indispensable. Usually, however, a t h i r d body's importance in t h e stabilization of AB s t a r t i n g from A B increases a s t h e third-body concentration increases, a n d after a certain concentration, t h e third-body-assisted recombination mechanism becomes predominant.
+
Appendix
Here we consider the feasibility of two- and three-body recumbination reactions using the theorems of conservation of linear moI1,C11t1111.
.tlltl I !
\\. rtrrrt. .I , i t
3tm,o.. \
tq:\ .L.
.\
ll,,,,
.md:
~ ' I ~ I I 1I ,
111
II..I,ICT~~I,
.... d ti \\,
l ~ l l . r, \ W L ~ ~ I 1. nS1 I t \ v t . ~ r .o ~ . d
.t--un~.
