On the Relation between Population Kinetics and State-to-State Rate Coefficients for Vibrational Energy Transfer

Transcription

1 Z. Phys. Chem. 05; 9(0 ): Elena I. Dasheskaya, Ilya Litin, and Egeny E. Nikitin* On the Relation between Population Kinetics and State-to-State Rate Coefficients for Vibrational Energy Transfer DOI 0.55/zpch Receied March 0, 05; accepted July 8, 05 Abstract: The relation between time-dependent population of ibrational state and collision-induced state-to-state rate coefficients is discussed within the Landau Teller kinetic equations for the relaxation of harmonic oscillators in a heat bath. In particular, the increase of the populations in the first and the second ibrational state of an initially cold oscillator shows a considerable ariety of its relation to a single Landau Teller state-to-state rate coefficient. It is suggested that this ariety should be kept in mind when experimental studies of the relaxation of specific leel are analyzed. Keywords: Reaction Kinetics. Dedicated to Prof. Dr. Dr. h.c. mult. Jürgen Troe on the occasion of his 75 th birthday Introduction Vibrational relaxation and dissociation of diatomic molecules in a heat bath proide a simple example for transient multileel kinetics. There are extensie solutions of the respectie master equations, employing arious models for collisioninduced state-to-state (StS) ibrational transitions and transitions to the dissocia- *Corresponding author: Egeny E. Nikitin, Schulich Faculty of Chemistry, Technion Israel Institute of Technology, 3000, Haifa, Israel; and Max-Planck-Institut für Biophysikalische Chemie, Am Fassberg, D Göttingen, Germany, enikiti@mpibpc.mpg.de Elena I. Dasheskaya: Schulich Faculty of Chemistry, Technion Israel Institute of Technology, 3000, Haifa, Israel; and Max-Planck-Institut für Biophysikalische Chemie, Am Fassberg, D Göttingen, Germany Ilya Litin: Institut für Physikalische Chemie, Uniersität Göttingen, Tammannstrasse 6, D Göttingen, Germany

2 56 E. I. Dasheskaya et al. tion continuum (see e.g. [ 7]). For not too high temperatures, relaxation and dissociation are decoupled, implying that the former establishes a non-equilibrium quasi-stationary distribution before a noticeable fraction of molecules has chance to arrie at the dissociation. Under this condition, the ibrational relaxation can be treated disregarding dissociation. The form of the kinetic relaxation equations and the general properties of their solution are well documented (see e.g. the texts [8 ]). Whileaariety ofapproximations for setting up the relaxationequations is aailable, a closer look at the simplest of these equations appears worthwhile, in particular when analytical solutions are possible. We are motiated for this work through our recent reinterpretation of the ibrational relaxation NO in Ar [, 3]. Originally, the experimental data were reported either in terms of the first-order (FO) rate laws [4, 5] or within a simplified two-state (TS) approximation [6] where thekineticswascharacterizedbyapparentstate-specific (SSp) rate coefficients. In order to arrie at a transparent relation between the StS and SSp rate parameters, in this article we adopt the Landau Teller (LT) approach to multi-state kinetics. We first consider some general features of TS and LT models (Section ), then discuss possible relations between the apparent SSp and the fundamental StS rate coefficients (Section 3). Finally we apply these relations to the relaxation of population of the first and second ibrational states of an initially cold harmonic oscillator (Section 4). Section 5 concludes the paper. Two state (TS) s. Landau Teller (LT) model The first relaxation models [7, 8] employed a TS concept (for the ground and the first excited ibrational states). The TS model allows one to derie well-defined apparent state-specific SSp rate coefficient that describes the exponential change of population of the excited ibrational state of a diatom, and relates this to the StS rate coefficient that describes the transfer of population from the excited to the ground state. The TS relaxation equation contains a single StS kinetic parameter, k TS 0 (T), that is related to the collisional deactiation of the first ibrational state in the 0transition upon collisions of the diatom with the atoms X at the heat bath of temperature T. An analytical solution for the time dependent population of the first leel, x TS (t) (and hence that of the zeroth leel, xts 0 (t), dueto the normalization x TS 0 (t) + xts (t)=)whichasymptotically approaches its final

3 On the Relation between Population Kinetics 563 equilibrium alue x TS f =xts f x TS f xts (t) x TS f xts i (T) starting from its initial alue xts i reads =F TS (t) = exp( [X]kTS (T)t) () Here k TS is a SSp relaxation rate coefficient which is related to the state-to-state rate coefficient k TS 0 by k TS (T)=kTS 0 (T) (+b(t)) () where b(t) is the Boltzmann factor for leel =aboe the leel =0with the energy difference ΔE 0, b(t) = exp( ΔE 0 /k B T).Equation() defines the SSp rate coefficient k TS (T) and proides the relation to the StS rate coefficient kts 0 (T) by Equation (). After papers [7, 8], it was quickly realized [9], with the citation of the Landau and Teller s work published later [0], that a two-leel scheme, except for special particular cases, is not adequate. The LT model represents a multileel ibrational manifold of harmonic oscillator states which are weakly coupled to a stochastic Boltzmann reseroir and which are only allowing the nearestneighbor transitions ±with linear dependence of the StS rate coefficient on the ibrational quantum number: k. =k 0 k. =k 0 b (3) Similar to the TS model, the LT model requires a single StS rate coefficient k 0 (T) only. Howeer, for this model, the relaxation kinetics for the population of a specific ibrational state is not exponential, and therefore well-defined SSp rate coefficients do not exist. Thus, the question arises whether ill-defined apparent SSp rate coefficients k (T) neertheless can be related to the well-defined StS rate coefficient k 0 (T). The LT relaxation equations for the populations x (t) employing StS rate coefficients from Equation (3), possess the remarkable property that the mean energy E LT of the ensemble of the harmonic oscillators (HO) exponentially relaxes from its initial alue E i towards its final, thermally equilibrium, alue E f.this is described by the LT expression (also cited sometimes as Bethe Teller equation [, ]): F LT E EHO f (t) E HO f ELT (t) EHO i = exp( [X]k E t) (4)

4 564 E. I. Dasheskaya et al. Here where k E is an energy relaxation rate coefficient which is related to the stateto-state rate coefficient k 0 by k E =k 0 ( b). (5) In Equation (4) E i =E LT (t),andeho t=0 f =ΔE 0b/( b).notethatequation(4) is alid irrespectie of an initial distribution of populations, x HO i,ofibrational states that defines the quantity E i by E i = ΔE 0 x HO i. =0 The eolution of the population of indiidual states, x LT (t), is described by analytical solution of the LT kinetic equations [3], and it depends on the initial distribution x HO i. A particular case, as discussed below, is the relaxation starting from an initial Boltzmann distribution of temperature T 0,forwhich x HO i =( b 0 )b 0, E i =ΔE 0 b 0 /( b 0 ) (6) with b 0 =exp( ΔE 0 /k B T 0 ). The Boltzmann-to-Boltzmann (from T 0 to T) relaxation of the harmonic oscillators within the LT model then occurs through a set of Boltzmann distributions (the so-called canonical inariance): x LT (τ, b 0,b)=( a(τ,b 0, b))a (τ, b 0,b) (7) where τ is the reduced time, τ = [X]k E t, related to the rate of the energy relaxation, see Equation (4). The explicit form for the function a(τ, b 0,b)reads [3]: a(τ, b 0,b) = ( b 0) exp( τ)( b 0 /b) ( b 0 )/b exp( τ)( b 0 /b) (8) with the initial and final alues gien by The expression x HO f a(τ, b 0,b) =b τ=0 0, a(τ, b 0,b) τ= =b. (9) xlt (τ, b 0,b) x HO f xho i F LT (τ, b 0,b) (0) with x HO f =( b)b then is the the LT counterpart of Equation () for an initial Boltzmann distribution. Howeer, the expression in Equation (0), in contrast to Equation (), does not hae exponential form.

5 On the Relation between Population Kinetics Apparent SSp rate coefficients for LT model Though F LT (τ, b 0,b)in Equation (0) is not exponential in time, one may try to fit it by an exponential in analogy to first order (FO) kinetics F LT (τ, b 0,b) F LTFO (τ, b 0,b)=exp( κ LTFO (b 0,b)τ). () Here τ should coer a reasonable range dictated by the experimental conditions and it corresponds to the trial alue of the apparent rate coefficient, κ LTFO (b 0,b), which can be regarded as SSp rare coefficient for the LT model. This implies that κ LTFO will depend, beside the heat bath parameters b 0 and b, alsoonacertain parameter p that goerns the choice of the aboe trial alue, i.e. κ LTFO = κ LTFO (b 0,b,p).Onceκ LTFO is chosen, it relates a SSp rate coefficient k to a StS rate coefficient k 0 by requiring that F LTFO from Equation () coincides with its SSp counterpart F SSp (τ, b 0,b)=exp( κ (b 0, b)τ). () In this way one gets an expression k (b 0,b,p)=κ LTFO (b 0,b,p)k 0 (b)( b), (3) which relates the multitude of experimentally measured rates k (b 0,b,p) to a single LT rate coefficient k 0 (b). A possibility of such a fitting is illustrated by Figure which shows the drop of the function F LT (τ, b 0,b),forb 0 = /0, b = /,fromto 0.,i.e.byoneorder of magnitude such as releant to the experimental conditions. The approximately linear dependence of log F LT (τ, b 0,b)on τ across this interal is obsered for =,, 3 but noticeable non-linear incubation periods are clearly apparent for higher ; these are due to the late arrial of populations at higher leels from the lower ones. The quality of the exponential fitting is illustrated by Figure for b 0 = /0, b = /. It clearly shows the difficulties with this type of fit for =3;for=4, the exponential fitting appears inadequate. Howeer, one can improe the LTFO fitting by introducing a delayed LTFO exponential (DLTFO) which corresponds to a first-order kinetic equation as F DLTFO (τ, τ d,b 0,b) exp[ κ DLTFO (b 0,b,τ d )(τ τ d )Θ(τ τ d )] (4) df DLTFO /dτ = κ DLTFO (b 0,b,τ d )Θ(τ τ d )F DLTFO (5) where Θ is the step function, and τ d is a delay time. We see from Figure that, for =4, the DLTFO graph noticeably differs from the LTFO graph but reproduces reasonably well the LT graph.

6 566 E. I. Dasheskaya et al. Figure : Single-leel LT relaxation cures (log scale), F LT (τ, b 0,b)s. τ (labeled by quantum numbers )forb 0 = /0, b = / (full lines). Open circles correspond to the exponential relaxation of the oscillator mean energy, F LT E (τ, b 0,b)s. τ. Interestingly, the population in leel =3relaxes approximately at the same rate as the mean energy (for these alues of b 0 and b). One way to choose SSp rate coefficients and to inspect the corresponding accuracy of the exponential approximation consists in the following procedure: one forces Equation () to pass through the point F LTFO =pat the time τ = τ (b 0,b,p), such as found from the requirement that the solution of the LT kinetic equation describes the drop of F LT by the same factor p during the time period τ =τ (b 0,b,p). In this way, the SSp rate coefficient k (b 0,b,p)isexpressed through StS rate coefficient k 0 as k (b 0,b,p)= K (b 0,b,p)k 0 (b) (6) with ( b) ln p K (b 0,b,p)= τ (b 0,b,p). (7) Here, τ is found from the equation F LT (τ, b 0,b) τ=τ =p. (8) (b 0,b,p)

7 On the Relation between Population Kinetics 567 Figure : Single-leel LT relaxation cures (log scale), F LT (τ, b 0,b)(full cures), their FO fitting F LTFO (τ, b 0,b)(dashed straight lines) for b 0 = /0, b = / (labeled by quantum numbers ). Fitted rate coefficients are κ LT (b 0,b)=.8, κ LT (b 0,b)=.3, κ LT 3 (b 0,b)=, κ LT 4 (b 0,b)=0.8.For=4, the DLTFO fitting F DLTFO 4 (τ, b 0,b,τ d ) (diamonds) corresponding to κ DLTFO 4 (b 0,b,τ d )=and τ d =0.3 is also shown. The ariety of κ LTFO (b 0,b,p) found from a ariation of p across a reasonable range (as determined by the experimental conditions) characterizes the accuracy of the exponential approximation. 4 Relaxation of initially cold oscillator We illustrate the preceding analysis by the case of an initially cold oscillator with b 0,b(typical conditions for relaxation in shock waes). Then, Equation (8) becomes a(τ, b 0,b) b0,b = exp( τ) /b exp( τ) (9)

8 568 E. I. Dasheskaya et al. and the expressions for F LT (τ, b 0,b) are simplified. For instance, F LT (τ, b 0,b) b0 F LT b, (τ, b) becomes: F LT (τ,b)= exp( τ) ( b exp( τ)) (0) In this case, equation Equation (8) can be soled analytically yielding, together with Equation (7), the following expression for K (b 0,b,p) b0 b, K (b, p): ( b) ln p K (b, p) = ln M(b, p) M(b, p) = b( p) b ( p) b( p) + ( b ( p) ) p + b ( p) The graphs of K (b, p) are shown in Figure 3 for three alues of p: p=/, p=/eand p=/5. Had the decay been exponential, all three cures would coincide. The difference between cures for arious alues of p characterize the () () Figure 3: Coefficients K (p, b) in Equation ()sb for different alues of p (solid lines). Also shown is the result for a TS model (dashed line). The upper abscissa axis refers to temperatures for NO molecule with ΔE 0 /k B = 700 K.

9 On the Relation between Population Kinetics 569 Figure 4: Similar to Figure 3 but for K (p, b). accuracy of the exponential approximation. For instance, with T=ΔE 0 /k B (i.e. b=exp( )=0.368), the difference in K (b, p) for p=/and p=/5equals 0.0, while the mean alue of K for these two alues of p is about.35.alsoshown in this Figure is K TS (b)=(+b), the respectie quantity for the two-states approximation,see Equation (). The conergence of K (b, p) to K TS (b) with decreasing b is a manifestation of progressiely better performance of the TS approximation when transitions to (and from) higher states are ignored. On the other hand, a sharp increase of K (b, p) for b about / and higher is related to the oershoot phenomenon (see below). Similar plots, but for K (b, p), are shown in Figure 4. Herethecureforthe TS model is absent since it is not applicably to the case =. As a practical example, we consider now the ibratioinal relaxation of NO in Ar behind a shock wae studied experimentally in Ref. [5, 6] inawidetempera- ture range. For relaxation in =state, the relation between the measured rate coefficients k [5, 6] and the fundamental rate coefficients k 0,aspredictedby the aboe treatment, is gien in Table as a list of conersion factors K =k /k 0 for different temperatures and parameters p that determines the point at which

10 570 E. I. Dasheskaya et al. Table : Conersion factors K between SSp rate coefficients k and StS rate coefficients k 0 for the relaxation of NO ( =) for different temperatures T and arious alues of the fitting parameters p. T 800 K 00 K 400 K 700 K 3000 K p=/ p=/e p=/ the exponential LTFO kinetics (see Equation ()) is fitted to the non-exponential LT kinetics. For a fixed alue of p, the temperature dependence of K demonstrates the manifestation of the multiple-state relaxation kinetics in an approximate first order single state kinetics. For a fixed temperature, the dependence of K on p indicates uncertainties in the approximation of non-exponential kinetics by an exponential decay. Turning now to relaxation in =and =states, we show in Figure 5 the predicted time dependence of populations of the ibrational states =and = of NO behind a shock wae at T = 700 K in a heat bath which was initially at room temperature. For this case, b 0 is about 3 0 4,andb = Thefigure shows graphs of F LT (τ, b) and F LT (τ, b) (full lines) which clearly demonstrate the change of slopes in the former (dashed lines) and the appearance of an incubation period in the latter (open symbols). Also shown are the graphs for F LTFO (τ, b) and F LTFO (τ, b) with κ LTFO =.7,κ LTFO =..For=, the straight line is defined by the condition that it connects the correct initial and final points of F LT (τ, b). For =, the straight line is defined by the condition that it starts at the correct initial point of F LT (τ, b) and then asymptotically runs parallel to F LT (τ, b).theratio κ LTFO /κ LTFO,equals0.647, while the ratio of the experimental rate coefficients k /k is 0.65 (see Ref. [5], Table ). If one goes to lower temperatures, the difficulties with FO interpretation of the relaxation increases due to the longer induction period for =state. One more interesting feature should be mentioned. For high enough temperature T, the populations of some lying leels pass, in the relaxation course, through a single maximum (the oershoot phenomenon), so that the function (τ, b) changes its sign before disappearing at equilibrium. For a gien alue of, the oershoot occurs under condition b>/(+). For=, the asymptotic (for τ )formoff LT (τ, b) is F LT F LT (τ, b) = ( b) exp( τ) + (b τ>> 3b ) exp( τ) + O(exp( 3τ)). (3)

11 On the Relation between Population Kinetics 57 Figure 5: Single-leel LT relaxation cures (NO + Ar, log scale), F LT (τ, b) and F LT (τ, b) for b (room temperature before the shock front) and b = (T = 700 K behind the shock front) for ΔE 0 /k B = 700 K (solid cures). Dashed lines show the change of slopes in the linear approximation of the exponent for =, and symbols indicate the existence of the induction period for =. Also shown are the graphs for F LTFO (τ, b) and F LTFO (τ, b) with κ LTFO =.7,κ LTFO =.(dotted lines). For =, the straight line is defined by the condition that it connects the correct initial and final points of F LT (τ, b).for=, the straight line is defined by the condition that it starts at the correct initial point of F LT (τ, b) and then asymptotically runs parallel to F LT (τ, b). Here, the threshold alue of b that corresponds to the oershoot is /.Forb</, the function F LT (τ, b) approaches zero from aboe, and for b>/from below (oershoot). In both cases, the time dependence of F LT (τ, b) is goerned by the lowest eigenalue of the LT kinetic matrix (factor of unity in front of τ in the exponent of the first term in the r.h.s. of Equation (3)). For b=/,thefirstterm in Equation (3) anishes, and the long-time decay of F LT (τ, b) is goerned by the second eigenalue (factor of two in front of τ in the remaining term in the r.h.s. of Equation (3)). Figure 6 shows examples of F LT (τ, b) for the alues of b=/as well as below (b = /4) and aboe (b =3/4) it. Also shown are possible exponential approximations to these three cases. Finally we emphasize that our analysis refers to Boltzmann-to-Boltzmann relaxation (shock wae conditions) and not to a relaxation of an initial non- Boltzmann distribution (e.g. an optical excitation of a single ibrational state).

12 57 E. I. Dasheskaya et al. Figure 6: Illustration of the oershoot for =.PlotsofF (τ, b) s. τ for b=/4, /, 3/4 (solid lines) and approximations of F (τ, b) by exponentials exp( βτ) with β =.6, 3.68, 7.5 respectiely (dashed lines). These exponentials are fitted by the requirement that they coincide with F (τ, b) at points where F (τ, b) = and F (τ, b) = 0.5. The exponential with β=(open circles) correspond to the energy relaxation. 5 Conclusion The interpretation of the ibrational relaxation kinetics in terms of state-specific relaxation rate coefficient meets some difficulties. First, this quantity cannot be strictly defined on the basis of a relaxation master equation since the population change of a single state is not exponential. Second, if it is defined on the basis of experimental data as an effectie rate coefficient, one should carefully indicate the time interal across which the change of the population was measured. The latter information can be used in forcing non-exponential relaxation to be represented by an exponential, thus allowing one to relate SSp rate coefficients to StS rate coefficients which enter into the master equations and which can, in principle, be calculated theoretically. The outlined concept has no adantage compared to a direct numerical solution of the releant rate equations proided they contain well-defined state-tostate rate coefficients. Howeer, if the latter are taken in a trial numerical form,

13 On the Relation between Population Kinetics 573 it is difficult to adjust them to a single effectie rate coefficient deried from the experimental kinetics of the population change. Moreoer, such an adjustment (many unknown to one known quantity) will suffer from seeral uncertainties. We therefore decided to use an analytical solution of Landau Teller rateequation that contains a single state-to-state rate coefficient. This allows to see the accuracy and possibility of adjustment of an exponentially-fitted experimental kinetics of a single-leel population to the theoretical non-exponential kinetics. In this respect the present approach is superior (compared to numerical solution of kinetic equations with fitted state-to-state rate coefficients) for an ealuation of experiments since it is more transparent. Its performance was demonstrated in Section 3 by introducing a concept of the delayed first-order relaxation, and in Section 4 by a oershoot phenomenon. Howeer, a simple case of single-leel relaxation during thermal heating of the ensemble of initially cold oscillators (e.g. behind the shock wae front) shows ambiguities in the use of SSp rate coefficients for the single-state relaxation kinetics. Fortunately, for the first ibrational state, the StS and a set of related SSp rate coefficients do not differ much and sometimes their difference falls into the accuracy range of the experiment proided that the heat bath temperature is noticeably below the oershoot threshold. The situation can be completely different for higher ibrational states where the population kinetics in the Boltzmann-to-Boltzmann relaxation shows an incubation delay. Similar difficulties arise when the initial state does not correspond to a Boltzmann distribution (e.g. it is prepared by an optical excitation). Acknowledgement: This work acknowledges many stimulating discussions with Professor Juergen Troe and his continuing interest in our work. References. V. A. LoDato, D. L. S. McElwain, and H. O. Pritchard, J. Am. Chem. Soc. 9 (969) J. E. Doe and D. G. Jones, J. Chem. Phys. 55 (97) J. E. Doe and J. Troe, Chem. Phys. 35 (978). 4. H. Teitelbaum, Can. J. Chem. 6 (983) H. Teitelbaum, Can. J. Chem. 6 (983) H. Teitelbaum, Can. J. Chem. 6 (983) H. Teitelbaum,Chem.Phys. 4 (988) I. V. Adamoich, S. O. Macheret, J. W. Rich, and C. E. Treanor, AIAA J 33 (995) M. Capitelli (Ed.), Nonequilibrium Vibrational Kinetics, Springer, Berlin Heidelberg (986). 0. M. Capitelli, C. M. Ferreira, B. F. Gordiets, and A. I. Osipo, Plasma kinetics in Atmospheric Gases, Springer, Berlin Heidelberg New York (000).

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