Deactivation rate, unimolecular reaction

The effective rate law correctly describes the pressure dependence of unimolecular reaction rates at least qualitatively. This is illustrated in figure A3,4,9. In the lunit of high pressures, i.e. large [M], becomes independent of [M] yielding the high-pressure rate constant of an effective first-order rate law. At very low pressures, product fonnation becomes much faster than deactivation. A j now depends linearly on [M]. This corresponds to an effective second-order rate law with the pseudo first-order rate constant Aq ... 

Consider the simple unimolecular reaction of Eq. (15.3), where the objective is to compute the forward rate constant. Transition-state theory supposes that the nature of the activated complex. A, is such that it represents a population of molecules in equilibrium with one another, and also in equilibrium with the reactant, A. That population partitions between an irreversible forward reaction to produce B, with an associated rate constant k, and deactivation back to A, with a (reverse) rate constant of kdeact- The rate at which molecules of A are activated to A is kact- This situation is illustrated schematically in Figure 15.1. Using the usual first-order kinetic equations for the rate at which B is produced, we see that... 

The molecules which have reached Ti will now react with a rate constant kr (unimolecular reaction) or [N] (bimolecular reaction with a ground state partner N) in competition with radiative (phosphorescence of rate constant P), non-radiative (A ) deactivations as well as quenching processes ( q[Q]) so that the final reaction quantum yield of the primary process is... 

How do molecules in unimolecular reactions attain their energy of activation Lindemann (1922) suggested its answer by pointing out that the behaviour of unimolecular reactions can be explained on the basis of bimo-lecular collisions provided we postulate that a time lag exists between activation and reaction during which activated molecules may either react or be deactivated to ordinary molecules. Thus, the rate of reaction will not be proportional to all the molecules activated, but only to those which remain active. Lindemann suggested that the above reaction takes place as follows ... 

The field of unimolecular reaction rates had an interesting history beginning around 1920, when chemists attempted to understand how a unimolecular decomposition N2Os could occur thermally and still be first-order, A — products, even though the collisions which cause the reaction are second-order (A + A— products). The explanation, one may recall, was given by Lindemann [59], i.e., that collisions can produce a vibrationally excited molecule A, which has a finite lifetime and can form either products (A — products), or be deactivated by a collision (A + A— A + A). At sufficiently high pressures of A, such a scheme involving a finite lifetime produces a thermal equilibrium population of this A. The reaction rate is proportional to A, which would then be proportional to A and so the reaction would be first-order. At low pressures, the collisions of A to form A are inadequate to maintain an equilibrium population of A, because of the losses due to reaction. Ultimately, the reaction rate at low pressures was predicted to become the bimolecular collisional rate for formation of A and, hence, second-order. 

The fall-off of the unimolecular rate constant as a function of pressure arises because of the way in which the competition between reaction and deactivation depends on pressure. The rate of reaction is equal to /t2[A ], and so its pressure dependence follows the pressure dependence of [A ], given by Eq. (8). As the pressure is reduced, the importance of k2 in the denominator relative to A i[M] increases, and so the steady-state concentration [A ] is reduced. A useful way of looking at this is that [A ] is depleted by reaction, and is lower than its Boltzmann value, A i[A]/fc i. This reactive depletion is most important at low pressures, where the collisional activation is too slow to replenish it. 

The effect of the competition is maximized when the rates of reaction and deactivation are equal, k [M] = k2, at this point the unimolecular rate coefficient is half of fcoo. At the transition pressure... 

Fig. 2.12. Schematic diagram of a 2-D potential energy curve for a unimolecular reaction. k and are the bimolecular rate coefficients for activation and deactivation respectively by a bath gas M. k, is the rate coefficient for the unimolecular reaction of an energized molecule, A, to give product, P.

The Hinshelwood-Lindemann model [5], in which molecules are activated and deactivated by collisions, is well accepted for describing the temperature T and pressure P dependence of thermal unimolecular reactions. The unimolecular rate constant. 

Prior to about 1970, classical experiments, in which the samples were in a thermal heat bath, resulted in an uncertain energy content of the molecule, as well as uncertain collisional deactivation rates. Although RRKM theory provided an excellent framework for a qualitative, as well as semiquantitative, understanding of the unimolecular reactions, the experiments failed to provide firm evidence for the fundamental correctness of the statistical assumptions. 

Now, let us find a little flaw in the theory equation (2-86) predicts only first-order behavior for the unimolecular reaction, something we know in fact is not true at low pressures. The reason for this failure in TST is the assumption of universal equilibrium between reactants and the transition state complex. At low pressures the collisional deactivation process becomes very slow, since collisions are infrequent, and the rate of decomposition becomes large compared to deactivation. In such an event, equilibrium cannot be established nearly every molecule which is activated will decompose to product. However, the magnitude of the rate of decomposition of the transition complex is much larger than the decomposition of the activated molecule in the collision theory scheme, so one must resist the temptation to equate the two. Since the transition state complex represents a configuration of the reacting molecule on the way from reactants to products, the activated molecule must be a precursor of the transition state complex. 

The first factor gives the rate constant for the collisional activation which initiates unimolecular reaction. The second factor in brackets is the fraction of A which then reacts by step (1.6). As processes (1.5) and (1.6) compete, this fraction may be less than unity. At low pressures collisional deactivation is much rarer than reaction, <5C the fraction of dissociated A becomes 1 and... 

The ratio k lk is the equilibrium constant for the reaction (Eq. 8), and dk ( )/ is the probability of A being in a state with energy E, P(E). The k [M] factor is the collision frequency for deactivation that is usually denoted by co. The unimolecular reaction rate constant can be obtained by integrating the effective rate constant over all energies higher than the activation energy. 

Generally speaking, the three unimolecular processes (radiative deactivation, radiationless deactivation, and chemical reaction) described above compete for deactivation of any excited state of a molecule. Therefore their individual specific rates and the kinetics of their competition in each excited state are of utmost importance in determining the actual behavior of the excited molecule. 

This equation shows that the reaction rate is neither first-order nor second-order with respect to species A. However, there are two limiting cases. At high pressures where [A] is lar e, the bimolecular deactivation process is much more rapid than the unimolecular decomposition (i.e., /c2[A][A ] /c3[A ]). Under these conditions the second term in the denominator of equation 4.3.20 may be neglected to yield a first-order rate expression. 

Rate constants of unimolecular processes can be obtained from spectral data and are useful parameters in photochemical kinetics. Even the nature of photoproducts may be different if these parameters change due to some perturbations. In the absence of bimolecular quenching and photochemical reactions, the following reaction steps are important in deactivating the excited molecule back to the ground state. 

Rates of Gas-Phase Reactions. Reaction rates have been reported for only a few CVD gas-phase reactions, and most reports are primarily for the silane system. Because of the high temperatures and low pressures used in CVD, the direct use of reported gas-phase rate constants must be done with care. In addition to mass-transfer and wall effects, process pressure may be another factor affecting reaction rates. Process pressure affects major CVD processes, such as the deposition of Si from SiH4 and GaAs from Ga(CH3)3, reactions that involve unimolecular decomposition. The collisional activation, deactivation, and decomposition underlying these reactions can be summarized qualitatively by the following reactions (139, 140) ... 

This effect of N08 ion is quantitatively consistent with a reaction mechanism (43) in which N08 interacts with an electronically excited water molecule before it undergoes collisional deactivation by a pseudo-unimolecular process (the NOs effect is temperature independent (45) and not proportional to T/tj (37)). Equation 1, according to this mechanism, yields a lifetime for H20 of 4 X 10 10 sec., based on a diffusion-controlled rate constant of 6 X 109 for reaction with N08 Dependence of Gh, on Solute Concentration. Another effect of NOa in aqueous solutions is a decrease in GH, with increase in N08 concentration (5, 25, 26, 38, 39). This decrease in Gh, is generally believed to result from reaction of N08 with reducing species before they combine to form H2. These effects of N08 on G(Ce+3) and Gh, raise the question as to whether or not they are both caused by reaction of N08 with the same intermediate. 

Both these processes can be considered to occur in several distinct stages as follows (i) formation of precursor state where the reacting centers are geometrically positioned for electron transfer, (ii) activation of nuclear reaction coordinates to form the transition state, (iii) electron tunneling, (iv) nuclear deactivation to form a successor state, and (v) dissociation of successor state to form the eventual products. At least for weak-overlap reactions, step (iii) will occur sufficiently rapidly (< 10 16s) so that the nuclear coordinates remain essentially fixed. The "elementary electron-transfer step associated with the unimolecular rate constant kel [eqn. (10)] comprises stages (ii)—(iv). 

We have eliminated the quantity jS which appears in D(M,y) by setting it equal to unity. If the over-all reaction is thermoneutral, then /S = 2 and a factor of 2 should appear in the denominators of Eqs. (XI.4.5) and (XI.4.6). This arises from the fact that this equation was derived for the unimolecular isomerization equation (XI. 1.9) and we have set the rates of isomerization kx A b and also the rates of deactivation k2 = ki-At these veiy low pressures we have the result that A and B are in equilibrium with each other but not with processes of activation or deactivation, which are much slower. Thus only half of the active species reaches the final state B. If kx were much greater or much smaller than /Itb (which occurs if the reaction is very energetic), then the factor of 2 disappears and j9 = 1 as implied above. 

Nucleophilic capture of the spirooctadienyl cation opens the 3-member ring. This behavior characterizes many reactions of many other cyclopropane-containing carbocations, as well, y-radiolysis of perdeuterated propane forms CsD ions, most of which either transfer D or form isopropyl adducts. As the propane pressure is raised from 1000 mbar to 2000 mbar, however, the isopropyl/ -propyl adduct ratio falls from 30 1 to about 5.5 1. This implies the formation of corner-protonated cyclopropane, which reacts with nucleophiles as though it were an -propyl cation. With increased pressure, vibrationally excited protonated cyclopropane experiences more frequent nonreactive collisions, which deactivate it and slow down its rate of unimolecular isomerization to isopropyl cation. 

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