Unimolecular decay exponential

In the above discussion it was assumed that the barriers are low for transitions between the different confonnations of the fluxional molecule, as depicted in figure A3.12.5 and therefore the transitions occur on a timescale much shorter than the RRKM lifetime. This is the rapid IVR assumption of RRKM theory discussed in section A3.12.2. Accordingly, an initial microcanonical ensemble over all the confonnations decays exponentially. However, for some fluxional molecules, transitions between the different confonnations may be slower than the RRKM rate, giving rise to bottlenecks in the unimolecular dissociation [4, ]. The ensuing lifetime distribution, equation (A3.12.7), will be non-exponential, as is the case for intrinsic non-RRKM dynamics, for an mitial microcanonical ensemble of molecular states. 

Recently, Miller and co-workers have obtained a generalized form of the distribution of unimolecular decay rates for the case of coupled open channels contributing with unequal partial half-widths [139]. Further results have also recently been obtained in the statistical theory of reactions where the possibility of algebraic decay besides the RRKM exponential decay has been discussed [140]. ... 

The classical unimolecular dynamics is ergodic for molecules like NO2 and D2CO, whose resonance states are highly mixed and unassignable. As described above, their unimolecular dynamics is identified as statistical state specific. The classical dynamics for these molecules are intrinsically RRKM and a microcanonical ensemble of phase space points decays exponentially in accord with Eq. (3). The correspondence found between statistical state specific quantum dynamics and quantum RRKM theory is that the average of the N resonance rate constants fe,) in an energy window E + AE approximates the quantum RRKM rate constant k E) [27,90]. Because of the state specificity of the resonance rates, the decomposition of an ensemble of the A resonances is non-exponential, i.e. 

Note that 4T/h has units of s and that the exponential is dimensionless. Thus, the expression in (3.1.17) is dimensionally correct for a first-order rate constant. For a second-order reaction, the equilibrium corresponding to (3.1.11) would have the concentrations of two reactants in the denominator on the left side and the activity coefficient for each of those species divided by the standard-state concentration, C, in the numerator on the right. Thus, C no longer divides out altogether and is carried to the first power into the denominator of the final expression. Since it normally has a unit value (usually 1 M ), its presence has no effect numerically, but it does dimensionally. The overall result is to create a prefactor having a numeric value equal to 4T/h but having units of M s as required. This point is often omitted in applications of transition state theory to processes more complicated than unimolecular decay. See Section 2.1.5 and reference 5. 

Intrinsic non-RRKM behavior occurs when an initial microcanonical ensemble decays nonexponentially or exponentially with a rate constant different from that of RRKM theory. The former occurs when there is a bottleneck (or bottlenecks) in the classical phase space so that transitions between different regions of phase space are less probable than that for crossing the transition state [fig. 8.9(e)]. Thus, a micro-canonical ensemble is not maintained during the unimolecular decomposition. A limiting case for intrinsic non-RRKM behavior occurs when the reactant molecule s phase space is metrically decomposable into two parts, for example, one part consisting of chaotic trajectories which can decompose and the other of quasiperiodic trajectories which are trapped in the reactant phase space (Hase et al., 1983). If the chaotic motion gives rise to a uniform distribution in the chaotic part of phase space, the unimolecular decay will be exponential with a rate constant k given by... 

After the laser pulse, the transient decays. For a unimolecular decay process, the concentration of the excited species decays exponentially, according to... 

N denotes the number of active (growing) nuclei. The time y represents the time the nucleus got activated. The exponent m gives the dimension of nuclei growth. The law of nucleation can be postulated in various ways, such as unimolecular decay law. The left-hand side of the equation origins from Avrami s treatment for the nuclei overly. It gives the relation between the extended rate of conversion and the true rate of conversion. The pre-exponential coefficient includes several constants grouped together. 

A situation that arises from the intramolecular dynamics of A and completely distinct from apparent non-RRKM behaviour is intrinsic non-RRKM behaviour [9], By this, it is meant that A has a non-random P(t) even if the internal vibrational states of A are prepared randomly. This situation arises when transitions between individual molecular vibrational/rotational states are slower than transitions leading to products. As a result, the vibrational states do not have equal dissociation probabilities. In tenns of classical phase space dynamics, slow transitions between the states occur when the reactant phase space is metrically decomposable [13,14] on the timescale of the imimolecular reaction and there is at least one bottleneck [9] in the molecular phase space other than the one defining the transition state. An intrinsic non-RRKM molecule decays non-exponentially with a time-dependent unimolecular rate constant or exponentially with a rate constant different from that of RRKM theory. 

Fig. 2.21. (a) Time-resolved LIF decay profiles for two closely spaced rotational levels of vibrationally excited CH3O (X). The solid line is an exponential fit for the decay convoluted with the dump laser pulse shape function, (b) Measured state specific unimolecular dissociation rate constants for CH3O (X) compared to calculated k E, J) curves without and with tunneling corrections. 

If only unimolecular processes are involved then ESi decays according to an exponential function and the lifetime of ESi is defined as... 

Thus if a molecule s unimolecular decomposition is in accord with RRKM theory, the t = 0 intercept of its P t) will equal k E), and its P t) will be exponential with a decay constant equal to k(E) (Bunker, 1966). 

For many of the model molecules studied by the trajectory simulations, the decay of P t) was exponential with a decay constant equal to the RRKM rate constant. However, for some models with widely disparate vibrational frequencies and/or masses, decay was either nonexponential or exponential with a decay constant larger than k E) determined from the intercept of P(f). This behavior occurs when some of the molecule s vibrational states are inaccessible or only weakly coupled. Thus, a micro-canonical ensemble is not maintained during the molecule s decomposition. These studies were a harbinger for what is known now regarding inelficient intramolecular vibrational energy redistribution (IVR) in weakly coupled systems such as van der Waals molecules and mode-specific unimolecular dynamics. 

The random lifetime assumption is perhaps most easily tested by classical trajectory calculations (Bunker, 1962 1964 Bunker and Hase, 1973). Initial momenta and coordinates for the Hamiltonian of an excited molecule can be selected randomly, so that a microcanonical ensemble of states is selected. Solving Hamilton s equations of motion, Eq. (2.9), for an initial condition gives the time required for the system to reach the transition state. If the unimolecular dynamics of the molecule are in accord with RRKM theory, the decomposition probability of the molecule versus time, determined on the basis of many initial conditions, will be exponential with the RRKM rate constant. That is, the decay is proportional to exp[-k( )t]. The observation of such an exponential distribution of lifetimes has been identified as intrinsic RRKM behavior. If a microcanonical ensemble is not maintained during the unimolecular decomposition (i.e., IVR is slower than decomposition), the decomposition probability will be nonexponential, or exponential with a rate constant that differs from that predicted by RRKM theory. The implication of such trajectory studies to experiments and their relationship to quantum dynamics is discussed in detail in chapter 8. 

In order to estimate the stability of triplet carbenes (19) under ambient conditions, laser flash photolysis ( LFP) [26] was carried out on the precursor diazomethanes (18) in solution at room temperature. The transient absorption bands formed upon the flash were recorded by a multi-channel detector. These bands were assigned to the triplet carbenes (19) by comparison with those obtained in matrix at low temperature. The kinetic information was then available by monitoring the decay of the transient absorption with oscillographic tracer. When triplet carbenes decayed unimolecularly, which is often so, lifetime (x) can be determined. However when the decay did not follow a single exponential, which is sometimes the case, x cannot be determined. In this case, a half-life (ti/2) is estimated from the decay curve as a rough measure of the stability. 

A ). Highly non-ergodic bdiaviour was found, at least for low energies. This approach has also been used for a study of unimolecular fragmentation, resulting in the possibility of strongly non-exponential decay behaviour. The model systems investigated are, however, still quite far from typical molecular reality. 

It is also possible that the use of equation (2.25) to describe the internal relaxation may have some value for certain unimolecular reaction problems - for example, in cases where it is not admissible to assume exponential decay of the populations in the unperturbed relaxation -but I have not yet had time to examine such a reformulation to see whether any tractable results ensue. 

It is known that the thermal decay of these compounds at not very high concentrations (<10 mol/1) is a unimolecular reaction with activation energy of -120 kj/mol and pre-exponential factor of lO -lO s [52]. The kinetics of the 320 nm band decrease in the temperature range of 323 K to 346 K are also described by a first-order equation with the rate constant kj = 10 exp(-114 4// r)s- [50]. 

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