The Rate of Unimolecular Processes

If the spontaneous emission of radiation of the appropriate energy is the only pathway for a return to the initial state, the average statistical time that the molecule spends in the excited state is called the natural radiative lifetime. For an individual molecule the probability of emission is time-independent and the total intensity of emission depends on the number of molecules in the excited state. In a system with a large number of particles, the rate of decay follows a first-order rate law and can be expressed as 

The radiative lifetime is nearly temperature independent, but depends to some extent on the environment. Thus, the solvent can influence the radiative lifetime either by means of its effect on the transition moment (cf. Section 2.7.2) or through its refractive index. (See Fdrster, 1951.) 

From the above estimate of the lifetime of a singlet state the rate constant of fluorescence kf - I/tq is obtained as kf = W-IO s , depending on the value of the oscillator strength /. Phosphorescence, on the other hand, is a spin-forbidden process with a much smaller rate constant, k - I0 -I0  

Each process competing with spontaneous emission reduces the observed lifetime x relative to the natural lifetime x . In the case where only unimo-lecular processes i with rate constant A , compete with emission, one has 

The rate of internal conversion (IC), a radiationless transition between isoenergetic levels of different states of the same multiplicity, may be of the same order of magnitude or even faster than vibrational relaxation. It depends, however, on the energy separation AEqo between the zero-vibrational levels of the electronic states involved (energy gap law, see Section 5.2.1). Similar relations hold for intersystem crossing transitions between states of different multiplicity, which are slower by 4-8 orders of magnitude. 

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Reactants AB+ + CD are considered to associate to form a weakly bonded intermediate complex, AB+ CD, the ground vibrational state of which has a barrier to the formation of the more strongly bound form, ABCD+. The reactants, of course, have access to both of these isomeric forms, although the presence of the barrier will affect the rate of unimolecular isomerization between them. Note that the minimum energy barrier may not be accessed in a particular interaction of AB+ with CD since the dynamics, i.e. initial trajectories and the detailed nature of the potential surface, control the reaction coordinate followed. Even in the absence (left hand dashed line in Figure 1) of a formal barrier (i.e. of a local potential maximum), the intermediate will resonate between the conformations having AB+ CD or ABCD+ character. These complexes only have the possibilities of unimolecular decomposition back to AB+ + CD or collisional stabilization. In the stabilization process,... 

In Section II. H and III we shall consider not only spectral intensities but also rates of unimolecular processes. If the operator T is interpreted as the transition operator, then S(E ) in (2.4) is (proportional to) the rate of... 

As in the case of unimolecular processes, it is certainly possible for a reaction to be neither second nor third order, and a more detailed model of the pressure dependence must be used. For example, the apparent rate constant can be expressed as a function of the air density (M) and the temperature T by the empirical Troe s relation (Troe, 1979)... 

Several experimental variables impact the relative rates of isomerization and nucleophilic attack in the allylic substitution reaction in Figure 14.15. For example, an increase in the rate of interconversion of the ir-allyl intermediates, such that the rate of this process is faster than nucleophilic attack, will change the enantioselectivity-determining step, because the diastereomeric complexes now equilibrate rapidly. Because isomerization of the diastereomeric ir-allyl complexes is a unimolecular process, but nucleophilic attack on the ir-allyl intermediates is a bimolecular process, reactions under dilute conditions will decrease the rate of nucleophilic attack, relative to the unimolecular ir-u-ir isomerization, and will help to achieve Curtin-Hammett conditions. 

The overall rate of deactivation is given by the sum of the rates of unimolecular and bimolecular processes... 

The qualitative features of reaction mechanisms in solutions are substantially different from those in gases. Unimolecular processes still occur via collisional activation. However, solvent molecules which can affect activation are always in high concentration. In terms of a rate law such as (5.22) the experiments are being carried out under conditions where kJ[M] kd or A [A] and A /[M] A +[A] here [M] represents the concentration of solvent, always in large excess. There is significant short-range order in liquids as solvent molecules are loosely bound to one another and form transient structures which reduce the mobility of the products of a decomposition. Thus the rate at which products separate by diffusion must limit the rate of unimolecular reaction in solution.Unimolecular decomposition may still be considered a two-step process ... 

Chain uncoiling, and the converse process of coiling, is conveniently considered as a unimolecular chemical reaction. It is assumed that the rate of uncoiling at any time after application of a stress is proportional to the molecules still coiled. The deformation Dhe(0 at tinie t after application of stress can be shown to be related to the equilibrium deformation Dhe( ) by the equation... 

When initiators are decomposed thermally, the rates of initiator disappearance (/rj) show marked temperature dependence. Since most conventional polymerization processes require that kj should lie in the range 10 6-1 O 5 s 1 (half-life ca 10 h), individual initiators typically have acceptable >fcd only within a relatively narrow temperature range (ca 20-30 °C). For this reason initiators are often categorized purely according to their half-life at a given temperature or vice For initiators which undergo unimolecular decomposition, the half-life is... 

The equation for the decay of a nucleus (parent nucleus - daughter nucleus + radiation) has exactly the same form as a unimolecular elementary reaction (Section 13.7), with an unstable nucleus taking the place of a reactant molecule. This type of decay is expected for a process that does not depend on any external factors but only on the instability of the nucleus. The rate of nuclear decay depends only on the identity of the isotope, not on its chemical form or temperature. 

Collisions at low ion energies (where Equation 1 can be applied) lead to a short-lived complex between the ion and the molecule—i.e., both collision partners move with the same linear velocity in the direction of the incident ion. The decay of the complex may be described by the theory of unimolecular rate processes if its excess energy can fluctuate between the various internal degrees of freedom. For example, the isotope effect in the reaction of Ar+ with HD may be explained by the properties of... 

Inspection of the death curves obtained from viable count data had early ehcited the idea that because there was usually an approximate, and under some circumstances a quite excellent, linear relationship between the logarithm of the number of survivors and time, then the disinfection process was comparable to a unimolecular reaction. This imphed that the rate of killing was a function of the amount of one of the participants in the reaction only, i.e. in the case of the disinfection process the number of viable cells. From this observation there followed the notion that the principles of first-order... 

Many molecules of interest to chemistry and biology contain H atoms. For the unimolecular dissociation of the excited states of such molecules, it is instructive to think of elimination of a particle or a group of particles. Thus, we can think of elimination of an electron, an H atom, an H2 molecule, an alkane group (in suitable cases), and so forth. Other factors remaining the same, the rate of the process may... 

If radicals are produced in the reactions of unimolecular hydroperoxide decomposition and the reaction of ROOH with hydrocarbon whose concentration at the initial stages of oxidation is virtually constant, the production of radicals from ROOH can be regarded as a pseudo-monomolecular process occurring at the rate V = [ROOH] = + iRH[RH]). The... 

Now, let us discuss the rate equations embodied in eq.(74). To do this, there is need of a statistical analysis. If the system is kept coupled to a thermostat at absolute temperature T, and assuming that w(i - >if) contains effects to all orders in perturbation theory, the rate of this unimolecular process per unit (state) reactant concentration k + is obtained after summation over the if-index is carried out with Boltzman weight factors p(if,T) ... 

If the surface complex is the chromophore, then the photochemical reductive dissolution occurs as a unimolecular process alternatively, if the bulk iron(III)(hydr)-oxide is the chromophore, then it is a bimolecular process. Irrespective of whether the surface complex or the bulk iron(IIl)(hydr)oxide acts as the chromophore, the rate of dissolved iron(II) formation depends on the surface concentration of the specifically adsorbed electron donor e.g. compare Eqs. (10.11) and (10.18). It has been shown experimentally with various electron donors that the rate of dissolved iron(II) formation under the influence of light is a Langmuir-type function of the dissolved electron donor concentration (Waite, 1986). 

Thus it has been possible to show that in the bromination of acetone, a process which has been found to be unimolecular, not the normal keto-form, but the tautomeric enol-form reacts. The enol-form is present, in equilibrium with the keto-form, in amount too small to be measured. As soon as this amount has reacted a further quantity is formed and the process is repeated. That the reaction is unimolecular follows from the fact that it is the rate of rearrangement (I) which is measured, whilst the reaction of the enol with bromine (II) takes place with immeasurable rapidity (Lapworth). 

This effect, in and of itself, tends to increase the yield of tar (and therefore of total volatiles), for the reason discussed earlier. However, increasing the ambient pressure also shifts the vapor-liquid equilibrium of the tar species to smaller tar species (with higher vapor pressures) and thus tends to diminish the overall release of tar. Wire-mesh experiments with well-controlled particle heating rates show a significant reduction in the yield of tar and total volatiles as the pressure is increased. The rate of devolatilization, however, is nearly insensitive to pressure, as would be expected for unimolecular reaction processes. 

It is convenient initially to classify elementary reactions either as energy-transfer-limited or chemical reaction-rate-limited processes. In the former class, the observed rate corresponds to the rate of energy transfer to or from a species either by intermolecular collisions or by radiation, or intramolecular-ly due to energy transfer between different degrees of freedom of a species. All thermally activated unimolecular reactions become energy-transfer-limited at high temperatures and low pressures, because the reactant can receive the necessary activation energy only by intennolecular collisions. 

In order to measure the magnitude of the chemical interactions between various ions and buffer gases, approaches that are based on the measurements of either equilibrium or rate constants for ionic processes can be envisioned. An example of a kinetic method is described in the following. The unimolecular kinetic process known as thermal electron detachment (TED) for negative ions (NT -> M + e), should be particularly sensitive to a chemical effect of the buffer gas. This is because the rate of TED will be given by = constant x where the electron... 

Elementary reactions (also termed monomolecular reactions) that involve only a single entity in the formation of an activated complex. Unimolecular rate constants, k, are concentration-independent and are typically expressed in units of sUnimolecular reactions are expected to be first order (i.e., -dc/dt = kc where c is the concentration and t is time). Examples of unimolecular processes include radioactive disintegrations, isomeriza-tions, disassociations, and decompositions. Reactions in solution are unimolecular only if the solvent is not covalently incorporated into the product(s). 

In spreading the drop does not extend as a homogeneous oil phase in the form of a thin lamina but takes place by a process of superficial solution of the oil in a thin film (Hardy s primary layer) which we shall note is unimolecular in thickness consideration of the mechanism of such spreading will be deferred to a subsequent section. If the area of the water on which the drop is placed is limited in extent the oil continues to spread until the rate of superficial solution is balanced by the rate of return of molecules from the surface film into the lens. The oil of the lens may be regarded... 

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