Rate constants of unimolecular reactions

Collision processes can result in the formation of the active molecule A (e), whose energy e on the internal degrees of freedom is higher than some threshold Eo value necessary for the decomposition of the molecule or rearrangement of atoms. Such active molecules can spontaneously undergo chemical transformations. These chemical reactions are named unimolecular reactions. 

The active molecules can be formed by collision processes of energy exchange (thermal activation), bimolecular association reactions of the type R + Ri - RR i(e) (chemical activation) and absorption of photons (photoactivation). 

The spontaneous unimolecular reaction obeys two laws conservation of energy and conservation of total angular moment J. The rate of spontaneotis unimolecular reactions is described by the expression 

The coefficient k( , J) is named the microscopic rate constant of unimolecular reaction. Sometimes this rate constant is named microcanonical because all states with equal e and J values are assumed to the equiprobable. If active molecules A ( e) are formed with some distribution fi[e, J) over the states with the energy e and angular moment J, the averaged rate constant k(8, J) is described by the equation 

It should be emphasized that, in principle, the Eo value (potential barrier) depends on J. 

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According to the transition state theory, the rate constant of unimolecular reaction (at high pressure in the gas phase) is the following [5] ... 

The rate constants of unimolecular reactions have the dimension per second (s" ). This means the process can happen that often per second, e.g., k = 6.3 x lO s being equivalent to 1.6 x 10 s per fragmentation on the average. It is important... 

Therefore a key problem is the separation of concentration and medium effects of micelles for bimolecular reactions, because if micelles affect rate constants of unimolecular reactions they could also affect those of bimolecular reactions. The... 

This computer program calculates the influence the pressure and the temperature have on the rate constants of unimolecular reactions and of association reactions. 

Fig. 26. Relative distribution (M) of pre-exponents Aqo in the rate constants of unimolecular reactions koo = Aqo exp (—E/RT). Curve plotted from data for 600 reactions [232]. M is given as percentage of the number of reactions studied...

QET is not the sole theory in the field indeed, there are several apparently competitive statistical theories for describing rate constants of unimolecular reactions [10,48]. However, none of these theories has been able to quantitatively describe all reactions of a given ion. QET, however, is well established and even in its simplified form allows sufficient insight into the behavior of isolated ions. Thus, we start out the chapter from the basic assumptions of QET. Along this scheme we will be led from the neutral molecule to ions, and from transition states and reaction rates to fragmentation products and thus, through the basic concepts and definitions of gas phase ion chemistry. 

This model idealizes the reaction ais motion on a one-dimensional harmonic potential surface, and describes solvent friction by means of a single (constant) parameter, thus it is clear that the chemical identity of the reactive solute and of the solvent is not taken into account. However, the principal merit of such a model is that it provides analytical ei ressions for rate constants of unimolecular reactions in solutions. These expressions provide inportant information on the influence of the solvent. 

In the real experiment, the consumption rate of A due to process 2 is often detected. The macroscopic rate constant of unimolecular reactions 1 for thermal reactions is determined as... 

Fig. 2.4. The plot of the rate constant of unimolecular reactions vs. concentration of the buffer gas M.

In Section 2.2.4 we obtained expression (2.76) for the rate constant on the basis of the activated complex meAod. This expression is applicable for the calculation of equilibrium rate constants of unimolecular reactions, in particular, K>(T) under the conditions when the temperature of reactants is constant (T = const). 

The macroscopic rate constant of monomolecular reactions can be calculated if k(e, J) and the distribution function f(8, J) are known. So, the rate constant of unimolecular reactions in the limit of high concentrations [M] can be calculated by formula (2.36) if the distribution functionMBtJ) is accepted in the form... 

As pointed out before kuni is a pseudo first order rate constant. Since kuni/[M] is independent of [M], kuni/[M] is a second order rate constant at low pressure. It is significant and important for consideration of isotope effects that this second order rate constant for unimolecular reactions depends only on the energy levels of reactant molecules A and excited molecules A, and on the minimum energy Eo required for reaction. It does not depend on the energy levels of the transition state. There will be further discussion of this point in the following section. 

Class (3) reactions include proton-transfer reactions of solvent holes in cyclohexane and methylcyclohexane [71,74,75]. The corresponding rate constants are 10-30% of the fastest class (1) reactions. Class (4) reactions include proton-transfer reactions in trans-decalin and cis-trans decalin mixtures [77]. Proton transfer from the decalin hole to aliphatic alcohol results in the formation of a C-centered decalyl radical. The proton affinity of this radical is comparable to that of a single alcohol molecule. However, it is less than the proton affinity of an alcohol dimer. Consequently, a complex of the radical cation and alcohol monomer is relatively stable toward proton transfer when such a complex encounters a second alcohol molecule, the radical cation rapidly deprotonates. Metastable complexes with natural lifetimes between 24 nsec (2-propanol) and 90 nsec (tert-butanol) were observed in liquid cis- and tra 5-decalins at 25°C [77]. The rate of the complexation is one-half of that for class (1) reactions the overall decay rate is limited by slow proton transfer in the 1 1 complex. The rate constant of unimolecular decay is (5-10) x 10 sec for primary alcohols, bimolecular decay via proton transfer to the alcohol dimer prevails. Only for secondary and ternary alcohols is the equilibrium reached sufficiently slowly that it can be observed at 25 °C on a time scale of > 10 nsec. There is a striking similarity between the formation of alcohol complexes with the solvent holes (in decalins) and solvent anions (in sc CO2). 

A major source of acceleration in enzymic reactions is approximation, that is to say, the bringing together of two or more reactants in the active site. Once the reagents are in contact, the subsequent reaction is intra- rather than intermolecular. Comparisons of the rates of intermolecular and intramolecular reactions are, however, difficult because the rate constants for bimolecular reactions have the units of M "1 s-1, whereas rate constants for unimolecular reactions have the units of s l. The best one can do in comparing them is to state the molarity at which the reactants would have to be present in the bimolecular reaction to achieve the rate of the unimolecular process when the effective molarity is large-say 1000 M or more-one has some measure of the power of approximation to accelerate chemical reaction. 

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. 

The measured rate constant for unimolecular reactions, association reactions, and certain bi-molecular reactions to be considered in the next section can have a complex dependence on total pressure, in addition to the strong temperature dependence of Eq. 9.83. This section introduces the theory of the pressure-dependence of the rate constant kmj the same theory follows to yield the pressure dependence of kassoc. Because kuni and kassoc are related by the equilibrium constant, which is independent of pressure, for a given reaction... 

The second point to be noted is that kf and kr cannot be assigned without a knowledge of the amplitude factor. This basic symmetry in the relaxation times occurs in many cases, and, in general, the rate constants for unimolecular reactions cannot be assigned unless the concentrations of A and B at equilibrium may... 

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... 

RRKM theory, an approach to the calculation of the rate constant of indirect reactions that, essentially, is equivalent to transition-state theory. The reaction coordinate is identified as being the coordinate associated with the decay of an activated complex. It is a statistical theory based on the assumption that every state, within a narrow energy range of the activated complex, is populated with the same probability prior to the unimolecular reaction. The microcanonical rate constant k(E) is given by an expression that contains the ratio of the sum of states for the activated complex (with the reaction coordinate omitted) and the total density of states of the reactant. The canonical k(T) unimolecular rate constant is given by an expression that is similar to the transition-state theory expression of bimolecular reactions. 

Hence, the pre-exponential factor of the rate constant for unimolecular reaction is equal, in order of magnitude, to the universal frequency of transition-state theory. This conclusion is supported by a vast amount of experimental data. Exceptions to this rule can be ascribed to important changes of structure taking place in the transition state. It is still usually difficult to foresee such exceptions. 

Harcourt reported that the observed rate constant of a reaction doubled with every 10° increase in temperature, and this trend is sometimes offered as a "rule of thumb" in kinetics. Use the Arrhenius equation to evaluate the validity of the "rule" for a unimolecular reaction occurring over temperature ranges from 0°C to 100°C. Does the accuracy of the generalization depend on the activation energy for the reaction ... 

The direct meastirements of the rate constants of the reaction of hydroxymethyl cyclohexadienyl radical with O2 has been made using the UV absorption method (Sect. 5.2.10), and the values of 2.5 x 10 cm molecule for benzene (Bohn and Zetzsch 1999 Grebenkin and Krasnoperov 2004 Raoult et al. 2004 Nehr et al. 2011), 6.0 x 10 cm molecule s for toluene (Knispel et al. 1990 Bohn 2001) are reported. Therefore, most hydroxymethyl cyclohexadienyl radicals are thought to react solely with O2. Also, the unimolecular decomposition rate of cyclohexadienyl radical from benzene back to CgHg + OH has been reported as (3.9 1.3) s at 298 K (Nehr et al. 2011). 

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 ... 

Now ku < 0.8 X 109 sec.-1, only slightly smaller than the upper limit 9 < 1.1 X 109 sec.-1 Apparently the unimolecular dissociation rate constants of all secondary complexes are less than ca. 5 X 107 sec.-1, those of the tertiary complexes less than 109 sec.-1, and those of the quaternary complexes probably of the order of 1010 sec.-1 These conclusions substantiate the view 16) that the mass spectrometrically observed tertiary ions arise predominantly from dissociation of the intermediate addition complexes C6Hi2+, C6Hn+, and C6Hi0+. Higher order ions, however, should arise principally from reactions of the dissociation products of the above complexes 62). 

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