Boltzmann equilibrium, activation energy

In Eq. (1.36), Nj is the equilibrium number of point defects, N is the total number of atomic sites per volume or mole, Ej is the activation energy for formation of the defect, is Boltzmann s constant (1.38 x 10 J/atom K), and T is absolute temperature. Equation (1.36) is an Arrhenius-type expression of which we will see a great deal in subsequent chapters. Many of these Arrhenius expressions can be derived from the Gibbs free energy, AG. 

Any alteration in AG will thus affect the rate of the reaction. If AG is increased, the reaction rate will decrease. At equilibrium, the cathodic and anodic activation energies are equal (AG 0 = AG 0) and the probability of electron transfer will be the same in both directions. A, known as the frequency factor, is given as a simple function of the Boltzmann constant k and the Planck constant, h ... 

Arrhenius law (1889) describing the dependence of a chemical reaction rate constant on temperature T is one of the most fundamental laws of chemical kinetics. The law is based on the notion that reacting particles overcome a certain potential barrier with height E , called the activation energy, under the condition that the energy distribution of the particles remains in Boltzmann equilibrium relative to the environment temperature T. When these conditions are satisfied, the Arrhenius law states that the rate constant K is proportional to exp[ —E /Kgr], where Kg is the Boltzmann constant. It follows that, for E > 0, K tends to zero as T 0. 

More importantly, a molecular species A can exist in many quantum states in fact the very nature of the required activation energy implies that several excited nuclear states participate. It is intuitively expected that individual vibrational states of the reactant will correspond to different reaction rates, so the appearance of a single macroscopic rate coefficient is not obvious. If such a constant rate is observed experimentally, it may mean that the process is dominated by just one nuclear state, or, more likely, that the observed macroscopic rate coefficient is an average over many microscopic rates. In the latter case k = Piki, where ki are rates associated with individual states and Pi are the corresponding probabilities to be in these states. The rate coefficient k is therefore time-independent provided that the probabilities Pi remain constant during the process. The situation in which the relative populations of individual molecular states remains constant even if the overall population declines is sometimes referred to as a quasi steady state. This can happen when the relaxation process that maintains thermal equilibrium between molecular states is fast relative to the chemical process studied. In this case Pi remain thermal (Boltzmann) probabilities at all times. We have made such assumptions in earlier chapters see Sections 10.3.2 and 12.4.2. We will see below that this is one of the conditions for the validity of the so-called transition state theory of chemical rates. We also show below that this can sometime happen also under conditions where the time-independent probabilities Pi do not correspond to a Boltzmann distribution. 

Rapid (non-radiative) internal conversion leads to the lowest vibronic excitation level of the Si manifold. Subsequent transition from this level to one of the vibronic excitation levels of the So manifold is radiative and corresponds to either spontaneous or stimulated emission, SE. In terms of a simple model, stimulated emission is generated through the interaction of the excited molecules with other photons of equal energy. This process can only become important with, respect to other competitive processes, such as spontaneous emission, when the concentration of excited states is very high, i.e. when the population of the upper state exceeds that of the lower state, a situation denoted by the term population inversion. In other words, the Boltzmann equilibrium of states must be disturbed. Notably, the lasing transition relates to energy levels that are not directly involved in the optical pumping process. The laser potential of an active material is characterized by Eq. (6-3). 

We are using three different symbols for the energy distribution function /(E) denotes the Boltzmann (equilibrium) distribution function, 3(E) refers to an unspecified distribution function, and /)(E) describes the overall energy distribution of reactants in chemically activated systems. 

E is always greater than the activation energy E for ionization of water, and the trend in Table 2 may be understood if vibrationally excited molecules may either enter the decomposition channel or relax to Boltzmann equilibrium with other degrees of freedom. By combining RRK theory with an exponential relaxation model Zimmerman (7) showed that (I) should account for the variation of with A in such a situation. 

Evidence for strong non-equilibrium effects has first been obtained in the investigation of unimolecular reactions at low pressures. Here, the transition from first- to second-order kinetics is caused by perturbations of the equilibrium distribution of molecules over energies close to the activation energy (see Section V.17). Furthermore, it stimulated theoretical investigations on similar effects in bimole-cular reactions. However, the study of simple models has shown that non-equilibrium effects are not very marked and corresponding corrections to the equilibrium rate constants (i.e. rate constants calculated under the assumption of the Maxwell-Boltzmann distribution) are of the order of several per cent only [339]. Yet, this conclusion is based on the assumption that the reaction cross section depends solely on the translational energy which readily relaxes. 

Therefore, E can easily be calculated, plotting the experimentally measured In k versus 1/T. According to the simple collision theory, an act of chemical reaction can only occur if colliding molecules have the kinetic energy which exceeds the activation barrier, The frequency factor. A, is the number of collisions of reacting molecules per unit time. The exponential term in (2.21) determines a portion of those collisions which can lead to the chemical transformation. Note that (2.21) postulates the fulfillment of the Boltzmann equilibrium distribution of molecular energies in the reaction mixture. 

An interesting observation is that the decay of the excited planar form is clearly non exponential, while the decay of the twisted state is exponential, at least for "long" times, i.e. after the solvent has relaxed. Thus the deactivation of the twisted state is a "classical" reaction, because it occurs from a species which has had time enough to equilibrate with its surrounding. The twisted state corresponds to a local minimum in a potential-energy-versus nuclear-and-solvent-coordinates diagram. By contrast, the deactivation of the planar excited state is an example of ultrafast reaction for which the Boltzmann equilibrium is not reached (Note that the conversion to the twisted form involves almost no activation energy). 

The first condition is rather obvious it should be possible to actually define a smooth reaction coordinate, by which we mean a motion of the nuclei on an adiabatic electronic surface which leads from the reactants to the products. In addition, for that coordinate, an activation energy must be obtainable. Secondly, the reactants in the reactant well are supposed to be in an equilibrium Maxwell-Boltzmann distribution and remain so even though occasionally some of the reactants escape over the barrier. Basically, this means that equihbration of the reactants upon such a disturbance is fast enough so as not to be a rate-determining step. In that case, the probabihty of reaching the top of the barrier can be found from equihbrium... 

It is easy to see that the free energy of the activated complex, G, is higher than that of the initial state. K is the equilibrium constant of the activated complex, and k is the Boltzmann constant. Then, we can write equations describing both AG and the activated state as ... 

Various statistical treatments of reaction kinetics provide a physical picture for the underlying molecular basis for Arrhenius temperature dependence. One of the most common approaches is Eyring transition state theory, which postulates a thermal equilibrium between reactants and the transition state. Applying statistical mechanical methods to this equilibrium and to the inherent rate of activated molecules transiting the barrier leads to the Eyring equation (Eq. 10.3), where k is the Boltzmann constant, h is the Planck s constant, and AG is the relative free energy of the transition state [note Eq. (10.3) ignores a transmission factor, which is normally 1, in the preexponential term]. 

In complete equilibrium, the ratio of the population of an atomic or molecular species in an excited electronic state to the population in the groun d state is given by Boltzmann factor e — and the statistical weight term. Under these equilibrium conditions the process of electronic excitation by absorption of radiation will be in balance with electronic deactivation by emission of radiation, and collision activation will be balanced by collision deactivation excitation by chemical reaction will be balanced by the reverse reaction in which the electronically excited species supplies the excitation energy. However, this perfect equilibrium is attained only in a constant-temperature inclosure such as the ideal black-body furnace, and the radiation must then give -a continuous spectrum with unit emissivity. In practice we are more familiar with hot gases emitting dis-... 

Here the pre-exponential factor At is the product of a temperature-dependent constant (ksT/h) = 2 X 10 °Ts where ke and h are the Boltzmann and Planck constants, and a solvent-specific coefficient that relates to both the solvent viscosity and to its orientational relaxation rate. This coefficient may be near unity for very mobile solvent molecules but may be considerably less than unity for viscous or orientationally hindered highly stractured solvents. The exponential factor involves the activation Gibbs energy that describes the height of the barrier to the formation of the activated complex from the reactants. It also describes temperature and pressure dependencies of the reaction rate. It is assumed that the activated complex is in equilibrium with the reactants, but that its change to form the products is rapid and independent of its environment in the solution (de Sainte Claire et al., 1997). 

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