Non-equilibrium distribution

By examining the expression for Q ( equation (B1.16.4)). it should now be clear that the nuclear spin state influences the difference in precessional frequencies and, ultimately, the likelihood of intersystem crossing, tlnough the hyperfme tenn. It is this influence of nuclear spin states on electronic intersystem crossing which will eventually lead to non-equilibrium distributions of nuclear spin states, i.e. spin polarization, in the products of radical reactions, as we shall see below. 

One such systematic generalization was obtained by Cohen,8 whose method is now given the point of departure was the expansion in clusters of the non-equilibrium distribution functions. This procedure is formally analogous to the series expansion in the activity where the integrals of the Ursell cluster functions at equilibrium appear in the coefficients. Cohen then obtained two expressions in which the distribution functions of one and two particles are given in terms of the solution of the Liouville equation for one particle. The elimination of this quantity between these two expressions is a problem which presents a very full formal analogy with the elimination (at equilibrium) of the activity between the Mayer equation for the concentration and the series... 

The point of departure of this method is the "cluster expansion of the non-equilibrium distribution functions ... 

X can be regarded as the local non-equilibrium distribution of the species a at a position rj. As the system approaches equilibrium, x becomes -small. In a like manner, the doublet density fjf3 can be decomposed into an equilibrium term, two contributions from the local non-equilibrium departures of the particle a and (3 separately and then their mutual effect... 

From frequency dependent dielectric loss measurements, the transitions associated with solvent dipole reorientations occur on a timescale of 10-n -10-13 s. By contrast, the time response of the electronic contribution to the solvent polarization is much more rapid since it involves a readjustment in electron clouds . The difference in timescales for the two types of polarization is of paramount importance in deciding what properties of the solvent play a role in electron transfer. The electronic component of the polarization adjusts rapidly and remains in equilibrium with the charge distribution while electron transfer occurs. The orientational component arising from solvent dipoles must adopt a non-equilibrium distribution before electron... 

Spectroscopic methods naturally focus attention on the distribution of products over internal energy states. It seems clear that virtually every elementary reaction produces products in a non-equilibrium distribution. Even phase space models3, which assume a strong coupling complex in a certain region of phase space, give... 

When the sample is excited, a photoinduced non-equilibrium distribution of the population induces a change in T and thus also in A ... 

The weak bond model assumes a non-equilibrium distribution of weak bonds arising from the disorder of the a-Si H network. It has been proposed that the shapes of the band tails are themselves a consequence of thermal equilibrium of the structure (Bar-Yam, Adler and Joannopoulos 1986). The formation energy of a tail state is assumed proportional to the difference in the one-electron energies, so that the energy, required to create a band tail state of energy Ey from the valence band mobility edge is... 

The above considerations on drug partitioning mainly apply to the equilibrium of drug distribution. However, it is important to realize that it may take some time until a drug that is applied rapidly (e.g., by injection or inhalation) actually reaches equilibrium. A practically important example of non-equilibrium distribution is provided by the drug thiopental, which is a barbiturate used for short-duration narcosis (Figure 2.13). 

Diffusion. The primary step results in the establishment of a non-equilibrium distribution of defects between the surface and the interior of the solid phase. The concentration gradient established is the driving force of the second step, that of diffusion. For the reaction to proceed at a significant rate the defects must be mobile in the bulk lattice, and this usually requires a large activation energy temperatures above the Tammann point must be attained. 

Thus the equilibrium ensemble has the minimum softness, or the maximum hardness. This result is valid for all nearly non-equilibrium distributions which obey Equations (4.8) and (4.7). Presumably it includes all cases of interest, since these are common assumptions in linear response theory. 

A more general case is that of a non-equilibrium distribution that is close to an equilibrium distribution. The parameters p, 77, E, I, v and A are all allowed to change, but (A) = N° is maintained. It is then found that d— (cr) increases as E — E increases, where the overbars indicate averages over the nonequilibrium ensemble. An interesting restriction is that /lA > i.e., the... 

Section 4 will be devoted to polyatomic molecules. A selection of papers is made to illustrate the point of view of the authors on the influence of non equilibrium distributions on the rates of plasmochemical dissociation reactions. The material presented has mostly been derived from the authors group investigations directly related to it. 

Non-Equilibrium Distribution of Adsorbing Ions Along the Diffuse Layer and Quasi-Equilibrium Distribution of Counterions... 

These results have been explained in terms of two models in which account is taken of a non-equilibrium distribution over vibrational states the truncated harmonic oscillator and the Morse oscillator with all transitions allowed [83]. The dissociation may take place from any vibrational level. It is shown that as the temperature is increased, the contribution to the decomposition process from the high vibrational levels is severely diminished and it is the lower states that make the major contribution. It is the reduction in the number of reactive states that is... 

A third model, a truncated harmonic oscillator in which dissociation taRes place only from the top vibrational level, leads to values of the activation energy that increase with temperature. None of the three models exhibits sufficient variance in the activation energy with increasing temperature when an equilibrium condition is imposed on all vibrational levels. Thus, two of the necessary ingredients for a successful explanation of low activation energies appear to be dissociation events from all vibrational levels and a non-equilibrium distribution of reactant molecules among the vibrational states. 

Pina, CM., Putnis, A. (2002). The kinetics of nucleation of sohd solutions from aqueous solutions A new model for calculating non-equilibrium distribution coefficients. Geochimica et Cosmochimica Acta, 66, 185-192. 

Let us assume in equation (1.41) that the values c, x and a have the characteristic relaxation times t, and U, respectively. Here, t should be as same as particle interaction time t r y. If we will also assume that in the initial indignant state of a chemical system a non-equilibrium distribution of energy takes place, then values k and V in equation (1.41) should be considered as variables with the characteristic relaxation time ti, the same as the time of relaxation of a single-partial function to an equilibrium Maxwell-Boltzmann distribution of energy. 

The influence of non-equilibrium distributions, gas compressibility and space inhomogeneity on the reaction rates for different processes is discussed. 

The solution (54) represents the local equilibrium Maxwell-Boltzmann distribution over the velocity and rotational energy levels with the temperature T and strongly non-equilibrium distribution over chemical species and vibrational energy levels. The distribution functions... 

In this Chapter, the theoretical models for non-equilibrium chemical kinetics in multi-component reacting gas flows are proposed on the basis of three approaches of the kinetic theory. In the frame of the one-temperature approximation the chemical kinetics in thermal equilibrium flows or deviating weakly from thermal equilibrium is studied. The coupling of chemical kinetics and fluid dynamics equations is considered in the Euler and Navier-Stokes approximations. Chemical kinetics in vibrationaUy non-equilibrium flows is considered on the basis of the state-to-state and multi-temperature approaches. Different models for vibrational-chemical coupling in the flows of multi-component mixtures are derived. The influence of non-equilibrium distributions on reaction rates in the flows behind shock waves and in nozzle expansion is demonstrated. 

Section 8.2.2 deals with radical-molecule reactions. Particularly relevant in this area is a recent report [81Skel] indicating that two different forms (cr and a) of the acetoxyl radical may exist, and that their interconversion may be slow enough that they can react from non-equilibrium distributions. Would these ideas prove of general applicability, then, some of the data in this Section (and in 8.2.1 as well) may require critical reevaluation. This would be particularly true of photogenerated radicals. Studies of this problem are at present at a very preliminary stage and their relevance in connection with the data presented in this Section cannot be truly assessed. It should also be noted that most of the data for acetoxyl refers to reactions that occur within the primary solvent cage. 

Consider now the relaxation of the mean energy using Eq. (8.36). It is expressed by the non-equilibrium distribution... 

When condition (dai/dt)react (dai/dt)rei is not satisfied, the perturbation of equilibrium distribution is substantial. However, in this case realization of the quasi-steady-state condition is possible, provided the overall rate dai/dt is low compared to partial rates (daj/dt)rei and (dai/dt)i.eact Then, the microscopic kinetic equation can be solved by the quasi-steady-state approximation. The approximation implies that the non-equilibrium distribution functions depend on time implicitly via the total concentration of reactants rather than explicitly. This also means that the macroscopic reaction rates are low compared to microscopic reaction and relaxation rates. Since the distribution functions in this approximation depend on the total concentration only, the reaction rates, according to Eq. (8.50) also depend on the total concentration. Hence, we come to macroscopic kinetic equations that involve only the total concentration of reactants and certain combinations of microscopic rate constants that have the meaning of macroscopic constants. Note that these macroscopic equations need not be consistent with the macroscopic kinetic law as, besides elementary reactive processes, they involve unreactive processes. 

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