Thermal equilibrium the Boltzmann distribution

It is possible to solve Eq. (1.10) numerically for the nuclear motion associated with chemical reactions and to calculate the reaction probability including detailed state-to-state reaction probabilities (see Section 4.2). However, with the present computer technology such an approach is in practice limited to systems with a small number of degrees of freedom. 

For practical reasons, a quasi-classical approximation to the quantum dynamics described by Eq. (1.10) is often sought. In the quasi-classical trajectory approach (discussed in Section 4.1) only one aspect of the quantum nature of the process is incorporated in the calculation the initial conditions for the trajectories are sampled in accord with the quantized vibrational and rotational energy levels of the reactants. 

Obviously, purely quantum mechanical effects cannot be described when one replaces the time evolution by classical mechanics. Thus, the quasi-classical trajectory approach exhibits, e.g., the following deficiencies (i) zero-point energies are not conserved properly (they can, e.g., be converted to translational energy), (ii) quantum mechanical tunneling cannot be described. 

Finally, it should be noted that the motion of the nuclei is not always confined to a single electronic state (as assumed in Eq. (1.5)). This situation can, e.g., occur when two potential energy surfaces come close together for some nuclear geometry. The dynamics of such processes are referred to as non-adiabatic. When several electronic states are in play, Eq. (1.10) must be replaced by a matrix equation with a dimension given by the number of electronic states (see Section 4.2). The equation contains coupling terms between the electronic states, implying that the nuclear motion in all the electronic states is coupled. 

Statistical mechanics gives the relation between microscopic information such as quantum mechanical energy levels and macroscopic properties. Some important statistical mechanical concepts and results are summarized in Appendix A. Here we will briefly review one central result the Boltzmann distribution for thermal equilibrium. 

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Eq. (4.28) implies that the absorption band-shape for m -> n is Lorentzian with half-width ynm. In thermal equilibrium, the Boltzmann distributions can be used for and In this case Eq. (4.28) becomes... 

At thermal equilibrium, the Boltzmann distribution determines the populations in various energy levels. For any two quantum states, the ratio of populations between the higher energy state and the lower energy state at equilibrium will always be ... 

In a chemical reaction system at thermal equilibrium, the Boltzmann distribution of molecular energies must be taken Into account In obtaining the average transmission probability. The tunneling factor Is tisually defined as the ratio of this averaged transmission probability to that obtained with the classical values... 

In thermal equilibrium, the Boltzmann distribution ultimately determines the macroscopic behaviour. In most cases one may avoid the issue of probability distribution but instead may concentrate only on the first few, may be the first two, moments or cumulants. For example, one needs to know the average energy, entropy etc and the various response... 

Specialized to thermal equilibrium, the velocity distributions for the molecules are the Maxwell-Boltzmann distribution (a special case of the general Boltzmann distribution law). The expression for the rate constant at temperature T, k(T), can be reduced to an integral over the relative speed of the reactants. Also, as a consequence of the time-reversal symmetry of the Schrodinger equation, the ratio of the rate constants for the forward and the reverse reaction is equal to the equilibrium constant (detailed balance). 

Here, we will compare the threshold line model with the TCE chemistry model. These two DSMC chemistry models were used to calculate dissociation rate coefficients under conditions where the translational, rotational, and vibrational modes are in thermal equilibrium with Boltzmann distributions. An isothermal heat bath is simulated that consists of 100 000 particles. Energy is exchanged between the various modes during collision, but chemical reactions are not processed. Instead, the average dissociation probability is evaluated over all collisions and then converted into a rate coefficient. Results for O2-O2 dissociation are shown in Fig. 5. The measured rate is that reported by Byron ... 

Let us now look for relations between the three Einstein coefficients Bn, B21, and A21. The total number (V of aU molecules per unit volume is distributed among the various energy levels Ei of population density IV, such that J], Ni = N. AA thermal equilibrium the population distribution Ni Ei) is given by the Boltzmann distribution... 

Standardizing the Method Equation 10.34 shows that emission intensity is proportional to the population of the excited state, N, from which the emission line originates. If the emission source is in thermal equilibrium, then the excited state population is proportional to the total population of analyte atoms, N, through the Boltzmann distribution (equation 10.35). 

When nuclei with spin are placed in a magnetic field, they distribute themselves between two Zeeman energy states. At thermal equilibrium the number (N) of nuclei in the upper (a) and lower (j8) states are related by the Boltzmann equation (1) where AE=E — Ep is the energy difference between the states. In a magnetic field (Hq), E = yhHo and... 

As long as the system is in thermal equilibrium the individual molecules in a sample are distributed among the 15 accessible states according to Boltzmann statistics ... 

Consider a simple two energy level system, e.g., an S - 1/2 system. For a sample of realistic size at thermal equilibrium n0 molecules are in the lowest energy state and / in the highest state according to the Boltzmann distribution... 

Transition State Theory [1,4] is the most frequently used theory to calculate rate constants for reactions in the gas phase. The two most basic assumptions of this theory are the separation of the electronic and nuclear motions (stemming from the Bom-Oppenheimer approximation [5]), and that the reactant internal states are in thermal equilibrium with each other (that is, the reactant molecules are distributed among their states in accordance with the Maxwell-Boltzmann distribution). In addition, the fundamental hypothesis [6] of the Transition State Theory is that the net rate of forward reaction at equilibrium is given by the flux of trajectories across a suitable phase space surface (rather a hypersurface) in the product direction. This surface divides reactants from products and it is called the dividing surface. Wigner [6] showed long time ago that for reactants in thermal equilibrium, the Transition State expression gives the exact... 

At thermal equilibrium, according to the Boltzmann distribution (see Section 2.3), the population of atoms and molecules in the excited state can never exceed the population in the ground state for a simple two-level system. 

At thermal equilibrium the population of any series of energy levels is described by the Boltzmann distribution law. If N0 molecules are in the ground state then the number Ni in any higher energy level is given by the equation ... 

We will consider dipolar interaction in zero field so that the total Hamiltonian is given by the sum of the anisotropy and dipolar energies = E -TEi. By restricting the calculation of thermal equilibrium properties to the case 1. we can use thermodynamical perturbation theory [27,28] to expand the Boltzmann distribution in powers of This leads to an expression of the form [23]... 

The Boltzmann distribution describes the relative populations of different states al thermal equilibrium. If equilibrium exists (which is not true in the blue cone of a flame but is probably true above the blue cone), the relative population (N /N0) of any two states is... 

The Boltzmann distribution applies to a system at thermal equilibrium. 

All the work just mentioned is rather empirical and there is no general theory of chemical reactions under plasma conditions. The reason for this is, quite obviously, that the ordinary theoretical tools of the chemist, — chemical thermodynamics and Arrhenius-type kinetics - are only applicable to systems near thermodynamic and thermal equilibrium respectively. However, the plasma is far away from thermodynamic equilibrium, and the energy distribution is quite different from the Boltzmann distribution. As a consequence, the chemical reactions can be theoretically considered only as a multichannel transport process between various energy levels of educts and products with a nonequilibrium population20,21. Such a treatment is extremely complicated and - because of the lack of data on the rate constants of elementary processes — is only very rarely feasible at all. Recent calculations of discharge parameters of molecular gas lasers may be recalled as an illustration of the theoretical and the experimental labor required in such a treatment22,23. ... 

For reactants in complete thermal equilibrium, the probability of finding a BC molecule in a specific quantum state, n, is given by the Boltzmann distribution (see Appendix A.l). Thus, in the special case of non-interacting molecules the probability PBC(n)y °f finding a BC molecule in the internal (electronic, vibrational, and rotational) quantum states with energy En is... 

The reactants as well as the activated complex are assumed to be distributed among their states in accordance with the Boltzmann distribution. This also implies (see Appendix A. 1.2) that the concentration of the activated complex molecules is related to concentrations of the reactants by a thermal equilibrium constant. 

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