Reaction probabilities Boltzmann constant

Where AG is the activation energy of the process, and T are the Boltzmann constant and the absolute temperature, respectively, v is the nuclear frequency factor, and is the transmission coefficient, a parameter that expresses the probability of the system to evolve from the reactant to the product configuration once the crossing of the potential energy curves along the reaction coordinate has been reached (Fig. 17.5). 

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 ... 

Where k is the transmission factor, < x >xs is the average of the absolute value of the velocity along the reaction coordinate at the transition state (TS), and P = l/keT ( vhere ke is the Boltzmann constant and T the absolute temperature). The term AG designates the multidimensional activation free energy that expresses the probability that the system vill be in the TS region. The free energy reflects enthalpic and entropic contributions and also includes nonequilibrium solvation effects [4] and, as will be shown below, nuclear quantum mechanical effects. It is also useful to comment here on the common description of the rate constant as... 

The selected vibration-rotation states k of the QTS that contribute to the reaction are those having non-zero FC overlap integral. In the case where one sums over the k -states, let us replace AE by the average thermal energy available to the system, namely, k T. T is the absolute temperature and kB is the Boltzmann constant. The minimal time during which a transition is possible will be At = h / k T and we take this as the measure of time (clock) for the conversion from QTS to P1+P2. Then, let T32 be the total probability to convert the QTS into P1+P2, so that the ratio T3 2/At is the rate of conversion per QTS molecule. If [QTS] is the concentration of QTS molecules at time t, the rate measured in the laboratory will be proportional to [QTS] T3 JAt. A phenomenological rate is given by ... 

Thermal rate constants are obtained by Boltzmann averaging the reaction probability over the initial rotational and translational energy distribution of... 

The energetic factor is of fundamental importance in the determination of reaction probabilities and, hence, rate constants. Once more, we start from the MaxweU-Boltzmann disttibution (eq. 1.10) to calculate this factor. In practice, in collision theory, we normally... 

The rate constant in this expression can be interpreted loosely as some characteristic attempt frequency multiplied by a Boltzmann factor, which represents the probability of occupying the initial states that lie just above the top of the barrier. The Arrhenius law predicts that even for the lowest barrier still satisfying Eq. (1.1) the rate constant vanishes at sufficiently low temperature. For instance, even for a very fast reaction with k0 = 1013s-1, V0 = 1.2 kcal/mol, = 1012s-1 at 300 K, the rate constant decreases to 10-9s-1 at T = 10 K. Such a low value of k completely precludes the possibly of measuring any conversion on a laboratory time scale. 

On the other hand, the Boltzmann method of calculating the most probable distribution, used in the theoretical model, precludes an explicit consideration of actual values of transition probabilities (rate constants). This would only be possible if plasma reactions are considered as a multi-channel transport problem. However, the knowledge of a large number of various transition probabilities is necessary for such... 

AB, corresponding to the left and right sides of Equation (10.14). If the probability of complex existing (the relative amount of time the system is found in the AB state) is p, then the Boltzmann probability law says that the energy difference between the two possible states is —ksT In As we shall see, the probability p is a function of V, the volume of the system. We define Ka = which will be shown to be related to (but not equal to) the equilibrium constant for the reaction. 

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. 

We consider an ensemble of reactant molecules with quantized energy levels to be immersed in a large excess of (chemically) inert gas which acts as a constant temperature heat bath throughout the reaction. The requirement of a constant temperature T of the heat bath implies that the concentration of reactant molecules is very small compared to the concentration of the heat bath molecules. The reactant molecules are initially in a MaxweD-Boltzmann distribution appropriate to a temperature T0 such that T0 < T. By collision with the heat bath molecules the reactants are excited in a stepwise processs into their higher-energy levels until they reach "level (2V+1) where they are removed irreversibly from the reaction system. The collisional transition probabilities per unit time Wmn which govern the rate of transfer of the reactant molecules between levels with energies En and Em are functions of the quantum numbers n and m and can, in principle, be calculated in terms of the interaction of the reactant molecules with the heat bath. 

It is important to clarify here that the description of PT processes by curve crossing formulations is not a new approach nor does it provide new dynamical insight. That is, the view of PT in solutions and proteins as a curve crossing process has been formulated in early realistic simulation studies [1, 2, 42] with and without quantum corrections and the phenomenological formulation of such models has already been introduced even earlier by Kuznetsov and others [47]. Furthermore, the fact that the fluctuations of the environment in enzymes and solution modulate the activation barriers of PT reactions has been demonstrated in realistic microscopic simulations of Warshel and coworkers [1, 2]. However, as clarified in these works, the time dependence of these fluctuations does not provide a useful way to determine the rate constant. That is, the electrostatic fluctuations of the environment are determined by the corresponding Boltzmann probability and do not represent a dynamical effect. In other words, the rate constant is determined by the inverse of the time it takes the system to produce a reactive trajectory, multiplied by the time it takes such trajectories to move to the TS. The time needed for generation of a reactive trajectory is determined by the corresponding Boltzmann probability, and the actual time it takes the reactive trajectory to reach the transition state (of the order of picoseconds), is more or less constant in different systems. 

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