Factor probability

A typical value of the collision number is 10 °s in gases at one atmosphere pressure and room temperature, and the number of successful collisions which can bring about the chemical reaction is equal to this number multiplied by the Anhenius or probability factor, exp(— /f 7 ), where E is the activation energy, the critical collision energy needed for reaction to occur. 

The sequences involving the i-th component may be separated from the other sequences and this probability factored from the sum of products as shown in equation 2.8-9, where is the cutset with factored out. This property relates the importance measures. Substituting equation 2.8-9 into equation 2.8-2 gives equation... 

The perturbing step may be taken either from the last minimum found, or from all the previous found minima, weighted by a probability factor which is such that low energy minima are used more often than high energy structures. 

Finally, the probability factor rj, which is taken to be coverage-independent in the model of a homogeneous surface with no lateral interactions between adsorbed particles, will be expressed by means of the Arrhenius formalism based on the Boltzmann distribution, viz. 

The pre-exponential factor for the H -i- H2 reaction has been determined to be approximately 2.3 x lO " mol cm s . Taking the molecular radii for H2 and H to be 0.27 and 0.20 nm, respectively, calculate the value of the probability factor P necessary for agreement between the observed rate constant and that calculated from collision theory at 300 K. 

It is sometimes informative to separate AG into hypothetical enthalpic and entropic terms, and then the Arrhenius factors may be related to the transition state activation parameters by Eqs 10.4 and 10.5. Thus, the Arrhenius activation energy can be approximately related to the potential energy of a transition state, and the preexponential A value includes probability factors. 

Perry, R.L., and J.J. Brady Changes in photoelectric probability factor... 

Hinshelwood (51) used reasoning based on statistical mechanics to show that the energy probability factor in the kinetic theory expressions (e E,RT) is strictly applicable only to processes for which the energy may be represented in two square terms. Each translational and rotational degree of freedom of a molecule corresponds to one squared term, and each vibrational degree of freedom corresponds to two squared terms. If one takes into account the energy that may be stored in 5 squared terms, the correct probability factor is... 

For the case where S = 2 this expression reduces to the simple exponential form of Arrhenius. For values of S greater than 2, it yields a much larger probability of reaction than one would obtain from the normal Arrhenius form. The enhancement may be several orders of magnitude. For example, when S = 10 and E/RT = 30, the ratio of the probability factor predicted by Hinshelwood s approach to that predicted by the conventional Arrhenius method is (30)4/4 = 3.375 x 104. The drawback of the approach is that one cannot accurately predict S a priori. When one obtains an apparent steric factor in excess of unity, this approach can often be used in interpretation of the data. 

Pilbrow, J.R. 1969. Anisotropic transition probability factor in E.S.R. Molecular Physics 16 307-309. 

The description of the product distribution for an FT reaction can be simplified and described by the use of a single parameter (a value) determined from the Anderson-Schulz-Flory (ASF) plots. The a value (also called the chain growth probability factor) is then used to describe the total product spectrum in terms of carbon number weight fractions during the FT synthesis. In the case... 

As indicated in Figure 10.2, there is a distinct change in the slope of the line at carbon numbers 8 to 12, and this has also been observed by other researchers.2-3 This change in the slope cannot be explained by the ASF model, which is based on the premise that the chain growth probability factor (a) is independent from the carbon number. Some further developments of the ASF model by Wojciechowski et al.3 made use of a number of abstract kinetic parameters for the calculation of a product spectrum. Although it still predicts a straight line for the a plot, they suggested that the break in the line is due to different mechanisms of chain termination and could be explained by the superposition of two ideal distributions. This bimodal distribution explained by two different mechanisms... 

Why must the invariants be absorbed into the action specs They must be absorbed because it s the pre- and postconditions that are the real behavioral spec An invariant is only a way of factoring out common assumptions made by all those within its own context. Outside that context, the same common assumptions might not apply (see Section 3.5.5), so we must make them specific in any pre- and postcondition we want to move out of the context. After all the actions have been combined, you can probably factor out a common invariant that occurs in all the combined operation specs. 

If we consider that equimolar quantities of reactants AA and BB react and that the reactivity of groups A or B remains unchanged at all stages of conversion. Then the degree of polymerisation changes with the extent of polymerisation that has occurred and the probability factor can give an idea about the Molecular weight distribution at a particular conversion p as under. 

On considering the collision frequency, energy and the probability factors discussed above, the rate of reaction by collision theory is given as... 

No attempt has been made to correlate the value of probability factor p with the structure and properties of the reacting molecules and it is also not possible to interpret the abnormally high rates that we sometimes observe. The value of p has been found to vary from unity to 10 9 when we move from simple atomic reaction to reactions involving complex molecules. There is no explanation in terms of collision theory for such a wide variation in the value of p. 

Entropy of activation (AA ), also called the probability factor. Not all collisions between molecules possessing the requisite AW result in reaction. Often collisions between molecules must also occur in a certain orientation, reflected by the value of AA. The more organized or less random the required orientation of the colliding molecules, the lower the entropy of activation and the slower the reaction. 

Reactivity of X. The more reactive Cl- is less selective and more influenced by the probability factor. The less reactive Br- is more selective and less influenced by the probability factor. As summarized by the reactivity-selectivity principle If the attacking species is more reactive, it will be less selective, and the yields will be closer to those expected from the probability factor. 

Calculate the theoretical % yield based only on the probability factor and then compare the theoretical and observed % yields. 

The high yield of cyclopropane indicates that a favorable probability factor outweighs the ring instability. For rings with more than six C s the ring stability effect is outweighed by the highly unfavorable probability... 

One can write the rate of the electron-transferprocess as proportional to the probability factors formulated originally by Gurney. Thus one obtains... 

This involves the diffusion coefficients DA and DB for the two colliding partners and the encounter radii RAB = aA + Cb sum of the interaction radii (Figure 6.1), p is a probability factor per encounter. The diffusion coefficients are given by the Stokes-Einstein equation... 

On further assumption that the interaction radius and the kinetic radius are equal, i.e. a = r, and the probability factor p = 1, we have the final form of the equation for efficient reaction,... 

We shall restrict our considerations here to physical systems in which the joint probabilities factorize if the times. .., tr are each widely separated from each other. Thus, e.g., p2 xu t2 x2, t2) j-rZ, p2(xu t p2 x2, oo) this implies that... 

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