Distribution equations

In the above discussion it was assumed that the barriers are low for transitions between the different confonnations of the fluxional molecule, as depicted in figure A3.12.5 and therefore the transitions occur on a timescale much shorter than the RRKM lifetime. This is the rapid IVR assumption of RRKM theory discussed in section A3.12.2. Accordingly, an initial microcanonical ensemble over all the confonnations decays exponentially. However, for some fluxional molecules, transitions between the different confonnations may be slower than the RRKM rate, giving rise to bottlenecks in the unimolecular dissociation [4, ]. The ensuing lifetime distribution, equation (A3.12.7), will be non-exponential, as is the case for intrinsic non-RRKM dynamics, for an mitial microcanonical ensemble of molecular states. 

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 conditions (2.195) are satisfied we also have the following distribution equations ... 

Since in most practical circumstances at temperatures where vapour transport is used and at around one atmosphere pressure, die atomic species play a minor role in the distribution of atoms, it is simpler to cast the distribution equations in terms of the elemental molecular species, H2, O2 and S2, tire base molecules, and the derived molecules H2O, H2S, SO2 and SO3, and eliminate any consideration of the atomic species. In this case, let X, be tire initial mole fraction of each atomic species in the original total of atoms, aird the variables Xi represent the equilibrium number of each molecular species in the final number of molecules, N/. Introducing tire equilibrium constants for the formation of each molecule from tire elemental atomic species, with a total pressure of one aurros, we can write... 

The particle size distribution of ball-milled metals and minerals, and atomized metals, follows approximately the Gaussian or normal distribution, in most cases when the logarithn of die diameter is used rather dran the simple diameter. The normal Gaussian distribution equation is... 

Analytieal solutions to equation 4.32 for a single load applieation are available for eertain eombinations of distributions. These coupling equations (so ealled beeause they eouple the distributional terms for both loading stress and material strength) apply to two eommon eases. First, when both the stress and strength follow the Normal distribution (equation 4.38), and seeondly when stress and strength ean be eharaeterized by the Lognormal distribution (equation 4.39). 

In eertain instanees, the eoneentrations of reaetion partieipants in the rate and produet distribution equation are expressed in terms... 

The Reactor Safety Study extensively used the lognormal distribution (equation 2.5-6) to represent the variability in failure rates. If plotted on logarithmic graph paper, the lopnormal distribution is normally distributed. 

The Bayes conjugate is the gamma prior distribution (equation 2.6-11). When equations 2.6-9 and... 

To obtain the confidence bounds, the posterior distribution (equation 2.6-12) is integrated from zero to A, , where A is the upper... 

A lrLL[uently encountered problem requires estimating a failure probability based on the number of failures, M, in N tests. These updates are assumed to be binomially distributed (equation 2.4-10) as p r N). Conjugate to the binomial distribution is the beta prior (equation 2.6-20), where / IS the probability of failure. 

In the introduction to this section, two differences between "classical" and Bayes statistics were mentioned. One of these was the Bayes treatment of failure rate and demand probttbility as random variables. This subsection provides a simple illustration of a Bayes treatment for calculating the confidence interval for demand probability. The direct approach taken here uses the binomial distribution (equation 2.4-7) for the probability density function (pdf). If p is the probability of failure on demand, then the confidence nr that p is less than p is given by equation 2.6-30. 

If the system distribution is the product of the component distributions (equation 2.7-23), the mean is given by equation 2.7-24 which, for uncorrelated variables becomes equation 2.7-25... 

The reason for calling equation 8.3-1 a "Gaussian diffusion model" is because it has the form of the normal/Gaussian distribution (equation 2.5-2). Concentration averages for long time intervals may be calculated by averaging the concentrations at grid elements over which the plume passes. 

So far we have discussed mainly stable configurations that have reached an equilibrium. What about the evolution of a system from an arbitrary initial state In particular, what do we need to know in order to be assured of reaching an equilibrium state that is described by the Boltzman distribution (equation 7.1) from an arbitrary initial state It turns out that it is not enough to know just the energies H ct) of the different states a. We also need to know the set of transition probabilities between ail pairs of states of the system. 

One can actually prove a stronger result all nondeterministic LG models that satisfy semi-detailed balance and possess no spurious conservation laws have universal equilibrium solutions whose mean populations are given by the Fermi-Dirac distribution (equation 9.93) [frishc87]. 

A critical difference between the transient and CW measurements is that while the CW probe source uniformly illuminates the sample, both the transient pump and probe beams have Gaussian distributions. Equation (7.7) can be rewritten for the transient case as ... 

Bibliography, of element determinations, 328-331 of x-ray literature, 40, 41 Binomial distribution, equation and discussion, 271-273... 

Figure 10.7 The population of excited energy states relative to that of the ground state for the CO molecule as predicted by the Boltzmann distribution equation. Graph (a) gives the ratio for the vibrational levels while graph (b) gives the ratio for the rotational levels. Harmonic oscillator and rigid rotator approximations have been used in the calculations. The dots represent ratios at integral values of v and 7. which are the only allowed values.

Suppose is a function of a alone and that neither dSt Ajda nor d Alda change sign over the range of concentrations encountered in the reactor. Then, for a system having a fixed residence time distribution. Equations (15.48) and (15.49) provide absolute bounds on the conversion of component A, the conversion in a real system necessarily falling within the bounds. If d S A/dc > 0, conversion is maximized by maximum mixedness and minimized by complete segregation. If d 0i A/da < 0, the converse is true. If cf- A/da = 0, micro-mixing has no effect on conversion. 

The size distribution equations rest squarely on the validity of the principle of equal reactivity. Should they be proved appreciably in error in a given instance, this principle would have to be modified, or abandoned, for the polymerization process concerned. Since the principle seems well established, the likelihood of need for such qualifications seems remote. 

It will be observed that both of the distribution equations, (19) and (27) for the mole and weight fraction distributions, respectively, contain factors cox and Since a is limited to values less than Q c = l/(/— 1), is always much less than unity (for/=3 the maximum value of is j3c = l/4), and the factors o x and change in opposite directions as x increases. The decrease of the latter outweighs the increase of the former for all permissible values of see p. 366,... 

Figure 18. Theoretical weight sequence distributions (Equation 8) for samples of Table V (Wn,i, and W j/Wn,i are shown beside each curve)...

Initial Ca and final Ca f vapor concentrations are calculated for each interval using the phase distribution equations in the absence of NAPL (see Table 14.3). 

The second stage of treatment is assumed to follow an exponential decrease in removal rates. Applying the approach of Kuo, this stage is divided into two time intervals, T2A and T2 2, representing the successive removal of equivalent amounts of toluene, Miem2A = Mrem2 2 = 2.3151. The initial theoretical concentration in the gas phase for the time interval T2A is equal to the vapor pressure of toluene, Ca = 109 mg/L. The final vapor concentration for this interval Ca f can be calculated from the total residual concentration Ctf and the phase distribution equations 5 and 7-9 in Table 14.3 ... 

The cumulative molar concentration of polymeric species P 0 may be evaluated from the population density distribution, Equation 16. The first two rate expressions represent monomeric additions which do not change the molar concentration of polymeric molecules. A rate constant that describes functionality as a separable function of the molecule s degree of polymerization satisfies this constraint. The simplest, realistic function is the linear expression... 

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