Second central moment

An example is the variance, which is the second central moment p2 ... 

With roi the particle s center of gravity, this equation defines R2gi by the second central moment of the density distribution of the particle projected on a line extending in r direction. The equation is simplified (ro = 0) if the origin of the coordinate system is chosen to rest in the center of gravity. 

The dispersion of this waiting time distribution, i.e., its second central moment, is a measure that we can use to define a homogenization time scale on which the dispersion is equal to that of a homogeneous (Poisson) system on a time scale given by the torsional autocorrelation time. The homogenization time scale shows a clear non-Arrhenius temperature dependence and is comparable with the time scale for dielectric relaxation at low temperatures.156... 

The second central moment which provides the varicuice of the distribution has the form ... 

In the Monte Carlo calculations described below we have checked the accuracy of the calculation by comparing the calculated first and second central moments for the linear equation with these exact expressions. 

It is also possible to calculate the second central moment which is defined as... 

Table I. Contributions to the Second Central Moment— Adsorption on Silica Gel at 50°C.

P2 second central moment of the residence time distribution... 

We note that the second central moment (or the dimensionless variance) of the dispersion curve is given by... 

Models with varying degrees of complexity have been employed to analyze the experimental results by a variety of techniques. The most comprehensive models include terms to account for axial dispersion in the packed bed, external mass transfer, intraparticle diffusion in both macropore and micropore regions of the pellet and a finite rate of adsorption. Of the several methods of analysis, the most popular ones are based on the moments of the response curve. The first moment of the chromatogram is defined by Equation 5.25 in which the concentration now is taken at the outlet of the column. The second central moment is calculated from equation... 

For a short pulse input of a nonadsorbable tracer and a catalyst characterized by a unidisperse pore structure, the following relationships are obtained for the first moment and second central moment of the effluent curve of the bed [24] ... 

The mean-square value, is the second moment about zero. The variance is the second moment about the mean, also termed the second central moment, and conventionally is denoted by... 

The second central moment gives the spread or dispersion of the distribution which is the variance in this moment notation the variance can be calculated using ... 

For asymmetric peaks the mean retention time of the chromatogram is calculated by the first absolute moment p,t (Eq. 2.27). The variance ot2 is determined by the second central moment according to Eq. 2.28. 

HETP and plate number The determination of these parameters, which characterize the efficiency, is more complex. The basic equations are explained in Section 2.4.2. The problem is to determine the peak width or variance from a measured chromatogram. Especially, if the second central moment has to be calculated the result depends very much on the extension of the base line, which has to be chosen for the integration interval. Figure 2.22 gives an example for a slightly asymmetric peak. 

Since the stochastic models gives the corrected retention time in the form of = UTs and the retention time as tp = fo + we can write the second central moment as... 

The first moment, the second central moment, and the column efficiency that are obtained with a nondestructive detector are as follows... 

For the first moment and the second central moment of the peak shape we get, respectively [8] ... 

Equation (4.7) suggests that the fluctuations in an equilibrium system will offer many examples of the minimum (maximum) principle. These fluctuations are usualy presented as the variance, or second central moment, since the average, or first moment, is zero. The variance is always positive, as required, and its magnitude is usually taken as an inverse measure of goodness. That is, we want the variance of a measured or calculated variable to be as small as possible. It seems entirely reasonable that the equilibrium system would have the smallest variance. 

This treatment is based on the two differential material balance equations on a fixed-bed and packed particles. Although these equations, together with an adsorption isotherm and suitable boundary and initial conditions, describe precisely the performance of chromatography, the complexity of their mathematical treatment sometimes makes them insolvable. If the distribution coefficient K given by Equation 11.1 is constant, then Hs can be correlated to several parameters by use of the first absolute moment and the second central moment, as given by Equation 11.20 [3]. 

The second central moment of /(z) is relevant to the rate of convergence of a Monte Carlo computation of , and is given by... 

SO from equations (9-112) and (9-115) we can get explicit expressions for the moments in terms of model parameters. Of particular interest is the first absolute moment, /rj, and the second central moment (or variance), fij- These are... 

Accordingly, the variance aj is determined by the second central moment applying Equation 2.39 ... 

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