Third moment

From this definition, we see it is the ratio of the third moment of the distribution about the molecular weight origin to the second moment about the origin. 

It can be demonstrated that the total number of crystals, the total length, the total area, and the total volume of crystals, all in a unit of system volume, can be evaluated from the zero, first, second, and third moments of the population density function. 

The magma density Mj- (mass of crystals per unit volume of slurry or Hquor) may be obtained from the third moment of the population density function and is given by... 

In practice, the calculation of peak skew for highly tailing peaks is rendered difficult by basehne errors in the calculation of third moments. The peak asymmetry factor, A, = b/a, at 10 percent of peak height (see Fig. 16-32) is thus frequently used. An approximate relationship between peak skew and A, for taihng peaks, based on data in Yau et al. is Peak skew= [0.51 -t- 0.19/(A, — 1)] . Values of A, < 1.25... 

Skewness The third moment of a residence time distribution ... 

Higher moments of the distribution are often of interest too, especially when noniso-tropic or anhannonic processes are being studied. The third moment of the distribution reflects the skewness defined as... 

The third moment is taken about the mean and is known as the skewness, y. It is defined by... 

In the case of preservation of the zeroth and third moments (number and mass respectively) the equations for the agglomeration term can be written in the following form... 

Assuming steady state is reaehed, size-independent growth and that the third moment (mass) is eonserved in the agglomeration and disruption proeess, i.e. 

The crystal growth rates can be directly determined from the second and third moment as described above. The calculated rates for calcium oxalate here are in the range 0.75 x 10 to 4.7 x 10 m/s. Literature values for the growth rate of calcium oxalate monohydrate vary considerably 1.08 x 10 m/s (Kavanagh, 1992), 3.4 X 10 to 5.0 x 10 m/s (Garside etal., 1982) and 2.8 x lO to 1.11 X 10 m/s (Nielsen and Toft, 1984). The values obtained from the experiments are therefore within the range of the literature data. It should be borne in... 

The third moment of the CSD is reiated to totai particie voiume. It thus couples with the mass baiance thereby providing ciosure... 

The solids hold-up. Mi, is related to the CSD by the third moment equation... 

The skew, the third moment about the mean, is a measure of symmetry of distribution and can be denoted by y (population) or g (sample). It is given for a continuous random variable by... 

This term is obviously the original mass of the powder sample, and p3 is the third moment of the probability... 

Moment analysis is one of the simplest types of analysis and is useful for measuring the performance of the chromatography. Moments can be used to measure the same things that are measured in ID chromatographic systems these include the first, second, and third moments, which are more accurate than the related peak maximum, peak width, and peak asymmetry. In 2D, however, these values each have a component in each dimension and this can be easily determined in software-based measurement systems. 

Accurate values of the higher moments are not. easy to obtain. The substantial sensitivity of an RTD curve to small differences in a third moment is brought out in problem P5.02.07A. 

Find the effect of relatively small changes in the value of the third moment on the RTD of problem P5.02.07. 

Calculated values of the third moment are often uncertain because of the possible inaccuracy of the measured (C, t) data. The table and the graphs cover a range of possible values of the third moment. It appears that some... 

Equation 4.76 is written for a non-linear molecule with 3N - 6 vibrations. For linear molecules (including diatomics), there are 3N - 5 vibrations, two of the moments of inertia, say Ia and Ib, are equal, and there is no third moment of inertia. Using u = hv/kT, we obtain a more compact formula. 

The third important average molecular weight is the third moment or the z average molecular weight, M. This average is calculated from the molecular weight distribution as follows ... 

However, melt elasticity is more closely related to the third moment known as the z-average molecular weight M. The values of are most often determined using either light-scattering photometry or ultracentrifugation. It is shown mathematically as ... 

The resulting equations show that the first-order polarizabilities atJ depend on the second moments of the distribution, while the second-order polarizabilities pijk are functions of both the second and third moments of the polarizable body, as in... 

As a means of choosing between models it has been suggested that some of the higher moments of the C curve could be used. The skewed-ness, or third moment, of either the dispersion or tanks-in-series model is uniquely determined by the value of the dispersion coeflScient or the number of tanks. Thus, the parameter is determined from the second moment and then used to calculate the third moment. This is compared with the third moment computed from the experimental data whichever model has the closest third moment would be chosen. Unfortunately, this method has two drawbacks that severely limit its usefulness. One is that experimental data is not good enough to give meaningful third moments. The other is that the third moments as calculated from the different models have almost the same values, as was shown by van Deemter (V2). 

The degree of skewness is measured by the dimensionless third moment, fi3/fll12, where the normalized moment fip = npfot0 so that fio—l. The ratio gives a measure of unimodal versus bimodal behaviour in the eigenspectrum, as we will show later. 

---