Derived mean sizes

If the number size distribution of a particulate system is found to be lognormal, equations (2.77) to (2.80) can be used to determine other mean sizes. For example, the volume-moment mean size is the mean size of a volume (mass) distribution  

Similarly, the mean size of a surface distribution is given by  

Using this equation, the volume specific surface of the particulate system may be determined since  

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Property mean sizes are derived by calculation from the actual distribution using a defined weighting for each size interval. 

This method is commonly nsed on spectral data to correct for multiplicative variations between spectra. In spectroscopy, snch variations often originate from nnintended or uncontrolled differences in sample path length (or effective path length, in the case of reflectance spectroscopy), caused by variations in sample physical properties (particle size, thickness), sample preparation, sample presentation, and perhaps even variations in spectrometer optics. Snch variations can be particularly problematic because they are confounded with mnltiplicative effects from changes in component concentrations, which often constitute the signal in qnantitative applications. It is important to note that multiplicative variations cannot be removed by derivatives, mean-centering or variable-wise scaling. 

The above derivation means that, once the composition and grain size are provided. 

Several parameters can be used to express the mean size of the milk fat globules. These parameters are derived from the so-called moments of the size distribution function the th moment of the distribution function (Sn) is equal to ... 

Using this theory, a kinetic expression for the mean size of the a regates, X(f), after the initial burst of nuclei can be derived ... 

Fumed silica aggregates are obviously linear and branched particle structures with a mean size of about 100 to 200 nm. By TEM we derive the size of the partially fused primary particles of about 10 run. This very small particle size correlates well with the high surfaces area of fumed silica which usually is larger than 100 m g as determined by nitrogen adsorption at 78 K according to BET [5]. Adsorption techniques and electron microscopy provide very close values of surface areas. This indicates that fumed silica exhibits a smooth particle surface in the range of nanometers, apparently its surface is free of micropores. 

Recall from Chapter 2 that the universal unit system used hy scientists is called Le Systeme Internationale d Unit6s or SI. It is a metric system based on seven base units—meter, second, kilogram, kelvin, mole, ampere, and candela—from which all other units are derived. The size of a unit in a metric system is indicated by a prefix related to the difference between that unit and the base unit. For example, the base unit for length in the metric system is the meter. One tenth of a meter is a decimeter where the prefix deci- means one tenth. And, one thousand meters is a kilometer. The prefix kilo- means one thousand. 

As discussed in Section 2, the atom-randomized reference state used in calculations for simple fluid systems cannot be generalized to the connected, finite-atomic-size system of protein-ligand complexes. Therefore, researchers attempted to make corrections to the simple random reference state in order to improve the derived mean-force potentials. 

However, the symmetric component gives something more awkward. The limit has different third divided differences on left and right, which means that there is a discontinuity of third derivative. The size of this discontinuity is given by the fourth divided difference of the original polygon. 

The Ag particles of the reduced sol-gel derived catalyst Ag/Si02-SG1/A had a mean size of about 8 nm as determined from the line broadening of the Ag(l 11) reflection during the XRD... 

The kinetic parameters (kg and g) in Eq. (10.27) can be related to the mean size (L) and total surface area (Ar) of the seed crystals and zero-time values of the supersaturation (Aco) and its time derivatives (Acoi and Aco2)... 

Tavare and Garside (1982) used the solution of the above equation to derive its parameters in terms of the differences in initial and final mean size, AL = Lf — Lo, and variance, Acr ... 

Various statistical parameters such as median and mean size, deviation, skewness and kur-tosis can be calculated from data derived from cumulative curves. The median or mean size permits the determination of the grade of gravel, sand or silt, or their lithified equivalents. Deviation affords a measure of sorting. However, the latter can be quickly and simply estimated by visual examination of the curve in that the steeper it is, the more uniform the sorting of the sediment. 

Mean size Xk r can be understood as the Mi root of the expectation value of the Mi power of X (power mean) when weighted by a given type of quantity (indicated by r). Mean values can be computed from the size distribution, which requires the knowledge of the density function q x) for the entire size range, or may be derived... 

Equations 2.33, 2.34 and 2.35 are derived from equations 2.1, 2.2, 2.3 and 2.24. As the lines in Figure 2.9 are parallel, equations 2.33, 2.34 and 2.35 must also apply to all mean sizes and to the mode the steepness constant b (or geometric standard deviation tXg) remain the same and only Xm (or Xg) varies in equation 2.24 (or 2.21) so that the expressions for the four size distributions can be written easily from one set of experimental data (bearing in mind the... 

The derived pore size distribution indicates a mean pore size of 5.1 nm, with a standard deviation of 1.1 nm. This is a pore of dimensions suitable for separating a... 

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