Partition probability curve

For any separator with a size-dependent performance, such as a hydrocyclone, a sedimenting centrifuge or a settling vessel, the grade efficiency varies with particle size, and a graphical representation of this is called the grade efficiency curve (see section 3.2.2). As the value of the grade efficiency has the character of probability, it is sometimes referred to as the partition probability the curve then becomes the partition probability curve or Tromp curve. 

Theoretical efforts a step beyond simply fitting standard statistical curves to fragment size distribution data have involved applications of geometric statistical concepts, i.e., the random partitioning of lines, areas, or volumes into the most probable distribution of sizes. The one-dimensional problem is reasonably straightforward and has been discussed by numerous authors... 

Figure 24. Lattice strain model applied to zircon-melt partition coefficients from Hinton et al. (written comm.) for a zircon phenocryst in peralkaline rhyolite SMN59 from Kenya. Ionic radii are for Vlll-fold coordination (Shannon 1976). The curves are fits to Equation (1) at an estimated eraption temperature of 700°C (Scaillet and Macdonald 2001). Note the excellent fit of the trivalent lanAanides, with the exception of Ce, whose elevated partition coefficient is due to the presence of both Ce and Ce" in the melt, with the latter having a much higher partition coefficient into zircon. The 4+ parabola cradely fits the data from Dj, and Dy, through Dzi to Dih, but does not reproduce the observed DuIDjh ratio. We speculate that this is due to melt compositional effects on Dzt and (Linnen and Keppler 2002), and possibly other 4+ cations, in very silicic melts. Because of its Vlll-fold ionic radius of 0.91 A (vertical line), Dpa is likely to be at least as high as Dwh, and probably considerably higher.

Fig. 3. Functions in the integrand of the partition function formula Eq. (6). The lower solid curve labeled Pq AU/kT) is the probability distribution of solute-solvent interaction energies sampled from the uncoupled ensemble of solvent configurations. The dashed curve is the product of this distribution with the exponential Boltzmann factor, e AJJ/kT r the upper solid curve. See Eqs. (5) and (6).

Very often it is not possible a priori to separate contaminated and uncontaminated soils at the time of sampling. The best that can be done in this situation is to assume the data comprise several overlapping log-normal populations. A plot of percent cumulative frequency versus concentration (either arithmetic or log-transformed values) on probability paper produces a straight line for a normal or log-normal population. Overlapping populations plot as intersecting lines. These are called broken line plots and Tennant and White (1959) and Sinclair (1974) have explained how these composite curves may be partitioned so as to separate out the background population and then estimate its mean and standard deviation. Davies (1983) applied the technique to soils in England and Wales and thereby estimated the upper limits for lead content in uncontaminated soils. 

A full description of drug distribution can be complex, whether it is based on a knowledge of tissue perfusion and partition of drug from plasma to tissue, or whether it is based on a kinetic analysis of plasma concentration-time curves. One of the major descriptive parameters, which is probably of most interest to the toxicologist, is the volume of distribution (F ). This is the amount of drug in the body A) divided by the plasma concentration (c) after distribution equilibrium has been established. 

Palytoxin is a white, amorphous, hydroscopic solid that has not yet been crystallized. It is insoluble in nonpolar solvents such as chlorophorm, ether, and acetone sparingly soluble in methanol and ethanol and soluble in pyridine, dimethyl sulfoxide, and water. The partition coefficient for the distribution of palytoxin between 1-butanol and water is 0.21 at 25°C based on comparison of the absorbance at 263 nm for the two layers. In aqueous solutions, palytoxin foams on agitation, like a steroidal saponin, probably because of its amphipathic nature. The toxin shows no definite melting point and is resistant to heat but chars at 300°C. It is an optically active compound, having a specific rotation of -i-26° 2° in water. The optical rotatory dispersion curve of palytoxin exhibits a positive Cotton effect with [a]25o being -i-700° and [a]2,j being +600° (Moore and Scheuer 1971 Tan and Lau 2000). 

I he simplest is the partition of a solute between two immiscible solvents. In this case [0] /[Z)], = K, where K is the partition coefficient. This equilibrium is often referred to as the Nernst distribution. When [Z)], is plotted against [Z)], at constant temperature the curve is a straight line which terminates at the point when both the fibre and the dyebath are saturated. There are slight deviations from the linearity of the curve, particularly as the solutions become more concentrated. This system is probably exhibited in its ideal form when dyeing cellulose acetate rayon from an alcoholic dye solution, but it is also essentially true in the case of the application of disperse dyes in aqueous suspension to cellulose acetate, because the dyes are all soluble in water to a very limited extent and the undissolved particles act as a reservoir to maintain the concentration of the solution. The curve for this isotherm is shown in Fig. 12.14. 

Compared with chromatographic methods, this countercurrent distribution method is probably less efficient, the apparatus required is very much more complicated, and the labor involved is considerably greater nevertheless, it possesses several definite advantages over chromatographic methods. The chief of these is the absence of any solid phase, which may act as an adsorbent for the solutes. In partition chromatographic methods this adsorption may often lead to distortion of the bands and render fractionation very inefficient. The behavior of a solute on a countercurrent distribution depends only on its partition coefficient, which in most cases is constant, so that it is possible to calculate the exact theoretical distribution curve and this may be used as a very sensitive test for purity. 

FIGURE 22 (a) Partition curve plotted on linear graph paper and (b) its anamorphosis plotted on a probability net. 

The Tromp curve from the mathematical point of view is a cumulative distribution curve and as such can be linearized on probability graph paper. Such anamorphosis is produced by plotting the partition coefficients on a probability scale versus specific gravity on a linear scale (Fig. 22b) for dense-media separation, and versus log 8 - 1) for jigs. 

Considering that in agreement with the integrated version of Equation 10.16 ff(XA) = 0, and E, = [F(Xa)]/[Fc(Xa)]/ the screen efficiency can be measured without knowing the total efficiency avoiding, thus, the need to measure the solids flow rate. If the recovery index is determined for any particle size, and not only for the cut size, a curve known as probability cumulative partition curve is obtained. Such curve is used in determining the recovery index in classification of powders. 

Table XV shows that the two methods used (equilibrium/ultracentrifugation and sedimentation/diffusion) give comparable results. The sodium taurocholate micelle is swollen appreciably by the presence of even small amounts of potassium oleate. As the weight ratio increases the micelle increases in size. It is not possible to say whether there are two different species of micelle present, although this seems unlikely since the schlieren sedimentation curve was symmetrical and showed no shoulders or second bumps that would suggest polydispersity. It is probable that sodium taurocholate and potassium oleate form a mixed micelle that increases in size as more oleate is added. Since both these compounds are soluble amphiphiles (42) they will be present in both the micelle and as monomers. At present it is impossible to know how the species are partitioned. If one assumes that the micelle composition is similar to that of the whole solution (a valid assumption at high micelle concentrations) then the number of molecules of each...

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