Log-probability plots

If a precipitate is allowed to undergo Ostwald ripening, or is sintered, or is caused to enter into a solid state reaction of some kind, it will often develop into a distribution which has a size limit to its growth. That is, there is a maximum, or minimum limit (and sometimes both) which the particle distribution approaches. The distribution remains continuous as it approaches that limit. The log-probability plot then has the form shown in 5.8.2. on the next page. 

Log probability plots are particularly useful when the distribution is bimodal, that is, when two separate distributions are present. Suppose we have a distribution of very small particles, say in suspension in its mother liquor. Ey an Ostwald ripening mechanism, the small particles redissolve and reprecipitate to form a distribution of larger particles. This would give us the distribution shown in 5.8.5. on the next page. 

However, an important difference also emerges from this analogy. The quantities that are normally distributed are logarithms of variables, not the variables themselves. This means that the mean and standard deviation obtained from log-probability plots are geometric averages... 

Example 7.7 Given the following impactor data, using a log probability plot, determine MMAD and aK. 

Figure ZS Log probability plot using the data from Example 2.1.

This modified form of the log-normal equation simplifies parameter determination from log-probability plots of experimental data. The graph paper may be furnished with additional scales of b and C both being determined by drawing a line parallel to the distribution through the pole (0.25 pm, 50%). 

Fig. 2.27 Log-probability plots of mixtures of two non-intersecting lognormal distributions.

Figure 2.26 presents relative plots of blends of two non-intersecting lognormal distributions with medians of 8 and 13 pm and standard deviations of 1.3 and 1.5 respectively. The areas under the two quite distinct curves give the proportions of the two components. On a log-probability plot (Figure 2. 27) the mixtures are asymptotic to both parents and have a point of inflection where the two distributions overlap. 

Using the raw data provided in Problem STT.2, generate an arithmetic-probability plot and a log-probability plot. 

It has already been stated that erg is identical for plots of the distribution of any parameter related to particle diameter, e.g. mass or surface area. Log-probability plots of count distribution,... 

The latest Coulter Counter instruments incorporate a microprocessor and the operator can specify the form of the output data to be obtained. The instrument converts count-data to practically any format desired, with the exception of log-probability plots. 

Figure 3.18 shows the particle size distribution, by mass, of the solids used in the tests (chalk), as obtained with the Ladal Pipette Centrifuge and the Andreasen Pipette Method. As can be seen from the log-probability plot, the distribution is very nearly log-normal and thus suitable for testing the performance of the hydrocyclone. Furthermore, the medium size of the chalk (3.9 microns) is within the range of cut sizes expected from the hydrocyclone (2 to 4 microns) which is a requirement for effective separator testing. 

For all the instrumental methods employed, geometric mean diameter (dg ) and geometric standard deviation (og) were obtained on a volume basis from log probability plots of cumulative volume distributions. 

The geometric standard deviation, being the ratio of two sizes, has no units and must always be greater than or equal to 1.0. The CMD (S0% cumulative size) and the GSD can be determined directly from a log-probability plot and completely define a lognormal distribution. 

Frequently, in aerosol sampling there is an aerodynamic size above which particles are aerodynamically unable to enter the sampling apparatus. This is called the aerodynamic cutoff size and means that the cumulative line on the log-probability plot will curve near its upper end, so that it never exceeds the cutoff size. 

A similar limit may exist for the lower end of the size distribution if sizing is done by optical microscopy. Because of the optical limitations explained in Section 20.2, particles less than about 0.3 m in diameter are not included in the size distribution. This 0.3- m limit, sometimes called the optical cutoff, curves the lower end of the size distribution line so that it never goes below 0.3 pm. An example of data having both cutoffs is shown in Fig. 4.13. These cutoffs are artifacts of the sampling and measurement system. If the cutoffs affect only a small fraction of the distribution, it is acceptable to ignore them when fitting a straight line to the data on a log-probability plot. 

The cumulative log-probability plot assumes that the data cover the size range fi om zero to infinity. If there are significant numbers of particles lost by truncation at either end of the size range, the ciunulative plot should not be used, but a frequency histogram can still be constructed without bias. 

---