 # Particle Size Result Interpretation: Number vs. Volume Distributions

Interpreting results of a particle size measurement requires an understanding of which technique was used and the basis of the calculations. Each technique generates a different result since each measures different physical properties of the sample. Once the physical property is measured a calculation of some type generates a representation of a particle size distribution. Some techniques report only a central point and spread of the distribution, others provide greater detail across the upper and lower particle size detected. The particle size distribution can be calculated based on several models: most often as a number or volume/mass distribution.

## Number vs. Volume Distribution

The easiest way to understand a number distribution is to consider measuring particles using a microscope. The observer assigns a size value to each particle inspected. This approach builds a number distribution—each particle has equal weighting once the final distribution is calculated. As an example, consider the nine particles shown in Figure 1. Three particles are 1 μm, three are 2 μm, and three are 3 μm in size (diameter). Building a number distribution for these particles will generate the result shown in Figure 2, where each particle size accounts for one third of the total. If this same result were converted to a volume distribution, the result would appear as shown in Figure 2 where 75% of the total volume comes from the 3μm particles, and less than 3% comes from the 1μm particles. Figure 2: Number and volume distributions of 1, 2, and 3 micron particles

When presented as a volume distribution it becomes more obvious that the majority of the total particle mass or volume comes from the 3 μm particles. Nothing changes between the left and right graph except for the basis of the distribution calculation.

Another way to visualize the difference between number and volume distributions is supplied courtesy of the City of San Diego Environmental Laboratory. In this case beans are used as the particle system. Figure 3 shows a population where there are 13 beans in each of three size classes, equal on a number basis. The same figure shows these beans placed in volumetric cylinders where it becomes apparent that the larger beans represent a much larger total volume than the smaller ones.

Figure 4 shows a population of beans where it may not be intuitively obvious, but there is an equal volume of each size, despite the wide range of numbers present. It becomes apparent that when the beans are placed in volumetric cylinders that each volumes are equal. Figure 4: (Click to enlarge) Equivalent volume of beans

## Transforming Results

Results from number based systems, such as microscopes or image analyzers construct their beginning result as a number distribution. Results from laser diffraction construct their beginning result as a volume distribution. The software for many of these systems includes the ability to transform the results from number to volume or vice versa. It is perfectly acceptable to transform image analysis results from a number to volume basis. In fact the pharmaceutical industry has concluded that it prefers results be reported on a volume basis for most applications (ref. 1). On the other hand, converting a volume result from laser diffraction to a number basis can lead to undefined errors and is only suggested when comparing to results generated by microscopy. Figure 5 below shows an example where a laser diffraction result is transformed from volume to both a number and a surface area based distribution. Notice the large change in median from 11.58 μm to 0.30 μm when converted from volume to number. Figure 5: (Click to enlarge) A volume distribution converted to area and number will include conversion errors

## Conclusions

Knowing a little bit about how calculation bases affect results will help users make informed choices. Number distributions are ideally chosen when the technique measures individual particles. Volume distributions are the default choice for many ensemble light scattering techniques including laser diffraction.

## References

1. Burgess, J., Duffy, E., Etzler, F., Hickey, A., Particle Size Analysis: AAPS Workshop Report, Cosponsored by the Food and Drug Administration and the United States Pharmacopeia, AAPS Journal 2004; 6 (3) Article 20 (http://www.aapsi.org)