NIST Researchers Develop More Accurate Formula for Measuring Particle Concentration

Researchers can use a metric called the particle number concentration (PNC) to calculate the number of particles in a sample, such as the number of marbles in a jar.

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S. Kelley/Inform Studio

Researchers at the National Institute of Standards and Technology (NIST) have developed a new mathematical formula to calculate the concentration of particles suspended in a solution. The new approach, which yields more accurate results than current methods, can be used to deliver the correct drug dosage to patients, measure the amount of nanoplastics in ocean water, and help ensure the correct level of additives in food products, among other applications.

The researchers published their findings in Analytical Chemistry.

“This new formula has the potential to help advance nanotechnology applications, such as food packing and preservation, and the fabrication of microchips and electronic devices,” said NIST engineer Elijah Petersen, who helped test the new formula. “It improves upon current methods by correcting for a common bias that assumes that particles are uniform in size.”

The particle number concentration measures how many particles are present in a given volume of gas or liquid. It is usually expressed in particles per cubic centimeter.

One way to find the concentration of particles suspended in a solution is to use two variables: the total mass of particles in solution and their size. However, the particles’ size can vary. This is referred to as their size distribution.

Current mathematical approaches work well when the particles are nearly uniform in size. However, they can produce inaccurate results when the particles vary significantly in size.

To understand why size distribution is important, imagine a contest where you have to estimate how many candies are in a jar. If the candies are all the same size, say a bunch of M&Ms, then you can use quick calculations to estimate the total number of candies in the jar. But what if the candies are different sizes? Let’s say there are full-size Kit Kats and Reese’s peanut butter cups mixed in. In that case, you would be better off with a different approach.

The new formula, which was derived by former NIST researcher Natalia Farkas, accounts for the variation in particle size to give a more accurate result.

To test the new formula, researchers applied it to samples of gold nanoparticles in water. NIST scientists had previously characterized these samples using multiple laboratory methods, providing highly accurate measurements of their actual particle number concentrations. The results showed that while the previous formulas overestimated the particle number concentration by about 6%, the new formula was accurate to within 1% of the directly measured value.

Researchers then applied the new formula to a more practical example: an anti-caking agent used in food production. Unlike the gold nanoparticles, which have a relatively narrow size distribution, the particles in this material varied widely in size. In this case, the estimates from the new and old formulas differed by as much as 36%.

“There are different ways to calculate the particle number concentration,” Petersen said. “Choosing the right formula can make a big difference.”

Paper: Natalia Farkas, John A. Kramar, Antonio R. Montoro Bustos, George Caceres, Monique Johnson, Matthias Roesslein and Elijah J. Petersen. Derivation of Particle Number Concentration from the Size Distribution: Theory and Applications. Analytical Chemistry. Published online May 16, 2025. DOI: 10.1021/acs.analchem.4c05990