Derivation of Particle Number Concentration from the Size Distribution: Theory and Applications

The particle number concentration (PNC) in a suspension is a key measurand in nanotechnology. A common approach for deriving PNC is to divide the total mass concentration by the per-particle mass, calculated as density times volume. The volume is most frequently derived from the arithmetic mean diameter (AMD) of the size distribution. The harmonic mean volume (HMV) has also been used. Given a known size distribution, we show that the correct PNC is obtained by using the arithmetic mean volume (AMV). The AMD-based volume results in an overestimate in PNC that increases superlinearly with increasing coefficient of variation (CV), reaching 12% at CV = 0.2 for a normal distribution. HMV would yield a much greater overestimate, exceeding 50%. The error in the AMD-derived PNC shows only weak skew dependence, suggesting a simple approximate correction as a function of CV in the common situation where AMD and CV are known but the overall size distribution is unknown. Using published data sets of gold nanoparticles, we demonstrate an overall consistency of ±1.1% in comparing the PNC directly determined by single-particle inductively coupled plasma-mass spectrometry (spICP-MS) and the PNCs derived from AMV using size distributions independently measured by high-resolution scanning electron microscopy and spICP-MS. We further compare AMV and AMD-derived PNCs for well-characterized polystyrene nanoparticle standards, illustrating sensitivity to distributional characteristics along with common errors to avoid. Nanoparticles in environmental samples, food additives, and nanomedicines often have CVs greater than 0.3, for which uncorrected AMD-derived PNC errors can exceed 35%.