Powder technology – converting mass to particle count For a crushed solid sample, the total number of particles N can be expressed in terms of the total sample mass m, the volume of one particle Vp, and the particle density ρ. Which expression is correct?

Introduction / Context:
In particle technology and size-reduction analysis, it is often necessary to relate bulk measurements (like total mass) to microscopic counts (number of particles). This conversion is foundational for population balance modeling, breakage kinetics, and estimating specific surface area from size distributions.

Given Data / Assumptions:

• Total mass of all particles in the sample: m.
• Volume of a single representative particle: Vp.
• True (material) density of the particles: ρ.
• Particles are compositionally identical (same ρ) and Vp represents the volume per particle used in the expression.

Concept / Approach:
The mass of one particle mp equals its density multiplied by its volume. The number of particles N is then the ratio of the total sample mass to the mass per particle. No special geometry is required for this mass–count relationship; only density and per-particle volume are needed.

Step-by-Step Solution:

Verification / Alternative check:
Check dimensional consistency. Units: m in kg, ρ in kg/m^3, Vp in m^3. Then ρ * Vp has units kg, so m/(ρ * Vp) is dimensionless, as expected for a count. The other options either put ρ and Vp in the numerator incorrectly or invert the relationship, yielding wrong dimensions.

Why Other Options Are Wrong:

• m * ρ / Vp: Has units kg * kg/m^3 / m^3 = kg^2/m^6, not dimensionless.
• m * Vp / ρ: Reduces to m^4/kg when using SI base units; physically meaningless for a count.
• Vp/(m * ρ): Inverts the ratio; would decrease as mass increases, contradicting physics.

Common Pitfalls:
Confusing bulk (apparent) density with true density, and interpreting Vp as geometric volume without accounting for porosity. For porous particles, use skeletal density for ρ or redefine Vp accordingly.

Final Answer:
m/(ρ * Vp)