Specific surface of spherical particles (per unit mass)\nFor a collection of identical spherical particles, what is the specific surface area (surface area per unit mass) in terms of particle diameter D and particle density rho?

Introduction / Context:
Specific surface area (SSA) strongly influences reaction rates, dissolution, heat transfer, and filtration performance. For ideal spheres, SSA can be expressed analytically from geometry, offering a quick check on particle size analyses and mass–transfer estimates.

Given Data / Assumptions:

• Particles are perfect spheres of uniform diameter D.
• Particle density is rho (mass per unit volume).
• SSA is defined as total surface area divided by total particle mass.

Concept / Approach:
For one sphere: surface area A = pi * D^2. Volume V = (pi/6) * D^3. Mass m = rho * V = rho * (pi/6) * D^3. Specific surface S = A / m = (pi * D^2) / (rho * (pi/6) * D^3) = 6 / (rho * D). This compact formula is widely used for monodisperse spheres and provides a consistency check; if D halves, SSA doubles for the same material.

Step-by-Step Solution:

Verification / Alternative check:
Dimensional check: D has length, rho has mass/volume, so S units are area/mass; 6/(D*rho) has (1/length) / (mass/volume) = area/mass → consistent.

Why Other Options Are Wrong:

• 4/(D*rho) or 3/(D*rho): incorrect geometric factors.
• 6*D*rho: wrong dependence; SSA decreases with larger D.
• pi*D^2/rho: missing normalization by volume, not per mass.

Common Pitfalls:
Confusing specific surface per unit volume vs. per unit mass; always check definitions and units.

Final Answer:
S = 6 / (D * rho)