Specific surface of spheres: how does specific surface vary with particle diameter Dp for spherical particles?

Introduction / Context:
Specific surface (surface area per unit mass or per unit volume) is a key parameter in reaction rates, drying, and sintering. For spheres, a simple geometric relation links specific surface and particle diameter.

Given Data / Assumptions:

• Spherical particles of diameter Dp.
• Specific surface per unit volume: S_v = surface area / volume.

Concept / Approach:
For a sphere, area A = πDp^2 and volume V = πDp^3/6. Thus S_v = A/V = 6/Dp. Per unit mass, S_m = S_v/ρ = 6/(ρ Dp). In both cases, specific surface is inversely proportional to Dp.

Step-by-Step Solution:
Write sphere geometry: A = πDp^2; V = πDp^3/6.Compute S_v = A/V = (πDp^2) / (πDp^3/6) = 6/Dp.Therefore, S ∝ 1/Dp.

Verification / Alternative check:
Dimensional analysis confirms inverse-length scaling; halving the diameter doubles specific surface.

Why Other Options Are Wrong:
Dp or Dp^2: imply larger particles have larger specific surface, contrary to geometry.1/Dp^2: corresponds to area per unit something else; not applicable for specific surface definitions.

Common Pitfalls:
Confusing specific surface per mass with per volume; both maintain 1/Dp dependence but differ by the density factor.

Final Answer:
1/Dp