For a cubical particle, if the “equivalent diameter” is taken equal to the cube height (edge length), what is the sphericity value (use the standard ratio of surface area of volume-equivalent sphere to actual surface area when equivalent diameter equals edge)?

Introduction / Context:Sphericity (ψ) is a dimensionless shape descriptor. While the rigorous definition uses volume-equivalent diameter, exam variations sometimes ask for special cases using a specified “equivalent diameter.” Here, the equivalent diameter is taken as the edge length of a cube, simplifying the ratio to a familiar constant.

Given Data / Assumptions:

• Cube of edge a; equivalent diameter is set to a.
• Surface area of cube = 6a^2.
• Surface area of sphere with diameter a = πa^2.

Concept / Approach:Using the given convention, sphericity ψ = (surface area of sphere of diameter equal to equivalent diameter) / (actual surface area of particle). With deq = a, numerator = πa^2 and denominator = 6a^2, giving ψ = π/6 ≈ 0.523, commonly rounded to 0.5 in multiple-choice settings.

Step-by-Step Solution:Compute cube area: 6a^2.Compute sphere area for diameter a: πa^2.Form ratio ψ = πa^2 / 6a^2 = π/6 ≈ 0.523 ≈ 0.5.

Verification / Alternative check:Note that the strict volume-equivalent definition would give a different number; the question expressly fixes deq to the edge length, making the π/6 result applicable and commonly approximated as 0.5.

Why Other Options Are Wrong:1, 2, 3: inconsistent with derived ratio.π/6 as an option might be considered, but numeric MCQs often expect the rounded 0.5 value.

Common Pitfalls:Confusing volume-equivalent diameter definition with the special equivalence stated in the problem.

Final Answer:0.5