Correct the statement for comminution laws: According to Bond’s law, the specific energy required to reduce a material from size D_f to D_p varies approximately with which expression?

Introduction / Context:
Comminution energy laws provide empirical relationships between energy input and particle size change. Three classical laws—Kick, Bond, and Rittinger—apply over different size ranges and mechanisms. Accurately recalling their mathematical forms is vital for ball mill sizing and power estimation.

Given Data / Assumptions:

• D_f: characteristic feed size.
• D_p: characteristic product size.
• Homogeneous material; bond work index concept applies.

Concept / Approach:
Bond’s law states that the energy required is proportional to the new crack length generated and correlates with the reciprocal of the square root of size. The commonly used form is: E = k_B * (1/√D_p − 1/√D_f), where k_B includes the Bond work index for the material and machine constants. By contrast, Kick’s law gives E ∝ ln(D_f/D_p) (size ratio), and Rittinger’s law gives E ∝ new surface area, often represented as E ∝ 1/D_p − 1/D_f for uniform particles.

Step-by-Step Solution:
Identify target law: Bond.Recall dependence: inverse square-root of size.Select expression with (1/√D_p − 1/√D_f).

Verification / Alternative check:
Ball mill design references and Bond’s original correlation consistently use the square-root reciprocal form.

Why Other Options Are Wrong:
(S/V) difference: aligns with Rittinger’s surface-area concept, not Bond’s.ln(D_f/D_p): Kick’s law, appropriate for coarse crushing.1/D_p − 1/D_f: a common Rittinger-type representation.

Common Pitfalls:
Mixing up the three laws; remember: Kick → log, Bond → inverse square root, Rittinger → inverse size (surface).

Final Answer:
E ∝ (1/√D_p) − (1/√D_f)