Question

The given equation represents which law: \[ E = K_k \ln \frac{d_1}{d_2} \]

• A. Rittinger’s law
• B. Bond’s law
• C. Fick's law
• D. Kick’s law

Hint:

Kick’s law is used for coarse grinding, assuming that energy required is proportional to the logarithm of size reduction ratio.

Answer 1

The Correct Option is D

Step 1: Understanding the given equation. The given equation: \[ E = K_k \ln \frac{d_1}{d_2} \] relates energy (\( E \)) to the size reduction of particles (\( d_1 \) and \( d_2 \)). This equation is derived from Kick’s law, which states that the energy required for size reduction is proportional to the logarithm of the ratio of initial to final particle sizes. 

Step 2: Explanation of Kick’s Law. Kick’s law is expressed as: \[ E = K_k \ln \frac{d_1}{d_2} \] where: 
- \( E \) = Energy required for size reduction, 
- \( K_k \) = Kick’s constant, 
- \( d_1 \) and \( d_2 \) = Initial and final particle sizes. 

Step 3: Why other options are incorrect.
 - (A) Rittinger’s law: States that energy required is proportional to the new surface area created, using \( E = K_R \left( \frac{1}{d_2} - \frac{1}{d_1} \right) \). 
- (B) Bond’s law: Uses an empirical equation to calculate energy consumption in size reduction. 
- (C) Fick's law: Describes diffusion, unrelated to particle size reduction.

Answer 2

The question gives the equation \(E = K_k \ln \frac{d_1}{d_2}\) and asks which law it represents. Each size reduction law has its own characteristic mathematical form, so matching the structure of this equation, specifically the natural log of a diameter ratio, to each candidate identifies the answer.

1. Rittinger's law: This law states that the energy needed for size reduction is proportional to the new surface area created, written as \[ E = K_R \left( \frac{1}{d_2} - \frac{1}{d_1} \right) \] The presence of reciprocal diameter terms rather than a logarithm rules this law out for the given equation.
2. Bond's law: This law relates energy to the square root of the reciprocal diameters, written as \[ E = K_B \left( \frac{1}{\sqrt{d_2}} - \frac{1}{\sqrt{d_1}} \right) \] Its square root form does not match the logarithmic form given in the question.
3. Fick's law: This law describes the rate of diffusion of a substance across a concentration gradient and has nothing to do with particle size reduction or energy input during milling, so it is unrelated to the given equation.
4. Kick's law: This law states that the energy required to reduce a particle from an initial size \(d_1\) to a final size \(d_2\) is proportional to the natural logarithm of the ratio of the two sizes, written exactly as \[ E = K_k \ln \frac{d_1}{d_2} \] This matches the given equation term for term.

The logarithmic ratio form of the equation is the defining signature of Kick's law and does not appear in Rittinger's or Bond's equations.

Therefore, the correct answer is Kick's law.