Unsteady leaching from porous solids: A long cylinder (5 cm diameter, flat ends sealed) and a sphere (5 cm diameter) are made from the same porous material and equally saturated with NaCl solution. Both are dipped for a short, equal time in a large, well-agitated water tank. If Xc and Xs are the fractions of salt remaining in the cylinder and sphere, which relation holds?

Introduction / Context:For brief extraction times from porous solids into a well-mixed liquid, the controlling factor is typically external mass transfer across the wetted external area, not deep internal diffusion. Thus, the surface-area-to-volume ratio (A/V) largely dictates how quickly a given body loses dissolved species initially. This question compares a long sealed-ended cylinder and a sphere of the same diameter.

Given Data / Assumptions:

• Same porous material and initial salt loading per unit volume.
• Both bodies have the same diameter (5 cm), but the cylinder is long and its flat ends are sealed (no mass transfer from the ends).
• Immersion time is short; tank is large and well agitated (external mass transfer controls initially).

Concept / Approach:At short times, the rate of salt removal is approximately proportional to external area A (and the mass transfer coefficient), while total salt content scales with volume V. The relevant metric is A/V. For a sphere of radius r, A/V = 3/r. For a long cylinder with sealed ends (radius r), only the curved surface transfers: A/V = 2/r. With equal r (2.5 cm), the sphere has larger A/V than the long cylinder, so it loses salt faster, leaving a smaller fraction remaining after the same time.

Step-by-Step Solution:

Verification / Alternative check:Dimensional analysis of early-time leaching (or lumped mass-transfer models) shows m(t)/m0 ≈ exp(−kA t / V) at short times; greater A/V decreases m faster.

Why Other Options Are Wrong:

• Equality (b): would require identical A/V, which is false here.
• 'Xc < Xs' (c): reverses the A/V effect.
• 'Depends on cylinder length' (d): for a long cylinder with ends sealed, A/V tends to 2/r and becomes length-independent.
• 'Insufficient information' (e): given assumptions are enough to decide.

Common Pitfalls:Including end areas for the cylinder (they are sealed); assuming internal diffusion dominates at very short times; forgetting A/V scaling.

Final Answer:Xc > Xs