Considering spherical gas bubbles of different diameters dispersed in a liquid, which bubble size provides the largest interfacial area per unit gas volume (a = surface area/volume)?

Introduction / Context:
Interfacial area per unit volume is a key determinant of gas–liquid mass transfer (k_La). For a given gas holdup, smaller bubbles create more interface area, increasing transfer rates—vital for aerobic bioprocesses and chemical absorbers.

Given Data / Assumptions:

• Spherical bubbles with diameters 1, 5, 15, and 50 mm.
• We compare a = surface area / volume for individual bubbles, which scales to interfacial area density for a swarm.
• Neglect coalescence and deformation effects.

Concept / Approach:
For a sphere, area A = π * d^2 and volume V = π * d^3 / 6. Thus a = A / V = 6 / d. Therefore, a is inversely proportional to diameter: the smaller the bubble, the greater the interfacial area per unit volume of gas.

Step-by-Step Solution:
Write a = 6 / d for spherical bubbles.Compare diameters: 1 mm yields a = 6 mm^-1; 5 mm yields 1.2 mm^-1; 15 mm yields 0.4 mm^-1; 50 mm yields 0.12 mm^-1.Identify the maximum: 1 mm bubble provides the largest a.Conclude smallest diameter → largest interfacial area per volume.

Verification / Alternative check:
Empirical mass-transfer studies show higher k_La with fine-pore spargers that produce small bubbles, consistent with the 6/d relationship.

Why Other Options Are Wrong:
Larger diameters reduce a due to the inverse proportionality; thus 5, 15, and 50 mm offer progressively less area.

Common Pitfalls:
Confusing total area at fixed gas flow with area per bubble; overlooking that very small bubbles can coalesce without surfactants or with high ionic strength, changing the actual a achieved.

Final Answer:
A bubble with a diameter of 1 mm