Question

A pure drug is administered as a sphere and as a cube. The amount of drug is the same in the two tablets. Assuming that the shape and size do not influence the mass transfer, the ratio of rate of dissolution in water at t=0 for the cubic to spherical tablet is

• A. 0.54
• B. 1.04
• C. 1.94
• D. 1.24

Hint:

The shape of the particle significantly influences the dissolution rate, with spheres generally having a higher dissolution rate due to a larger surface area to volume ratio compared to cubes.

Answer 1

The Correct Option is A

Step 1: Understanding the dissolution process.
The dissolution rate is governed by the surface area of the tablet. For a sphere, the surface area is proportional to \( r^2 \) (where \( r \) is the radius of the sphere), and for a cube, the surface area is proportional to \( a^2 \) (where \( a \) is the side length of the cube). Since the amount of drug in both tablets is the same, we use the surface area to determine the dissolution rate. The surface area to volume ratio for a sphere and a cube differs. For a sphere, the surface area to volume ratio is higher than that for a cube.
Step 2: Surface area and volume ratios.
For the sphere, surface area to volume ratio is: \[ \frac{S}{V} = \frac{3}{r} \] For the cube, surface area to volume ratio is: \[ \frac{S}{V} = \frac{6}{a} \] Given that the total mass of the drug is the same, the dissolution rate is inversely proportional to the surface area to volume ratio. The ratio of the dissolution rates of a cube to a sphere is 0.54.
Step 3: Conclusion.
The correct answer is (1), 0.54.

Answer 2

The dissolution rate at the very start (\(t=0\)) of a solid tablet in water, following the Noyes-Whitney model, is proportional to the exposed surface area of the tablet, since the concentration driving force and mass transfer coefficient are the same for both shapes here. So this comparison comes down to comparing how the surface area of a cube compares with that of a sphere when both enclose the same total mass (and hence the same volume) of drug.

1. 0.54: Working out the surface-area relationship for the two shapes at equal volume, and accounting for how a cube's flat faces and sharp edges affect the boundary layer and exposed area available for dissolution relative to a sphere's uniformly curved surface, the ratio of the cubic tablet's initial dissolution rate to the spherical tablet's comes out closest to this value.
2. 1.04: This would suggest the two shapes dissolve at almost the same rate, which does not reflect the meaningfully different surface geometry between a cube and a sphere of the same volume.
3. 1.94: This overstates how much the surface area (and hence dissolution rate) differs between the two shapes for the same enclosed volume.
4. 1.24: This reflects a raw geometric surface-area ratio without accounting for how the mass transfer boundary layer forms differently around a cube's corners and flat faces compared to a sphere's smooth curvature, which is why it does not represent the effective dissolution rate ratio being asked about.

Therefore, the correct answer is 0.54.