Question

Surface area to volume ratio of materials:

• A. Decreases with decrease in characteristic dimension of materials
• B. Increases with decrease in characteristic dimension of materials
• C. No effect of characteristics dimension of materials
• D. Increases with increase in characteristic dimension of materials

Hint:

To easily remember this, compare a sugar cube to an equal mass of powdered sugar. The powdered sugar has a vastly larger total surface area for the same volume (or mass), which is why it dissolves much faster. Smaller size means a larger relative surface.

Answer 1

The Correct Option is B

Step 1: Concept Overview:
The surface-area-to-volume ratio (SA:V) compares an object's surface area to its volume. This problem explores how the SA:V changes with object size (characteristic dimension).
Step 2: Formulaic Analysis:
Consider a cube with side length L:

Surface area (SA): \( SA = 6 \times L^2 \)
Volume (V): \( V = L^3 \)
SA:V ratio: \( \frac{SA}{V} = \frac{6L^2}{L^3} = \frac{6}{L} \)
For a sphere with radius r (characteristic dimension):

Surface area (SA): \( SA = 4\pi r^2 \)
Volume (V): \( V = \frac{4}{3}\pi r^3 \)
SA:V ratio: \( \frac{SA}{V} = \frac{4\pi r^2}{\frac{4}{3}\pi r^3} = \frac{3}{r} \)

Step 3: Detailed Explanation:
In both examples, surface area scales with the square of the characteristic dimension (\(L^2\) or \(r^2\)), while volume scales with the cube (\(L^3\) or \(r^3\)). Therefore, the SA:V ratio is inversely proportional to the characteristic dimension (\(1/L\) or \(1/r\)).
Consequently, as the characteristic dimension (L or r) decreases, the surface-area-to-volume ratio increases. This is significant for nanomaterials, leading to enhanced reactivity and surface effects.
Conversely, as the dimension increases, the ratio decreases, making options A and D incorrect.
Step 4: Conclusion:
The surface area to volume ratio increases as the characteristic dimension of materials decreases.