Question

As the size of a material reduces to the nanoscale, the surface area-to-volume ratio

• A. \( \text{Increases significantly} \)
• B. \( \text{Decreases} \)
• C. \( \text{Remains constant} \)
• D. \( \text{Becomes zero} \)

Hint:

Surface Area/Volume ratio scales inversely with size (\( \propto \frac{1}{L} \)). Smaller size always guarantees an exponentially larger relative surface profile.

Answer 1

The Correct Option is A

Concept: Consider a generic cubic particle with side length \( L \).
• Total Surface Area (\( A \)) = \( 6L^2 \)
• Total Volume (\( V \)) = \( L^3 \)
• Surface area-to-volume ratio: \[ \frac{A}{V} = \frac{6L^2}{L^3} = \frac{6}{L} \]

Step 1: Mathematical Scaling Analysis
As the spatial scale of the material shrinks, the value of the denominator \( L \) becomes extremely small (\( L \to 10^{-9} \text{ m} \)). Mathematically: \[ \lim_{L \to \text{small}} \frac{6}{L} = \text{Very Large Value} \] Therefore, reducing the size leads to a geometric explosion in surface area relative to its internal volume. A massive percentage of the material's atoms end up living directly on its boundaries rather than locked within its crystal interior, driving heightened chemical reactivity and catalytic behavior.