Surface area to volume ratio

This does not mean that one small particle has a greater total surface area than one large particle. It means that the small particle's surface area is large compared with its own volume. If a fixed quantity of material is divided into many small particles, their combined surface area is also much greater than the surface area of the original lump.

Using cubes to see the relationship

For a cube of side length $l$:

$\text{surface area}=6l^2$

$\text{volume}=l^3$

Therefore:

$\frac{\text{surface area}}{\text{volume}}=\frac{6l^2}{l^3}=\frac{6}{l}$

The ratio is inversely proportional to the side length. Halving $l$ doubles the surface area to volume ratio; reducing $l$ by a factor of 10 increases the ratio by a factor of 10.

For example, a cube with sides of 100 cm has a surface area of $60\,000\text{ cm}^2$ and a volume of $1\,000\,000\text{ cm}^3$. Its surface area to volume ratio is $0.06:1$. A cube with sides of 10 cm has a surface area of $600\text{ cm}^2$ and a volume of $1\,000\text{ cm}^3$, giving $0.6:1$. The smaller cube has only one tenth of the side length, but ten times the surface area to volume ratio.

The same mathematical pattern applies to other similarly shaped particles: as their characteristic size decreases, their surface area to volume ratio increases. The exact formula depends on shape, but the relationship remains the same.

Why a high ratio changes properties

Chemical reactions involving a solid take place at its surface, where its particles can meet the reacting substance. Material buried inside a large particle is not immediately available for reaction. In a much smaller particle, a greater proportion of the material lies at or near the surface.

A high surface area to volume ratio therefore provides more exposed material and more possible reaction sites relative to the amount of substance present. Small particles can consequently react more readily than larger particles of the same material. Nanoparticles have an especially high surface area to volume ratio, so their properties can differ from those of the same substance in bulk.

This relationship helps explain why nanoparticles can make excellent catalysts. A small mass can expose a large surface on which reactant particles can meet and react. The important chain of reasoning is:

smaller particle → higher surface area to volume ratio → greater proportion exposed at the surface → more available reaction sites → altered reactivity or catalytic behaviour.