Surface-area-to-volume ratio

The surface-area-to-volume ratio also called the surface-to-volume ratio and variously denoted sa/vol or SA:V, is the amount of surface area per unit volume of an object or collection of objects. The surface-area-to-volume ratio is measured in units of inverse distance. A cube with sides of length a will have a surface area of 6a² and a volume of a³. The surface to volume ratio for a cube is thus shown as $SA:V = \frac{6a^2}{a^3} = \frac{6}{a}$ .

For a given shape, SA:V is inversely proportional to size. A cube 2 m on a side has a ratio of 3 m⁻¹, half that of a cube 1 m on a side. On the converse, preserving SA:V as size increases requires changing to a less compact shape.

1 Physical chemistry
2 Biology
3 Examples
4 See also
5 References
6 External links

Physical chemistry

In involving a solid material, the surface-area-to-volume ratio is an important factor for the reactivity, that is, the rate at which the chemical reaction will proceed. Materials with large surface area to volume ratios (e.g., very small diameter, or very porous or otherwise not compact) react at much faster rates than monolithic materials, because more surface is available to react. Examples include grain dust; while grain is not typically flammable, grain dust is explosive. Finely ground salt dissolves much more quickly than coarse salt. It is same-case-applicable to a multiparticulate system or any system that has a surface coating, a very important parameter to be consider while performing coating for pharmaceutical solid oral-dosage form.

High surface-area-to-volume ratio provides a strong "driving force" to speed up thermodynamic processes that minimize thermodynamic free energy.

Biology

The ratio between the surface area and volume of cells and organisms has an enormous impact on their biology. For example, many aquatic microorganisms have increased surface area to increase their drag in the water.hta This reduces their rate of sink and allows them to remain near the surface with less energy expenditure. Humans and other large animals cannot rely on diffusion for absorption and rejection of respiratory gases for their whole body; however, animals such as flatworms and leeches can, as they have more surface area per unit volume. For similar reasons, surface to volume ratio places a maximum limit on the size of a cell.

An increased surface area to volume ratio also means increased exposure to the environment. The many tentacles of jellyfish and anemones provide increased surface area for the acquisition of food. Greater surface area allows more of the surrounding water to be sifted for nutrients.

Individual organs in animals are often shaped by requirements of surface area to volume ratio. The numerous internal branchings of the lung increase the surface area through which oxygen is passed into the blood and carbon dioxide is released from the blood. The intestine has a finely wrinkled internal surface, increasing the area through which nutrients are absorbed by the body.

Smaller single celled organisms need to have a high surface area to volume ratio in order to survive. This is because they rely on oxygen diffusing into the cell. The higher the SA:Volume ratio they have, the more efficient this process can be.

A wide and thin cell, such as a nerve cell, or one with membrane protrusions such as microvilli has a greater surface-area-to-volume ratio than a spheroidal one.

Increased surface area can also lead to biological problems. More contact with the environment through the surface of a cell or an organ (relative to its volume) increases loss of water and dissolved substances. High surface-area-to-volume ratios also present problems of temperature control in unfavorable environments.

Examples

Shape	Length $a$	Area	Volume	SA/V ratio	SA/V ratio for unit volume
Tetrahedron	side	$\sqrt{3} a^2$	$\frac{\sqrt{2}a^3}{12}$	$\frac{12\sqrt{3}}{\sqrt{2}a} \approx \frac{14.697}{a}$	7.21
Cube	side	$6a^2$	$a^3$	$\frac{6}{a}$	6
Octahedron	side	$2\sqrt{3}a^2$	$\frac{1}{3} \sqrt{2}a^3$	$\frac{6\sqrt{3}}{\sqrt{2} a} \approx \frac{7.348}{a}$	5.72
Dodecahedron	side	$3\sqrt{25%2B10\sqrt{5}} a^2$	$\frac{1}{4} (15%2B7\sqrt{5}) a^3$	$\frac{12\sqrt{25%2B10\sqrt{5}}}{(15%2B7\sqrt{5})a} \approx \frac{2.694}{a}$	5.31
Icosahedron	side	$5\sqrt{3}a^2$	$\frac{5}{12} (3%2B\sqrt5)a^3$	$\frac{12 \sqrt{3}}{(3%2B\sqrt{5})a} \approx \frac{3.970}{a}$	5.148
Sphere	radius	$4\pi a^2$	$\frac{4\pi a^3}{3}$	$\frac{3}{a}$	4.836

Example of Cubes of varying size
Side	Area of Face	Total Surface Area	Volume of Cube	Surface Area to Volume Ratio
1 m	1 m²	6 m²	1 m³	6.0 m⁻¹
2 m	4 m²	24 m²	8 m³	3.0 m⁻¹
4 m	16 m²	96 m²	64 m³	1.5 m⁻¹
6 m	36 m²	216 m²	216 m³	1.0 m⁻¹
8 m	64 m²	384 m²	512 m³	0.75 m⁻¹
12 m	144 m²	864 m²	1728 m³	0.5 m⁻¹
20 m	400 m²	2400 m²	8000 m³	0.3 m⁻¹

References

Schmidt-Nielson, K (1984). Scaling: Why is Animal Size so Important?. New York, NY: Cambridge University Press.
Vogel, S (1988). Life's Devices: The Physical World of Animals and Plants. Princeton, NJ: Princeton University Press.

External links

Sizes of Organisms: The Surface Area:Volume Ratio