Biology:Surface-area-to-volume ratio

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Short description: Surface area per unit volume
Graphs of surface area, A against volume, V of the Platonic solids and a sphere, showing that the surface area decreases for rounder shapes, and the surface-area-to-volume ratio decreases with increasing volume. Their intercepts with the dashed lines show that when the volume increases 8 (2³) times, the surface area increases 4 (2²) times.

The surface-area-to-volume ratio or surface-to-volume ratio (denoted as SA:V, SA/V, or sa/vol) is the ratio between surface area and volume of an object or collection of objects.

SA:V is an important concept in science and engineering. It is used to explain the relation between structure and function in processes occurring through the surface and the volume. Good examples for such processes are processes governed by the heat equation,[1] that is, diffusion and heat transfer by thermal conduction.[2] SA:V is used to explain the diffusion of small molecules, like oxygen and carbon dioxide between air, blood and cells,[3] water loss by animals,[4] bacterial morphogenesis,[5] organisms' thermoregulation,[6] design of artificial bone tissue,[7] artificial lungs[8] and many more biological and biotechnological structures. For more examples see Glazier.[9]

The relation between SA:V and diffusion or heat conduction rate is explained from flux and surface perspective, focusing on the surface of a body as the place where diffusion, or heat conduction, takes place, i.e., the larger the SA:V there is more surface area per unit volume through which material can diffuse, therefore, the diffusion or heat conduction, will be faster. Similar explanation appears in the literature: "Small size implies a large ratio of surface area to volume, thereby helping to maximize the uptake of nutrients across the plasma membrane",[10] and elsewhere.[9][11][12]

For a given volume, the object with the smallest surface area (and therefore with the smallest SA:V) is a ball, a consequence of the isoperimetric inequality in 3 dimensions. By contrast, objects with acute-angled spikes will have very large surface area for a given volume.

For solid spheres

Plot of the surface-area:volume ratio (SA:V) for a 3-dimensional ball, showing the ratio declining inversely as the radius of the ball increases.

A solid sphere or ball is a three-dimensional object, being the solid figure bounded by a sphere. (In geometry, the term sphere properly refers only to the surface, so a sphere thus lacks volume in this context.)

For an ordinary three-dimensional ball, the SA:V can be calculated using the standard equations for the surface and volume, which are, respectively, SA=4πr2 and V=(4/3)πr3. For the unit case in which r = 1 the SA:V is thus 3. For the general case, SA:V equals 3/r, in an inverse relationship with the radius - if the radius is doubled, the SA:V halves (see figure).

For n-dimensional balls

Balls exist in any dimension and are generically called 'n-balls or hyperballs, where n is the number of dimensions. The same reasoning can be generalized to n-balls using the general equations for volume and surface area, which are:

V=rnπn/2Γ(1+n/2)
SA=nrn−1πn/2Γ(1+n/2)

So the ratio equals SA/V=nr−1. Thus, the same linear relationship between area and volume holds for any number of dimensions (see figure): doubling the radius always halves the ratio.

Dimension and units

The surface-area-to-volume ratio has physical dimension inverse length (L−1) and is therefore expressed in units of inverse metre (m−1) or its prefixed unit multiples and submultiples. As an example, a cube with sides of length 1 cm will have a surface area of 6 cm2 and a volume of 1 cm3. The surface to volume ratio for this cube is thus

SA:V=6cm21cm3=6cm−1.

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

Applications

Physical chemistry

See also: Chemistry:Dust explosion

Materials with high surface area to volume ratio (e.g. very small diameter, very porous, or otherwise not compact) react at much faster rates than monolithic materials, because more surface is available to react. An example is grain dust: while grain is not typically flammable, grain dust is explosive. Finely ground salt dissolves much more quickly than coarse salt.

A high surface area to volume ratio provides a strong "driving force" to speed up thermodynamic processes that minimize free energy.[13]

Biology

Cells lining the small intestine increase the surface area over which they can absorb nutrients with a carpet of tuftlike microvilli.


An increased surface area to volume ratio also means increased exposure to the environment. The finely-branched appendages of filter feeders such as krill provide a large surface area to sift the water for food.[14]

Individual organs like the lung have numerous internal branchings that increase the surface area; in the case of the lung, the large surface supports gas exchange, bringing oxygen into the blood and releasing carbon dioxide from the blood.[15][16] Similarly, the small intestine has a finely wrinkled internal surface, allowing the body to absorb nutrients efficiently.[17]

Cells can achieve a high surface area to volume ratio with an elaborately convoluted surface, like the microvilli lining the small intestine.[18]


The surface to volume ratios of organisms of different sizes also leads to some biological rules such as Allen's rule, Bergmann's rule[19][20][21] and gigantothermy.[22]

Fire spread

In the context of wildfires, the ratio of the surface area of a solid fuel to its volume is an important measurement. Fire spread behavior is frequently correlated to the surface-area-to-volume ratio of the fuel (e.g. leaves and branches). The higher its value, the faster a particle responds to changes in environmental conditions, such as temperature or moisture. Higher values are also correlated to shorter fuel ignition times, and hence faster fire spread rates.

Planetary cooling

A body of icy or rocky material in outer space may, if it can build and retain sufficient heat, develop a differentiated interior and alter its surface through volcanic or tectonic activity. The length of time through which a planetary body can maintain surface-altering activity depends on how well it retains heat, and this is governed by its surface area-to-volume ratio. For Vesta (r=263 km), the ratio is so high that astronomers were surprised to find that it did differentiate and have brief volcanic activity. The Moon, Mercury and Mars have radii in the low thousands of kilometers; all three retained heat well enough to be thoroughly differentiated although after a billion years or so they became too cool to show anything more than very localized and infrequent volcanic activity. As of April 2019, however, NASA has announced the detection of a "marsquake" measured on April 6, 2019, by NASA's InSight lander.[23] Venus and Earth (r>6,000 km) have sufficiently low surface area-to-volume ratios (roughly half that of Mars and much lower than all other known rocky bodies) so that their heat loss is minimal.[24]

Mathematical examples

Shape Image Characteristic
length a 		SA/V ratio SA/V ratio for
unit volume
Tetrahedron 60px edge 66a≈14.697a 7.21
Cube 70px edge 6a 6
Octahedron 70px edge 36a≈7.348a 5.72
Dodecahedron 70px edge 1225+105(15+75)a≈2.694a 5.31
Capsule 100x100px radius (R) 125a 5.251
Icosahedron 70px edge 123(3+5)a≈3.970a 5.148
Sphere 70px radius 3a 4.83598
Examples of cubes of different sizes
Side of
cube		 Side2 Area of a
single face		 6 × side2 Area of
entire cube
(6 faces)		 Side3 Volume Ratio of
surface area
to volume
2 2×2 4 6×2×2 24 2×2×2 8 3:1
4 4×4 16 6×4×4 96 4×4×4 64 3:2
6 6×6 36 6×6×6 216 6×6×6 216 3:3
8 8×8 64 6×8×8 384 8×8×8 512 3:4
12 12×12 144 6×12×12 864 12×12×12 1,728 3:6
20 20×20 400 6×20×20 2,400 20×20×20 8,000 3:10
50 50×50 2,500 6×50×50 15,000 50×50×50 125,000 3:25
1,000 1,000×1,000 1,000,000 6×1,000×1,000 6,000,000 1,000×1,000×1,000 1,000,000,000 3:500

See also

References

Specific
  1. ↑ Planinšič, Gorazd; Vollmer, Michael (February 20, 2008). "The surface-to-volume ratio in thermal physics: from cheese cube physics to animal metabolism". European Journal of Physics 29 (2): 369–384. doi:10.1088/0143-0807/29/2/017. Bibcode: 2008EJPh...29..369P. https://iopscience.iop.org/article/10.1088/0143-0807/29/2/017/meta. Retrieved 9 July 2021. 
  2. ↑ Planinšič, Gorazd (2008). "The surface-to-volume ratio in thermal physics: from cheese cube physics to animal metabolism". European Journal of Physics 29 (2): 369–384. doi:10.1088/0143-0807/29/2/017. Bibcode: 2008EJPh...29..369P. https://iopscience.iop.org/article/10.1088/0143-0807/29/2/017/meta. 
  3. ↑ Williams, Peter; Warwick, Roger; Dyson, Mary; Bannister, Lawrence H. (2005). Gray's Anatomy (39 ed.). Churchill Livingstone. pp. 1278–1282. 
  4. ↑ Jeremy M., Howard; Hannah-Beth, Griffis; Westendorf, Rachel; Williams, Jason B. (2019). "The influence of size and abiotic factors on cutaneous water loss". Advances in Physiology Education 44 (3): 387–393. doi:10.1152/advan.00152.2019. PMID 32628526. 
  5. ↑ Harris, Leigh K.; Theriot, Julie A. (2018). "Surface Area to Volume Ratio: A Natural Variable for Bacterial Morphogenesis". Trends in Microbiology 26 (10): 815–832. doi:10.1016/j.tim.2018.04.008. PMID 29843923. 
  6. ↑ Louw, Gideon N. (1993). Physiological Animal Ecology. Longman Pub Group. 
  7. ↑ Nguyen, Thanh Danh; Olufemi E., Kadri; Vassilios I., Sikavitsas; Voronov, Roman S. (2019). "Scaffolds with a High Surface Area-to-Volume Ratio and Cultured Under Fast Flow Perfusion Result in Optimal O2 Delivery to the Cells in Artificial Bone Tissues". Applied Sciences 9 (11): 2381. doi:10.3390/app9112381. 
  8. ↑ J. K, Lee; H. H., Kung; L. F., Mockros (2008). "Microchannel Technologies for Artificial Lungs: (1) Theory". ASAIO Journal 54 (4): 372–382. doi:10.1097/MAT.0b013e31817ed9e1. PMID 18645354. 
  9. ↑ 9.0 9.1 Glazier, Douglas S. (2010). "A unifying explanation for diverse metabolic scaling in animals and plants". Biological Reviews 85 (1): 111–138. doi:10.1111/j.1469-185X.2009.00095.x. PMID 19895606. https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1469-185X.2009.00095.x. 
  10. ↑ Alberts, Bruce (2002). "The Diversity of Genomes and the Tree of Life". Molecular Biology of the Cell, 4th edition. New York: Garland Science. ISBN 0-8153-4072-9. ISBN 0-8153-3218-1. https://www.ncbi.nlm.nih.gov/books/NBK26866/#_A44_. 
  11. ↑ Adam, John (2020-01-01). "What's Your Sphericity Index? Rationalizing Surface Area and Volume". Virginia Mathematics Teacher 46 (2). https://digitalcommons.odu.edu/mathstat_fac_pubs/174. 
  12. ↑ Okie, Jordan G. (March 2013). "General models for the spectra of surface area scaling strategies of cells and organisms: fractality, geometric dissimilitude, and internalization". The American Naturalist 181 (3): 421–439. doi:10.1086/669150. ISSN 1537-5323. PMID 23448890. Bibcode: 2013ANat..181..421O. 
  13. ↑ Whittingham, M (May 1989). "Basic solid state chemistry Anthony R. West, John Wiley & Sons, New York (1988), 415 pages £13.95 ($32.95)". Solid State Ionics 34 (3): 213. doi:10.1016/0167-2738(89)90043-x. ISSN 0167-2738. https://doi.org/10.1016/0167-2738(89)90043-x. 
  14. ↑ Kils, U.: Swimming and feeding of Antarctic Krill, Euphausia superba - some outstanding energetics and dynamics - some unique morphological details. In Berichte zur Polarforschung, Alfred Wegener Institute for Polar and Marine Research, Special Issue 4 (1983): "On the biology of Krill Euphausia superba", Proceedings of the Seminar and Report of Krill Ecology Group, Editor S. B. Schnack, 130-155 and title page image.
  15. ↑ Tortora, Gerard J.; Anagnostakos, Nicholas P. (1987). Principles of anatomy and physiology (Fifth ed.). New York: Harper & Row, Publishers. pp. 556–582. ISBN 978-0-06-350729-6. https://archive.org/details/principlesofan1987tort. 
  16. ↑ Williams, Peter L; Warwick, Roger; Dyson, Mary; Bannister, Lawrence H. (1989). Gray's Anatomy (Thirty-seventh ed.). Edinburgh: Churchill Livingstone. pp. 1278–1282. ISBN 0443-041776. 
  17. ↑ Romer, Alfred Sherwood; Parsons, Thomas S. (1977). The Vertebrate Body. Philadelphia, PA: Holt-Saunders International. pp. 349–353. ISBN 978-0-03-910284-5. 
  18. ↑ Krause J. William (July 2005). Krause's Essential Human Histology for Medical Students. Universal-Publishers. pp. 37–. ISBN 978-1-58112-468-2. https://books.google.com/books?id=cRayoldYrcUC&pg=PA37. Retrieved 25 November 2010. 
  19. ↑ Meiri, S.; Dayan, T. (2003-03-20). "On the validity of Bergmann's rule". Journal of Biogeography 30 (3): 331–351. doi:10.1046/j.1365-2699.2003.00837.x. Bibcode: 2003JBiog..30..331M. 
  20. ↑ Ashton, Kyle G.; Tracy, Mark C.; Queiroz, Alan de (October 2000). "Is Bergmann's Rule Valid for Mammals?". The American Naturalist 156 (4): 390–415. doi:10.1086/303400. PMID 29592141. Bibcode: 2000ANat..156..390A. 
  21. ↑ Millien, Virginie et al. (May 23, 2006). "Ecotypic variation in the context of global climate change: Revisiting the rules". Ecology Letters 9 (7): 853–869. doi:10.1111/j.1461-0248.2006.00928.x. PMID 16796576. Bibcode: 2006EcolL...9..853M. 
  22. ↑ Fitzpatrick, Katie (2005). "Gigantothermy". Davidson College. http://www.bio.davidson.edu/people/midorcas/animalphysiology/websites/2005/Fitzpatrick/Gigantothermy.htm. 
  23. ↑ "Marsquake! NASA's InSight Lander Feels Its 1st Red Planet Tremor". 23 April 2019. https://www.space.com/insight-mars-lander-first-marsquake.html. 
  24. ↑ "Archived copy". http://www.astro.uvic.ca/~venn/A201/maths.6.planetary_cooling.pdf. 



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