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| The '''Lamm equation'''<ref>O Lamm: (1929) "Die Differentialgleichung der Ultrazentrifugierung"'' Arkiv för matematik, astronomi och fysik'' '''21B No. 2''', 1–4</ref> describes the sedimentation and diffusion of a [[solution|solute]] under [[ultracentrifuge|ultracentrifugation]] in traditional [[Circular sector|sector]]-shaped cells. (Cells of
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| other shapes require much more complex equations.) It was named after [[Ole Lamm]], later professor of physical chemistry at the [[Royal Institute of Technology]], who derived it during his Ph.D. studies under [[Svedberg]] at [[Uppsala University]].
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| The Lamm equation can be written:<ref name=Rubinow>{{cite book |title=Introduction to mathematical biology |author = SI Rubinow |url=http://books.google.com/books?id=3j0gu63QWmQC&pg=PA250&dq=Lamm+equation&sig=mH9T2kyIE0PIqMGzh6WF-KGyApc#PPA235,M1 |isbn=0-486-42532-0 |year=2002 (1975) |publisher=Courier/Dover Publications |pages=235–244}}</ref><ref name=Mazumdar>{{cite book |title=An Introduction to Mathematical Physiology and Biology |url=http://books.google.com/books?id=mqD6qSYGM-QC&pg=PA33&dq=Lamm+equation&lr=&sig=qiZ-6RAfQsRqDAzyuOJsiz4gOlE |page= 33 ff |author= [http://www.maths.adelaide.edu.au/jagan.mazumdar/ Jagannath Mazumdar] |year=1999 |publisher=Cambridge University Press |location=Cambridge UK |isbn=0-521-64675-8 }}</ref>
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| :<math>
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| \frac{\partial c}{\partial t} =
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| D \left[ \left( \frac{\partial^{2} c}{\partial r^2} \right) +
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| \frac{1}{r} \left( \frac{\partial c}{\partial r} \right) \right] -
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| s \omega^{2} \left[ r \left( \frac{\partial c}{\partial r} \right) + 2c \right]
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| </math>
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| where ''c'' is the solute concentration, ''t'' and ''r'' are the time and radius, and the parameters ''D'', ''s'', and ''ω'' represent the solute diffusion constant, sedimentation coefficient and the rotor [[angular velocity]], respectively. The first and second terms on the right-hand side of the Lamm equation are proportional to ''D'' and ''sω''<sup>2</sup>, respectively, and describe the competing processes of [[diffusion]] and [[sedimentation]]. Whereas [[sedimentation]] seeks to concentrate the solute near the outer radius of the cell, [[diffusion]] seeks to equalize the solute concentration throughout the cell. The diffusion constant ''D'' can be estimated from the [[hydrodynamic radius]] and shape of the solute, whereas the buoyant mass ''m''<sub>''b''</sub> can be determined from the ratio of ''s'' and ''D'' | |
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| :<math> | |
| \frac{s}{D} = \frac{m_b}{k_B T}
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| </math>
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| where ''k''<sub>''B''</sub>''T'' is the thermal energy, i.e.,
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| [[Boltzmann constant|Boltzmann's constant]] ''k''<sub>''B''</sub> multiplied by
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| the [[temperature]] ''T'' in [[kelvin]]s.
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| [[Solution|Solute]] [[molecules]] cannot pass through the inner and outer walls of the
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| cell, resulting in the [[boundary condition]]s on the Lamm equation
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| :<math>
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| D \left( \frac{\partial c}{\partial r} \right) - s \omega^2 r c = 0
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| </math>
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| at the inner and outer radii, ''r''<sub>''a''</sub> and ''r''<sub>''b''</sub>, respectively. By spinning samples at constant [[angular velocity]] ''ω'' and observing the variation in the concentration ''c''(''r'', ''t''), one may estimate the parameters ''s'' and ''D'' and, thence, the (effective or equivalent) buoyant mass the solute.
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| ==Derivation of the Lamm equation==
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| {{Empty section|date=July 2010}}
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| ==Faxén solution (no boundaries, no diffusion)==
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| {{Empty section|date=September 2010}}
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| ==References and notes==
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| <references/>
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| {{DEFAULTSORT:Lamm Equation}}
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| [[Category:Laboratory techniques]]
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| [[Category:Partial differential equations]]
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