The Lamm equationO Lamm: (1929) "Die Differentialgleichung der Ultrazentrifugierung"'' Arkiv för matematik, astronomi och fysik'' 21B No. 2, 1–4 describes the sedimentation and diffusion of a solute under ultracentrifugation in traditional sector-shaped cells. (Cells of 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 PhD studies under

Svedberg In chemistry, a Svedberg unit or svedberg (symbol S, sometimes Sv) is a non- SI metric unit for sedimentation coefficients. The Svedberg unit offers a measure of a particle's size indirectly based on its sedimentation rate under acceleration ...

Uppsala University Uppsala University (UU) () is a public university, public research university in Uppsala, Sweden. Founded in 1477, it is the List of universities in Sweden, oldest university in Sweden and the Nordic countries still in operation. Initially fou ...

. The Lamm equation can be written: :

\frac = 
D \left \left( \frac \right) + 
\frac \left( \frac \right) \right - 
s \omega^ \left r \left( \frac \right) + 2c \right

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 In physics, angular velocity (symbol or \vec, the lowercase Greek letter omega), also known as the angular frequency vector,(UP1) is a pseudovector representation of how the angular position or orientation of an object changes with time, i ...

, respectively. The first and second terms on the right-hand side of the Lamm equation are proportional to ''D'' and ''sω''², respectively, and describe the competing processes of

diffusion Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...

and

sedimentation Sedimentation is the deposition of sediments. It takes place when particles in suspension settle out of the fluid in which they are entrained and come to rest against a barrier. This is due to their motion through the fluid in response to th ...

. Whereas

seeks to concentrate the solute near the outer radius of the cell,

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''_''b'' can be determined from the ratio of ''s'' and ''D'' :

\frac = \frac

where ''k''_B''T'' is the thermal energy, i.e., the

Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative thermal energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin (K) and the ...

''k''_B multiplied by the absolute temperature ''T''. Solute

molecules A molecule is a group of two or more atoms that are held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions that satisfy this criterion. In quantum physics, organic chemistry ...

cannot pass through the inner and outer walls of the cell, resulting in the boundary conditions on the Lamm equation :

D \left( \frac \right) - s \omega^2 r c = 0

at the inner and outer radii, ''r''_''a'' and ''r''_''b'', respectively. By spinning samples at constant

''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 of the solute.

References and notes

{{DEFAULTSORT:Lamm Equation Laboratory techniques Partial differential equations