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 In chemistry, a solution is a special type of homogeneous mixture composed of two or more substances. In such a mixture, a solute is a substance dissolved in another substance, known as a solvent. If the attractive forces between the solvent ...

under ultracentrifugation in traditional

sector Sector may refer to: Places * Sector, West Virginia, U.S. Geometry * Circular sector, the portion of a disc enclosed by two radii and a circular arc * Hyperbolic sector, a region enclosed by two radii and a hyperbolic arc * Spherical sector, a po ...

-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 The KTH Royal Institute of Technology ( sv, Kungliga Tekniska högskolan, lit=Royal Institute of Technology), abbreviated KTH, is a public research university in Stockholm, Sweden. KTH conducts research and education in engineering and technolo ...

, who derived it during his Ph.D. studies under

Svedberg 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 (i.e. how fast a p ...

Uppsala University Uppsala University ( sv, Uppsala universitet) 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 opera ...

. 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 or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an objec ...

, 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 the ...

. 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 The hydrodynamic radius of a macromolecule or colloid particle is R_. The macromolecule or colloid particle is a collection of N subparticles. This is done most commonly for polymers; the subparticles would then be the units of the polymer. R_ ...

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.,

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

''k''_''B'' multiplied by the

temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied o ...

''T'' in

kelvin The kelvin, symbol K, is the primary unit of temperature in the International System of Units (SI), used alongside its prefixed forms and the degree Celsius. It is named after the Belfast-born and University of Glasgow-based engineer and phys ...

Solute In chemistry, a solution is a special type of homogeneous mixture composed of two or more substances. In such a mixture, a solute is a substance dissolved in another substance, known as a solvent. If the attractive forces between the solvent ...

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

cannot pass through the inner and outer walls of the cell, resulting in the

boundary condition In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to th ...

s 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