Lamm equation
Encyclopedia
The Lamm equation 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 Ph.D. studies under Svedberg
at Uppsala University
.
The Lamm equation can be written:
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 sω2, 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 mb can be determined from the ratio of s and D
where kBT is the thermal energy, i.e.,
Boltzmann's constant kB multiplied by
the temperature
T in kelvin
s.
Solute
molecules cannot pass through the inner and outer walls of the
cell, resulting in the boundary conditions on the Lamm equation
at the inner and outer radii, ra and rb, 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.
Solution
In chemistry, a solution is a homogeneous mixture composed of only one phase. In such a mixture, a solute is dissolved in another substance, known as a solvent. The solvent does the dissolving.- Types of solutions :...
under ultracentrifugation
Ultracentrifuge
The ultracentrifuge is a centrifuge optimized for spinning a rotor at very high speeds, capable of generating acceleration as high as 2,000,000 g . There are two kinds of ultracentrifuges, the preparative and the analytical ultracentrifuge...
in traditional sector
Circular sector
A circular sector or circle sector, is the portion of a disk enclosed by two radii and an arc, where the smaller area is known as the minor sector and the larger being the major sector. In the diagram, θ is the central angle in radians, r the radius of the circle, and L is the arc length of the...
-shaped cells. (Cells of
other shapes require much more complex equations.) It was named after Ole Lamm
Ole Lamm
Ole Albert Lamm, born December 25, 1902 in Gothenburg, died August 14, 1964 in Stockholm, was a Swedish physical chemist whose research included diffusion and sedimentation phenomena....
, later professor of physical chemistry at the Royal Institute of Technology
Royal Institute of Technology
The Royal Institute of Technology is a university in Stockholm, Sweden. KTH was founded in 1827 as Sweden's first polytechnic and is one of Scandinavia's largest institutions of higher education in technology. KTH accounts for one-third of Sweden’s technical research and engineering education...
, who derived it during his Ph.D. studies under Svedberg
Svedberg
A svedberg is a non-SI physical unit used for sedimentation coefficients. It characterizes the behaviour of a particle type in sedimentation processes, notably centrifugation. The svedberg is technically a measure of time, and is defined as exactly 10-13 seconds A svedberg (symbol S, sometimes...
at Uppsala University
Uppsala University
Uppsala University is a research university in Uppsala, Sweden, and is the oldest university in Scandinavia, founded in 1477. It consistently ranks among the best universities in Northern Europe in international rankings and is generally considered one of the most prestigious institutions of...
.
The Lamm equation can be written:
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
Angular velocity
In physics, the angular velocity is a vector quantity which specifies the angular speed of an object and the axis about which the object is rotating. The SI unit of angular velocity is radians per second, although it may be measured in other units such as degrees per second, revolutions per...
, respectively. The first and second terms on the right-hand side of the Lamm equation are proportional to D and sω2, respectively, and describe the competing processes of diffusion
Diffusion
Molecular diffusion, often called simply diffusion, is the thermal motion of all particles at temperatures above absolute zero. The rate of this movement is a function of temperature, viscosity of the fluid and the size of the particles...
and sedimentation
Sedimentation
Sedimentation is the tendency for particles in suspension to 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 forces acting on them: these forces can be due to gravity, centrifugal acceleration...
. Whereas sedimentation
Sedimentation
Sedimentation is the tendency for particles in suspension to 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 forces acting on them: these forces can be due to gravity, centrifugal acceleration...
seeks to concentrate the solute near the outer radius of the cell, diffusion
Diffusion
Molecular diffusion, often called simply diffusion, is the thermal motion of all particles at temperatures above absolute zero. The rate of this movement is a function of temperature, viscosity of the fluid and the size of the particles...
seeks to equalize the solute concentration throughout the cell. The diffusion constant D can be estimated from the hydrodynamic radius
Hydrodynamic radius
The hydrodynamic radius of a macromolecule or colloid particle has two meanings. Some books use it as a synonym for the Stokes radius. Section 6.4 page 290....
and shape of the solute, whereas the buoyant mass mb can be determined from the ratio of s and D
where kBT is the thermal energy, i.e.,
Boltzmann's constant kB multiplied by
the temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
T in kelvin
Kelvin
The kelvin is a unit of measurement for temperature. It is one of the seven base units in the International System of Units and is assigned the unit symbol K. The Kelvin scale is an absolute, thermodynamic temperature scale using as its null point absolute zero, the temperature at which all...
s.
Solute
Solution
In chemistry, a solution is a homogeneous mixture composed of only one phase. In such a mixture, a solute is dissolved in another substance, known as a solvent. The solvent does the dissolving.- Types of solutions :...
molecules cannot pass through the inner and outer walls of the
cell, resulting in the boundary conditions on the Lamm equation
at the inner and outer radii, ra and rb, respectively. By spinning samples at constant angular velocity
Angular velocity
In physics, the angular velocity is a vector quantity which specifies the angular speed of an object and the axis about which the object is rotating. The SI unit of angular velocity is radians per second, although it may be measured in other units such as degrees per second, revolutions per...
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.