# Sedimentation coefficient

The sedimentation coefficient s of a particle is used to characterize its behaviour in sedimentation processes, notably centrifugation. It is defined as the ratio of a particle's sedimentation velocity to the acceleration that is applied to it (causing the sedimentation).

${\displaystyle s={\frac {v_{t}}{a}}}$

The sedimentation speed ${\displaystyle v_{t}}$ (in ms−1) is also known as the terminal velocity. It is constant because the force applied to a particle by gravity or by a centrifuge (measuring typically in multiples of tens of thousands of gravities in an ultracentrifuge) is cancelled by the viscous resistance of the medium (normally water) through which the particle is moving. The applied acceleration a (in m/s2) can be either the gravitational acceleration g, or more commonly the centrifugal acceleration ${\displaystyle \omega ^{2}r}$. In the latter case, ${\displaystyle \omega }$ is the angular velocity of the rotor and r is the distance of a particle to the rotor axis (radius).

The viscous resistance is given by the Stokes' law: 6πηr0v where η is the viscosity of the medium, r0 is the radius of the particle and v is the velocity of the particle. This law applies only for large spheres in an infinite sea of the fluid.

The centrifugal force is given by the familiar equation: mrω2. Here, r is the distance of the particle from the axis of rotation. When the two forces (viscous force and the centrifugal force) balance (they are oppositely directed), the particle moves with constant velocity called the terminal velocity. Hence the terminal velocity is given by the following equation.

${\displaystyle {v_{t}}={\frac {mr\omega ^{2}}{6\pi \eta r_{0}}}}$

Rearranging this equation we get the final formula:

${\displaystyle s={\frac {v_{t}}{r\omega ^{2}}}={\frac {m}{6\pi \eta r_{0}}}}$

The sedimentation coefficient has the dimensions of a unit of time and is expressed in svedbergs. One svedberg is defined as exactly 10−13 s. Essentially the sedimentation coefficient serves to normalize the sedimentation rate of a particle by the acceleration applied to it. The resulting value is no longer dependent on the acceleration, but depends only on the properties of the particle and the medium in which it is suspended. Sedimentation coefficients quoted in literature usually pertain to sedimentation in water at 20°C.

Larger particles sediment faster and have higher sedimentation coefficients (svedberg, or S values). Sedimentation coefficients are, however, not additive. The sedimentation rate does not only depend on the mass or volume of a particle. When two particles bind together, there is inevitably a loss of surface area. Thus, when measured separately they will have svedberg values that may not add up to that of the bound particle. This is notably the case with the ribosome. Ribosomes are most often identified by their sedimentation coefficient. For instance, the 70 S ribosome that comes from bacteria has actually a sedimentation coefficient of 70 svedberg, although it is composed of a 50 S subunit and a 30 S subunit.