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Stokes–Einstein–Sutherland equation, for diffusion of spherical particles through a liquid with low Reynolds number: = Here q is the electrical charge of a particle; μ q is the electrical mobility of the charged particle; η is the dynamic viscosity;
The Stokes radius or Stokes–Einstein radius of a solute is the radius of a hard sphere that diffuses at the same rate as that solute. Named after George Gabriel Stokes , it is closely related to solute mobility, factoring in not only size but also solvent effects.
Knowing the terminal velocity, the size and density of the sphere, and the density of the liquid, Stokes' law can be used to calculate the viscosity of the fluid. A series of steel ball bearings of different diameters are normally used in the classic experiment to improve the accuracy of the calculation.
For instance, a 20% saline (sodium chloride) solution has viscosity over 1.5 times that of pure water, whereas a 20% potassium iodide solution has viscosity about 0.91 times that of pure water. An idealized model of dilute electrolytic solutions leads to the following prediction for the viscosity μ s {\displaystyle \mu _{s}} of a solution: [ 57 ]
The Stokes-Einstein equation describes a frictional force on a sphere of diameter as = where is the viscosity of the solution. Inserting this into 9 gives an estimate for k D {\displaystyle k_{D}} as 8 R T 3 η {\displaystyle {\frac {8RT}{3\eta }}} , where R is the gas constant, and η {\displaystyle \eta } is given in centipoise.
The three viscosity equations now coalesce to a single viscosity equation = = because a nondimensional scaling is used for the entire viscosity equation. The standard nondimensionality reasoning goes like this: Creating nondimensional variables (with subscript D) by scaling gives
The Einstein field equations (EFE) may be written in the form: [5] [1] + = EFE on a wall in Leiden, Netherlands. where is the Einstein tensor, is the metric tensor, is the stress–energy tensor, is the cosmological constant and is the Einstein gravitational constant.
This expression (which is a normal distribution with the mean = and variance = usually called Brownian motion ) allowed Einstein to calculate the moments directly. The first moment is seen to vanish, meaning that the Brownian particle is equally likely to move to the left as it is to move to the right.