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For the diffusion equation this formula gives
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For and diffusion
For higher values, the surface diffusion of deposited atomic species would allow for the formation of crystallites with long range atomic order.
For an attractive interaction between particles, the diffusion coefficient tends to decrease as concentration increases.
For a repulsive interaction between particles, the diffusion coefficient tends to increase as concentration increases.
For fluorophores in solution this means that the intensity and polarization of the emitted light is dependent on rotational diffusion.
For biological molecules the diffusion coefficients normally range from 10 < sup >− 11 </ sup > to 10 < sup >− 10 </ sup > m < sup > 2 </ sup >/ s.
For the case of diffusion in two or more dimensions Fick's Second Law becomes
For more detail on diffusion length, see these examples.
For the inhomogeneous anisotropic media these two forms of the diffusion equation should be combined in
For anisotropic multicomponent diffusion coefficients one needs 4-index quantities, for example,, where i, j are related to the components and α, β = 1, 2, 3 correspond to the space coordinates.
For similar reasons, and also due to the small size of helium atoms, helium's diffusion rate through solids is three times that of air and around 65 % that of hydrogen.
For dilute gases, kinetic molecular theory relates the diffusion coefficient D to the particle density n = N / V, the molecular mass M, the collision cross section, and the absolute temperature T by
For example, gravitational convection can be seen in the diffusion of a source of dry salt downward into wet soil due to the buoyancy of fresh water in saline.
For another solute, bicarbonate, dialysis solution level is set at a slightly higher level than in normal blood, to encourage diffusion of bicarbonate into the blood, to act as a pH buffer to neutralize the metabolic acidosis that is often present in these patients.
For diffusion of heat ( thermal diffusion ), the Péclet number is defined as:
For diffusion of particles ( mass diffusion ), it is defined as:
For the latter portions of the process, boundary and lattice diffusion from the boundary become important.
For solids, on the other hand, diffusion of atoms to the surface might not be sufficient and the surface tension can vary with an increase in surface area.
For instance, the net rate at which a chemical dissolved in a fluid moves toward or away from some point is proportional to the Laplacian of the chemical concentration at that point ; expressed symbolically, the resulting equation is the diffusion equation.
For localized movement, cytoplasmic streaming within cells and slow diffusion of ions and molecules between cells are utilized.
For example, G841 is a dark blue made by Great American Market ( GAM ), and R02 is a light amber made by Rosco and L216 is a diffusion filter made by Lee Filters.
For a given amount of confusion and diffusion, an SP network has more " inherent parallelism "
For this reason path integrals were used in the study of Brownian motion and diffusion a while before they were introduced in quantum mechanics.
For and equation
For an engine speed of 5000 rpm, a gear ratio of 4.00 to 1, and a tire radius of 13 inches, the equation would look like this: Af
For example, on the basis of the regression equation, a child with an I.Q. of 120 in this sample would be expected to earn an achievement test score of 4.8 ( grade equivalent ).
For example, the number of solutions of an equation over a finite field reflects the topological nature of its solutions over the complex numbers.
For example, the equation y = x corresponds to the set of all the points on the plane whose x-coordinate and y-coordinate are equal.
For flow through a tube with one inlet ( state 1 ) and exit ( state 2 ) as shown in the figure in this section, the continuity equation may be written and solved as:
; Associativity: For all a, b and c in A, the equation ( a • b ) • c = a • ( b • c ) holds.
For non-integer α, the functions J < sub > α </ sub >( x ) and J < sub >− α </ sub >( x ) are linearly independent, and are therefore the two solutions of the differential equation.
For example, most ab initio calculations make the Born – Oppenheimer approximation, which greatly simplifies the underlying Schrödinger equation by assuming that the nuclei remain in place during the calculation.
For a steady flow of charge through a surface, the current I ( in amperes ) can be calculated with the following equation:
For a fully relativistic treatment, the equation of state used will interpolate between the equations for small ρ and for large ρ.
For an axially loaded straight column with any end support conditions, the equation of static equilibrium, in the form of a differential equation, can be solved for the deflected shape and critical load of the column.
For example, the condition number associated with the linear equation
For example, a circle of radius 2 may be described as the set of all points whose coordinates x and y satisfy the equation x < sup > 2 </ sup > + y < sup > 2 </ sup > = 4.
For other initial conditions, the equation of motion is given by the exponential of a matrix: for an initial point x < sub > 0 </ sub >,
For an ellipse in canonical position ( center at origin, major axis along the X-axis ), the equation simplifies to
For example, the equation has 2 sets of solutions: ( with any x ) and ( with any y ).
For this equation, − 1 is a root of multiplicity 2.
For example, one can use it to determine, for any polynomial equation, whether it has a solution by radicals.
For current cryptographic purposes, an elliptic curve is a plane curve which consists of the points satisfying the equation
For example, the Cobb-Douglas equation — often used in economics — is nonlinear:
For an economic model described by more than one equation, simultaneous-equation methods may be used to remedy similar problems, including two IV variants, Two-Stage Least Squares ( 2SLS ), and Three-Stage Least Squares ( 3SLS ).
For example, if is uncorrelated with years of education, then the equation can be estimated with ordinary least squares.
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