Định luật Fick về khuếch tán phân tử

Fick's law of diffusion
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Fick's laws of diffusion describe [You must be registered and logged in to see this link.] and can be used to solve for the diffusion coefficient D. They were derived by [You must be registered and logged in to see this link.] in the year [You must be registered and logged in to see this link.].



First law
Fick's first law is used in steady-state [You must be registered and logged in to see this link.], i.e., when the concentration within the diffusion volume does not change with respect to time . In one (spatial) dimension, this is

where

• J is the diffusion flux in dimensions of [([You must be registered and logged in to see this link.]) length⁻² time^-1], example (mol/m².s)
  
• is the diffusion coefficient or [You must be registered and logged in to see this link.] in dimensions of [length² time⁻¹], example (m²/s)
  
• (for ideal mixtures) is the concentration in dimensions of [(amount of substance) length⁻³], example (mol/m³)
  
• is the position [length], example (m)
  

is proportional to the velocity of the diffusing particles, which depends on the temperature, [You must be registered and logged in to see this link.] of the fluid and the size of the particles according to the [You must be registered and logged in to see this link.].
In dilute aqueous solutions the diffusion coefficients of most ions are similar and have values that at room temperature are in the range of 0.6x10^-9 to 2x10^-9 m²/s. For biological molecules the diffusion coefficients normally range from 10^-11 to 10^-10 m²/s.
In two or more dimensions we must use , the [You must be registered and logged in to see this link.] or [You must be registered and logged in to see this link.] operator, which generalises the first derivative, obtaining

The driving force for the one-dimensional diffusion is the quantity which for ideal mixtures is the concentration gradient. In chemical systems other than ideal solutions or mixtures, the driving force for
diffusion of each species is the gradient of [You must be registered and logged in to see this link.] of this species. Then Fick's first law (one-dimensional case) can be written as:

where the index i denotes the ith species, c is the concentration (mol/m³), R is the [You must be registered and logged in to see this link.] (J/(K mol)), T is the absolute temperature (K), and μ is the chemical potential (J/mol).

Second law
Fick's second law is used in non-steady or continually changing state diffusion, i.e., when the concentration within the diffusion volume changes with respect to time.

Where

• is the concentration in dimensions of [(amount of substance) length^-3], [mol m^-3]
  
• is time [s]
  
• is the diffusion coefficient in dimensions of [length² time^-1], [m² s^-1]
  
• is the position [length], [m]
  

It can be derived from the Fick's First law and the mass balance:

Assuming the diffusion coefficient D to be a constant we can exchange the orders of the differentiating and multiplying by the constant:

and, thus, receive the form of the Fick's equations as was stated above.
For the case of diffusion in two or more dimensions the Second Fick's Law is:

which is analogous to the [You must be registered and logged in to see this link.].

If the diffusion coefficient is not a constant, but depends upon the coordinate and/or concentration, the Second Fick's Law becomes:

An important example is the case where φ is at a steady state, i.e. the concentration does not change by time, so that the left part of the above equation is identically zero. In one dimension with constant , the solution for the concentration will be a linear change of concentrations along . In two or more dimensions we obtain
which is [You must be registered and logged in to see this link.], the solutions to which are called [You must be registered and logged in to see this link.] by mathematicians.

References

• A. Fick, Phil. Mag. (1855), 10, 30.
  
• A. Fick, Poggendorff's Annel. Physik. (1855), 94, 59.
  
• W.F. Smith, Foundations of Materials Science and Engineering 3^rd ed., McGraw-Hill (2004)
  
• H.C. Berg, Random Walks in Biology, Princeton (1977)
  

Notes

1. [You must be registered and logged in to see this link.] [You must be registered and logged in to see this link.] [You must be registered and logged in to see this link.]
   

External links

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• [You must be registered and logged in to see this link.]
  
• [You must be registered and logged in to see this link.] - Rex Njoku & Dr.Anthony Steyermark
  

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