# Fick's law of diffusion

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Fick's laws of diffusion describe diffusion.

 Contents

## History

Fick's laws of diffusion were derived by Adolf Fick in the year 1858.

## Fick's First Law

Fick's First Law is used in steady state diffusion, i.e., when the concentration within the diffusion volume does not change with respect to time (Jin=Jout).

[itex]J = - D \frac{\partial c}{\partial x}[itex]

Where

• [itex]J[itex] is the diffusion flux in dimensions of [parts length-2 time-1]
• [itex]D[itex] is the diffusion coefficient in dimensions of [length2 time-1]
• [itex]c[itex] is the concentration in dimensions of [parts length-3]
• [itex]x[itex] is the position

## Fick's Second Law

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

[itex]\frac{\partial c}{\partial t} = D \frac{\partial^2 c}{\partial x^2}[itex]

Where

• [itex]c[itex] is the concentration in dimensions of [parts length-3]
• [itex]t[itex] is time
• [itex]D[itex] is the diffusion coefficient in dimensions of [length2 time-1]
• [itex]x[itex] is the position

## A Biological Perspective

The first law gives rise to the formula

[itex]\mathrm{Rate\ of\ diffusion} = \frac{K A (P_2 - P_1)}{D}[itex]

It states that the rate of diffusion of a gas across a membrane is

• Constant for a given gas at a given temperature by an experimentally determined factor, [itex]K[itex]
• Proportional to the surface area over which diffusion is taking place, [itex]A[itex]
• Proportional to the difference in partial pressures of the gas across the membrane, [itex]P_2 - P_1[itex]
• Inversely proportional to the distance over which diffusion must take place, or in other words the thickness of the membrane, [itex]D[itex].

Fick's first law is also important in radiation transfer equations. However, in this context it becomes inaccurate when the diffusion constant is low and the radiation becomes limited by the speed of light rather than by the resistance of the material the radiation is flowing through. In this situation, one can use a flux limiter.

The exchange rate of a gas across a fluid membrane can be determined by using this law together with Graham's law.

## References

• W.F. Smith, Foundations of Materials Science and Engieering 3rd ed., McGraw-Hill (2004)

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