# Fick's law

**Fick's law** relates the molecular (diffusive) flux to its mass fraction or mole fraction gradient in a multicomponent mixture.

## Fick's law on mass basis

In one dimension, Fick's law is expressed as:

(1) |

where

- is the mass diffusion flux, which is expressed as the mass of per unit area per unit time.
- is the mass density of the mixture.
- is the diffusion coefficient.
- is the mass fraction of A in the mixture.
- is position, the dimension of which is length.

In two or more dimensions we use , the gradient operator, which is independent of the chosen coordinate system. The generalized form of Fick's equation is

(2) |

where denotes the diffusion flux vector.

## Fick's law on a mole basis

Fick's law can also be written on a mole basis:

(3) |

where

- is the molar diffusion flux, which is expressed as the moles of per unit area per unit time.
- is the molar concentration of the mixture.
- is the mole concentration of A.

## Relationship between the convective flux, molecular flux, and species velocities

### Mass basis

The convective flux of on a mass basis is

(4) |

which represents the transport of species due to both bulk flow and diffusion. Here, is the mass-average velocity. In a multicomponent fluid with distinct species, the mass-average velocity is

(5) |

where

- is the weight fraction of species .
- is the velocity of species .
- is the total density of the mixture.

Hence, the molecular flux of species is related to the difference between the velocity of and the mass-average velocity of the fluid.

(6) |

### Mole basis

The convective flux of on a mole basis is

(7) |

which represents the transport of species due to both bulk flow and diffusion. Here, is the mole-average velocity. In a multicomponent fluid with distinct species, the mole-average velocity is

(8) |

where

- is the mole fraction of species .
- is the velocity of species .
- is the total mole concentration of the mixture.

Hence, the molecular flux of species is related to the difference between the velocity of and the mole-average velocity of the fluid.

(9) |