Molecular diffusion, often called simply diffusion, is the thermal motion of all (liquid or gas) particles at temperatures above absolute zero. The rate of this movement is a function of temperature, viscosity of the fluid and the size (mass) of the particles. Diffusion explains the net flux of molecules from a region of higher concentration to one of lower concentration, but it is important to note that diffusion also occurs when there is no concentration gradient. The result of diffusion is a gradual mixing of material. In a phase with uniform temperature, absent external net forces acting on the particles, the diffusion process will eventually result in complete mixing.
Diffusive equilibrium is reached when the concentrations of the diffusing substance in the two compartments becomes equal.
Consider two systems; S1 and S2 at the same temperature and capable of exchanging particles. If there is a change in the potential energy of a system; for example μ1>μ2 (μ is Chemical potential) an energy flow will occur from S1 to S2, because nature always prefers low energy and maximum entropy.
Though the different systems are at equilibrium, there is still water passing through the semipermeable membrane. So if food coloring is put in system A, eventually it would be of equal color to system B.
Molecular diffusion is typically described mathematically using Fick's laws of diffusion.
Read more about Molecular Diffusion: Applications, Significance, Tracer, Self- and Chemical Diffusion, Non-equilibrium System, Concentration Dependent "collective" Diffusion, Equimolecular Counterdiffusion
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