Diffusion

 

Diffusion is a process that takes place in fluids (gasses and liquids) whereby substances dispersed (or dissolved) in the fluid are transported from areas of high concentration to those of low concentration as a result of the differential energy provided by the concentration gradient. The process is mathematically expressed in the following equation known as Fick’s law.

 

 

where (Q) is the mass of solute transported per unit time across an area (A) normal to the direction of diffusion under the concentration gradient. The constant (K) (more often given the symbol (D) with a suitable subscript denoting the medium to which it applies i.e. a gas or a liquid) is called the Diffusivity of the solute. The Diffusivity of a substance is defined as the mass of solute transported across unit area in unit time under unit concentration gradient. (K) will be unique for any particular solute and will be a function of temperature.

 

The process of diffusion is important in many mass transfer processes in liquids and gasses and also to a limited extent in solids. Diffusion is particularly important in solute band dispersion a transport mechanism that takes place in all chromatographic processes.

The Diffusion Process in a Chromatographic System

 

Diffusion processes play important parts in peak dispersion. The process not only contributes to dispersion directly (i.e., longitudinal diffusion), but also helps to reduce the dispersion that results from solute transfer between the two phases. Consider the situation depicted in figure 11.

 

Figure 11. The Diffusion Process

 

Consider a sample of solute introduced in plane (A), (plane (A) having unit cross-sectional area). Solute will diffuse according to Fick's law in both directions (x) and, at a point (x) from the sample point, according, the mass of solute transported across unit area in unit time (m_x) according to Fick will be given by,

 

                                                                                                      

 

 where (D_m) is the Diffusivity of the solute in the fluid.

 

      and  is the concentration gradient at (x).

 

 

Now, mass of solute leaving the slice (dx) thick, at (x + dx), i.e., (m_x + dx), is,

 

                                       

Thus, the net change in mass per unit time in the slice (dx) thick will be,

 

 

or                                              

 

                    Now  as                          ,

 

  then                       

 

 

         or,                                              

 

Now, this is a standard differential equation and one solution to this equation, which can be proved by appropriate differentiation, takes the Gaussian form as follows:

 

 

Now, if  , the variance of the resulting Gaussian function will be.

                                                       

 

_{Thus, the longitudinal dispersion in a chromatographic column can be deduced.}