Diffusion Coefficient
The rate of diffusion of a substance is characterized by the diffusion coefficient.
In chemical kinetics we dealt with the charge of chemical species in a chemical reaction mixture with time. In diffusion studies, we deal with the change in the location of solute with time. The conceptually simplest illustration of the feature of this time dependence is given by the changes exhibited by an initially thin sheet of solute. As time progresses, the concentration of the solute corresponds to the curves shown. The rate, at which these curves develop, i.e. the rate of spreading out of the solute, depends, of rate given solvent and temperature, on the nature of the solute.
The driving force diffusion at any time is the rate of change of concentration along the diffusion direction, i.e. the concentration gradient or c is the concentration and x the one dimension we deal with here.
Experimental studies of diffusion show that proportionality exists between the rate of flow across any cross section with area A and the concentration gradient at that cross section. Thus we write:
Rate of diffusion in x direction ∝ A δc/δx
The diffusion in x direction = - DA δC/δx
The negative sign allows D, the diffusion coefficient, to take on positive values equation is based on observation of diffusion processes and is known as Fick’s first law of diffusion. The proportionality constant S depends on the nature of both the diffusing solute and solvent. If the rate of diffusion is expressed in terms of same quantity measure that is used in c, such as grams, molecules, or moles, then the units for D are, for example, square meter per second.
If the diffusion through a unit cross section area is considered in the equation becomes simply:
Rate of diffusion = D δc/δx
It is often more convenient to have a description of the effect of diffusion on the concentration in a volume element, as in fig. for this we need to calculate the net rate with which the diffusion substance accumulates in the volume in the volume element A dx. This can be done by calculating the rate of the substance enters the volume element at x and the rate with it leaves at x+ dx. The concentration gradient in general varies along x, and the rate with which it varies is:
δ/δx = (δc/ δx = δ^{2}c/ δx^{3}
Now rate of entry at x = -DA δc/δx
Rate of leaving at x + dx = -DA (δc/ δx +δ^{2}c / δxc^{2}dx)
Subtraction of these two rates gives:
Rate of accumulation = -DA δc/δx - [-DA (δc/ δx = δ^{2}c / δx^{2} dx)
= DA δ^{2}c/ δx^{2} dx
Example: how far can one expect a sucrose molecule; dissolve in water at 25˚C, to move from its original position as a result of its diffusional motion in a time of 1 h? The value of the diffusion coefficient D for sucrose in water at 25˚C is 4.0 × 10^{-6} cm^{2} s^{-1}, or 4.0 × 10^{-10} m^{2} s^{-1}.
Solution: a particular estimate of the average distance that the sucrose molecule might have moved. We obtain:
√x^{2} = [2 (4.0 × 10^{-10} m^{2} s^{-1}) (3600 s)]1/2 = 1.7 × 10^{-3} m = 1.7 mm
This example illustrates the slowness of the diffusion process. It is not an effective process for transport over appreciable distances. Biological systems make use of diffusion when very short distances are involved but depend on other mechanisms for transport over longer distances.
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