The chemical diffusion coefficient and its derivation for special cases

The chemical diffusion coefficient and its derivation for special cases 

The chemical diffusion coefficient D is necessary and sufficient for a phenomenological description of binary diffusion. For one-dimensional diffusion, in which a constant diffusional cross-section is assumed, the average particle velocities Vi (z = 1, 2) in an arbitrary coordinate system are given as  

Ni is the mole fraction of particles of type z, and is the molar volume. In the general case, is dependent upon the mole fraction. The chemical diffusion coefficient must be independent of the chosen coordinate system. The quantity (vi — V2), which is the average velocity of particles of type 1 relative to the velocity of particles of type 2, is also independent of the coordinate system. Therefore, this quantity may be used to give a suitable definition of 5. According to eq. (5-32), Vf is proportional to j]. Therefore, by Pick s first law, Vf is also proportional to the concentration gradient. The first step, then, is to divide (t i — U2) by dN2jdx, Finally, in order to make D conform to the diffusion coefficient in Pick s laws, it is necessary to multiply (vi —1 2) by Ni N2 9 so that the definition of 3 becomes [19]  

The coordinate x in eq. (5-33) appears only as a differential, so that the choice of the origin of the coordinate system for this definition is unimportant. Also, no assumption of a constant molar volume has been made. For ideal dilute solutions, 1, A2 0, and is constant. It then follows from eq. (5-33) that is equal to -5 dc2ldx. This is in agreement with eq. (5-29). It can be seen that for ideal dilute solutions the component diffusion coefficient Di and the chemical diffusion coefficient 3 are identical. 

An expression for the chemical diffusion coefficient for binary diffusion by means of vacancies can be derived in a straightforward way. Assume that the molar volume is independent of concentration. Since transport occurs via vacancies, there will be a flux of vacancies in addition to the fluxes of the components 1 and 2 in the lattice system. If the jump frequency r of particles of type 1 into the vacancies is greater than that of particles of type 2, then a local flux of vacancies will occur towards the region of higher concentration of component 1. Under the assumption of internal thermodynamic equilibrium, these vacancies are removed from the crystal at sites of repeatable growth (i.e. dislocations, grain boundaries). Because of this flux 

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