# (p.268) Appendix F Mutual Diffusion and Self-Diffusion

# (p.268) Appendix F Mutual Diffusion and Self-Diffusion

In this appendix we want to discuss the difference between mutual diffusion and self-diffusion from the point of view of the thermodynamics of irreversible processes. A fuller discussion of this topic can be found in Tyrrell and Harris (1984).

# F.1 Mutual Diffusion

We first consider mutual diffusion in a two component mixture. Component 0 denotes the solute and component 1 denotes the solvent. Let J_{i} denote the diffusion flux of component *i* in the barycentric coordinate system. That is, the flux is measured relative to the local center of mass motion of the fluid. The basic description of diffusion is Fick’s law

Because we are in the barycentric coordinate system, J_{0} + J_{1} = 0 and only one of the fluxes is independent. It then follows that *L* _{00} = *-L* _{01}. The relation between the flux J_{0} and the thermodynamic forces, the gradients of chemical potential, is then

Using the Gibbs-Duhem relation *c* _{0}∇µ_{0} + *c*1_{∇µ} _{1} = 0, eqn (F.2) can be written

where *x* _{1} is the mole fraction of component 1. Consequently the relationship between the phenomenological coefficient *L* _{00} and the mutual diffusion coefficient, is

# (p.269) F.2 Self-Diffusion

Now we consider the same system except that now some of the solute molecules (component 0) are tagged somehow. The tagged mo;ecules are denoted as component *α* and the untagged solute molecules as component *β*. The fractions of tagged and untagged molecules are *y _{α}* and

*y*respectively.

_{β}*c*+

_{α}*c*=

_{β}*c*

_{0}. It is assumed that molecules of type

*α*and of type

*β*are identical except for tagging.

The fluxes of the solute species can be written as

In other words, the tagged system is treated as a three component mixture. The *l* coefficients are the phenomenological coefficients appropriate to the discussion of the three component system. The chemical potentials of the tagged species are

Here, *µ* _{0} is the chemical potential of component zero. It is not a standard state chemical potential; there is no implication that the solution is ideal.

In a self diffusion experiment, the concentrations of species 0 and 1 are uniform. Thus *∇ _{µ}*

_{0}=

*∇*

_{µ}_{1}= 0. Since

*y*+

_{α}*y*= 1,

_{β}*∇*=

_{yα}*-∇*. Consequently

_{yβ}Writing J_{β} = -D_{s}∇_{cβ} as the definition of the self-diffusion coefficient, *D _{s}* we find that

# F.3 Relation between *D*_{m} and *D* _{s}

_{m}

Now we return to the general situation where components 0 and 1 are not necessarily uniform in space. Since J_{0} = J_{α} + J_{β}, we have

where we have used the Onsager relation *l _{αβ}* =

*l*. For this equation to be compatible with eqn (F.1.3) the coefficient of ▽

_{βα}_{µ}

_{0}must be

*L*

_{00}/

*x*

_{1}and the coefficient of ▽

_{yα}must vanish. These conditions yield

where we have again used the Onsager relations. The second of eqns (F.3.2) can be used to eliminate *l _{αβ}* from the first, and we find

The equation for *D _{s}* becomes

To go any further it is necessary to know more about the *y* dependence of the *l* coefficients. Irreversible thermodynamics is silent on the subject of the dependence of phenomenological coefficients on thermodynamic state. It just assumes that the coefficients are independent of gradients so that the theory is linear. It is clear, however, that there is no a priori reason why eqns (F.3.3) and (F.3.4) should be the same. First of all, (F.3.3) contains the thermodynamic derivative (*∂µ* _{0}/*∂c* _{0})*T, P* as a factor while (F.3.4) does not. Secondly, the remaining factors are not necessarily the same. In the limit *c* _{0} → 0, however, the thermodynamic derivative does approach *RT*/*c* _{0}. Nevertheless, non thermodynamic arguments are needed to conclude that *D _{m}* →

*D*as

_{s}*c*

_{0}→ 0, that is, at infinite dilution of solute.