Mean chemical potential

If in an electrolyte solution a molecule dissociates into v+ pieces of cations each with a valence z+ and v. pieces of anions each with a valence z., the relation of z,v+ +z v =0 holds. We can then define the mean chemical potential fi of the dissociated ions in general by Eq. 8.44 ... 

The mean chemical potential of a pair of cations and anions can be estimated from the ionic dissociation equilibrium shown as an example in Eqs. 8.42 and 8.45 ... 

Beside the chemical potential of an electrolyte B (p%) the mean chemical potential (p ), i.e., the average part of... 

What is the significance of these quantities fi, x, and It is obvious they are all average quantities—the mean chemical potential the mean standard chemical potential the mean ionic mole fraction x, and the mean ionic-activity coefficient f. In the case of and fi% the arithmetic mean (half the sum) is taken because free energies are additive, but in the case of x and f , the geometric mean (the square root of the product) is taken because the effects of mole fraction and activity coefficient on free energy are multiplicative. 

Mean Chemical Potentials and Mean Ionic Activity Coefficients. 

H cv ( yP) + cr ycr j The activity coefficients, just like the chemical potentials, can only occur in certain combinations, and never alone in equilibrium equations. Thus the mean chemical potential of the Na+ and Cl ions is... 

In general, corresponding to the mean chemical potential (27.19) there corresponds a mean ionic activity coefficient given by... 

As explained in Section 4.8.2, our bulk model depends on four adjustable parameters, namely T, /jl, A/x, and Ewa- Among these the mean chemical potential //. determines mainly gas-liquid phase eoexistence. Because the bulk experimental mixture is always in the liquid state, /x should be larger than —2.0 (in our dimensionless units) so that Awe are dealing with a (bulk) liquid state in the model calculations for all considered temperatmes. The incremental chemical potential A/x, on the other hand, is the thermodynamic variable conjugate to the composition cast here in terms of for... 

In ionic systems, since the positive and negative ions always come in pairs, physically it is not possible to measure the chemical potentials j,Ag+ nnd pci-separately only the sum can be measured. A similar problem arises for the definition of enthalpy and Gibbs free energy of formation. For ions, these two quantities are defined with respect to a new reference state based on the H" " ions, as described in Box 8.1. For the chemical potential, a mean chemical potential is defined by... 

The activity coefficients y of electrolytes are defined with respect to ideal solutions. For example, the mean chemical potential for AgCl is written as... 

The existence of this process demonstrates that the structure of the amorphous intercrystalline layers in a semicrystalline polymer is different from a polymer melt. The reason can be easily seen All the chain sequences are fixed with their ends in the crystallites and, furthermore, the concentration of entanglements is enhanced. As a consequence, the mean chemical potential of the units is higher than in a melt and varies with the layer thickness. The direction of change is obvious. The numbers of entanglements and points of chain entry into the crystallites are constant. The motional restrictions thus become diminished if the layer thickness increases, which implies a decrease in the chemical potential. Under such conditions each change in temperature leads to a new local equilibrium between crystallites and amorphous regions, via a surface crystallization or melting process. 

Strictly speaking it is not simply possible to introduce a charged particle into the volume, on account of the high energy that is connected with the violation of electroneutraJity. Thus, we preferably speak of (/Xd+ + jUd ) as the chemical potential of the (separated) pair or of twice the mean chemical potential of the individual defects. 

This sum conesponds to the work done in transferring 2 moles of ions (one mole of K and one mole of Cl ) from infinity into the solution. In order to refer to the chemical potential of 1 mole of transferred ions, one is talking about a mean chemical potential and mean activity coefficients/+, and Eq. (A7) must be divided by two ... 

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