Temperature Dependence of Chemical Potential and Drive

Introduction To begin, let us consider as a typical example the change with temperature in the chemical potential of table salt /t(NaCl) (Fig. 5.1). For comparison, the graphic also shows the temperature dependence of the chemical drive of table salt to decompose into the elements j (NaCl — Na + 2 2)- 

It is striking that the chemical potential falls more and more steeply with increasing temperature. Except for a very few exceptions of dissolved substances (e.g., Ca jw), all substances exhibit this behavior. The tendency of a substance to transform generally decreases when it is put into a warmer environment. 

The chemical drive A T), which is calculated from the temperature-dependent potentials, exhibits a noticeably more linear gradient than the p T) curves. Both curves intersect at the standard temperature T because the chemical potential of a substance at standard conditions corresponds to the drive to decompose into the elements (here sodium and chlorine). 

The initial value of the length is represented by /q and e represents the temperature coefficient. 

To indicate the change of chemical potential as a result of warming, we proceed exactly in the same manner  

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When an electric field is applied to a system consisting of droplets of liquid phase B present in liquid A, electrocapillary forces can bring about the movement of these droplets. These forces can be used to recover trace metals and metal mattes from waste pyrometallurgical slags [47]. The driving force in thermocapillary flows is (dy/dT) i.e. the temperature dependence of the surface tension. Whereas in electrocapillarity the driving force is (dy/dE) where E is the electrical potential at constant chemical potential X, and (dy/dE) is equal [47] to the surface excess charge density (qi) at the droplet interface (Equation 18). 

A membrane-based separation of gas mixtures is schematically shown in Figure 1. The gas mixture (feed) is passed over the membrane. The component that permeates preferentially is concentrated in the permeate, whereas the retained component is enriched in the retentate. Which component is preferentially permeating depends on the solubility of the components in the polymer matrix and the diffusion rate of the components through the membrane. The driving force for permeation is given by the difference of chemical potentials of the components on feed and permeates side and depends on pressure, temperature, and concentration difference. 

The above equation correctly implies that the driving force of a flux is the gradient of chemical potential that represent the gradient of the Gibbs free energy of 1 and not just the gradient in concentration. Both Li and Pi depend on Ci, temperature, and pressure (or stress). At constant T and p, one can express Equation 11.9 as... 

In summary, the chemical potential of a substance depends on its concentration, the pressure, the electrical potential, and gravity. We can compare the chemical potentials of a substance on two sides of a barrier to decide whether it is in equilibrium. If fij is the same on both sides, we would not expect a net movement of species / to occur spontaneously across the barrier. The relative values of the chemical potential of species / at various locations are used to predict the direction for spontaneous movement of that chemical substance (toward lower /a ), just as temperatures are compared to predict the direction for heat flow (toward lower T). We will also find that Afij from one region to another gives a convenient measure of the driving force on species /. 

The quantity of primary interest in our thermodynamic construction is the partial molar Gihhs free energy or chemical potential of the solute in solution. This chemical potential depends on the solution conditions the temperature, pressure, and solution composition. A standard thermodynamic analysis of equilibrium concludes that the chemical potential in a local region of a system is independent of spatial position. The ideal and excess contributions to the chemical potential determine the driving forces for chemical equilibrium, solute partitioning, and conformational equilibrium. This section introduces results that will be the object of the following portions of the chapter, and gives an initial discussion of those expected results. 

Conversely, the correct approach to formulate the diffusion of a single component in a zeolite membrane is to use the MaxweU-Stefan (M-S) framework for diffusion in a nonideal binary fluid mixture made up of species 1 and 2 where 1 and 2 stands for the gas and the zeohtic material, respectively. In the M-S theory it is recognized that to effect relative motions between the species 1 and 2 in a fluid mixture, a force must be exerted on each species. This driving force is the chemical potential gradient, determined at constant temperature and pressure conditions [68]. The M-S diffiisivity depends on coverage and fugacity, and, therefore, is referred to as the corrected diffiisivity because the coefficient is corrected by a thermodynamic correction factor, which can be determined from the sorption isotherm. 

The three main driving forces which have been used within diffusion models (moisture content, partial pressure of water vapor, and chemical potential) will now be discussed. Attempts to predict diffusion coefficients theoretically will also be reviewed, together with experimental data for fitted diffusion coefficients and their dependence on temperature and moisture content. 

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