Liquid chemical potential

Thermal difiusivity Chemical potential Chemical potential in bulk liquid Chemical potential in bubble Density... 

In view of the aforementioned assumption, it is sufficient to consider the liquid content in drying material only. The gravitational force was already assumed as insignificant, so the moisture flux depends on the gradient of liquid chemical potential 0, e) expressed as follows... 

If we neglect the pressure dependence of the liquid chemical potential, and assume the vapor phase to behave ideally, we obtain for the homogeneous nucleation rate in a supercooled vapor. 

So the average driving force on the tray is only a function of the chemical potential of the incoming vapour and liquid chemical potential ... 

As always indicates the pure A and xa + xb = 1. In addition xa > xb- Equation (3.142) predicts a downward shift of the liquid chemical potential of A, which is shown as long-dashed line in Fig. 3.20. We note that we are interested only in the immediate vicinity of the boiling temperature Tb. Therefore this line and the corresponding solid line are parallel to good approximation. Furthermore we assume that the amount of B in the gas phase is negligible or causes only a negligible shift of the gas phase chemical potential. Therefore we find that the intersection of the chemical potentials of A in the liquid phase and the gas phase has shifted to a higher temperature Tb +ST. 

The coexisting densities below are detennined by the equalities of the chemical potentials and pressures of the coexisting phases, which implies that tire horizontal line joining the coexisting vapour and liquid phases obeys the condition... 

Swope W C and Andersen H C 1995 A computer simulation method for the calculation of chemical potentials of liquids and solids using the bicanonical ensemble J. Chem. Phys. f02 2851-63... 

At equilibrium, in order to achieve equality of chemical potentials, not only tire colloid but also tire polymer concentrations in tire different phases are different. We focus here on a theory tliat allows for tliis polymer partitioning [99]. Predictions for two polymer/colloid size ratios are shown in figure C2.6.10. A liquid phase is predicted to occur only when tire range of attractions is not too small compared to tire particle size, 5/a > 0.3. Under tliese conditions a phase behaviour is obtained tliat is similar to tliat of simple liquids, such as argon. Because of tire polymer partitioning, however, tliere is a tliree-phase triangle (ratlier tlian a triple point). For smaller polymer (narrower attractions), tire gas-liquid transition becomes metastable witli respect to tire fluid-crystal transition. These predictions were confinned experimentally [100]. The phase boundaries were predicted semi-quantitatively. 

Ultimately, the surface energy is used to produce a cohesive body during sintering. As such, surface energy, which is also referred to as surface tension, y, is obviously very important in ceramic powder processing. Surface tension causes liquids to fonn spherical drops, and allows solids to preferentially adsorb atoms to lower tire free energy of tire system. Also, surface tension creates pressure differences and chemical potential differences across curved surfaces tlrat cause matter to move. 

The chemical potential of a curved surface is extremely critical in ceramic processing. It detemiines reactivity, tlie solubility of a solid in a liquid, tire rate of liquid evaporation from solid surfaces, and material transport during sintering. 

In general there are two factors capable of bringing about the reduction in chemical potential of the adsorbate, which is responsible for capillary condensation the proximity of the solid surface on the one hand (adsorption effect) and the curvature of the liquid meniscus on the other (Kelvin effect). From considerations advanced in Chapter 1 the adsorption effect should be limited to a distance of a few molecular diameters from the surface of the solid. Only at distances in excess of this would the film acquire the completely liquid-like properties which would enable its angle of contact with the bulk liquid to become zero thinner films would differ in structure from the bulk liquid and should therefore display a finite angle of contact with it. 

At the junction of the adsorbed film and the liquid meniscus the chemical potential of the adsorbate must be the resultant of the joint action of the wall and the curvature of the meniscus. As Derjaguin pointed out, the conventional treatment involves the tacit assumption that the curvature falls jumpwise from 2/r to zero at the junction, whereas the change must actually be a continuous one. Derjaguin put forward a corrected Kelvin equation to take this state of affairs into account but it contains a term which is difficult to evaluate numerically, and has aroused little practical interest. 

Figure 3.10 is a plot of potential against distance from the wall for a liquid in a capillary of sufficient width for its middle A to be outside the range of forces from the wall. Since the capillary condensate is in equilibrium with the vapour, its chemical potential (=p represented by the horizontal line GF, will be lower than that of the free liquid the difference in chemical potential of the condensate at A, represented by the vertical distance AF, is brought about entirely by the pressure drop, Ap = 2y/r , across the meniscus (cf. Equation (3.6)) but at some point B. say, nearer the wall, the chemical potential receives a contribution represented by the line BC, from the adsorption potential. Consequently, the reduction Ap in pressure across the meniscus must be less at B than at A, so that again... 

Fig. 3.10 Contributions to the lowering of chemical potential of the condensed liquid in a capillary, arising from adsorption forces (c) and meniscus curvature (Ap). The chemical potential of the free liquid is , and that of the capillary condensed liquid is (= ) z is the distance from the capillary wall. (After Everett. )...

Here d/l is the additional wall area exposed when the uptake diminishes by dn moles through evaporation from the capillary p." is the chemical potential of the capillary condensate and p° that of the bulk liquid adsorptive. The negative sign is necessary because the area A exposed increases as the uptake diminishes. If the adsorptive vapour behaves as a perfect gas,... 

Picture the transfer, under equilibrium conditions, of dn mole of adsorptive from the bulk liquid where its chemical potential is /i , to a... 

Thermodynamics of Liquid—Liquid Equilibrium. Phase splitting of a Hquid mixture into two Hquid phases (I and II) occurs when a single hquid phase is thermodynamically unstable. The equiUbrium condition of equal fugacities (and chemical potentials) for each component in the two phases allows the fugacitiesy andy in phases I and II to be equated and expressed as ... 

Mass transfer Irreversible and spontaneous transport of mass of a chemical component in a space with a non-homogeneous field of the chemical potential of the component. The driving force causing the transport can be the difference in concentration (in liquids) or partial pressures ( in gases) of the component. In biological systems. 

To test the results of the chemical potential evaluation, the grand canonical ensemble Monte Carlo simulation of the bulk associating fluid has also been performed. The algorithm of this simulation was identical to that described in Ref. 172. All the calculations have been performed for states far from the liquid-gas coexistence curve [173]. 

Panagiotopoulos et al. [16] studied only a few ideal LJ mixtures, since their main objective was only to demonstrate the accuracy of the method. Murad et al. [17] have recently studied a wide range of ideal and nonideal LJ mixtures, and compared results obtained for osmotic pressure with the van t Hoff [17a] and other equations. Results for a wide range of other properties such as solvent exchange, chemical potentials and activity coefficients [18] were compared with the van der Waals 1 (vdWl) fluid approximation [19]. The vdWl theory replaces the mixture by one fictitious pure liquid with judiciously chosen potential parameters. It is defined for potentials with only two parameters, see Ref. 19. A summary of their most important conclusions include ... 

P. Bryk, A. Patrykiejew, O. Pizio, S. Sokolowski. The chemical potential of Lennard-Jones associating fluids from osmotic Monte Carlo simulations. Mol Phys 92 949, 1997 A method for the determination of chemical potential for associating liquids. Mol Phys 90 665, 1997. 

J. D. Weeks, D. Chandler, H. C. Andersen. Role of repulsive forces in determining the equilibrium structure of simple liquids. J Chem Phys 54 5237, 1971. R. L. Rowley, M. W. Schuck, J. Perry. A direct method for determination of chemical potential with molecular dynamics simulations. 2. Mixtures. Mol Phys 55 125, 1995. 

Here rj is the viscosity of the dewetting liquid. Note that a relaxational term proportional to a has been added, with fi(j)) being the chemical potential of the vapor. This term alone guarantees that a homogeneous liquid film will relax to its equilibrium value hooip) by evaporation or condensation. For h = hooip) this term vanishes. 

Thermodynamic information can also be obtained from simulations. Currently we are measuring the differences in chemical potential of various small molecules in dimethylimidazolium chloride. This involves gradually transforming one molecule into another and is a computationally intensive process. One preliminary result is that the difference in chemical potential of propane and dimethyl ether is about 17.5 kj/mol. These molecules are similar in size, but differ in their polarity. Not surprisingly, the polar ether is stabilized relative to the non-polar propane in the presence of the ionic liquid. One can also investigate the local arrangement of the ions around the solute and the contribution of different parts of the interaction to the energy. Thus, while both molecules have a favorable Lennard-Jones interaction with the cation, the main electrostatic interaction is that between the chloride ion and the ether molecule. 

It is not unusual for the full chemical potential of a reaction to be diminished by slower transport processes (i.e., to be transport limited). In fast liquid phase enzyme reactions, mechanical stirring rates can have a strong influence on the observed kinetics that may be limited by the rate of contacting of the reactants and enzymes. Most heterogeneous catalytic reactions take... 

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