Simple system chemical potential

An empirical formula, due to Pick, shows that, under simple cucumstances where the chemical potential of a component in a system is dehned by the equation... 

Let us consider a simple model of a quenched-annealed system which consists of particles belonging to two species species 0 is quenched (matrix) and species 1 is annealed, i.e., the particles are allowed to equlibrate between themselves in the presence of 0 particles. We assume that the subsystem composed of 0 particles has been a usual fluid before quenching. One can characterize it either by the density or by the value of the chemical potential The interparticle interaction Woo(r) does not need to be specified for the moment. It is just assumed that the fluid with interaction woo(r) has reached an equlibrium at certain temperature Tq, and then the fluid has been quenched at this temperature without structural relaxation. Thus, the distribution of species 0 is any one from a set of equihbrium configurations corresponding to canonical or grand canonical ensemble. We denote the interactions between annealed particles by Un r), and the cross fluid-matrix interactions by Wio(r). 

We close these introductory remarks with a few comments on the methods which are actually used to study these models. They will for the most part be mentioned only very briefly. In the rest of this chapter, we shall focus mainly on computer simulations. Even those will not be explained in detail, for the simple reason that the models are too different and the simulation methods too many. Rather, we refer the reader to the available textbooks on simulation methods, e.g.. Ref. 32-35, and discuss only a few technical aspects here. In the case of atomistically realistic models, simulations are indeed the only possible way to approach these systems. Idealized microscopic models have usually been explored extensively by mean field methods. Even those can become quite involved for complex models, especially for chain models. One particularly popular and successful method to deal with chain molecules has been the self-consistent field theory. In a nutshell, it treats chains as random walks in a position-dependent chemical potential, which depends in turn on the conformational distributions of the chains in... 

By virtue of the function (3.6), concentrations, which are readify determined parameters, can be used instead of chemical potentials in the thermodynamic equations for ideal systems. The simple connection between the concentrations and chemical potentials is lost in real systems. To facilitate the changeover from ideal to nonideal systems and to avoid the use of two different sets of equations in chemical thermodynamics,... 

In real mixtures and solutions, the chemical potential ceases to be a linear function of the logarithm of the partial pressure or mole fraction. Consequently, a different approach is usually adopted. The simple form of the equations derived for ideal systems is retained for real systems, but a different quantity a, called the activity (or fugacity for real gases), is... 

Widom test-particle method. Provides the chemical potential in various ensembles. Relatively easy to implement and can be used as an additional measurement in standard MC ensembles (and also MD). Computational overhead is small. Yields good accuracy in simple systems, although less reliable in very dense or complex systems (i.e., chain molecules). 

Equilibrium between simple salts and aqueous solutions is often relatively easily demonstrated in the laboratory when the composition of the solid is invariant, such as occurs in the KCI-H2O system. However, when an additional component which coprecipitates is added to the system, the solid composition is no longer invariant. Very long times may be required to reach equilibrium when the reaction path requires shifts in the composition of both the solution and solid. Equilibrium is not established until the solid composition is homogeneous and the chemical potentials of all components between solid and aqueous phases are equivalent. As a result, equilibrium is rarely demonstrated with a solid solution series. 

Studies of H2 have proven the feasibility of using the LEPS formalism to study gas-surface reactions, and have indicated that relationships between the potential surface and chemical dynamics derived from gas-phase studies can be generalized to reactions with surfaces. Reactions of H2, however, represent simple systems compared even to other diatomic molecules, and extensions to other more complicated reactions are rare. A few studies of other diatomic... 

Earlier, Gavach et al. studied the superselectivity of Nafion 125 sulfonate membranes in contact with aqueous NaCl solutions using the methods of zero-current membrane potential, electrolyte desorption kinetics into pure water, co-ion and counterion selfdiffusion fluxes, co-ion fluxes under a constant current, and membrane electrical conductance. Superselectivity refers to a condition where anion transport is very small relative to cation transport. The exclusion of the anions in these systems is much greater than that as predicted by simple Donnan equilibrium theory that involves the equality of chemical potentials of cations and anions across the membrane—electrolyte interface as well as the principle of electroneutrality. The results showed the importance of membrane swelling there is a loss of superselectivity, in that there is a decrease in the counterion/co-ion mobility, with greater swelling. 

The research carried out on these very simple potential forms has revealed that they capture many of the physical/rheological features present in real systems. By focusing on such potentials, which have one or two adjustable parameters one can gain insights into generic effects that span many examples of chemical system. These potentials are also, on occasions (e.g. the potential given in Equation (1)) amenable to simple analytic treatments. 

For ideal multicomponent systems, a simple linear relationship exists between the chemical potential fii) and the logarithm of the mole fraction of solvent and solute, respectively. 

Figure 3.8 Conceptualization of the potential functions in a hydrostatic system and in a simple chemical system, (a) In the unequilibrated hydrostatic system, water will flow from reservoir 2 of higher hydrostatic potential (=gh2, where g is the acceleration due to gravity and h2 is the observable height of water in the tank) to reservoir 1 of lower hydrostatic potential total water volumes (i.e., total potential energies W [ and W2) do not dictate flow. Similarly, benzene molecules move from liquid benzene to the head space in the nonequilibrated chemical system, not because there are more molecules in the flask containing the liquid, but because the molecules initially exhibit a higher chemical potential in the liquid than in the gas. (b) At equilibrium, the hydrostatic system is characterized by equal hydrostatic potentials in both reservoirs (not equal water volumes) and the chemical system reflects equal chemical potentials in both flasks (not equal benzene concentrations).

Equation (6.35b) shows that the new intensive variable, chemical potential pi, as introduced in this chapter, is actually superfluous for the case c = 1, because its variations can always be expressed in terms of the old variations dT dP. Thus, as stated in Inductive Law 1 (Table 2.1), only two degrees of freedom (independently variable intensive properties) suffice to describe the thermodynamic variability of a simple c = 1 system. This confirms (as expected) that chemical potential pu only becomes an informative thermodynamic variable when chemical change is possible, that is, for c > 2 chemical components. 

With an open system to which electrodes are attached, we can study the stability of interface morphology in an external electric field. A particularly simple case is met if the crystals involved are chemically homogeneous. In this case, Vfij = 0, and the ions are essentially driven by the electric field. Also, this is easy to handle experimentally. The counterpart of our basic stability experiment (Fig. 11-7) in which the AO crystal was exposed to an oxygen chemical potential gradient is now the exposure of AX to an electric field from the attached electrodes. In order to define the thermodynamic state of AX, it is necessary to apply electrodes with a predetermined... 

Equation D3.5.13 illustrated that the free energy of an interfacial system can be expressed in terms of the interfacial tension and chemical potential of the overall system. A simple differentiation or alternatively the reutilization of the definition of the interfacial tension used in Equation D3.5.7 at constant pressure and temperature yields ... 

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