Chemical potential at equilibrium

Now, we would like to investigate adsorption of another fluid of species / in the pore filled by the matrix. The fluid/ outside the pore has the chemical potential at equilibrium the adsorbed fluid / reaches the density distribution pf z). The pair distribution of / particles is characterized by the inhomogeneous correlation function /z (l,2). The matrix and fluid species are denoted by 0 and 1. We assume the simplest form of the interactions between particles and between particles and pore walls, choosing both species as hard spheres of unit diameter... 

This does not imply that all components will have the same chemical potential at equilibrium, only that the chemical potential for each component is the same in all phases. xThe independent components are those that are not related through chemical equilibrium reactions. See Section 1.2c for a discussion of components in a system. 

By combining these expressions for defect chemical potentials and coefficients with the relations between the chemical potentials at equilibrium (for example Eqs. (74)) explicit expressions are obtained for the defect concentrations at equilibrium which are quite analogous to the quasi-chemical results (Section IV- A) apart from the presence of the activity coefficients. We consider examples of these equations in later sections. 

Figure 5.72 Effects of an erroneous application of the principle of equality of chemical potentials at equilibrium (from Connolly, 1992).

Uniform chemical potential at equilibrium assumes that the component conveys no other work terms, such as charge in an electric field. If other other energy-storage mechanisms are associated with a component, a generalized potential (the diffusion potential, developed in Section 2.2.3) will be uniform at equilibrium. 

We now consider more on the equality of chemical potentials at equilibrium. 

It can readily be shown (see Box 3.5) that the chemical potential of a component in a two-phase system (for example, oil and water), at equilibrium at a fixed temperature and pressure, is identical in both phases. Because of the need for equality of chemical potential at equilibrium, a substance in a system which is not at equilibrium will have a tendency to diffuse spontaneously from a phase in which it has a high chemical potential to another in which it has a low chemical potential. In this respect the chemical potential resembles electrical potential hence its name is an apt description of its nature. 

From the chemical potential at equilibrium follows a condition for 02,/ that the system can reach equilibrium and Ostwald ripening will stop (see equation (8.17)). Here the subscript 1 stands for initial conditions, and stands for the initial average droplet size ... 

Equating the two chemical potentials at equilibrium, and noting that the fractional loading is... 

Note that in (4.194), we use the nonconventional standard chemical potential. At equilibrium between the two phases, we have... 

The flow of species between the subsystems results in the equality of chemical potentials at equilibrium, and therefore the equilibrium is classified as chemical equilibrium. 

It is possible to calculate the free energy of both isotropic (high y) and anisotropic phases (low y) and to plot them as a function of v. From this plot it is possible to obtain the biphasic region by drawing a common tangent (the two phases have the same chemical potential at equilibrium see construction of the spinodal in Chapter 4). 

This is the fundamental equation of chemical equilibrium. It applies to any chemical system (not just isolated systems) at constant temperature and pressure, that is, the normal working conditions of chemistry. To understand how it works, consider the simple example of a transformation of a pure substance between states A and B, with d A = -d B equation 7.49 says that ( Ta - Tb) d A < 0, so if Ta > l B then a must decrease - hence p, as chemical potential. At equilibrium, Ta = ItB so that the sign of d A is irrelevant the system has no driving force to evolve either way. 

The activity coefficient of the solvent remains close to unity up to quite high electrolyte concentrations e.g. the activity coefficient for water in an aqueous solution of 2 m KC1 at 25°C equals y0x = 1.004, while the value for potassium chloride in this solution is y tX = 0.614, indicating a quite large deviation from the ideal behaviour. Thus, the activity coefficient of the solvent is not a suitable characteristic of the real behaviour of solutions of electrolytes. If the deviation from ideal behaviour is to be expressed in terms of quantities connected with the solvent, then the osmotic coefficient is employed. The osmotic pressure of the system is denoted as jz and the hypothetical osmotic pressure of a solution with the same composition that would behave ideally as jt. The equations for the osmotic pressures jt and jt are obtained from the equilibrium condition of the pure solvent and of the solution. Under equilibrium conditions the chemical potential of the pure solvent, which is equal to the standard chemical potential at the pressure p, is equal to the chemical potential of the solvent in the solution under the osmotic pressure jt,... 

At equilibrium, the chemical potential for a given molecular species is constant throughout the system. The two terms on the right-hand side of (11.4) can vary in space, however, so as to add up to a constant. In an inhomogeneous system, the number density and excess chemical potential adjust so as to yield the same constant chemical potential. Due to the local nature of the excess chemical potential, it is reasonable to define an excess chemical potential at a single point in space and/or for a single molecular conformation [29]. That excess chemical potential then determines... 

In everyday chemical usage, the word equilibrium means that a reaction has stopped, e.g. because it has reached its position of minimum chemical potential or because one reactant has been consumed completely. In this electroanalytical context, however, we say that we are making a measurement of potential at equilibrium , yet the system has clearly not reached a true equilibrium because as soon as the voltmeter is replaced with a connection having zero resistance, a cell reaction could commence. What then do we mean by equilibrium in this electroanalytical context ... 

B) is constant. The chemical potential of product phase (D) is equal to its Gibbs free energy of formation. The chemical potential of (A), which is the combination of the electrochemical potential of (A ) and (e ) according to Eqn. 6, is fixed at location (III) at equilibrium. It is further assumed that the chemical potential of (A) at (I) is greater than its chemical potential at (III) in this PEVD system. 

The concept of the chemical potential is introduced here because this property plays an important role in reacting systems. In this context one may consider that a reaction moves in the direction of decreasing chemical potential, reaching equilibrium only when the potential of the reactants equals that of the products [3], Thus, from Eq. (16) the criterion for equilibrium for combustion products of a chemical system at constant T and P is... 

Figure 1. Nitrogen in a 10 (Di teniai=3.3 nm) cylindrical pore of MCM-41 at 77.4 K. The chemical potential of equilibrium transition BF, j,e- io=-1.42 kT, is obtained from the Maxwell s rule and also corresponds to the intersection point of the Grand Potential (solid line). Lines CG and EA, which bound the hysteresis loop, correspond to the spinodal condensation and desorption, respectively.

This description has to be compared with that proposed by non-equilibrium thermodynamics in terms of only two states, corresponding to the melted and crystallized phases in the example we are discussing, from which only one may account for the linear domain, when the chemical potentials at the wells are not very different. This feature imposes serious limitations in the application of NET to activation processes since that condition is rarely encountered in experimental situations and has therefore restricted its use to only transport processes. The mesoscopic version of non-equilibrium thermodynamics, on the contrary, circumvents the difficulty offering a promising general scenario useful in the characterization of the wide class of activated processes, which appear frequently in systems outside equilibrium of different nature. 

An iteration scheme is used to numerically solve this minimization condition to obtain Peq(r) at the selected temperature, pore width, and chemical potential. For simple geometric pore shapes such as slits or cylinders, the local density is a function of one spatial coordinate only (the coordinate normal to the adsorbent surface) and an efficient solution of Eq. (29) is possible. The adsorption and desorption branches of the isotherm can be constructed in a manner analogous to that used for GCMC simulation. The chemical potential is increased or decreased sequentially, and the solution for the local density profile at previous value of fx is used as the initial guess for the density profile at the next value of /z. The chemical potential at which the equilibrium phase transition occurs is identified as the value of /z for which the liquid and vapor states have the same grand potential. 

Application of NLDFT to adsorption of fluids in porous media is usually carried out at constant temperature and pressure (constant chemical potential). The equilibrium state of the grand canonical ensemble corresponds to the minimum of the following thermodynamic grand potential ... 

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