Chemical potential minima

As with all determinations of thermodynamic stability, we comihehce by defining all stable phases possible, and their standard, chemical, potentials. For inost, metals there are many such phases, including oxides, hydroxides and dissolved ions. For brevity, here, only the minimum number of phases is Considered. The siriiplest system is a metal, ilf, which can oxidise lo form a stable dissolved pro,duct, (qorrpsipn), or to form a stable oxide MO (passivation), lit aqueous environments thfbe equilibria Can thereby be... 

If we could prevent the mixture from separating into two phases at temperatures below Tc, we would expect the point of inflection to develop into curves similar to those shown in Figure 8.17 as the dotted line for /2, with a maximum and minimum in the fugacity curve. This behavior would require that the fugacity of a component decreases with increasing mole fraction. In reality, this does not happen, except for the possibility of a small amount of supersaturation that may persist briefly. Instead, the mixture separates into two phases. These phases are in equilibrium so that the chemical potential and vapor fugacity of each component is the same in both phases, That is, if we represent the phase equilibrium as... 

For below one monolayer the deposit may consist of an inhomogeneous mixture of a bare substrate and clusters approximately one monolayer thick. The state with the lowest chemical potential is represented by a line through the origin and tangent to the E[N) curve near the first local minimum, at Ni Nq. The slope of this tangent defines a chemical potential Thus, as N is increased from 0 to the fraction of the sub-... 

A different scenario occurs when the bonding between the substrate and film is relatively weak. For WKl the first minimum in E N)—NfJLp is positive, and the chemical potential is greater than jjlq. In that case... 

The tools for calculating the equilibrium point of a chemical reaction arise from the definition of the chemical potential. If temperature and pressure are fixed, the equilibrium point of a reaction is the point at which the Gibbs free energy function G is at its minimum (Fig. 3.1). As with any convex-upward function, finding the minimum G is a matter of determining the point at which its derivative vanishes. 

The equilibrium constant expresses the point of minimum free energy for a chemical reaction, as set forth in Equation 3.3, in terms of the chemical potential functions above. The criterion for equilibrium becomes,... 

In the open molecule coupled to an external electron reservoir, which fixes the system chemical potential, they combine the minimum-energy responses in the system number of electrons and the remaining nuclear coordinates to a unit displacement of Qs. The associated MECs,... 

By taking the minimum in Eq. (69) subject to the restrictions specified and using Lagrange undetermined multipliers (see, for example, Ref. 6), one finds a set of relationships satisfied by the defect chemical potentials. The results for the three basic types of intrinsic lattice disorder are as follows ... 

In [25, 26] it is shown that at given pq the diquark gap is independent of the isospin chemical potential for Pi ) < Pic(Pq), otherwise vanishes. Increase of isospin asymmetry forces the system to pass a first order phase transition by tunneling through a barrier in the thermodynamic potential (2). Using this property we choose the absolute minimum of the thermodynamic potential (2) between two /3-equilibrium states, one with and one without condensate for the given baryochemical potential Pb = Pu + 2pd-... 

No symbol has been approved by the IUPAC for dissociation energy in the chemical thermodynamics section [13]. Under Atoms and Molecules, either El or D is indicated. The latter is more common, and IUPAC recommends Do and De for the dissociation energy from the ground state and from the potential minimum, respectively. Because the bond energy concept will be omnipresent in this book and can be explored in a variety of ways, some extra names and symbols are required. This matter will be handled whenever needed, but for now we agree to use DUP for a standard bond dissociation internal energy and DHj for a standard bond dissociation enthalpy, both at a temperature T. In cases where it is clear that the temperature refers to 298.15 K, a subscript T will be omitted. 

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 ... 

The phase rule states that, when equilibrium conditions are sustained, a minimum number of intensive properties of the (subsurface) system can be used to calculate its remaining properties. An intensive property is a property that is independent of the amount of substance in the domain. Examples of intensive properties include temperature (7), pressure (P), density (p), and chemical potential (p), which is a relative measure of the potential energy of a chemical compound. The phase rule specifies the minimum number of intensive properties that must be determined to obtain a comprehensive thermodynamic depiction of a system. 

Linear regions (constant slope, hence constant potentials) define the composition interval over which a two-phase assemblage is stable. Because the minimum Gibbs free energy curve of the system is never convex, the chemical potential of any component will always increase with the increase of its molar proportion in the system. 

Polymorphism occurs whenever a given component exists under different aggregation states as a function of P and T. The stable state requires that the chemical potential of the component (hence, the Gibbs free energy of the phase for a phase composed of a single component) be at minimum at equilibrium. Figure 2.4 shows examples of G-T plots for Al2Si05 in various P conditions. 

Referring to Figure 3, evidence exists for placement of the Fermi levels (chemical potentials) of the redox reactions involving Hzr H2O and O2 roughly at the positions shown relative to the energies of the conduction band minimum and valence band maximum of the semiconductor, E and E, respectively. This picture takes the electron in a vacuum at infinity as the zero of energy. On this basis, the Fermi level for the reaction... 

This critical position of equilibrium can be defined in two fundamentally different ways. The first is that the system A-B with composition xo has reached an equilibrium where its Gibbs energy is at a minimum. The second definition is that phases ai and Q2 with compositions x and x are in equilibrium because the chemical potentials of A and B are equal in both phases. The importance of these two definitions becomes clearer if we consider how it would be possible to write a computer programme to find x and xf. ... 

Using umbrella sampling, Tieleman and Marrink [18] determined a PMF for transferring a DPPC lipid from water to the center of a DPPC bilayer (Figure 3B). The DPPC PMF has a deep minimum at its equilibrium position and a steep slope in free energy as it moved into bulk water. The free energy of desorption (AGdesorb) was 80 kj/mol, and is directly related to its excess chemical potential in the bilayer compared to water. 

Solid-solid reactions are as a rule exothermic, and the driving force of the reaction is the difference between the free energies of formation of the products and the reactants. A quantitative understanding of the mechanism of solid-solid reactions is possible only if reactions are studied under well-defined conditions, keeping the number of variables to a minimum. This requires single-crystal reactants and careful control of the chemical potential of the components. In addition, a knowledge of point-defect equilibria in the product phase would be useful. 

Figure 15-2A gives a calculated isotherm below the critical temperature. Point a is selected arbitrarily along the liquid part of the isotherm. Point d is selected arbitrarily along the gas part of the isotherm. Points b and c are minimum and maximum points on the van der Waals loop. Points e represent the points along the loop for which the chemical potentials are equal. Point f is the point along line be which has the same pressure as points e. 

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