Gibbs energy chemical potential relationship

The relationships of the type (3.1.54) and (3.1.57) imply that the standard electrode potentials can be derived directly from the thermodynamic data (and vice versa). The values of the standard chemical potentials are identified with the values of the standard Gibbs energies of formation, tabulated, for example, by the US National Bureau of Standards. On the other hand, the experimental approach to the determination of standard electrode potentials is based on the cells of the type (3.1.41) whose EMFs are extrapolated to zero ionic strength. 

When the adsorbed components are electrically charged, then the partial molar Gibbs energy of the charged component depends on the charge of the given phase, and thus the chemical potentials in the above relationships must be replaced by the electrochemical potentials. The Gibbs adsorption isotherm then has the form... 

An electrical potential difference between the electrodes of an electrochemical cell (called the cell potential) causes a flow of electrons in the circuit that connects those electrodes and therefore produces electrical work. If the cell operates under reversible conditions and at constant composition, the work produced reaches a maximum value and, at constant temperature and pressure, can be identified with the Gibbs energy change of the net chemical process that occurs at the electrodes [180,316]. This is only achieved when the cell potential is balanced by the potential of an external source, so that the net current is zero. The value of this potential is known as the zero-current cell potential or the electromotive force (emf) of the cell, and it is represented by E. The relationship between E and the reaction Gibbs energy is given by... 

Figure 2.1 Relationships between chemical potentials of two components and Gibbs free energy of a binary phase.

Figure 2,2 Relationships between chemical potential and Gibbs free energy in a two-phase system with partial miscibility...

Practically in every general chemistry textbook, one can find a table presenting the Standard (Reduction) Potentials in aqueous solution at 25 °C, sometimes in two parts, indicating the reaction condition acidic solution and basic solution. In most cases, there is another table titled Standard Chemical Thermodynamic Properties (or Selected Thermodynamic Values). The former table is referred to in a chapter devoted to Electrochemistry (or Oxidation - Reduction Reactions), while a reference to the latter one can be found in a chapter dealing with Chemical Thermodynamics (or Chemical Equilibria). It is seldom indicated that the two types of tables contain redundant information since the standard potential values of a cell reaction ( n) can be calculated from the standard molar free (Gibbs) energy change (AG" for the same reaction with a simple relationship... 

Of the three quantities (temperature, energy, and entropy) that appear in the laws of thermodynamics, it seems on the surface that only energy has a clear definition, which arises from mechanics. In our study of thermodynamics a number of additional quantities will be introduced. Some of these quantities (for example, pressure, volume, and mass) may be defined from anon-statistical (non-thermodynamic) perspective. Others (for example Gibbs free energy and chemical potential) will require invoking a statistical view of matter, in terms of atoms and molecules, to define them. Our goals here are to see clearly how all of these quantities are defined thermodynamically and to make use of relationships between these quantities in understanding how biochemical systems behave. 

A particularly interesting relationship between the chemical potential and the Gibbs free energy, G, is found from Eq. 1 and G = H TS = U - - PV - TS ... 

Gibbs-Duhem Relationship The partial molar properties of a multicomponent phase cannot be varied independently (the mole fractions, jc, = ,/E of the components total unity). For example, for the chemical potentials, /i, the Gibbs-Duhem relationship is En, dni = 0 (for details, see e.g., Atkirs, 1990 Blandamer, 1992 Denbigh, 1971). Similar constraints apply to the partial molar volumes, enthalpies, entropies, and heat capacities. For pure substances, the partial molar property is equal to the molar property. For example, the chemical potential of a pure solid or liquid is its energy per mole. For gaseous, liquid, or solid solutions, X, = X,(ny), that is, the chemical potentials and partial molar volumes of the species depend on the mole fractions. 

For mixtures, the relationship between the Gibbs free energy of a reaction and the composition of the system is obtained by substituting the expression for the chemical potential in terms of the activity of a species /... 

The chemical potential provides the fundamental criteria for determining phase equilibria. Like many thermodynamic functions, there is no absolute value for chemical potential. The Gibbs free energy function is related to both the enthalpy and entropy for which there is no absolute value. Moreover, there are some other undesirable properties of the chemical potential that make it less than suitable for practical calculations of phase equilibria. Thus, G.N. Lewis introduced the concept of fugacity, which can be related to the chemical potential and has a relationship closer to real world intensive properties. With Lewis s definition, there still remains the problem of absolute value for the function. Thus,... 

Understand the relationships among Gibbs free energy, chemical potential, reaction quotients (Q), the equilibrium constant, and the saturation index SI). 

The relationship between the surface tension, the chemical potential and the Gibbs surface energy is given by the general relation... 

The thermodynamic relationship between standard state chemical potential differences and the position of chemical equihbrium can be shown graphically. Figure 3.2 illustrates what happens to the Gibbs free energy G when the solute is partitioned between an aqueous phase in contact with an immiscible organic phase, diethyl ether in this example. The hypothetical plots of G versus the mole fraction, denoted by X, of solute i dissolved in the ether phase are superimposed for comparison. When there is no solute in the ether phase, a standard state chemical potential, can be realized. In the other extreme, when 100% of all of the mass of solute is in the ether phase... 

The same relationships can be derived in a more formal (and more usual) way by beginning with the equations for the chemical potential of electrolyte solutes. Because the total Gibbs energy of the solute must equal the sum of its parts (cations + anions). 

The chemical potentials are calculated in J kg" in the anode and cathode channel C-fields of the model. The three transformers shown in the effort-activated bonds around the 1 junction have factors of 1000/Mj in order to convert the chemical potentials to J mol". The 1 junction shown in Fig. 10.4 enforces the following relationship, which defines the negative of the change in Gibbs free energy per mole of fuel for the reaction ... 

In this simpler form, the Gibbs equation gives the relationship between the surface excess and the surface energy of an interface. We recall that the chemical potential of the solute is given by... 

The driving force for sintering (a reduction in excess surface free energy) is translated into a driving force that acts at the atomic level (thus resulting in atomic diffusion) by means of differences in curvature that inherently occur in different parts of the three-dimensional compact. These differences in curvature create chemical potential and vacancy concentration differences, and thus control the direction of matter transport. The relationship that links surface energy, curvature and concentration differences is the Gibbs-Thomson equation ... 

For two or more phases in equilibrium there is a separate equilibrium relationship for each of the chemical species present, and that relationship is that the partial molar Gibbs energy, called the chemical potential, for species i is the same in all of the phases (1, 2,. ..). The same statement applies to species j, k, and so on. 

The Gibbs energy is related to the chemical potentials by the following relationship ... 

The chemical potential is thus the amount by which the Gibbs energy of the system would change if an additional particle was introduced, with the pressure and temperature fixed, while surface energy excess defines the relationship between the amount of work performed in enlarging a surface and the surface area created. TOP shows a strong dependence on the cluster size, which manifests some sort of a relationship between the chemical potentials and the cluster dimensions. 

Discuss the relationships among chemical potential, activity and the Gibbs energy of a mixture. 

In Section 6.3, we used the Gibbs—Duhem equation to provide a relationship between the partial molar properties of different species in a mixture. We can use this equation to relate the activity coefficients of different species in a mixture as well. We begin by writing Equation (6.19) in terms of partial molar Gibbs energy, that is, chemical potential ... 

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