Gibbs free energy definition

It is sometimes convenient to reformulate the generalized diffusional driving forces dg either in terms of mass or molar functions, using the partial mass Gibbs free energy definition, Gg = hg —Tsg, and the chain rule of partial differentiation assuming that the chemical potential (i.e., /Xg = Gg) is a function of temperature, pressure and concentration (Slattery [89], sect. 8.4). dg can then be expressed in several useful forms as listed below. Expressing the thermodynamic functions on a mass basis we may write ... 

Then, by use of the chain rule of partial differentiation on p,g we get (2.460), and thereafter eliminating the partial derivatives therein by (2.461) and (2.462), introducing the activity through (2.468), and eliminating the standard state potential by (2.471) as well as the Gibbs free energy definition ... 

From the Gibbs free energy definition and the equilibrium condition (5.46) it is easy to found that = H -H )/T. Replacing it in the relation (5.47) leads to the... 

The Gibbs free energy definition is employed in order to express the standard state chemical potential /x° and entropy potentials and introduce the entropy potential... 

We divide by Avogadro s number to convert the partial molar Gibbs free energy to a molecular quantity, and the minus sign enters because the force and the gradient are in opposing directions. Recalling the definition of chemical potential [Eq. (8.13)], we write jUj + RT In aj = ii2 + RT In 7jC, where aj... 

The definitions of enthalpy, H, Helmholtz free energy. A, and Gibbs free energy, G, also give equivalent forms of the fundamental relation (3) which apply to changes between equiUbrium states in any homogeneous fluid system ... 

The same relations (11) and (12) hold for the Gibbs free energy in the (N, p,T) ensemble. Equation (11) is also valid for a quanmm mechanical system. Note that for a linear coupling scheme such as Eq. (10), the first term on the right of Eq. (12) is zero the matrix of second derivatives can then be shown to be definite negative, so that the free energy is a concave function of the Xi. 

This expression shows that the maximum possible useful work (i.e., reversible work) that can be obtained from any process occurring at constant temperature and pressure is a function of the initial and final states only and is independent of the path. The combination of properties U + PV - TS or H - TS occurs so frequently in thermodynamic analysis that it is given a special name and symbol, F, the free energy (sometimes called the Gibbs Free Energy). Using this definition, Equation 2-143 is written... 

As equation 2.4.8 indicates, the equilibrium constant for a reaction is determined by the temperature and the standard Gibbs free energy change (AG°) for the process. The latter quantity in turn depends on temperature, the definitions of the standard states of the various components, and the stoichiometric coefficients of these species. Consequently, in assigning a numerical value to an equilibrium constant, one must be careful to specify the three parameters mentioned above in order to give meaning to this value. Once one has thus specified the point of reference, this value may be used to calculate the equilibrium composition of the mixture in the manner described in Sections 2.6 to 2.9. 

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. 

It is apparent from the definition of enthalpy H and the introduction of the concept of the Gibbs free energy G... 

In geology it is customary to consider systems in which the intensive variables pressure (P) and temperature (T) are characteristic of the ambient and, therefore, are prefixed and constant. In these conditions, the Gibbs free energy of the system (G) is at minimum at equilibrium. The treatments presented in this chapter are based on this fundamental principle. Let us first introduce in an elementary fashion some fundamental definitions. 

If the heat capacity of a chemically complex melt can be obtained by a linear summation of the specific heat of the dissolved oxide constituents at all T (i.e., Stebbins-Carmichael model), the melt is by definition ideal. The addition of excess Gibbs free energy terms thus implies that the Stebbins-Carmichael model calculates only the ideal contribution to the Gibbs free energy of mixing. 

The American physicist Josiah Gibbs introduced (ca. 1875) a thermodynamic quantity combining enthalpy and entropy into a single value called free energy (or Gibbs free energy). In honor of its inventor, it is usually symbolized as G. The definition of free energy is... 

We saw in Chapter 2 that an important thermodynamic quantity is the Gibbs free energy, AG. The specific functional relationship we use to describe the free energy will depend on whether we are studying a physical or a chemical transformation. For physical processes, such as phase transformations, the most useful form of the Gibbs free energy is its definition given in Chapter 2 ... 

The quantities G, //, and S are called extensive thermodynamic functions because the magnitude of the quantity in each case depends on the amount of substance in the system. The change in Gibbs free energy under addition of unit concentration of component / at constant concentrations of the other components is called the partial Gibbs free energy of the /-component, i.e., the chemical potential of the /-component in the system. The chemical potential is an intensive thermodynamic quantity, like temperature and concentrations. The formal definition is... 

The pressure dependence of the Gibbs free energy is needed to calculate G at conditions other than the standard state. From the definition of the free energy (Eq. 9.1) the total differential of G is... 

What is the physical nature of the Gibbs free energy, and what is free about it We can consider this question first from the viewpoint of fundamental thermodynamic definitions, with no microscopic molecular connotations. For a reversible change of state carried out under conditions of constant T and P, we can write... 

However, it is useful, to provide a thermodynamic definition of a first-order transition. Specifically, it is one in which there is a discontinuity in a first derivative of the Gibbs free energy. The advantage of this definition is the guidance it provides for the experimental study of phase transitions. A useful expression for the free energy in this regard is... 

By definition, the equilibrium constants do not depend on the concentrations of the reactants and products, and are related to the standard Gibbs free energy of the reaction per mol of reaction as ... 

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