Gibbs entropy function

The second law is more subtle and difficult to comprehend than the first. The full scope of the second law only became clear after an extended period of time in which (as expressed by Gibbs) truth and error were in a confusing state of mixture. In the present chapter, we focus primarily on the work of Carnot (Sidebar 4.1), Thomson (Sidebar 4.2), and Clausius (Sidebar 4.3), which culminated in Clausius clear enunciation of the second law in terms of the entropy function. This in turn led to the masterful reformulation by J. W. Gibbs, which underlies the modem theory of chemical and phase thermodynamics and is introduced in Chapter 5. 

It was the principal genius of J. W. Gibbs (Sidebar 5.1) to recognize how the Clausius statement could be recast in a form that made reference only to the analytical properties of individual equilibrium states. The essence of the Clausius statement is that an isolated system, in evolving toward a state of thermodynamic equilibrium, undergoes a steady increase in the value of the entropy function. Gibbs recognized that, as a consequence of this increase, the entropy function in the eventual equilibrium state must have the character of a mathematical maximum. As a consequence, this extremal character of the entropy function makes possible an analytical characterization of the second law, expressible entirely in terms of state properties of the individual equilibrium state, without reference to cycles, processes, perpetual motion machines, and the like. 

Gibbs criterion (I) In an isolated equilibrium system, the entropy function has the mathematical character of a maximum with respect to variations that do not alter the energy. 

According to the Gibbs criterion, the entropy function S is a maximum (with respect to certain allowed variations). Recall that for a general function f(x, y,...), the conditions that/be a maximum are ... 

Let us now attempt to re-express the Gibbs criterion of equilibrium in alternative analytical and graphical forms that are more closely related to Clausius-like statements of the second law. For this purpose, we write the constrained entropy function S in terms of its... 

As shown in the figure, the curvature of the entropy function always causes it to fall below its tangent planes. A mathematical object having such distinctive global curvature (such as an eggshell or an upside-down bowl) is called convex. Accordingly, we may restate the Gibbs criterion in terms of this intrinsic convexity property of the entropy function S = S(U,V,N) ... 

The standard partial molar Gibbs free energy of solution is related to the enthalpy and entropy functions at the column temperature T by the expression... 

The only two functions actually required in thermodynamics are the energy function, obtained from the first law of thermodynamics, and the entropy function, obtained from the second law of thermodynamics. However, these functions are not necessarily the most convenient functions. The enthalpy function was defined in order to make the pressure the independent variable, rather than the volume. When the first and second laws are combined, as is done in this chapter, the entropy function appears as an independent variable. It then becomes convenient to define two other functions, the Gibbs and Helmholtz energy functions, for which the temperature is the independent variable, rather than the entropy. These two functions are defined and discussed in the first part of this chapter. 

The AG of the reaction is then calculated in one of two ways (1) the appropriate addition and substraction of AGf for reactants and products, or (2) the calculation of Gibbs energy functions for reactants and products from enthalpy and entropy increments... 

Based upon experimentally observed spectroscopic data, statistical thermodynamic calculations provide thermodynamic data which would not be obtained readily from direct experimental measurements for the species and temperature of interest to rocket propulsion. If the results of the calculations are summarized in terms of specific heat as a function of temperature, the other required properties for a particular specie, for example, enthalpy, entropy, the Gibb s function, and equilibrium constant may be obtained in relation to an arbitrary reference state, usually a pressure of one atmosphere and a temperature of 298.15°K. Or alternately these quantities may be calculated directly. Significant inaccuracies in the thermochemical data are not associated generaUy with the results of such calculations for a particular species, but arise in establishing a valid basis for comparison of different species. 

Answer 2 given above invites, of course, another question Where do the fundamental thermodynamic relation h = h x) and the relation y = y x) come from An attempt to answer this question makes us to climb more and more microscopic levels. The higher we stay on the ladder the more detailed physics enters our discussion of h = h(x) and y = y(x). Moreover, we also note that the higher we are on the ladder, the more of the physics enters into y = y(x) and less into h = h x). Indeed, on the most macroscopic level, i.e., on the level of classical equilibrium thermodynamics sketched in Section 2.1, we have s = s(y) and y y. All the physics enters the fundamental thermodynamic relation s s(t/), and the relation y = y is, of course, completely universal. On the other hand, on the most microscopic level on which states are characterized by positions and velocities of all ( 1023) microscopic particles (see more in Section 2.2.3) the fundamental thermodynamic relation h = h(x) is completely universal (it is the Gibbs entropy expressed in terms of the distribution function of all the particles) and all physics (i.e., all the interactions among particles) enters the relation y = y(x). 

Oxides. Decomposition pressure measurements on the TbO system by Eyring and his collaborators (64) have been supplemented by similar and related studies on the PrO system (46) and on other lanthanide-oxygen systems (43, 44). Extensive and systematic studies of vaporization processes in lanthanide-oxide systems have been undertaken by White, et al. (6, 188,196) using conventional Knudsen effusion measurements of the rates of vaporization of the oxides into high vacuum. Combination of these data with information on the entropies and Gibbs energy functions of reactants and products of the reaction yields enthalpies of reaction. In favorable instances i.e., if spectroscopic data on the gaseous species are available), the enthalpies of formation and the stabilities of previously undetermined individual species are also derived. The rates of vaporization of 17 lanthanide-oxide systems (196) and the vaporization of lanthanum, neodymium, and yttrium oxides at temperatures between 22° and 2700°K. have been reported (188). 

The profound consequences of the microscopic formulation become manifest in nonequilibrium molecular dynamics and provide the mathematical structure to begin a theoretical analysis of nonequilibrium statistical mechanics. As discussed earlier, the equilibrium distribution function / q contains no explicit time dependence and can be generated by an underlying set of microscopic equations of motion. One can define the Gibbs entropy as the integral over the phase space of the quantity /gq In / q. Since Eq. [48] shows how functions must be integrated over phase space, the Gibbs entropy must be expressed as follows ... 

Key words critically evaluated data enthalpy enthalpy of formation entropy equilibrium constant of formation Gibbs energy function Gibbs energy of formation heat capacity thermochemical tables. 

The third-law method is based on a knowledge of the absolute entropy of the reactants and products. It allows the calculation of a reaction enthalpy from each data point when the change in the Gibbs energy function for the reaction is known. The Gibbs energy function used here is defined as... 

Gurvich et al. (9) calculated functions that differ by -1.3 cal K mol" in entropy and Gibbs energy function. 

The level at 15.254 cm" has a large effect on the heat capacity and entropy below 100 K. The heat capacity effect decreases to zero above 600 K where the 15.254 cm" level Is fully populated. The higher excited states affect the heat capacity values above 3000 K. The Gibbs energy function values up to 6000 K are essentially Independent of the cut-off procedure, the inclusion of levels for n>2, and the estimated missing levels (for n<39). 

In modem physics, there exist alternative theories for the equilibrium statistical mechanics [1, 2] based on the generalized statistical entropy [3-12]. They are compatible with the second part of the second law of thermodynamics, i.e., the maximum entropy principle [13-14], which leads to uncertainty in the definition of the statistical entropy and consequently the equilibrium probability density functions. This means that the equilibrium statistical mechanics is in a crisis. Thus, the requirements of the equilibrium thermodynamics shall have an exclusive role in selection of the right theory for the equilibrium statistical mechanics. The main difficulty in foundation of the statistical mechanics based on the generalized statistical entropy, i.e., the deformed Boltzmann-Gibbs entropy, is the problem of its connection with the equilibrium thermodynamics. The proof of the zero law of thermodynamics and the principle of additivity... 

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