Clausius Gibbs

Clausius great paper of 1850 can be recognized as a landmark in the development of thermodynamics. As remarked by Thomson in 1851, the merit of first establishing [Carnot s theorem] upon correct principles is entirely due to Clausius. In his 1889 eulogy of Clausius, Gibbs praised the 1850 paper in the following terms ... 

In contrast to Clausius, Gibbs did not discuss uncompensated heat, as he started directly with the total differential of entropy. Gibbs presentation appealed very much to De Donder, However, he wanted to find the meaning of this mysterious uncompensated heat. He considered a system whose physical conditions, such as pressure and temperature, were uniform and which was closed to the flow of matter. Chemical reactions, however, could go on inside the system. De Donder first introduced what he called the degree of advancement, , of the chemical reaction so that the reaction rate v is the time derivative of . 

See also Ampere, Andre-Marie Clausius, Rudolf Julius Emmanuel Electricity Electricity, History of Faraday, Michael Gibbs, Josiah Willard Magnetism and Magnets Molecular Energy Oersted, Hans Christian Thomson, William. 

See also Carnot, Nicholas Leonard Sadi Clausius, Rudolf Julius Emmanuel Gibbs, Josiah Willard Heat Transfer Helmoltz, Herman von Joule, James Prescott Ostwald, Wilhelm Thermodynamics,... 

Clausius-Clapeyron, 230,303-305 Gibbs-Helmholtz, 459,461 Henderson-Hasselbalch, 79-80 mass relation, 62-63 Nemst, 493-494 net ionic, 79-80 nonmetals, 575q nuclear, 513,530-531q Planck s, 135 redox, balancing, 90-92 Schrodinger, 140 thermochemical, 204... 

If Comte had lived long enough to see the development of thermodynamics and its applications, he might have retracted these words. However, he died well before the work of Black, Rumford, Hess, Carnot, Joule, Clausius, Kelvin, Helmholtz, and Nernst that established different aspects of the sciences, followed by the contributions of Gibbs, Lewis, and Guggenheim that unified the science into a coherent whole.a... 

Most methods for the determination of phase equilibria by simulation rely on particle insertions to equilibrate or determine the chemical potentials of the components. Methods that rely on insertions experience severe difficulties for dense or highly structured phases. If a point on the coexistence curve is known (e.g., from Gibbs ensemble simulations), the remarkable method of Kofke [32, 33] enables the calculation of a complete phase diagram from a series of constant-pressure, NPT, simulations that do not involve any transfers of particles. For one-component systems, the method is based on integration of the Clausius-Clapeyron equation over temperature,... 

The Clausius-Clapeyron equation provides a relationship between the thermodynamic properties for the relationship psat = psat(T) for a pure substance involving two-phase equilibrium. In its derivation it incorporates the Gibbs function (G), named after the nineteenth century scientist, Willard Gibbs. The Gibbs function per unit mass is defined... 

The Helmholtz and Gibbs energies are useful also in that they define the maximum work and the maximum non-expansion work a system can do, respectively. The combination of the Clausius inequality 7dS > dq and the first law of thermodynamics dU = dq + dw gives... 

There is another important law that follows from the classical theory of capillarity. This law was formulated by J. Thomson [16], and was based on a Clausius-Clapeyron equation and Gibbs theory, formulating the dependence of the melting point of solids on their size. The first known analytical equation by Rie [17], and Batchelor and Foster [18] (cited according to Refs. [19,20]) is... 

The mathematical basis of classic thermodynamics was developed by J. Willard Gibbs in his essay [1], On the Equilibrium of Heterogeneous Substances, which builds on the earlier work of Kelvin, Clausius, and Helmholtz, among others. In particular, he derived the phase mle, which describes the conditions of equilibrium for multiphase, multicomponent systems, which are so important to the geologist and to the materials scientist. In this chapter, we will present a derivation of the phase rule and apply the result to several examples. 

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. 

These capture various aspects of the more general and comprehensive statements of Carnot, Clausius, and Gibbs that are still to follow. 

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. 

In summary, Clausius states that entropy strives toward a maximum in isolated processes tending toward equilibrium, while Gibbs states that entropy is at a maximum in isolated equilibrium states. 

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... 

R.B. Cundall et al, "Vapor Pressure Measurements on Some Organic High Explosives , J-ChemSoc, Faraday Trans I, 74 (6), 1339—45 (1978) CA 89, 181933 (1978) [Equilibrium vap press were detd for various expls by the Knudson cell technique. The data for HMX follows the Clausius-Clapeyron eqtn. The values detd for the const A and B in the eqtn, log10p = A—(B/T), plus the std enthalpy, entropy and Gibbs energy of sublimation from the authors calcns are presented in Table 7 ... 

An alternative proof of the Clausius-Clapeyron Equation emerges (Frame 50) using the Gibbs Duhem Equation. 

The Gibbs-Duhem equation (50.6), derived below, proves to be a useful starting point for an alternative derivation of the Clausius Claperyron equation to that offered in Frame 26) and offers an alternative proof of the Phase Rule to that given in Frame 30. 

Derivation of Clausius-Clapeyrron Equation using the Gibbs-Duhem Equation. [ ln the form of Equation (26.5), Frame 26]... 

In the 19th century the variational principles of mechanics that allow one to determine the extreme equilibrium (passing through the continuous sequence of equilibrium states) trajectories, as was noted in the introduction, were extended to the description of nonconservative systems (Polak, 1960), i.e., the systems in which irreversibility of the processes occurs. However, the analysis of interrelations between the notions of "equilibrium" and "reversibility," "equilibrium processes" and "reversible processes" started only during the period when the classical equilibrium thermodynamics was created by Clausius, Helmholtz, Maxwell, Boltzmann, and Gibbs. Boltzmann (1878) and Gibbs (1876, 1878, 1902) started to use the terms of equilibria to describe the processes that satisfy the entropy increase principle and follow the "time arrow."... 

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