Clausius generalized

In an obituaiy talk given at the Physical Society of Berlin in 1889, Hermann Helmholtz stressed that Clausius s strict formulation of the mechanical heat theory is one of the most surprising and interesting achievements of the old and new physics, because of the absolute generality independent of the nature of the physical body and since it establishes new, unforeseen relations between different branches of physics. 

Entropy is often described as a measure of disorder or randomness. While useful, these terms are subjective and should be used cautiously. It is better to think about entropic changes in terms of the change in the number of microstates of the system. Microstates are different ways in which molecules can be distributed. An increase in the number of possible microstates (i.e., disorder) results in an increase of entropy. Entropy treats tine randomness factor quantitatively. Rudolf Clausius gave it the symbol S for no particular reason. In general, the more random the state, the larger the number of its possible microstates, the more probable the state, thus the greater its entropy. 

The second general principle is based on the properties of the entropy function, and is contained in the aphorism of Clausius... 

Of course, depending on the system, the optimum state identified by the second entropy may be the state with zero net transitions, which is just the equilibrium state. So in this sense the nonequilibrium Second Law encompasses Clausius Second Law. The real novelty of the nonequilibrium Second Law is not so much that it deals with the steady state but rather that it invokes the speed of time quantitatively. In this sense it is not restricted to steady-state problems, but can in principle be formulated to include transient and harmonic effects, where the thermodynamic or mechanical driving forces change with time. The concept of transitions in the present law is readily generalized to, for example, transitions between velocity macrostates, which would be called an acceleration, and spontaneous changes in such accelerations would be accompanied by an increase in the corresponding entropy. Even more generally it can be applied to a path of macrostates in time. 

All partitioning properties change with temperature. The partition coefficients, vapor pressure, KAW and KqA, are more sensitive to temperature variation because of the large enthalpy change associated with transfer to the vapor phase. The simplest general expression theoretically based temperature dependence correlation is derived from the integrated Clausius-Clapeyron equation, or van t Hoff form expressing the effect of temperature on an equilibrium constant Kp,... 

Some of the historical reluctance to assimilate the entropy concept into general scientific thinking, and much of the introductory student s bewilderment, might have been avoided if Clausius had defined entropy (as would have been perfectly legitimate to do) as... 

Generally, vapour pressure measurements are fitted to a form of the Clausius-Clapeyron equation ... 

The Clausius-Clapeyron equation describes the univariant equilibrium between crystal and melt in the P-Tfield. Because molar volumes and molar entropies of molten phases are generally greater than their crystalline counterparts, the two terms and AFfusion both positive and we almost invariably observe an... 

In 1901, H. Kamerlingh Onnes introduced a fundamentally new and improved description of real gas PVT properties in terms of the virial equation of state. [The word virial, deriving from the Latin word viris ( force ) was introduced into physics by R. Clausius, whom we shall meet later.] This equation expresses the compressibility factor Z(Vm, T) in terms of a general power series expansion in inverse molar volume Vm. The starting point for the virial expansion is the ideal limiting behavior (2.12), which can also be expressed as... 

The simple inequality (4.10) captures the essence of the second law. Its general consistency with universal inductive experience will be established in Section 4.4, and its further consequences (culminating in the final form of the second law as expressed by Clausius) will be developed in Sections 4.5-4.7. Thus, Carnot s remarkable principle provides virtually complete answers to the questions posed at the beginning of this chapter, although the relationship of (4.10) to these broader issues will certainly not become obvious until the following section. 

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

The general inequality (4.48) leads to the famous Clausius formulation of the second law ... 

The Clausius statement of the second law, although logically able to serve as a basis for the general equilibrium theory, was couched in terms of nonequilibrium processes that themselves lay outside the scope of such a theory. Attempts to derive the consequences of the Clausius statement were therefore tortuous and indirect, making further progress difficult. 

Let us first introduce a useful short-cut to the constrained optimization procedure employed in Section 5.2, based on the general Clausius inequality [cf. (4.43)] for spontaneous changes toward equilibrium ... 

In general, the molar enthalpy of vaporization is obtained from the Clausius-Clapeyron equation, representing the difference per mole of the enthalpy of the vapour and of the liquid at equilibrium with it ... 

A third statement of the second law is based on the entropy. In reversible systems all forces must be opposed by equal and opposite forces. Consequently, in an isolated system any change of state by reversible processes must take place under equilibrium conditions. Changes of state that occur in an isolated system by irreversible processes must of necessity be spontaneous or natural processes. For all such processes in an isolated system, the entropy increases. Clausius expressed the second law as The entropy of the universe is always increasing to a maximum. Planck has given a more general statement of the second law Every physical and chemical process in nature takes place in such a way as to increase the sum of the entropies of all bodies taking any part in the process. In the limit, i.e., for reversible processes, the sum of the entropies remains unchanged. 

Equation (9) is sometimes known as Clausius-Clapeyron equation and is generally spoken to as first latent heat equation. It was first derived by Clausius (1850) on the thermodynamic basis of Clapeyron equation. 

Boltzmann [3]. Boltzmann was led to thiB generalized formulation of the problem by some attempts he had undertaken (1866) 11] to derive from kinetic concepts the Camot-Clausius theorem about the limited convertibility of heat into work. In order to carry through such a derivation for an arbitrary thermal system (Boltzmann [5], (1871)) it was necessary to calculate, e.g., for a nonideal gas, bow in an infinitely slow change of the state of the system the added amount of heat is divided between the translational and internal kinetic energy and the various forms of potential energy of the gas molecule. It is just for this that the distribution law introduced above is needed. 

Clausius/Clapeyron equation, 182 Coefficient of performance, 275-279, 282-283 Combustion, standard heat of, 123 Compressibility, isothermal, 58-59, 171-172 Compressibility factor, 62-63, 176 generalized correlations for, 85-96 for mixtures, 471-472, 476-477 Compression, in flow processes, 234-241 Conservation of energy, 12-17, 212-217 (See also First law of thermodynamics) Consistency, of VLE data, 355-357 Continuity equation, 211 Control volume, 210-211, 548-550 Conversion factors, table of, 570 Corresponding states correlations, 87-92, 189-199, 334-343 theorem of, 86... 

An equation of fundamental importance which finds extensive application in the one-component, two-phase systems, was derived by Clapeyron and independently by Clausius, from the Second law of thermodynamics and is generally known as the Clapeyron-Clausius equation. The two phases in equilibrium may be any of the following types ... 

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