Clausius theorem

Liouville theorem and related forms The Helmholtz-Lagrange relation given in equ. (4.46) is related to many other forms which all state certain conservation laws (the Clausius theorem, Abbe s relation, the Liouville theorem). The most important one in the present context is the Liouville theorem [Lio38] which describes the invariance of the volume in phase space. The content of this theorem will be discussed and represented finally in a slightly different form which allows a new access to the luminosity introduced in equ. (4.14). 

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. 

Some of you may have heard about the thermodynamic time arrow. Gases escape from open containers and heat flows from a hot body to its colder environment. Never has spontaneous reversal of such processes been observed. We call these irreversible processes. The world is always heading forward in time. Mathematically this is expressed by Clausius theorem. 

This completes the heuristic derivation of the Boltzmann transport equation. Now we trim to Boltzmaim s argument that his equation implies the Clausius fonn of the second law of thennodynamics, namely, that the entropy of an isolated system will increase as the result of any irreversible process taking place in the system. This result is referred to as Boltzmann s H-theorem. 

The pressure is usually calculated in a computer simulation via the virial theorem ol Clausius. The virial is defined as the expectation value of the sum of the products of the coordinates of the particles and the forces acting on them. This is usually written iV = X] Pxi where x, is a coordinate (e.g. the x ox y coordinate of an atom) and p. is the first derivative of the momentum along that coordinate pi is the force, by Newton s second law). The virial theorem states that the virial is equal to —3Nk T. 

This very important theorem was recognised by R. Clausius in 1850, although he did not at the time give the very simple interpretation, in terms of the conception of intrinsic energy, which was brought forward by Lord Kelvin a year later. 

Planck (loc. cit. 276) has observed that the point on which the whole matter turns is the establishment of a characteristic equation for each substance, which shall agree with Nernst s theorem. For if this is known we can calculate the pressure of the saturated vapour by means of Maxwell s theorem ( 90). He further remarks that, although a very large number of characteristic equations (van der Waals, Clausius s, etc.) are in existence, none of them leads to an expression for the pressure of the saturated vapour which passes over into (9) 210, at very low temperatures. Another condition which must be satisfied is... 

Carnot stated that the efficiency of a reversible Camot engine depends only on the temperatures of the heat reservoirs and is independent of the nature of the working substance. This theorem can be proved by showing that the assumption of a reversible engine with any but the known efficiency of a reversible Camot engine leads to a contradiction of the Clausius statement of the second law. 

One hundred fifty years ago, the two classic laws of thermodynamics were formulated independently by Kelvin and by Clausius, essentially by making the Carnot theorem and the Joule-Mayer-Helmholtz principle of conservation of energy concordant with each other. At first the physicists of the middle 1800s focused primarily on heat engines, in part because of the pressing need for efficient sources of power. At that time, chemists, who are rarely at ease with the calculus, shied away from... 

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 ... 

It was initially appreciated by R. Clausius that Carnot s theorem (4.25) allows the second law to be reformulated in a profoundly improved form. Clausius recognized that (4.25) is nothing more than the exactness condition (1.16a) for the differential dqmY/Ti.e., that L = 1 /T is an integrating factor for the inexact differential state property, a conserved quantity that... 

Clausius proceeded to demonstrate the power of entropy to express the deep consequences of the second law. We begin by introducing the inequality of Clausius, which complements Carnot s theorem (4.25) for the irreversible case. 

Table 5.1 summarizes the various constraint conditions and the associated thermodynamic potentials and second-law statements for direction of spontaneous change or condition of equilibrium. All of these statements are equivalent to Carnot s theorem ( dq/T < 0) or to Clausius inequality ([Pg.164]

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... 

Sadi Carnot s principle. Generalization of this principle by Clausius.— In 1824 di Carnot published a short work on the mechanical effects of heat depending on the one hand upon the impossibility of perpetual motion, on the other hand upon the principle, then accepted without question, that aroimd a closed cycle a i stem undergoes losses and absorptions of heat which exactly compensate each other, he demonstrated a theorem of the greatest importance both for the theory of heat and for the applications of this science to heat-engines. 

It was Clausius who in an imperishable memoir reconciled Carnot s theorem with the principle of equivalence. But Clausius did not limit himself to the realization of this work, which alone would have assured him the admiration of physicists. He... 

Strictly speaking, the equation K =S is an extension of Boltzmann s theory, in so far as we have ascribed a definite value to the entropy constant. According to Boltzmann, the probabihty contains an undetermined factor, which cannot be evaluated without the introduction of new hypotheses. Boltzmann and Clausius suppose that the entropy may assume any positive or negative value, and that the change in entropy alone can be determined by experiment. Of late, however, Planck, in connection with Nemst s heat theorem, has stated the hypothesis that the entropy has always a finite positive value, which is characteristic of the chemical behaviour of the substance. The probabihty must then always be greater than unity, since its logarithm is a positive quantity. The thermodynamical probabihty is therefore proportional to, but not identical with, the mathematical probabihty, which is always a proper fraction. The definition of the quantity w on p. 15 satisfies these conditions, but so far it has not been shown that this definition is sufficient under all circumstances to enable us to calculate the entropy. 

In the original derivation of the classical virial theorem given by Clausius, an expression corresponding to eqn (5.30) is also obtained. In the classical case one argues that the time average of d(f p)/dt vanishes over a sufficiently long period of time or that the motion is periodic to obtain the equivalent of eqn (5.31). ... 

That this is indeed the differential form of the customary virial theorem is readily seen by multiplying Eq. (26) throughout by x and then integrating over all x from —oo to +00. Some elementary integrations by parts recovers the usual (integral) virial theorem of Clausius, in, of course, now fully quantum-mechanical form [54]. 

Changes of sublimation pressure with temperature can be predicted and quantitatively calculated by means of the theorem of Lc Chatclier and the Clausius-Clapeyron equation, in the same way as changes of vapour pressure of a liquid (pp. 18 and 19). 

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