Clausius analysis

Analysis by Clausius in 1849 showed that the ammonia molecules travel only some 0.001 cm. between collisions with air molecules at atmospheric pressure, in time intervals of about lO b sec. between collisions. This meemt... 

Jarzynski, C., Microscopic analysis of Clausius-Duhem processes, J. Stat. Phys. 1999, 96, 415-427... 

As the first law is sometimes referred to as the law that defines the fundamental thermodynamic property U, the internal energy of the system, the second law is considered to define the other fundamental property, the entropy S. Classical thermodynamics, via Clausius s thorough analysis [3] of thermodynamic cycles that extract work from available heat, has produced the relation between S and the heat added reversibly to the system at a temperature T ... 

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."... 

Of course, the more conceptual analysis and criticism of these foundations attracted no attention at first because of the rich yield of results that could be compared with experiment. These results were obtained in fast succession by A. Krdnig (1856), R. Clausius (since 1857), and J. C. Maxwell (1859) through the kinetic interpretation of the equation of state, diffusion, heat conduction, and viscosity.4... 

Bearing this restriction in mind, and assuming that the adsorbate phase is in equilibrium with the gas phase, a Clausius-Clapeyron analysis yields... 

Then, a similar Clausius-Clapeyron analysis leads to... 

There is a more or less generalized agreement that the isosteric adsorption heat is strongly affected by the microstructure of the adsorbent, particularly in the case of porous solids. This magnitude is better suited for structural analysis than other thermodynamic quantities. The use of the Clausius—Clapeyron equation to determine the isosteric adsorption heat has several limitations both theoretical and experimental, that are well known. 

In some problems, it is not certain in advance what variables should be used for a linear least-squares fit. In the vapor pressure case, we had the Clausius-Clapeyron equation, Eq. (11.43), which indicated that ln(P) and l/T were the variables that should produce a linear relationship. In the analysis of chemical rate data, it may be necessary to try two or more hypotheses to determine which gives the best fit. In a reaction involving one reactant, the concentration c of the reactant is given by Eq.(l 1.45) if there is no back reaction and if the reaction is a first-order reaction. If there is no back reaction and the reaction is a second-order reaction, the concentration of the reactant is given by... 

Heats of snrface reactions are directly obtainable from simple LEED observations. The nsnal application is to measure the enthalpy of adsorption of reversibly bound adsorbates. When the adsorbate produces a characteristic LEED pattern with extra beams, the mere existence of these beams, and not detailed intensity analysis, informs one of the presence of the characteristic adsorbed structure on the surface. At a given temperature there is a pressure at which this surface structure is just maintained, and the rates of evaporation and condensation into the structure are equal. Measurements of this pressure p as a function of absolute temperature T give the isosteric enthalpy of adsorption AH by application of the Clausius-Clapeyron equation for constant coverage... 

Thermodynamic analysis of the equilibrium between a condensed phase (solid or liquid) and the vapor is summarized by the Clausius-Clapeyron equation ... 

Another reason for Clausius s withdrawal from the foray may have been that many people still openly questioned the real existence of atoms, and in a molecular rationalization of entropy, atoms could not be ignored. Boltzmann however was ready to take the heat. He wrote, I am conscious of being only an individual struggling weakly against the stream of time. Considering a gas as a collection of particles, Boltzmann used statistical analysis to show that entropy is a manifestation of the naairal tendency of a system to seek the state of maximum disorder. 

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