Reversible and irreversible transformations

Similarly, we find the minimum work required for compression by setting the value of Pop at each stage just infinitesimally greater than the pressure of the gas Pop = p + dp. The argument will obviously yield Eq. (7.5) for the minimum work required for compression if and k are the initial and final volumes in the compression. Equation (7.5) is, of course, general and not restricted to gases. 

For the ideal gas, the maximum quantity of work produced in the expansion or the minimum destroyed in the compression is equal to the shaded area under the isotherm in Fig. 7.6. For the ideal gas the maximum or minimum work in an isothermal change in state is easily evaluated, since p = nRT/V. Using this value for the pressure in Eq. (7.5), we obtain 

Consider the same system as before a quantity of gas confined in a cylinder at a constant temperature T. We expand the gas from the state T,p, V to the state T, p2, V2 and then we compress the gas to the original state. The gas has been subjected to a cyclic transformation returning at the end to its initial state. Suppose that we perform this cycle by two different processes and calculate the net work effect W y for each process. 

Process . Single-stage expansion with P p = P2, then single-stage compression with - op Pi  

Since V2 — is positive, and p2 — Pi is negative, W y is negative. Net work has been destroyed in this cycle. The system has been restored to its initial state, but the surroundings have not been restored masses are lower in the surroundings after the cycle. 

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Quantitative predictions about the heats of formation of hydrides of intermetallic compounds have been discussed by Shilov etal. (1989) they studied reversible and irreversible transformations in intermetallic compound-hydrogen systems and observed that four basic types of PIT diagrams exist for these systems. 

Fig. 112. (a) Phase diagram of the Pb/Ge(100) system as determined using LEED-AES [93Z1]. The reversible and irreversible transformations induced by heating of RT phases are illustrated. The "3x6" denotes the (3,0)x(l,2) structure, (b) Phase diagram of the Pb/Ge(100) system as determined using RT STM observations [97F2]. 

Although supramolecular assemblies are ideally suited to study the physics of small systems under external load, only a few are appropriate for the study of reversible and irreversible transformations. Most examples are from biomolecules such as proteins and nucleic acids [93, 140]. Along these fines, Bustamante and coworkers were the first to establish a reversible model system to apply Jarzynki s relation to compute the free energy difference from out-of-equilibrium experiments [141]. 

Table 20.3 lists the reversible and irreversible processes that may be significant in the deep-well environment.3 The characteristics of the specific wastes and the environmental factors present in a well strongly influence which processes will occur and whether they will be irreversible. Irreversible reactions are particularly important. Waste rendered nontoxic through irreversible reactions may be considered permanently transformed into a nonhazardous state. A systematic discussion of mathematical modeling of groundwater chemical transport by reaction type is provided by Rubin.30... 

For a scientist, the primary interest in thermodynamics is in predicting the spontaneous direction of natural processes, chemical or physical, in which by spontaneous we mean those changes that occur irreversibly in the absence of restraining forces—for example, the free expansion of a gas or the vaporization of a hquid above its boiling point. The first law of thermodynamics, which is useful in keeping account of heat and energy balances, makes no distinction between reversible and irreversible processes and makes no statement about the natural direction of a chemical or physical transformation. 

It will apparently be possible to provide coordination between the capabilities of equilibrium models in (1) the analysis of perfection of the energy and substance transformation processes and (2) the analysis of different irreversible phenomena on the basis of dual interpretation of equilibrium processes as being both reversible and irreversible at a time. In the first case they are convenient for interpretation as reversible in terms of the system interaction with the environment and in the second case—as irreversible in terms of their inner content according to Gorban. It is clear that to explain the dual interpretation it is necessary to extend the analysis by Gorban to the nonisolated thermodynamic systems with other characteristic functions to be used along with entropy. 

Figure 2. Schematic representation of different possible movements in rotaxanes a) pirouetting, b) shuttling, and c) clipping, the last leading to the reversible (or irreversible) transformation of a [2] rotaxane into a [2] catenane...

The way in which the separation of the terms of the right hand side of the entropy equation into the divergence of a flux and a source term has been achieved may at first sight seem to be to some extent arbitrary The two groups of terms must, however, satisfy a number of requirements which determine this separation uniquely First, one such requirement is that the entropy source term totai must be zero if the thermodynamic equilibrium conditions are satisfied within the system. Another requirement the source term must satisfy is that it should be invariant under a Galilean transformation (e.g., [147]), since the notations of reversible and irreversible behavior must be invariant under such a transformation. The terms included in the source term satisfy this requirement [32]. 

In papers53,54 an attempt has been made to investigate the mechanism of cationic cyclic acetal polymerization of 1,3-dioxolane. The presence of an acetal bond in the monomer molecule decreases the stability of active centers which are subjected to various reversible and irreversible chemical transformations. 

In recent times, various modifications of the basic DSC method have been reported. They include the so-called modulated DSC, in which a sinusoidally changing amplitude that is governed, as in standard DSC, by the temperature measured at the sample position.The modification resembles in several aspects the features of Fourier transform spectroscopy. Its main advantage over standard DSC resides in its ability to differentiate between reversible and irreversible thermal effects. "... 

Clearly, the second process just described is a reversible process, while the first is irreversible. There is another important characteristic of reversible and irreversible processes. In the irreversible process just described, a single mass is placed on the piston, the stops are released, and the piston shoots up and settles in the final position. As this occurs the internal equilibrium of the gas is completely upset, convection currents are set up, and the temperature fluctuates. A finite length of time is required for the gas to equilibrate under the new set of conditions. A similar situation prevails in the irreversible compression. This behavior contrasts with the reversible expansion in which at each stage the opposing pressure differs only infinitesimally from the equilibrium pressure in the system, and the volume increases only infinitesimally. In the reversible process the internal equilibrium of the gas is disturbed only infinitesimally and in the limit not at all. Therefore, at any stage in a reversible transformation, the system does not depart from equilibrium by more than an infinitesimal amount. 

Silica, with the chemical formula Si02 and relative molar mass of 60.084, exhibits a complex polymorphism characterized by a large number of reversible and irreversible phase transformations (Figure 10.1) usually associated with important relative volume changes (AV/V). At low temperature and pressure beta-quartz (P-quartz) [14808-60-7] predominates, but above 573°C, it transforms reversibly into the high-temperature alpha-quartz (a-quartz) [14808-60-7] with a small volume change (0.8 to 1.3 vol.%) ... 

The cycloaddition reaction of ierf-butylcyanoketene 28 with 1,3-cyclohexadiene 27 gives the cyclobutanone 29 along with some of the ether 30 (Scheme 12). This cycloaddition was found to be reversible. Cyclobutanone 29 is thermally and irreversibly transformed into the bicycHc... 

Figure 3.6 Reversible and irreversible processes. (A) The system reaches the state X from the standard state O through a path I involving irreversible processses. It is assumed that the same transformation can be achieved through a reversible transformation R. (B) An example of an irreversible process is the spontaneous expansion of a gas into vacuum. The same change can be achieved reversibly through an isothermal expansion of a gas that occurs infinitely slowly so that the heat absorbed from the reservoir equals the work done on the piston. In a reversible isothermal expansion the change in entropy can be calculated using dS = dQ/T.

Dual Lanczos transformation theory is a projection operator approach to nonequilibrium processes that was developed by the author to handle very general spectral and temporal problems. Unlike Mori s memory function formalism, dual Lanczos transformation theory does not impose symmetry restrictions on the Liouville operator and thus applies to both reversible and irreversible systems. Moreover, it can be used to determine the time evolution of equilibrium autocorrelation functions and crosscorrelation functions (time correlation functions not describing self-correlations) and their spectral transforms for both classical and quantum systems. In addition, dual Lanczos transformation theory provides a number of tools for determining the temporal evolution of the averages of dynamical variables. Several years ago, it was demonstrated that the projection operator theories of Mori and Zwanzig represent special limiting cases of dual Lanczos transformation theory. 

Simple Parallel Reactions. The simplest types of parallel reactions involve the irreversible transformation of a single reactant into two or more product species through reaction paths that have the same dependence on reactant concentrations. The introduction of more than a single reactant species, of reversibility, and of parallel paths that differ in their reaction orders can complicate the analysis considerably. However, under certain conditions, it is still possible to derive useful mathematical relations to characterize the behavior of these systems. A variety of interesting cases are described in the following subsections. 

The extent to which the ion-radical pair suffers a subsequent (irreversible) transformation (with rate constant k characteristic of highly reactive cation radicals and anion radicals) that is faster than the reverse or back electron transfer (/cBET) then represents the basis for the electron-transfer paradigm that drives the coupled EDA/CT equilibria forward onto products (P)20 (equation 8). 

Microbes are ubiquitous in the subsurface environment and as such may play an important role in groundwater solute behavior. Microbes in the subsurface can influence pollutants by solubility enhancement, precipitation, or transformation (biodegradation) of the pollutant species. Microbes in the groundwater can act as colloids or participate in the processes of colloid formation. Bacterial attachment to granular media can be reversible or irreversible and it has been suggested that extracellular enzymes are present in the system. Extracellular exudates (slimes) can be sloughed-off and act to transport sorbed materials [122]. The stimulation of bacterial growth in the subsurface maybe considered as in situ formation of colloids. 

A clue to the direction that needs to be followed to reach a criterion of spontaneity can be obtained by noticing in Table 5.1 that Q and Ware equal to zero for the reversible cycle but are not zero for the irreversible cycle. In other words, it is changes in the surroundings as well as changes in the system that must be considered in distinguishing a reversible from an irreversible transformation. Evidently, then, we need to find a... 

We may contrast this result for A totai with that for Al/totai for an ideal gas, as mentioned in Section 5.1. In the irreversible expansion of an ideal gas, Allgys = 0 the surroundings undergo no change of state (Q and W are both equal to zero), and hence, A /total = 0- ff we consider the reversible expansion of the ideal gas, AUsys is also equal to zero and AUsun is equal to zero because Q = —W, so again A /total = 0- Clearly, in contrast to AS, AU does not discriminate between a reversible and an irreversible transformation. 

K > 10, quasireversible if 0.01 < if < 10 and irreversible if if < 0.01. In the quasi-reversible range AWp decreases with decreasing if. This is partly caused by the transformation of the backward component under the influence of increased frequency (see Fig. 2.5). The maximum of this component decreases faster than the absolute value of the minimum of the forward component. For the conditions of Fig. 2.5, the... 

Figure 5.3. Connections among r- and p-space densities, density matrices, and form factors. Two-headed arrows signify reversible transformations single-barbed arrows signify irreversible transformations. A Fourier transform is denoted by JF.

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