Thermodynamic Analogy

In addition to theoretical estimates of the rate constant based on postulated structures for the activated complex, the statistical theory of rate constants can be used to correlate kinetic trends in homologous series of reactions. For this purpose we construct a thermodynamic interpretation of the rate constant. 

The reaction rate, in terms of activated complex theory, is the number of A-B composite systems that traverse the pass per second. We computed it by treating separately two aspects of the problem. First local equilibrium was assumed and the number of A-B complexes calculated. Then progress along the reaction coordinate was considered and the rate of barrier crossing determined. From (9.10)-(9.12) this frequency (sec ) can be identified as kT Kyjh. Assuming k = 1, the reaction rate is 

The ratio n jn n has the form of an equilibrium constant for formation of activated complex. For systems in which the activity of the various species equals their concentration (dilute gases, dilute solutions of nonelectrolytes) (9.41) defines a totally empirical function, the activation equilibrium constant K iT), 

Continuing in this vein, the corresponding enthalpy, entropy, and Gibbs free energy of activation are 

These quasi-thermodynamic parameters can be related to the Arrhenius activation energy by means of (8.7) using (9.42) the result is 

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Tautomerism has been discussed in Section 4.04.1.5.2. It concerns prototropic tautomerism and the decreasing order of stability is (hydrazone) >A (azo)> A (enehydrazine). The isomerization A -> A occurs via a A -pyrazoline (65BSF769). Pyrazolidones and amino-A -pyrazolines exist as such. The only example of non-prototropic tautomerism deals with the isomerization (403) —> (404) (74CJC3474). This intramolecular process is another example (Section 4.04.1.5) of the thermodynamic analogy between prototropy and metallotropy. 

The formal thermodynamic analogy existing between an ideal rubber and an ideal gas carries over to the statistical derivation of the force of retraction of stretched rubber, which we undertake in this section. This derivation parallels so closely the statistical-thermodynamic deduction of the pressure of a perfect gas that it seems worth while to set forth the latter briefly here for the purpose of illustrating clearly the subsequent derivation of the basic relations of rubber elasticity theory. 

In 1982, Nalewajski and Parr took the thermodynamic analogy to its logical conclusion by extending the Legendre-transform structure of classical thermodynamics to DFT [8]. One of their results was the Maxwell relation for Equation 18.6,... 

These statements appear in Chapter XIII, where they are derived, in a different form than in Chapter XIV, where they are used in the discussion of the thermodynamical analogies. We will follow the latter formulation and the presentation of Lorentz [2]. 

A serious objection against such a thermodynamic analogy, however, for identifying Lj SjTgj with the molar cohesion energy is twofold ... 

A reversible process is the thermodynamic analog of frictionless motion in mechanics, When a process has been conducted reversibly, we can, by performing the inverse process in reverse, set the system back in precisely its initial state, with zero net expenditure of work in the overall process. The system and its surroundings are once again exactly as they were at the beginning. A reversible process is an idealization which constitutes a limit that may he approached but not attained in real processes.)... 

Phase trcmsitions in monolayers may be treated thermodynamically analogously to those in three-dimensional systems. As will be derived in sec. 3.4, the Clausius-Clapeyron equation, relating the variation of pressure with temperature for a two-dimensional situation, reads... 

Hejna, B. Information Capacity of Quantum Transfer Channels and Thermodynamic Analogies, Thermodynamics - Fundamentals and its Application in Science, Ricardo Mo-rales-Rodriguez (Ed.), ISBN 978-953-51-0779-8, InTech, 2012. Available from http // www.intechopen.com/articles/show/title/information-capacity-of-quantum-transfer-channels-and-thermodynamic-analogies... 

The basic thermodynamic data of the individual components, i.e. the temperatures and enthalpies of fusion, were mainly taken from the literature and are summarized in Table 3.2. Data published by Bottinga and Richet (1978) were taken into account as well. However, for some components, the values of the enthalpy of fusion were not known. In such cases, these data were estimated on the basis of thermodynamic analogy. Besides, the full chemical formula and the acronym of the component is given in the table. 

Redox parameters analogous to those for acid-base chemistry can be defined for all aqueous systems. The redox intensity factor pE is an energy parameter in non-dimensional form that describes the ratio of electron acceptors (oxidants) and donors (reductants) in a redox couple. The redox potential (Ej ) of the system is an alternative and equivalent intensity factor. Table I summarizes the complete thermodynamic analogy between pH and pE. An analogy between acid-base and redox systems can also be made for capacity factors. 

The time-dependent expectation value of a thermodynamic observable (i.e., property) is illustrated in terms of the Hamiltonian operator. The result provides an estimate of the total energy of the system, whose classical thermodynamic analog is the internal energy ... 

An equilibrium constant depends upon the initial and final states only, a velocity constant upon the intermediate stages through which molecules must pass on the way from one to the other. In particular, there is, for a chemical reaction, a transition state in which reacting substances and products are indistinguishable. The kinetic theory tells us a good deal about the attainment of this transition state. In the formulation of its properties, thermodynamic analogies are also found helpful in a way which will appear at a later stage. 

The thermodynamic analogy can be carried farther if the molecules in their transition state, that is, in the condition where they are on the point of changing into reaction products, are regarded as constituting a definite and special chemical species. This species may be imagined to possess properties which can be formulated in the same way as those of normal molecules. There then arises the possibility of applying the statistical formula for the absolute value of an equilibrium constant ... 

It was logical to extend the results of the thermostatic theory to temporal changes. Stochastic thermostatics adopts an intermediate level between statistical and phenomenological thermodynamics. Analogously, in principle the stochastic treatment of thermodynamics processes has an intermediate character between nonequilibrium statistical mechanics and phenomenolog-... 

We can consider Eq. (1), which has the standard form of a statistical mechanical average (where t plays the role of the energy ) as a basis for the thermodynamic analogy of the percolation problem [5]. It is suitable for numerical determination of cluster properties. 

Myler CA. Use of a thermodynamic analogy for pneumatic transport in horizontal pipes. Ph.D. diss.. University of Pittsburgh, Pittsburgh, PA, 1989. 

The thermodynamic analogy is not to be used as a method for establishing absolute values for the various activation parameters these are of limited significance since they derive from an equilibrium thermodynamic interpretation of intrinsically nonequilibrium properties. Furthermore, to accept such numbers uncritically ascribes a measure of definiteness to the activated complex which is unwarranted. On the other hand, trends and similarities may be useful in helping to characterize reaction mechanism. In Table 9.4 the values of and AHq calculated from (9.43) and (9.44) are given for a number of gas-phase reactions. For the bimolecular reactions the value of ASq depends upon the choice of standard state for rate constants in units of cm mol sec the natural standard state is a concentration of 1 mol cm. ... 

Alternatively, the thermodynamic analogy can be used to provide estimates of rate constants. In equilibrium thermodynamics one uses... 

An artifact of this kind of decomposition is that it raises the question of how, and on what scale, individual processes (atoms) (e.g., clouds, radiation, chemistry, etc.) interact, and hence the extent to which parameterizations must be coupled to one another, and not just to larger-scale processes. Thermodynamic analogies are useful to a point for instance, diffusion parameterizes molecular transport in fluids. However, any attempt to develop a kinetic theory capable of aggregating many small-scale processes is impeded by our lack of understanding of what exactly constitutes the atoms and the rules that govern their behavior. 

A recent account of the thermodynamic analogy of the present derivatives given in eqns (61), (63), (64) and (65) can be found in ref. 72. In these equations, E is the total energy of the system and F the Hohenberg-Kohn functional, the sum of the kinetic energy and electron-electron repulsion functional. 

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