Quantum thermodynamic definition

A quantum thermodynamic definition of electronegativity has been provided by Gyftopoulos and Hatsopoulos [27] by considering the atom or the molecule as a member of a grand canonical ensemble where the energy (E) and the number of electrons (N) are continuous functions and all other properties of the ensemble are written in terms of these two independent variables. The chemical potential of the ensemble can be written as ... 

Gyftopoulos and Hatsopoulos [267] proposed a quantum thermodynamic definition of electronegativity (j) of a system where the electronic chemical potential (fi) is being considered as the negative of the same. 

The material covered in this chapter is self-contained, and is derived from well-known relationships such as Newton s second law and the ideal gas law. Some quantum mechanical results and the statistical thermodynamics definition of entropy are given without rigorous derivation. The end result will be a number of practical formulas that can be used to calculate thermodynamic properties of interest. 

Several approximations that allow simple estimates of bond parameters are presented as a demonstration that predictions based on quantum potentials are of correct order, and not as an alternative to well-established methods of quantum chemistry. In the same spirit it is demonstrated that the fundamental thermodynamic definition of chemical equilibrium can be derived directly from known quantum potentials. The main advantage of the quantum potential route is that it offers a logical scheme in terms of which to understand the physics of chemical binding. It is only with respect to electron-density distributions in bonds that its predictions deviate from conventional interpretations in a way that can be tested experimentally. 

Thermodynamical calculations are helpful in deciding between various secondary mechanisms taking place in photochemical reactions. A given intermediate product can exist in equilibrium with the reactants only if there is a decrease in free energy, but even then the rate of formation of the intermediate compound may be too slow to be an appreciable factor. Another basis for reaching a decision as to intermediate steps and the mechanism of the reaction rests on the quantum calculations of reaction rate. It will be shown in Chapter IX that many reactions which appear possible on paper may be definitely excluded on the basis of these theoretical calculations. 

Equation (5.20) is the basis for calculation of absolute entropies. In the case of an ideal gas, for example, it gives the probability ft for the equilibrium distribution of molecules among the various quantum states determined by the translational, rotational, and vibrational energy levels of the molecules. When energy levels are assigned in accord with quantum mechanics, this procedure leads to a value for the energy as well as for the entropy. From these two quantities all other thermodynamic properties can be evaluated from definitions (of H. G,... 

In considering what we mean by short-hved molecules , we must be careful to distinguish between thermodynamic stability and kinetic inertness. A compound that is thermodynamically unstable need not be short-lived, provided a high activation barrier exists towards reaction. Such a molecule is said to be kinetically inert. A species that survives for only short periods is said to be labile. Thus most of those short-lived species that we will consider in this chapter have a degree of thermodynamic instability coupled with kinetic lability. An added complication arises when we consider photochemical systems. Here the abihty of a molecule to absorb hght at a particular wavelength and the quantum yield for reaction will determine, to some extent, its hfetime. The broad definition adopted in this article is that a short-hved molecule is one whose concentration decays, under ambient conditions, on a timescale ranging from nanoseconds or even picoseconds to a few seconds. 

The third law, like the two laws that precede it, is a macroscopic law based on experimental measurements. It is consistent with the microscopic interpretation of the entropy presented in Section 13.2. From quantum mechanics and statistical thermodynamics, we know that the number of microstates available to a substance at equilibrium falls rapidly toward one as the temperature approaches absolute zero. Therefore, the absolute entropy defined as In O should approach zero. The third law states that the entropy of a substance in its equilibrium state approaches zero at 0 K. In practice, equilibrium may be difficult to achieve at low temperatures, because particle motion becomes very slow. In solid CO, molecules remain randomly oriented (CO or OC) as the crystal is cooled, even though in the equilibrium state at low temperatures, each molecule would have a definite orientation. Because a molecule reorients slowly at low temperatures, such a crystal may not reach its equilibrium state in a measurable period. A nonzero entropy measured at low temperatures indicates that the system is not in equilibrium. 

Until now, our formulation of statistical thermodynamics has been based on quantum mechanics. This is reflected by the definition of the canonical ensemble partition function Q, which turns out to be linked to matrix elements of the Hamiltonian operator H in Eq. (2.39). However, the systems treated below exist in a region of thermodjniamic state space where the exact quantum mechanical treatment may be abandoned in favor of a classic dc.scription. The transition from quantum to classic statistics was worked out by Kirkwood [22, 23] and Wigner [24] and is rarely discussed in standard texts on statistical physics. For the sake of completeness, self-containment, and as background information for the interested readers we summarize the key considerations in this section. 

We refrain from entering here upon a thoroughgoing discussion of the preceding definition of temperature from the thermodynamic and axiomatic point of view (a complete treatment for tlie generalized statistics introduced, by the quantum theory is given in Appendix XXIX, p. 336), and merely add a brief remark on the units in which temperature is measured. 

The two previous secfions were devoted to modeling quantum resonances by means of effective Hamiltonians. From the mathematical point of view we have used two principal tools projection operators that permit to focus on a few states of interest and analytic continuation that allows to uncover the complex energies. Because the time-dependent Schrodinger equation is formally equivalent to the Liouville equation, it is attractive to try to solve the Liouville equation using the same tools and thus establishing a link between the dynamics and the nonequilibrium thermodynamics. For that purpose we will briefly recall the definition of the correlation functions which are similar to the survival and transition amplitudes of quantum mechanics. Then two models of regression of a fluctuation and of a chemical kinetic equation including a transition state will be presented. 

The definition of the gas-phase acidity through reaction (7.3) implies that this quantity is a thermodynamic state function. Thus, one could use quantum chemical approaches to obtain gas-phase acidities from the theoretically computed enthalpies of the species involved. However, two points must be noted before one proceeds A chemical bond is being broken and an anion is being formed. Thus, one may anticipate the need for a proper treatment of electronic correlation effects and also of basis sets flexible enough to allow the description of these effects and also of the diffuse character of the anionic species, what immediately rules out the semi-empirical approaches. Hence, our discussion will only consider ab initio (Hartree-Fock and post-Hartree-Fock) and DFT (density functional theory) calculations. 

Another aspect of the comparison between the calculated and experimental cohesive energy is important to recall here. It is related to the difference between the definition of cohesive energy and the crystal formation energy that is reported in thermodynamic tables, the main point probably being that quantum mechanical calculations refer to the static limit (T = 0 K and frozen nuclei), whereas experiments refer to some finite temperature. In fact, the... 

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