Thermodynamic analogy with

Equation (3.16) shows that the force required to stretch a sample can be broken into two contributions one that measures how the enthalpy of the sample changes with elongation and one which measures the same effect on entropy. The pressure of a system also reflects two parallel contributions, except that the coefficients are associated with volume changes. It will help to pursue the analogy with a gas a bit further. The internal energy of an ideal gas is independent of volume The molecules are noninteracting so it makes no difference how far apart they are. Therefore, for an ideal gas (3U/3V)j = 0 and the thermodynamic equation of state becomes... 

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. 

A more interesting possibility, one that has attracted much attention, is that the activation parameters may be temperature dependent. In Chapter 5 we saw that theoiy predicts that the preexponential factor contains the quantity T", where n = 5 according to collision theory, and n = 1 according to the transition state theory. In view of the uncertainty associated with estimation of the preexponential factor, it is not possible to distinguish between these theories on the basis of the observed temperature dependence, yet we have the possibility of a source of curvature. Nevertheless, the exponential term in the Arrhenius equation dominates the temperature behavior. From Eq. (6-4), we may examine this in terms either of or A//. By analogy with equilibrium thermodynamics, we write... 

That all actual processes are irreversible does not invalidate the results of thermodynamic reasoning with reversible processes, because the results refer to equilibrium states. This procedure is exactly analogous to the method of applying the principle of Virtual Work in analytical statics, where the conditions of equilibrium are derived from a relation between the elements of work done during virtual i.e., imaginary, displacements of the parts of the system, whereas such displacements are excluded by the condition of equilibrium of the system. 

Studies of the thermodynamics of formation of complexes of zinc with 2,2 -bipyridine in the presence of halide and thiocyanate ions in DMF has been carried out.205 The zinc complexes of 2,2 -bipyridine with A-(benzenesulfonyl)glycine and A-(toluene- -sulfonyl)glycine were synthesized and infrared data implied a structural analogy with the cadmium complexes. This suggests that the zinc promotes sulfonamide nitrogen deprotonation in the amino acid ligands in the solid state.206... 

The reductive coupling of guaiazulene, a fulvalene analog, with activated calcium in THF affords a 60 40 mixture of two isomers, 8,8 -147 (Figure 75) and 8,6 -(diguaiazulenide)bis(tetrahydrofuran)calcium, respectively. Thermolyses of each isomer revealed that the 8,8 -isomer is the thermodynamically preferred isomer, and it forms preferentially when the coupling is performed at 67 °C.342... 

By analogy with Hamilton s principle of least action, the simplest proposition that could solve the thermodynamic problem is that equilibrium also depends on an extremum principle. In other words, the extensive parameters in the equilibrium state either maximize or minimize some function. 

Once the cluster expansion of the partition function has been made the remaining thermodynamic functions can be obtained as cluster expansions by taking suitable derivatives. Of particular interest are the expressions for the equilibrium concentrations of intrinsic point defects for the various types of lattice disorder. Since the partition function is a function of Nx, N2, V, and T, it is convenient for the derivation of these expressions to introduce defect chemical potentials for each of the species in the set (Nj + N2) defined, by analogy with ordinary Gibbs chemical potentials (cf. Section I), by the relation... 

Zeolites are the main catalyst in the petrochemical industry. The importance of these aluminosilicates is due to their capacity to promote many important reactions. By analogy with superacid media (1), carbocations are believed to be key intermediates in these reactions. However, simple carbocationic species are seldom observed on the zeolite surface as persistent intermediates within the time-scale of spectroscopic techniques. Indeed, only some conjugated cyclic carbocations were observed as long living species, but covalent intermediates, namely alkyl-aluminumsilyl oxonium ions (2) (scheme 1), where the organic moiety is bonded to the zeolite structure, are usually thermodynamically more stable than the free carbocations (3,4). 

By analogy with the basicity towards protons (equation 5) the basicity towards monocharged metal ions (Table 4) is defined by means of the thermodynamics of the reaction ... 

This is defined as the surface free energy per unit area projected onto the low index facet plane. The use of the projected free energy allows a direct analogy with the thermodynamics of a liquid-vapor system. See, e.g., Williams et al. for a clear discussion. 

V.2.1 Centroid transition state theory. A third methodology, is to construct approximate theories for dynamical properties, which make use of only thermodynamic quantities. In analogy with classical TST, Gillan, Voth and coworkers have formulated and studied a quantum TST which is based on the centroid potential of mean force Wc (q) ... 

By analogy with allylic organometallic compounds (see Section . ), the possibility of achieving intramolecular related zinc-ene reactions involving allenylzinc species acting as ene-components has been investigated. Such reactions benefit from favorable thermodynamics and were thus expected to proceed more readily than the related intermolecular additions of allenylzincs to alkynes or alkenes. 

Another associated issue was the possibility of inactivating the LRES (lym-phoreticuloendothelial system). By analogy with other injectable systems, it could also be deduced that the injectable emulsion system needed to be sterile and apy-rogenic and free of acute or chronic toxicities from components or their associated degradation products. It also followed that the injectable system required to be stable, although how stability was to be determined and, more to the point, measured, has remained an issue to the present day. This is mainly because emulsions are thermodynamically unstable although their stability can be extended by formulation. As a result emulsion products are now available that are submicron in diameter, sterile, and stable for several years after preparation. In major part this has been due to the use of phospholipids as stabilizers and emulsifiers, in particular the mixed products identified as the lecithin of commerce. 

A well understood case is that of quinoline reaction at position 2 is kinetically favored as compared with reaction at position 4, but the adduct from the latter is thermodynamically more stable. This situation, where the site of attack leading to the more stable adduct is the y position, is analogous with those regarding the formation of Meisenheimer adducts from benzene and pyridine derivatives and RCT nucleophiles. Presumably, with quinoline kinetic control favors the position that is more strongly influenced by the inductive effect of the heteroatom. The fact that position 2 of quinoline is the most reactive toward nucleophilic reagents is probably related to the lower 71-electron density at that position.123 However, the predominance of the C-4 adduct at equilibrium can be better justified by the atom localization energies for nucleophilic attachment at the different positions of quinoline. Moreover, both 7t-electron densities and atom localization energies indicate position 1 of isoquinoline to be the most favored one for nucleophilic addition. 

By analogy with the thermodynamic expression for the pressure of Eq. 8.113, we can write... 

The Maxwell relations are powerful tools of thermodynamic derivation. With the help of these relations and derivation techniques analogous to those illustrated in Sidebars 5.3-5.6, the skilled student of thermodynamics can (in principle ) re-express practically any partial derivative in terms of a small number of base properties involving only PVT variables. Consider, for example, the eight most common variables... 

In view of the Hessian character (10.20) of the thermodynamic metric matrix M(c+2), the eigenvalue problem for M(c+2) [(10.23)] can be usefully analogized with normal-mode analysis of molecular vibrations [E. B. Wilson, Jr, J. C. Decius, and P. C. Cross. Molecular Vibrations (McGraw-Hill, New York, 1955)]. The latter theory starts from a similar Hessian-type matrix, based on second derivatives of the mechanical potential energy Vpot (cf. Sidebar 2.8) rather than the thermodynamic internal energy U. 

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