Thermodynamic stabilities

The four thermodynamically stable forms of carbon are diamond, graphite, Cgo, Buckminster fullerene, and carbon nanotubes. It would be a challenge to extend the experience gained in CNT to nanotubes made of other material than carbon. It would also be interesting to form stable spherical structures in the nanoscale dimensions without agglomeration. At what scale would the quantum analysis for atoms be applicable when compared with the Newtonian mechanics used to describe macro 

From a fundamental point of view, bridge phases comprising different numbers of molecular strata may be viewed as different thermodynamic phases. This interpretation is evident if one considers the thermodynamic potential defined in Eq. (2.71). Together with Eq. (143), we obtain 

Applying the arguments detailed in Section (1.6) and in view of the fact that the fluid substrate potential does not depend on y, we immediately conclude from Eq. (5.118) that 

To obtain a more concise picture of thermodynamic stability of different film morphologies, we plot j) as a, function of asxo in Fig. 5.25 for the same system anaJj zed in Figs. 5.22-5.24. In a sequence of MC simulations in the grand mixed isostress isostrain ensemble, we calculate (j directly from Eq. (5.119) using the molecular expression for Tyy [see Eq. (5.85)], which does not contain any fluid substrate contribution between the fluid substrate 

5 Phase behavior of shecir-deformed confined fiuids 

A confined fiuid may undergo phase transitions among thin gaseous films, liquid-like, and bridge phases similar to those observed for the confined lattice fiuid in Section 4.5.3. To demonstrate the close correspondence between the two models as far as the phase behavior is concerned, we calculate the average overall density defined in Eq. (5.74) for various substrate separations A z. A plot of p in Fig. 5.9 for c, = exhibits two discontinuities. By a parallel analysis of in Fig. 5.8, the one around. s 8.2 turns 

When the free enthalpy of reaction AG for the transformation of the structure of a compound to any other structure is positive, then this structure is thermodynamically stable. Since AG depends on the transition enthalpy AH and the transition entropy AS, and AH and AS in turn depend on pressure and temperature, a structure can be stable only within a certain range of pressures and temperatures. By variation of the pressure and/or the temperature, AG will eventually become negative relative to some other structure and a phase transition will occur. This may be a phase transition from a solid to another solid modification, or it may be a transition to another aggregate state. 

Higher pressures p favor structures that occupy a lower volume, i.e. that have a higher density. As their formation involves a decrease in the volume (negative AV), AH will attain a negative value. For instance, diamond (density 3.51 gem-3) is more stable than graphite (density 2.26 g cm-3) at very high pressures. 

The question of the relative stability of carbenium ions in the gas phase and in solution has been discussed in detail in various monogr hs devoted to the thermodynamics of formation of these spedes and we will only deal vrith recent important contributions in this area. 

In the gas phase, Lossing et have published studies of free-radicd ion- 

Hehre et 3 . who have used ion-cyclotron resonance techniques to obtain criteria of aliphatic and aromatic carbocation stabilities. 

Of course, the traditional problem of the lack of precise knowledge of the heats of solvation for the passage of these ions into solution, makes the above criteria of stability less valuable to the condensed-phase chemist. A major breakthrough in this classical impasse has been achieved by Arnett and coworkers who have recently carried out calorimetric measurements leading to reliable values of the enthalpy of ionisation of various alkyl, cycloalkyl and aiyl halides in solution. These determinations owe their validity to the use of superacid conditions and the NMR verification that the ions expected were in fact formed in those media without Ihe occurrence of secondary reactions. One of the most important conclusion of these studies is that on the whole the relative stabUities of carbenium ions are the same in the gas pha% and in the solvents used, i.e., electrostatic solvation effects do not alter the order of stability. The importance of this new experimental approach is quite obvious and one can except in the near firture considerable advances in the field of the thermodynamics of reactive carbenium ions in solution through the attmnment of a precise knowledge of AG° values for their formation in various media. 

The results of Arnett and Petro in methylene chloride are directly relevant to cationic polymerisation. The following values of AHj/kcal mole for the ionisation of RCl by SbFs are a good illustration of the relative stability of typical carbenium ions involved in polymerisations as possible chain carriers or as initiators (CH3)2CH, 7.5 (CH3)3C , -15.5 Ph(CH3)2C , -19.0 Ph2ClC+, -23.7 Ph3C+, -27.1. This series confirms an expected behaviour, but stands as the first experimental set of values directly related to the thermodynamics of initiaticwi. 

The energy required to proceed from reactants to products is AG, the free energy of activation, which is the energy at the transition state relative to the reactants. We develop the theoretical foundation for these ideas about reaction rates in Section 3.2. We first focus attention on the methods for evaluating the inherent thermodynamic stability of representative molecules. In Section 3.3, we consider general concepts that interrelate the thermodynamic and kinetic aspects of reactivity. In Section 3.4, we consider how substituents affect the stability of important intermediates, such as carbocations, carbanions, radicals, and carbonyl addition (tetrahedral) intermediates. In Section 3.5, we examine quantitative treatments of substituent effects. In the final sections of the chapter we consider catalysis and the effect of the solvent medium on reaction rates and mechanisms. 

The AHJ of a given compound is a physical constant and is independent of the process by which the compound is formed. Therefore, AHJ values are additive and can be calculated precisely for balanced chemical equations if all the necessary data are available. For example, it might be experimentally impossible to measure the A// of methane directly by calorimetry, but it can be calculated as the sum of the enthalpy for an equivalent reaction sequence, e.g  

Consider the replacement of a carbon atom in an odd AH anion by a heteroatom. Dewar has shown that if the heteroatom contributes one 7i-electron (e.g., 434- 435), the change in total 7i-bond energy E is given to a first-order approximation by Eq. (6). °  

If the heteroatom contributes two t-electrons (e.g., 434- 433) the difference in total rt-energy between the AH anion and the neutral heterocon-jugated system is given by Eq. (7). 

In the absence of precise thermochemical data it is difficult to draw firm conclusions about the relative stability of mesomeric betaines and their isomers (e.g., 213 and 215), but some estimate can be obtained by comparing their pK values. 

The pKj values of a number of betaines and related heterocycles are shown in Table 

Suppose that the free energy difference between these heterocycles (Table VI) and their conjugate acids can be equated to a constant term, which represents the dissociation energy of the OH bond (AEoh) plus the difference in the total n-bond energies of the two species (A ). For convenience, we will restrict attention to heterocycles isoelectronic with the a-naphthylmethyl anion. The equilibrium 436 437 is a representative example. 

We now assume that n-bond energies of the conjugate acids (e.g., 436) are 

A hydrocarbon with a large strain energy will always be susceptible to some highly exothermic reaction (certainly combustion), but it m or may not be kinet-ically unstable. Consider cubane and (the known [37]) tricyclo[3.3.0.0 ]octa-3,7-diene, 35 both are (CH)s molecules  

So what can we conclude about the stability of polyprismanes from strain calculations We carried out the above illustrative calculations on [3,4]prismane. Here are resrrlts of the homodesmotic strain calculations of Minyaev et al. [1] at the B3LYP/6-311G(2df,p) level [stack height, ring size], nomenclature of Fig. 12.1, SE and SE/C-C, in kJ moU  

A graphical presentation of these values. Fig. 12.16, clearly shows that the SE/C-C levels off for the cyclopropane series at about 100 kJ mol , and for the cyclohexane series at about 85 kJ mol .  

Another factor, besides strain, that can destabilize a molecule is an unfavorable electronic interaction, for example that manifested as antiaromaticity, in, say cyclobutadiene [39], but unlike strain, electronic interactions can be favorable and stabilizing. Minyaev et al. proposed that orbital interactions between the two capping (CH)n and the adjacent C moieties stabilize [3 n]- and [4,n]-prismanes by CT-overlap of rr-type ring orbitals, and precedent for snch stabihzation in simple 

16 Variation of strain energy per C-C bond with number of staeked rings, for the cyclopropane ([2,3]- to [7,3]prismane) and cyclohexane ([2,6]- to [7,6]prismane) prismane series. Calculated by homodesmotic reactions (cf Fig. 12.14) at the B3LYP/6-31 lG(2df,p) (for stack heights 2-4, [1]) and B3LYP/6-31G (for stack heights 5-7, this work) levels the two basis sets give essentially the same SE/C-C, see text. The SE/C-C levels off for the cyclopropane series at about 100 kJ mol , and for the cyclohexane series at about 85 kJ moG  

Equilibrium solubility measurements are not always technically feasible, however. For example, solution-mediated phase transformations may occur. 

Dissolution rate from constant surface area 

Rapid analysis, automated measurement of solution clear points 

Yields quantitative AG before form conversions occur in suspensions 

Solution-mediated phase transformations to other forms, labUe solvates, slow transformation kinetics due to poor solubility or when near T, Solution-mediated phase transformations, non-ambient temperature control throughout measurement Thermal lag with fast heating, solution-mediated phase transformations with slow heating Compression-induced phase transformations 

The essences of the generalized vdW-P theory proposed by Tanaka follows. A system consisting of n unit cells is in equilibrium with a gas phase of the guest. For the S-I hydrate, there are = 46 water molecules per unit 

Tse and D. D. Klug, Phys Rev. B, 73, 064203 (2006). Copyright 2006 American Physical Society. 

It is straightforward to extend the probability factor to multiple occupancy. For example, the number of ways to choose / singly occupied and doubly occupied large cage and singly occupied small cages becomes 

The canonical partition function is given by the free energy of the host lattice. The entropy arises from the proton disorder and from the occupation of the guests. The free energy is because of the guests vibrations in the cages. For single occupancy, 

The free energy of the system is given by the entropy arising from the occupancy of guest molecules and the free energy because of guests vibrations inside the cavities. 

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Frenkel D 1988 Thermodynamic stability of a smectic phase in a system of hard rods Nature 332 822-3... 

Mchale J M ef a/1996 Surface energies and thermodynamic stability in nanocrystalline aluminas Science 277 788... 

The composition of the products from the isomerization of an unsaturated compound under the influence of a catalytic amount of a base is governed by the relative thermodynamic stabilities of the starting compound and the product. Of particular synthetic interest are isomerizations in which there is an accumulation of an isomer in the isomerization sequence. Isolation of the desired intermediate in a reasonable state of purity is often a matter of careful selection of the base and the solvent. The following reactions are representative examples ... 

By protodetritiation of the thiazolium salt (152) and of 2 tritiothiamine (153) Kemp and O Brien (432) measured a kinetic isotope effect, of 2.7 for (152). They evaluated the rate of protonation of the corresponding yiides and found that the enzyme-mediated reaction of thiamine with pyruvate is at least 10 times faster than the maximum rate possible with 152. The scale of this rate ratio establishes the presence within the enzyme of a higher concentration of thiamine ylide than can be realized in water. Thus a major role of the enzyme might be to change the relative thermodynamic stabilities of thiamine and its ylide (432). 

Marin, D. Mendicuti, F. Polarographic Determination of Composition and Thermodynamic Stability Constant of a Complex Metal Ion, /. Chem. Educ. 1988, 65, 916-918. 

Of the legion of partially saturated six-membered ring heterocycles, an idea of their stability, or lack of it, can normally be gained by consideration of the thermodynamic stability of the various components which can be identified in them. Thus, those rings which contain ester or amide functions can be expected to possess the chemical reactivity and the... 

The deposition of metals directly from these halides would require high temperatures to be efficient, but because of the thermodynamic stabilities of the hydrogen halides, direct reduction can readily be carried out with hydrogen at lower temperamres. The general reaction... 

The temperature at which this reaction is canied out is limited by considerations of the possibility of re-evaporation of As2 molecules and gallium atoms from the GaAs him. The semiconduchng compounds are less susceptible to this problem than the separate elements because of the thermodynamic stabilities of diese compounds, as discussed above. 

Processes in which solids play a rate-determining role have as their principal kinetic factors the existence of chemical potential gradients, and diffusive mass and heat transfer in materials with rigid structures. The atomic structures of the phases involved in any process and their thermodynamic stabilities have important effects on drese properties, since they result from tire distribution of electrons and ions during tire process. In metallic phases it is the diffusive and thermal capacities of the ion cores which are prevalent, the electrons determining the thermal conduction, whereas it is the ionic charge and the valencies of tire species involved in iron-metallic systems which are important in the diffusive and the electronic behaviour of these solids, especially in the case of variable valency ions, while the ions determine the rate of heat conduction. 

As pointed out in Section 2.4, shock waves are such rapid processes that there is no time for heat to flow into the system from the surroundings they are considered to be adiabatic. By the second law of thermodynamics, the quantity (S — Sg) must be positive for any thermodynamic process in an isolated system. According to (2.54), this quantity can only be positive if the P-V isentrope is concave upward. Thus, the thermodynamic stability condition for a shock wave is... 

The thermodynamic stability of a protein in its native state is small and depends on the differences in entropy and enthalpy between the native state and the unfolded state. From the biological point of view it is important that this free energy difference is small because cells must be able to degrade proteins as well as synthesize them, and the functions of many proteins require structural flexibility. 

It has been found that there is often a correlation between the rate of deprotonation (kinetic acidity) and the thermodynamic stability of the carbanion (thermodynamic acidity). Because of this relationship, kinetic measurements can be used to construct orders of hydrocarbon acidities. These kinetic measurements have the advantage of not requiring the presence of a measurable concentration of the carbanion at any time instead, the relative ease of carbanion formation is judged from the rate at which exchange occurs. This method is therefore applicable to very weak acids, for which no suitable base will generate a measurable carbanion concentration. 

The relative stability of the intermediates determines the position of substitution under kinetically controlled conditions. For naphthalene, the preferred site for electrophilic attack is the 1-position. Two factors can result in substitution at the 2-position. If the electrophile is very bulky, the hydrogen on the adjacent ring may cause a steric preference for attack at C-2. Under conditions of reversible substitution, where relative thermodynamic stability is the controlling factor, 2-substitution is frequently preferred. An example of this behavior is in sulfonation, where low-temperature reaction gives the 1-isomer but at elevated temperatures the 2-isomer is formed. ... 

The toroidal and helical forms that we consider here are created as such examples these forms have quite interesting geometrical properties that may lead to interesting electrical and magnetic properties, as well as nonlinear optical properties. Although the method of the simulations through which we evaluate the reality of the structure we have imagined is omitted, the construction of toroidal forms and their properties, especially their thermodynamic stability, are discussed in detail. Recent experimental results on toroidal and helically coiled forms are compared with theoretical predictions. 

We showed the possible existence of various forms of helically coiled and toroidal structures based on energetic and thermodynamic stability considerations. Though the formation process of these structures is not the subject of this work, the variety of patterns in the outer and inner surface of the structures indicates that there exist many different forms of stable cage carbon structures[10-19]. The molecules in a onedimensional chain, or a two-dimensional plane, or a three-dimensional supermolecule are possible extended structures of tori with rich applications. 

A study of the lithium-ammonia reduction of 14-en-16-ones would extend our understanding of the configuration favored at C-14 in metal-ammonia reductions. Although several simple 14-en-16-ones are known, their reduction by lithium and ammonia apparently has not been described in the literature. Lithium-ammonia reduction of A-nortestosterone, a compound that structurally is somewhat analogous to a 14-en-16-one, affords roughly equal amounts of the 5a- and 5 -dihydro-A-nortestosterones. " This finding was interpreted as indicating that there is little difference in thermodynamic stability between the two stereoisomeric products. 

The yields of thermal rearrangements of some perfluorinated olefins are very low The fact that perfluorocyclobutene yields perfluoro-l,3-butadiene at 650 °C only in a 12% yield [7] is due to the higher thermodynamic stability of perfluoro cyclobutene compared with the-open chain product [72 ]... 

Rearrangement studies give an interesting insight into the specific effect of fluonne on the thermodynamic stability and rearrangement kinetics of fluonnated cyclopropanes Fluorine decreases the thermodynamic stability of the cyclopropyl nng, in contrast with the generally observed effect of fluonne increasing the stability of molecules to which it is introduced [124]... 

The greater thermodynamic stability of iodates enables iodine to displace CI2 and Bt2 from their halates ... 

In keeping with the thermodynamic stability of the ene-hydrazine, the mono-phenylhydrazone of cyclohexane-1,3-dione 38 provided only one of two possible isomers 39, or at least, one dominant product 39 on indolization. ... 

Two independent molecular orbital calculations (HMO method) of delocalization energies for isoindole and isoindolenine tautomers agree that the isoindole form should possess the more resonance stabilization. The actual difference calculated for isoindole-isoindolenine is about 8 kcal/mole, but increases in favor of the isoindole with phenyl substitution at position 1 (Table VI).Since isoindole and isoindolenine tautomers have roughly comparable thermodynamic stabilities, the tautomeric proce.ss is readily obser-... 

The increase in thermodynamic stability of 85 is achieved by easy ring opening (01H127). This knowledge allows one to control the regioselectivity of the oxidative amination of the 6-aryl-l,2,4-tiiazine 4-oxides 53, obtaining either (i) the 5-amino-1,2,4-triazine 4-oxides 56 in the reaction of 53 with amines at low temperature in the presence of the oxidant or (ii) the 3-amino-1,2,4-triazine 4-oxides 88, provided the reaction is carried out in two steps (addition and oxidation) at room temperature or higher. 

Molecular mechanics calculations with the molecular mechanics force field program were performed to compare thermodynamic stability among araguspongine B (17) (containing two cw-fused perhydropyrido[2,l-Z>] [l,3]oxazine bicycles), araguspongine D (18) (containing two tran -fused perhydropyrido[2,l-Z)][l,3]oxazine bicycles), and araguspongine E (19)... 

The activation energies were computed to 3.0 (toward 183), 0.3 (toward 182), and 21.8 kcal/mol (toward 184) at the B3-LYP/6-31G level, and thus the mechanism leading to 182 is the preferred one. The transition states of all three reactions belong to concerted but asynchronous reaction paths. The transacetalization of 177 with acylium cations results in the formation of the thermodynamically stabilized 187 (Scheme 121) [97JCS(P2)2105]. 186 is less stable than 187, and 185 is destabilized by 32.5 kcal/mol. Moreover, transacetalization of 177 with sulfinyl cations is not a general reaction. Further computational studies on dioxanes cover electrophilic additions to methylenedioxanes [98JCS(P2)1129] and the influence... 

The thermodynamic stabilities of three possible annular tautomers of the parent 1,2,4,5-thiatriazine 134 were compared using ab initio HF/6-31G calculations (OOJOC931). The 4H isomer 134a appears to be the most stable (it is more stable than 134b by 11.6 kcal/mol and more stable than 134c by 15.5 kcal/mol), presumably because it allows low-energy distortion from planarity and formation of the boat conformation. 

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