Entropy empirical

We note that the first sum in Eq. (45a) has the form of the negative of an entropy [see Eq. (39)]. This entropy will be called photon-site entropy because it involves the df sites zm. Likewise the second sum in Eq. (45a) is the negative of empirical entropy, in that it involves the empirical data mm. Hence, overall, estimator (45a) is one of minimum photon-site entropy plus minimum empirical entropy plus maximum Jaynes entropy (third sum). 

This is a principle of minimum empirical entropy. The solution hm to Eq. (47a) cannot be obtained by setting its derivative equal to zero. That would instead minimize Eq. (47a). In fact, the solution hm tends to be extreme and... 

This is a principle of minimum empirical entropy (first sum) plus maximum weighted-Burg entropy (second sum). Once again, differentiation does not always yield the solution. Taking the second derivative of Eq. (49a) shows that the first derivative attains a maximum if and only if... 

We have seen in Section 1.8 that under suitable conditions the performance of work can be related to a function of state, the energy. The question arises whether a similar option exists for the transfer of heat, again under suitable conditions. The answer is in the affirmative unfortunately, the correspondence is not so easily demonstrated. A fair amount of mathematical groundwork must be laid to establish the link between heat flow and a new function of state. Readers not interested in the mathematical niceties can assume the implication of the Second Law of Thermodynamics, namely that there does exist a function A which converts the inexact differential dQ into an exact differential through the relationship dQ/A — ds, where s is termed the empirical entropy function. The reader can then proceed to Section 1.13, beginning with Eq. (1.13.1), without loss of continuity. 

A similar argument is used to deal with s this is based on the empirical observation that in an adiabatic process involving a noble gas at low pressures, the product PV7 is virtually constant. Here 7 is a fixed quantity (which will later turn out to be the ratio of molar heat capacities at constant pressure and volume) whose exact significance is irrelevant at this stage near room temperature and for monatomic gases 7 has a value close to 5/3. We therefore use the product PV7 as a measure of the empirical entropy through the simple relation... 

Equation 3. We estimated the error to be expected in the use of Equation 3 by comparing empirical entropies with those calculated using contributions tabulated by Latimer O) without modification. The empirical data were taken from Hepler and Olofsson W, Robie, Hemingway and Fisher ( ) and the National Bureau of Standards Technical Note 270 series (11). Data for mercury are listed in Table I. Empirical and calculated entropies are compared... 

To estimate the error expected in predictive use of Equation 4, we attempted to fit a straight line to the empirical entropies of Hg(II) compounds as though they were functions of the entropies of the corresponding Pb(II) compounds, as required by Equation 4. The slope should be 1.0 and the intercept, 1.5R n(W]yj/Wjj), if the k term is small for pairs of compounds of similar properties ( ). ... 

We develop instead a shortcut for the Second Law along the following lines Just as we were able to link the performance of work under adiabatic conditions to the existence of a function of state, the energy, so we will postulate that the transfer of heat under reversible conditions is related to another functions of state, termed the empirical entropy, symbolized by s (or, later, the metrical entropy, S), which will then found to be useful to characterize various processes. In short, it is claimed as part of the Second Law ofThermodynamics that... 

A reversible incremental transfer of heat, dr Q, between a given system and its surroundings is related to a new incremental function of state, termed the empirical entropy, ds, through an integrating denominator, k, whose physical significance is to be established later. 

In addition to showing an approximately linear variation of In K with l/T (van t Hoff equation), an adsorption equilibrium constant should meet Boudart s four semi-empirical entropy criteria [8] ... 

The significant aspect of Eq. (31) is that there are two contributions to X an empirical entropy correction, a, and an enthalpic contri-butii, 0, related to the heat of mixing. In the second term of Eq. (31), 0 =[(Z-2)AWj2]/RT y = 2/(Z-2) relates the number of pairs of unlike neighbors in the lattice to the volume fractions of the two polymers and e = related to the probability of... 

AH and AS to various notional subprocesses such as bond dissociation energies, ionization energies, electron affinities, heats and entropies of hydration, etc., which themselves have empirically observed values that are difficult to compute ab initio. 

However, solubility, depending as it does on the rather small difference between solvation energy and lattice energy (both large quantities which themselves increase as cation size decreases) and on entropy effects, cannot be simply related to cation radius. No consistent trends are apparent in aqueous, or for that matter nonaqueous, solutions but an empirical distinction can often be made between the lighter cerium lanthanides and the heavier yttrium lanthanides. Thus oxalates, double sulfates and double nitrates of the former are rather less soluble and basic nitrates more soluble than those of the latter. The differences are by no means sharp, but classical separation procedures depended on them. 

Although these potential barriers are only of the order of a few thousand calories in most circumstances, there are a number of properties which are markedly influenced by them. Thus the heat capacity, entropy, and equilibrium constants contain an appreciable contribution from the hindered rotation. Since statistical mechanics combined with molecular structural data has provided such a highly successful method of calculating heat capacities and entropies for simpler molecules, it is natural to try to extend the method to molecules containing the possibility of hindered rotation. Much effort has been expended in this direction, with the result that a wide class of molecules can be dealt with, provided that the height of the potential barrier is known from empirical sources. A great many molecules of considerable industrial importance are included in this category, notably the simpler hydrocarbons. 

The most common states of a pure substance are solid, liquid, or gas (vapor), state property See state function. state symbol A symbol (abbreviation) denoting the state of a species. Examples s (solid) I (liquid) g (gas) aq (aqueous solution), statistical entropy The entropy calculated from statistical thermodynamics S = k In W. statistical thermodynamics The interpretation of the laws of thermodynamics in terms of the behavior of large numbers of atoms and molecules, steady-state approximation The assumption that the net rate of formation of reaction intermediates is 0. Stefan-Boltzmann law The total intensity of radiation emitted by a heated black body is proportional to the fourth power of the absolute temperature, stereoisomers Isomers in which atoms have the same partners arranged differently in space, stereoregular polymer A polymer in which each unit or pair of repeating units has the same relative orientation, steric factor (P) An empirical factor that takes into account the steric requirement of a reaction, steric requirement A constraint on an elementary reaction in which the successful collision of two molecules depends on their relative orientation. 

Trouton s rule The empirical observation that the entropy of vaporization at the boiling point (the enthalpy of vaporization divided by the boiling temperature) is approximately 85 J-K—1 -mol 1 for many liquids. 

The redox reactions of the actinide elements have been the subject of a recent and authoritative review by Newton and Baker . The net activation process concept is used to interpret the experimental data. Empirical correlations shown to exist include those between the entropies of the activated complexes and their charges, and, for a set of similar reactions, between AG and AG , and and A/f . The present state of the evidence for binuclear species is discussed. 

D. J. Very empirical treatment of solvation and entropy a force field derived from Log Po/w J. Comput.-Aided Mol. Des. 2001, 15, 381-393. 

---