Entropy Measures

Many substances exist in two or more solid allotropic fomis. At 0 K, the themiodynamically stable fomi is of course the one of lowest energy, but in many cases it is possible to make themiodynamic measurements on another (metastable) fomi down to very low temperatures. Using the measured entropy of transition at equilibrium, the measured heat capacities of both fomis and equation (A2.1.73) to extrapolate to 0 K, one can obtain the entropy of transition at 0 K. Within experimental... 

Measure Entropy In the same way as the information dimension, Dp generalizes the fractal dimension. Dp, of an attractor. 4, by taking into account the relative frequency with which the individual e-boxes of a partition are visited by points on the attractor, so too the finite set entropy generalizes to a finite measure entropy,... 

Although it is obviously impossible to enumerate all possible configurations for infinite lattices, so long as the values of far separated sites are statistically independent, the average entropy per site can nonetheless be estimated by a limiting procedure. To this end, we first generalize the definitions for the spatial set and spatial measure entropies given above to their respective block-entropy forms. 

Dimensions The limits of the set and measure entropies as T, B —> oo yield the spatio-temporal set and measure dimensions, respectively ... 

It is simplest to consider these factors as they are reflected in the entropy of the solution, because it is easy to subtract from the measured entropy of solution the configurational contribution. For the latter, one may use the ideal entropy of mixing, — In, since the correction arising from usual deviation of a solution (not a superlattice) from randomness is usually less than — 0.1 cal/deg-g atom. (In special cases, where the degree of short-range order is known from x-ray diffuse scattering, one may adequately calculate this correction from quasi-chemical theory.) Consequently, the excess entropy of solution, AS6, is a convenient measure of the sum of the nonconfigurational factors in the solution. 

Although the Boltzmann equation may appear simple, applying it to a molecular system always is challenging. The reason is that there are immense numbers of molecules in any realistic molecular system, so it is necessary to count huge numbers of possibilities to determine the value of W. Instead, scientists have found ways to measure entropy by analyzing energy dispersal. 

The present example does support some speculation on an outstanding puzzle for our conceptualization of hydrophobic effects. It is known that the sum of the standard hydration entropies of K+(aq) and Cl (aq) is about twice the standard hydration entropy of Ar(aq) [67]. The case of methanol as the solvent is qualitatively different. If hydrophobic effects are conceptualized on the basis of hydration entropies and specific hydration structures then this is paradoxical according to the measured entropies K+(aq) + Cl (aq) is about as hydrophobic as Ar(aq) + Ar(aq), but the hydration structures neighboring K+(aq), Cl (aq), and Ar(aq) should be qualitatively different. This paradox has not been given a satisfactory explanation. 

As was discussed earlier in Section 1.2.8 a complication arises in that two of these properties (solubility and vapor pressure) are dependent on whether the solute is in the liquid or solid state. Solid solutes have lower solubilities and vapor pressures than they would have if they had been liquids. The ratio of the (actual) solid to the (hypothetical supercooled) liquid solubility or vapor pressure is termed the fugacity ratio F and can be estimated from the melting point and the entropy of fusion. This correction eliminates the effect of melting point, which depends on the stability of the solid crystalline phase, which in turn is a function of molecular symmetry and other factors. For solid solutes, the correct property to plot is the calculated or extrapolated supercooled liquid solubility. This is calculated in this handbook using where possible a measured entropy of fusion, or in the absence of such data the Walden s Rule relationship suggested by Yalkowsky (1979) which implies an entropy of fusion of 56 J/mol-K or 13.5 cal/mol-K (e.u.)... 

As in the previous experiments, at each epoch we would like to select a waveform (or really the error covariance matrix associated with a measurement using this waveform) so that the measurement will minimize the uncertainty of the dynamic model of the target. We study two possible measures entropy of the a posteriori pdf of the models and mutual information between the dynamic model pdf and measurement history. Both of these involve making modifications to the LMIPDA-IMM approach that are described in [5]. Since we want to minimize the entropy before taking the measurement, we need to consider the expected value of the cost. To do this we replace the measurement z in the IMM equations by its expected value. In the case of the second measure, for a model we have... 

B.E.T. equation for multimolecular adsorption depend on the ability of a diatomic molecule to rotate on the surface. Kemball (8, 9) measured entropies of substances physically adsorbed on mercury and found a range of freedem from complete mobility for acetone to complete immobility with water. 

The entropy of fusion in cal deg-1 g-1 is easily computed from the data of Table 3 by dividing the heats of fusion by the absolute temperature of the melting point. For theoretical reasons in many cases it is interesting to know the entropy of fusion at constant volume. Thus, the total measured entropy of fusion at constant pressure and composition, ASf, must be corrected for the gain in entropy due to the volume increase on fusion. The fusion process may be imagined as occurring at... 

Although Equation (4.1) is useful, particularly for isentropic changes of state for which dS = 0, the entropy appears as an independent variable. This circumstance is inconvenient from an experimental point of view, because no meters or devices exist that measure entropy. A new function in which the temperature is an independent variable rather than the entropy is obtained by subtracting... 

The net entropy is the difference between the measured entropy and the magnetic entropy. 

The Adam-Gibbs equation (4-10) can be tested directly by using the calorimetrically measured entropy difference AS to compute the temperature-dependence of the relaxation time, with B then being a fitting parameter. This has been done, for example, with the data for o-terphenyl shown in Fig. 4-11, and the predicted temperature-dependence of the viscosity is found to be in qualitative, but not quantitative, agreement with the measured viscosity (see, for example. Fig 4-12). The main reason for the failure in Fig. 4-12 is that the temperature Tj at which the entropy extrapolates to zero for o-terphenyl lies below the VFTH temperature Tq required to fit the viscosity data hence the predicted viscosity does not vary as rapidly with temperature as it should. 

Claasen et al. ( ), observed two of the six vibrational levels In the vapor, and calculated the other four levels fro normal coordinate treatment of the molecule. They also obtained a measured entropy of BrF (g), from low temperature data... 

Skochdopole, Griff el, and Spedding 310) have compared measured entropies for the rare earths with theoretically predicted values. Although they do not predict a value for europium, they believe it is somewhat higher than its immediate periodic table neighbors. On this basis, we adopt a value of 17 e. u. for the entropy of europium at 298 K. Spedding and Daane 314) remark that europium is the most volatile of the rare earths. Landolt-Bornstein 208) report available spectroscopic terms from which we have calculated the thermodynamic properties of the ideal monatomic gas. The remaining data listed for this element are estimated and are consistent with the above known facts. These data are intended for use only until measured values become available. 

Let, in compliance with the solution of Gibbs Paradox, the integration constant S0 be the (change of) entropy AS which is added to the entropy a to figure out the measured entropy Sclaus of the equilibrium state of the system A (the final state of Gay-Lussac experiment) at a temperature 0. We have shown that without such correction, the less entropy o is evidenced,... 

We conclude by that, the diminishing of the measured entropy value about AS against S awaited, evidenced by Gibbs Paradox, does not originate in a watched system itself. Understood this way, it is a contradiction of a gnozeologic character based on not respecting real properties of any observation [6]. And, this means the following. 

As a result of a knowledge of the electronic density of states, it is then possible to assess the electronic contribution to the entropy. An example of these results is given in fig. 6.12 which shows not only the computed electronic entropies for Ti and Zr (the solid lines) but also the experimental values for these entropies as obtained by taking the measured total entropy and subtracting off the contributions due to vibrations. In addition to their ability to shed light on the entropies themselves, these results were also used to examine the relative importance of vibrations and electronic excitations to the hcp-bcc structural transformation in these metals. Unlike in the case of simple metals, it is found that the electronic contribution to the free energy difference is every bit as important as the vibrational terms. In particular, in the case of Ti, it is claimed that the measured entropy change... 

Availability and its related measure entropy become objective properties of matter and cease to represent the extent of knowledge of observers. For example, a single particle has an entropy associated with its state in the same sense that it has energy and momentum associated with that state. 

Neptunium. Here again the earlier work was compromised by the presence of NpO(g), which at the onset produces an NpO/Np ratio in the gas of about 7. The more recent work of Ackermann and Rauh (26, 27) gives excellent precision and second/third law agreement. The measured entropy for the solid of 12.06 is also plotted in Fig. 4, and reflects the even smaller radius of the orthorhombic room-temperature phase. 

It is known that the calorimetrically measured entropy is less than the entropy obtained from the spectroscopic data by 0.81 e.u. This is due to the residual entropy. The calculated entropy of liquid is compared with the spectroscopic entropy which may be evaluated from the difference between the spectroscopic entropy of gas and the entropy of vaporization. 

In order to discriminate between localized and mobile guest molecules, one can consider also the measured entropy of the sorbate and compare this quantity with the values obtained from theory for both models. For an energetically homogeneous sorbent, the calculation of the entropy of the sorbate is quite straightforward in principle. 

No evidence of any transition is shown below the melting temperature. In additirai, the measured entropy of fusion, ASj, is that expected for full orientational and positional disordering (see Table 3.1). It is thus clear fltat the torsional libration about the C3 axis must lead to occasional jumps into nei boiing symmetry positions that reach the NMR frequency of 10-100 kHz at about 115 K, M K below the melting temperature. Such rather large-scale jump-motion does not disturb the crystal symme-... 

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