Entropy evaluation

The configurational entropy evaluated above omits all mention of electron spin. Because there are two spin directions, a spin degeneracy of 2, these must be included in the possible configurations. When this term is included, the Heikes equation becomes the Chaikin-Beni equation ... 

Entropy evaluations from published cryothermal data on the lanthanide (III) oxides are summarized in Table II with an indication of the lowest temperature of the measurements and the estimated magnetic entropy increments below this temperature. Their original assignment of crystalline field levels from thermal data still appears to be in good accord with recent findings e.g., 17). Unfortunately, measurements on these substances were made only down to about 8°K. because the finely divided oxide samples tend to absorb the helium gas utilized to enhance thermal equilibration between sample and calorimeter. 

Figure 1. Correlation of lattice energy for A+B salts with formula unit volume. Table I. Entropy evaluations for dissociable fluoro-complexes...

Born Treatment. Several attempts have been made to evaluate absolute ionic entropies by means of the Born equation (eqn. 2.11.18) or the modified Born equation. The method consists mainly of employing the temperature coefficient of the ionic free energies of solvation as discussed in sect. 2.11.2 to obtain the entropy of solvation. The entropy of the individual gaseous ions can be calculated by the methods also discussed in sect. 2.11.2 and consequently ionic entropies evaluated. 

By a similar procedure, evaluate the standard molar entropy, the standard molar entropy of formation, and the standard molar Gibbs energy of formation of crystalline silver chloride at 298.15 K. You need the following standard molar entropies evaluated from spectroscopic and calorimetric data ... 

Thus, the given technique of entropy evaluation in a separate business structure allows establishing the norms of acceptable deviations from the most probable value of random processes in it. 

An important application of the Lipari-Szabo order parameters derived from relaxation studies is to estimate the conformational entropy. Genheden and co-workers discussed the issue of conformational entropy and order parameters using long MD simulations for several proteins as a starting point. They found that the order parameters and conformational entropies calculated over 10-100 ns windows were typically well-behaved on a per-residue level, while the total conformational entropy evaluated as the sum over residues, was more difficult to estimate. Kasinath et al. probed the microscopic origin of the link between conformational dynamics and conformational entropy using MD simulations for a number of proteins. They demonstrated that the motions of methyl-bearing side chains were sufficiently coupled to those of other side chains to he excellent reporters of the side chain conformational entropy. Marsh proposed the use of relative accessible surface area in monomeric proteins as an indicator of conformational entropy/flexibility. 

Table I. Entropy evaluations for dissociable fluoro-complexes...

For the construction of the functional in the above section, the choice to set the reference (the prior) density to that of a hypothetical noble gas atom, in analogy to Sanderson s electronegativity scale, was motivated and the particular choice lead to results which could be interpreted chemically. Following these findings one can see that it would be interesting to compare the information entropy, evaluated locally as... 

The thermodynamic values tabulated are derived from the full arsenal of methods available such as enthalpy or internal energy measurements by direct calorimetry, cryogenic heat capacity measurements from sufficiently low temperature to permit entropy evaluation at room temperature, as well as some... 

The standard molar entropy (S°) is the absolute entropy evaluated when one mole of a substance is in its standard state at a particular temperature. 

A novel approach for suppression of material noise in ultrasonic signals, based on noncoherent detector statistics and signal entropy, has been presented. Experimental evaluation of the technique, using ultrasonic images from samples with coarse material structure, has proven its high performance. 

The Shannon Equation Eq. (1)) [4] enables one to evaluate the information content, I (also known as the Shannon entropy), of the system. 

To characterize the state of the adsorbed phase, it is useful to evaluate its molar entropy, s , defined as the mean molar value for all the molecules adsorbed over the complete range of surface coverage up to the given amount adsorbed. The molar integral entropy of adsorption. As, is then defined as... 

Equation (3.7) gives a simple procedure for evaluating the entropy change accompanying a change of state. At the normal boiling point of a liquid, for example, the heat is absorbed reversibly and equals the heat of vaporization AH,. Since T is constant, the entropy of vaporization is AH,/T. For benzene, for example, AS, = (30.8 k J mol" )/353 = 87 J K mol. ... 

We express the calculated entropies of mixing in units of R. For ideal solutions the values of are evaluated directly from Eq. (8.28) ... 

Remember that the hump which causes the instability with respect to phase separation arises from an unfavorable AH considerations of configurational entropy alone favor mixing. Since AS is multiplied by T in the evaluation of AGj, we anticipate that as the temperature increases, curves like that shown in Fig. 8.2b will gradually smooth out and eventually pass over to the form shown in Fig. 8.2a. The temperature at which the wiggles in the curve finally vanish will be a critical temperature for this particular phase separation. We shall presently turn to the Flory-Huggins theory for some mathematical descriptions of this critical point. The following example reminds us of a similar problem encountered elsewhere in physical chemistry. 

The thermal conductivity of soHd iodine between 24.4 and 42.9°C has been found to remain practically constant at 0.004581 J/(cm-s-K) (33). Using the heat capacity data, the standard entropy of soHd iodine at 25°C has been evaluated as 116.81 J/ (mol-K), and that of the gaseous iodine at 25°C as 62.25 J/(mol-K), which compares satisfactorily with the 61.81 value calculated by statistical mechanics (34,35). 

More fundamental treatments of polymer solubihty go back to the lattice theory developed independentiy and almost simultaneously by Flory (13) and Huggins (14) in 1942. By imagining the solvent molecules and polymer chain segments to be distributed on a lattice, they statistically evaluated the entropy of solution. The enthalpy of solution was characterized by the Flory-Huggins interaction parameter, which is related to solubihty parameters by equation 5. For high molecular weight polymers in monomeric solvents, the Flory-Huggins solubihty criterion is X A 0.5. 

Substituting this result into equation 143, written for entropy, gives equation 148, where is evaluated at the mixture T and P ... 

The enthalpy and entropy are simple sums of the ideal gas and residual properties, which are evaluated separately. 

Similarly, the second-order derivative can be shown to be zero (see, Problems, Section 2.20). Evaluating the third term gives an expression for entropy gain along the Hugoniot... 

The temperature dependence of reaction rates permits evaluation of the enthalpy and entropy components of the free energy of activation. The terms in Eq. (4.4) corresponding to can be expressed as... 

Equation (5-43) has the practical advantage over Eq. (5-40) that the partition functions in (5-40) are difficult or impossible to evaluate, whereas the presence of the equilibrium constant in (5-43) permits us to introduce the well-developed ideas of thermodynamics into the kinetic problem. We define the quantities AG, A//, and A5 as, respectively, the standard free energy of activation, enthalpy of activation, and entropy of activation from thermodynamics we now can write... 

If the heat capacity can be evaluated at all temperatures between 0 K and the temperature of interest, an absolute entropy can be calculated. For biological processes, entropy changes are more useful than absolute entropies. The entropy change for a process can be calculated if the enthalpy change and free energy change are known. 

Where ortho effects and special entropy factors control the relation of the reaction rates, it seems more appropriate to evaluate relative activation from the energies of activation. 

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