Entropy derivatives

In Sections IVA, VA, and VI the nonequilibrium probability distribution is given in phase space for steady-state thermodynamic flows, mechanical work, and quantum systems, respectively. (The second entropy derived in Section II gives the probability of fluctuations in macrostates, and as such it represents the nonequilibrium analogue of thermodynamic fluctuation theory.) The present phase space distribution differs from the Yamada-Kawasaki distribution in that... 

The most likely state, which is denoted throughout by an overbar, is the one that maximizes the entropy. In this case the entropy derivative vanishes when the boundary temperatures of the subsystem equal that of the respective reservoirs,... 

DSC measurements with a microcalorimeter played a key role in tracing the origin of the step observed in the spin transition curve of [Fe(2-pic)3]-Cl2-EtOH [24]. The mixing entropy derived from the measured heat capacity data showed a significant reduction in the region of the step. This has been... 

First-order estimates of entropy are often based on the observation that heat capacities and thereby entropies of complex compounds often are well represented by summing in stoichiometric proportions the heat capacities or entropies of simpler chemical entities. Latimer [12] used entropies of elements and molecular groups to estimate the entropy of more complex compounds see Spencer for revised tabulated values [13]. Fyfe et al. [14] pointed out a correlation between entropy and molar volume and introduced a simple volume correction factor in their scheme for estimation of the entropy of complex oxides based on the entropy of binary oxides. The latter approach was further developed by Holland [15], who looked into the effect of volume on the vibrational entropy derived from the Einstein and Debye models. 

There is another entropy contribution which should always be included in a kinetic model, and that is the configurational entropy of an adlayer. This entropy derives from the various ways that adsorbates may be distributed over the sites. At high temperatures the adlayer will be disordered because this gives a high configurational entropy. At low temperatures the energy determines the structure, which is then most often ordered with a low entropy. 

To evaluate (dH/dP)T, we start from the expression (5.46b) for dH in terms of its natural variables S, P [or, equivalently, use the identity (1.13) to change the variable held constant from S to T, then use the Maxwell relation (5.49d) to replace the entropy derivative as follows ... 

It is of importance to note that, except for hydrogen and deuterium molecules, the entropy derived from heat capacity measurements, i.e., the thermal entropy, as it is frequently called, is equivalent to the practical entropy in other words, the nuclear spin contribution is not included in the former. The reason for this is that, down to the lowest temperatures at which measurements have been made, the nuclear spin does not affect the experimental values of the heat capacity used in the determination of entropy by the procedure based on the third law of thermodynamics ( 23b). Presumably if heat capacities could be measured right down to the absolute zero, a temperature would be reached at which the nuclear spin energy began to change and thus made a contribution to the heat capacity. The entropy derived from such data would presumably include the nuclear spin contribution of R In (2i + 1) for each atom. Special circumstances arise with molecular hydrogen and deuterium to which reference will be made below ( 24n). 

Show that although the partial molar heat content of the constituent of an ideal solution is independent of the composition ( 34a), this is not the case for the partial molar free energy and entropy. Derive expressions for (d/Lii/dN<)r p and (dSi/dNi)T,p for an ideal solution. 

It is seen that the general form for b derived above, taking into account both A and the entropy derivative /jlF, is not either identical with the experimental behavior at Hg where b = RT/p F- -K. The general case above is rather of the form R/ F/ T -I- K ) which is not in any way reducible to, or reconcilable with, the experimental Eq. (14) for b for the h.e.r. at Hg. 

Frenkel defects and impurity ions can diffuse through the silver halide lattice by a number of mechanisms. Silver ions can diffuse by a vacancy mechanism or by replacement processes such as collinear or noncollinear interstitialcy jump mechanisms [18]. The collinear interstitial mechanism is one in which an interstitial silver ion moves in a [111] direction, forcing an adjacent lattice silver ion into an interstitial position and replacing it The enthalpies and entropies derived from temperature-dependent ionic conductivity measurements for these processes are included in Table 4. The collinear interstitial mechanism is the most facile process at room temperature, but the other mechanisms are thought to contribute at higher temperatures. 

It now remains to evaluate the various entropy derivatives, so that the stability restrictions of Eqs. 7.2-7 can be put into a more usable form. Starting from... 

The uncertainty in the values of the entropy derived from thermochemical data usually does not exceed IJ mol and, hence, cannot practically affect the final results of the determination with the third-law method. If, however (in the absence of reliable data), one uses averaged values of known to within 9 J mol (see Sect.4.5) in the calculation, this factor can no longer be neglected, because the error in determination of the enthalpy becomes as high as 3%. Even in this case, however, this restriction can hardly be a serious obstacle to the use of this method. 

GENERIC tries to formulate a general time evolution equation by which the time evolution (derivative) of a state variable (which can be, e.g., mass density or fraction, momentum, energy) is determined by two potentials the total energy of the system and a dissipation function. Just the latter one introduces the irreversibility (and, in this way, the thermodynamics ) into consideration and description of the system behavior. The dissipation function or potential is a function of derivatives (with respect to the state variables) of a quantity which should have the physical meaning of the entropy of the system and this latter function is minimum at zero state variables, is zero at zero entropy derivatives just mentioned and a concave function. The general evolution equation can be reformulated by means of Poisson brackets. To apply the GENERIC formalism first one has to select suitable state variables for the problem or system which is to be modeled. The next step is to formulate... 

The remaining step is to express the entropy derivative in terms of P, V, T. This is provided by the Maxwell relationship in eg. fa.iQ ). Thus finally we have... 

Landauer R Inadequacy of entropy and entropy derivatives in characterizing steady-state. Phys RevA 1975, 12(2) 636-638. 

The H-F Eq. 18.6 has two parts the (j)-dependent configurational entropy derived from the lattice model without free volume and the enthalpic part taken from the Hildebrand s theory of regular solutions (Shinoda 1978 Reichart et al. 1997 Maranas et al. 1998). More recent version of Eq. 18.6 was used for the interpretation of SANS data, and it will be discussed in reference to the lattice cluster theory (LCT) (Freed and Dudowicz 2005). 

TABLE 1.3 Mean Force Constant, Mean Atomic Displacement, and Vibrational Entropy Derived from the... 

The researchers found that the rate constants for several intramolecular ester formations with a wide variety of E.M. values were directly correlated to the mole fraction of the reactants present as N ACs. The mole fractions were calculated using molecular dynamics simulations. When the ground state resides naturally in an NAC, then an E.M. of around 10 was achieved (the same value that we introduced earlier as the upper limit to proximity effects). To achieve an NAC, the reactant must be placed in a conformation that also has a higher enthalpy, because the reactants are within van der Waals distances. Therefore, the rate enhancement obtained by NAC formation is postulated to be also enthalpy derived, not solely entropy derived. 

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