Perfectly ordered crystal

Calculate the entropy of a tiny solid made up of four diatomic molecules of a compound such as carbon monoxide, CO, at T = 0 when (a) the four molecules have formed a perfectly ordered crystal in which all molecules are aligned with their C atoms on the left (top-left image in Fig. 7.7) and (b) the four molecules lie in random orientations (but parallel, any of the images in Fig. 7.7). 

The so-called Boson peak is visible as a hump in the reduced DOS, g(E)IE (Fig. 9.39b), and is a measure of structural disorder, i.e., any deviation from the symmetry of the perfectly ordered crystal will lead to an excess vibrational contribution with respect to Debye behavior. The reduced DOS appears to be temperature-independent at low temperatures, becomes less pronounced with increasing temperature, and disappears at the glass-liquid transition. Thus, the significant part of modes constituting the Boson peak is clearly nonlocalized on FC. Instead, they represent the delocalized collective motions of the glasses with a correlation length of more than 20 A. 

The H-bonding in the anhydrous 1 Im (Table 24) has topologic properties (Fig. 46) similar to those in the alcohol coordinatoclathrates of 1 with 1 2 host guest stoichiometry (cf. Fig. 17 a). Assuming a perfectly ordered crystal lattice, the resulting central loop of H-bonds should appear to have homodromic directionality with the donor/acceptor functions separated in space. This contrasts to the behavior in the dihydrated l Im where no such characteristic loops are formed. Involvement of the C—H hydrogen atoms of the imidazole molecule, however, is similar in both cases. 

Many examples of Ruddleston-Popper phases have been synthesized. The structure of the first member of the series, corresponding to n = 1, is adopted by a number of compounds, including the important phase La2Cu04 (Section 4.3.3) and is often referred to as the K2NiF4 structure. In practice, synthesis of A +1B C)3 +i phases frequently results in disordered materials in which random or partly ordered regions of 100 faults occur, and particular efforts have to be made to produce perfectly ordered crystals. 

Unlike simple inorganic compounds (e.g., NaCl or KC1), polymers do not have a perfectly ordered crystal lattice formation and are not completely crystalline. In fact, they contain both crystalline and amorphous regions. Hence, the X-ray diffractions from them are found to be a mixture of sharp as well as diffused patterns. 

The first theoretical considerations concerning n (p) and G (p) of concentrated 3-D emulsions and foams were based on perfectly ordered crystals of droplets [4,5,15-18]. In such models, at a given volume fraction and applied shear strain, all droplets are assumed to be equally compressed, that is, to deform affinely under the applied shear thus all of them should have the same shape. Princen [15,16] initially analyzed an ordered monodisperse 2-D array of deformable cylinders and concluded that G = Qiox(p < (/), and that G jumps to nearly the 2-D Laplace pressure of the cylinders at the approach of ( > = 100%, following a ( — dependence. 

The concept of structural dynamics is clearly demonstrated in the VPO catalyst system. The high-resolution TEM in Figure 1.10 shows an activated VPO catalyst (no promoters, made by the alcohol route) that exhibits a typical [215, 225, 244] termination with little structural order supported on a perfectly ordered crystal consistent with the pyrophosphate structure (and with other structures of the VPO family [245, 246]). The lack of long-range order is not seen due to the operation... 

Returning to the example (Frame 16, section 16.4) of the solid CO crystal. If we consider firstly the (hypothetical) perfectly ordered crystal (shown schematically in Figure 17.1(a)) then since there can be only one arrangement in which the dipoles C —> O are always completely aligned and in which perfect symmetry is found, then W = 1 in equation (17.1) and in this case ... 

For perfectly ordered crystals at absolute zero, solutions to the Schrodinger equation can be calculated on fast computers using density functional theory (DFT) based on the self-consistent local density approximation (LDA) simplifying procedures using different basis functions include augmented... 

The constraint of a collision in a given sequence in our simple chain model means that there is a shock front propagating through the system, a front which reverses its direction every time an end atom collides with the hard walls. When a perfectly ordered crystal hits a hard wall, one can understand how a dispersion-free propagation of a shock wave is possible. The new feature is that such a shock front was seen in full MD simulations of impact heated clusters, using realistic forces, and has been recently studied in more detail. ... 

The displacements of atoms may take several forms (Figure 13.2). These have been described by Jack Dunitz, Verner Schomaker, and Kenneth N. Trueblood as follows The perfectly ordered crystal would have every atom firmly fixed to its own perfectly defined site in each unit cell for the entire period of observation. There are, however, various types of disorder from unit cell to unit cell. If the atom jumps to a different site, that is one kind of disorder [a mixture of static and dynamic disorder ] if it moves to and fro, that is another kind of disorder [ dynamic disorder] if it is forever in one site in a certain unit cell and in a different site in another cell, that is still another kind [static disorder]. Each of these types of vibrations, displacements, and disorder has somewhat similar effects on the intensities of Bragg reflections the effect they have in common is that they reduce these intensities by an amount that increases with increasing scattering angle, 26, as shown in Figure 13.1. 

Some of the experimental aspects of crystal structure analyses can now be applied to compounds that are not truly crystalline, or that are variants of the perfectly ordered crystals we have been describing so far. 

Entropy Absolute or third-law entropies (relative to a perfectly ordered crystal at 0 K) of a compound in its standard state S or of an... 

The conventionally chosen zeros for S and H are of course irrelevant provided they are consistent. The convention actually used is S = 0 for each element at 0 in the state of a perfectly ordered crystal and if = 0 for each element in its stable form at 25 °C. 

For samples taken up-dip of the Setif and Medjounes areas the values range from +2.9 to -5.8%o (Tables 7.2,7.3 Fig. 7.2). The light carbon makes itself felt in the carbonates where we find traces of recrystallization or of other mineralogical neoformations and in particular the appearance of mixed-layer clay minerals with perfectly ordered crystal structures. In the same samples microfissures are filled by secondary calcite or dolomite with detrital carbonate cement frequently being present between the crystals. In other cases we observe entire fields of neoformed minerals (dolomite... 

Aust.. Phys. 1993, 46, 651. This paper includes energy-surface calculations for all three alloys, assuming perfectly ordered crystal structures. 

O A simplified representation of the same perfect crystal at a temperature above 0 K. Vibrations of the individual molecules within the crystal cause some dipoles to be oriented in directions other than those in a perfect arrangement. The entropy of such a crystalline solid is greater than zero b ause there is disorder in the crystal. Many different arrangements are possible for such a disordered array, so there is a greater dispersal of energy than in the perfectly ordered crystal. 

The third law of thermodynamics concerns the entropy of perfectly-ordered crystals at zero kelvins. 

According to this principle, every substance (element or compound) in a pure, perfectly-ordered crystal at 0 K, at any pressure, has a molar entropy of zero ... 

Suppose we wish to evaluate the entropy of an amount n of a pure substance at a certain temperature T and a certain pressure. The same substance, in a perfectly-ordered crystal at zero kelvins and the same pressure, has an entropy of zero. The entropy at the temperature... 

If the substance in the state of interest is a hquid or gas, or a crystal of a different form than the perfectly-ordered crystal present at zero kelvins, the heating process will include one or more equilibrium phase transitions under conditions where two phases are in equilibrium at the same temperature and pressure (Sec. 2.2.2). For example, a reversible heating process at a pressure above the triple point that transforms the crystal at 0 K to a gas may involve transitions from one crystal form to another, and also melting and vaporization transitions. 

As explained in Sec. 6.1, by convention the zero of entropy of any substance refers to the pure, perfectly-ordered crystal at zero kelvins. In practice, experimental entropy values depart from this convention in two respects. First, an element is usually a mixture of two or more isotopes, so that the substance is not isotopically pure. Second, if any of the nuclei have spins, weak interactions between the nuclear spins in the crystal would cause the spin orientations to become ordered at a very low temperature. 

The neglect of these two effects results in a practical entropy scale, or conventional entropy scale, on which the crystal that is assigned an entropy of zero has randomly-mixed isotopes and randomly-oriented nuclear spins, but is pure and ordered in other respects. This is the scale that is used for published values of absolute third-law molar entropies. The shift of the zero away from a completely-pure and perfectly-ordered crystal introduces no inaccuracies into the calculated value of AS for any process occurring above 1 K, because the shift is the same in the initial and final states. 

The other substances listed in Table 6.1 have residual entropies that are greater than zero within the uncertainty of the data. What is the meaning of this discrepancy between the calorimetric and spectroscopic results We can assume that the true values of 5 at 298.15 K are the spectroscopic values, because their calculation assumes the solid has only one microstate at 0 K, with an entropy of zero, and takes into account aU of the possible accessible microstates of the ideal gas. The calorimetric values, on the other hand, are based on Eq. 6.2.2 which assumes the solid becomes a perfectly-ordered crystal as the temperature approaches 0 K. ... 

We obtain the same result if a species present in one phase is totally excluded from another. For example, solvent molecules of a solution are not found in a pure perfectly-ordered crystal of the solute, undissociated molecules of a volatile strong acid such as HCl can exist in a gas phase but not in aqueous solution, and ions of an electrolyte solute are usually not found in a gas phase. For each such species absent from a phase, there is one fewer amount variable and also one fewer relation for transfer equilibrium on balance, the number of independent variables is still 2-1-5. 

The perfectly ordered crystal is one with the lowest free enCTgy. Since a certain amount of lattice disorder can be tolerated at equilibrium, it does not necessarily represent the crystal with pafect intmial orda-. 

Here 5i,c is the conformation entropy of the liquid and Sc,c that of the crystal. For the perfectly ordered crystal 5c.c is approximately zero, so that (A5 ) will equal 5i c. When there is an element of disorder in the crystal, as exists for some polymers, a finite value needs to be assigned to 5c,c and the value of A5c will be correspondingly reduced. The values of A Sc, calculated in this manner are listed in the sixth column of Table 6.5. In order to better compare different polymers, ASc per bond is given in the fifth column of the table. Among all the polymers listed, all but five have ASc values per bond that are between 1.0 and 1.6 e.u. With only this small range in values of ASc per bond for most polymers, it is extremely difficult to quantitatively relate the change in conformational entropy to the melting temperature. 

PE is a negative number, and —PE is the energy required to take Avogadro s number molecules, at rest and in contact with each other in a perfectly ordered crystal, into the gas phase, at rest. The reason why the condition m < n appears is that the interaction between molecules m and n must not be counted twice, once under the m-to-n form, and once under the n-to-m form. By comparing equations 8.10 and 8.12, one sees that in 8.10 each U(m, n) is counted twice. Therefore ... 

For a gas of non-interacting point particles (structureless) the curve would be a parabola with its vertical axis and its apex at the origin. The probability of finding a second atom at a given distance from a first one is dependent only on the volume a able, and therefore varies with the square of the distance. For a perfectly ordered crystal, it consists of a... 

The peak width generated by the finite size of a perfectly ordered crystal does not increase with the peak order and allows the estimation of the crystal size D based on the Scheuer equation, which, when the peak width is measured as an integral... 

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