Ordered closest packing

Alloys of lead and thallium have a structure based upon cubic closest packing from 0 to about 87-5 atomic percent thallium. The variation of the lattice constant with composition gives strong indication that ordered structures PbTl, and PbTl, exist. In the intermediate ranges, solid solutions of the types Pb(Pb,Tl)a and Pb(Pb,Tl)TlB exist. Interpretation of interatomic distances indicates that thallium atoms present in low concentration in lead assume the same valence as lead, about 2-14, and that the valence of thallium increases with increase in the mole fraction of thallium present, having the same value, about 2-50, in PbTls and PbTl, as in pure thallium. A theory of the structure of the alloys is presented which explains the observed phase diagram,... 

An important parameter for surface reactivity is the density of atoms in the surface. The general rule of thumb is that the more open the surface, the more reactive it is. We return to this effect in much more detail in Chapter 6. Note that (110) is the most open basal plane of an fee crystal, whereas (111) exhibits the closest packing. For bcc crystals the order is the opposite, i.e. (Ill) is the most open and (110) the most packed. 

Figure 9.32 The densest ordered packings of monospheres hep (a) and fee (b) octahedral (c) and tetrahedral (d) PBU/C closest packing ((a) or (b)), prismoidal PBU/C for packing with /7p=4.0 (e).

V, is the molar volume of polymer or solvent, as appropriate, and the concentration is in mass per unit volume. It can be seen from Equation (2.42) that the interaction term changes with the square of the polymer concentration but more importantly for our discussion is the implications of the value of x- When x = 0.5 we are left with the van t Hoff expression which describes the osmotic pressure of an ideal polymer solution. A sol vent/temperature condition that yields this result is known as the 0-condition. For example, the 0-temperature for poly(styrene) in cyclohexane is 311.5 K. At this temperature, the poly(styrene) molecule is at its closest to a random coil configuration because its conformation is unperturbed by specific solvent effects. If x is greater than 0.5 we have a poor solvent for our polymer and the coil will collapse. At x values less than 0.5 we have the polymer in a good solvent and the conformation will be expanded in order to pack as many solvent molecules around each chain segment as possible. A 0-condition is often used when determining the molecular weight of a polymer by measurement of the concentration dependence of viscosity, for example, but solution polymers are invariably used in better than 0-conditions. 

The suggestion that in metal crystals the atoms are arranged in closest packing was made by Barlow before the development of the x-ray technique, in order to account for the observations that many metals crystallize with cubic or hexagonal symmetry and that in the latter case many of the observed values of the axial ratio lie near the ideal value 2y/2/ v 3 = 1.633 for hexagonal closest packing. 

The famous Heusler type structure can be considered as a ternary ordered variant of the BiF3 type. On the other hand, the YPd2Pb structure can also be derived from a cubic closest packing of the lead atoms, where the octahedral voids are filled with yttrium (rocksalt substructure) and the tetrahedral voids are filled by the palladium atoms. 

The limitations of the adsorption method are those implied in the basic assumptions, namely the existence of a monomolecular layer as above determined, and the arrangement of closest packing assumed. However, results thus far obtained appear to warrant the assumptions made. There can be no question but that the method gives the correct order of surface measured. As in the solubility method previously described, the surface measured by the Emmett-Brunauer technique measures the surface at all indentations and infinitesimal cracks present. It possesses a marked advantage. over other methods since it can fee used with sponge-like particles possessing infinite labyrinth-like structures and is independent of chemical activity. 

In order to determine the thermodynamic properties by means of the perturbation theory, the thermodynamic properties of the reference system are needed. Here, the expressions for the equation of state and the radial distribution function of a system of hard spheres are included for both the fluid and solid reference states. A face-centred-eubic arrangement of the particles at closest packing is assumed for the solid phase. 

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