Identical units

W L Bragg [7] observed that if a crystal was composed of copies of identical unit cells, it could then be divided in many ways into slabs with parallel, plane faces whose distributions of scattering matter were identical and that if the pathlengths travelled by waves reflected from successive, parallel planes differed by integral multiples of the... 

The otiier type of noncrystalline solid was discovered in the 1980s in certain rapidly cooled alloy systems. D Shechtman and coworkers [15] observed electron diffraction patterns with sharp spots with fivefold rotational synnnetry, a syimnetry that had been, until that time, assumed to be impossible. It is easy to show that it is impossible to fill two- or tliree-dimensional space with identical objects that have rotational symmetries of orders other than two, tliree, four or six, and it had been assumed that the long-range periodicity necessary to produce a diffraction pattern with sharp spots could only exist in materials made by the stacking of identical unit cells. The materials that produced these diffraction patterns, but clearly could not be crystals, became known as quasicrystals. 

In all of these stmctures the atomic positions are fixed by the space group syimnetry and it is only necessary to detennine which of a small set of choices of positions best fits the data. According to the theory of space groups, all stmctures composed of identical unit cells repeated hi three dimensions must confomi to one of 230 groups tliat are fomied by coinbinmg one of 14 distinct Bmvais lattices with other syimnetry operations. 

Compounds of the type RO—Y—OR, where the two parent compounds are identical and contain a group having priority over ethers for citation as suffix, are named as assemblies of identical units. For example, HOOC—CH2—O—CH2CH2—O—CH2—COOH is named 2,2 -(ethylene-dioxy)diacetic acid. 

For large capacitor banks using a number of small identical units per phase the configuration of the units must be such that a fault or a failure in one unit of a phase should not result in an overvoltage of more than 10% across the remaining healthy units in that phase. This requirement becomes more specific when the capacitor units are used... 

The structure of the LH2 complex of R. acidophila is both simple and elegant (Figure 12.17). It is a ring of nine identical units, each containing an a and a P polypeptide of 53 and 41 residues, respectively, which both span the membrane once as a helices (Figure 12.18). The two polypeptides bind a total of three chlorophyll molecules and two carotenoids. The nine heterodimeric units form a hollow cylinder with the a chains forming the inner wall and the P chains the outer wall. The hole in the middle of the cylinder is empty, except for lipid molecules from the membrane. 

Small spherical viruses have a protein shell around their nucleic acid that is constructed according to icosahedral symmetry. Objects with icosahedral symmetry have 60 identical units related by fivefold, threefold, and twofold symmetry axes. Each such unit can accommodate one or severed polypeptide chains. Hence, virus shells are built up from multiples of 60 polypeptide chains. To preserve quasi-equivalent symmetry when packing subunits into the shell, only certain multiples (T = 1, 3, 4, 7...) are allowed. 

Figure 18.1 A crystal is built up from many billions of small identical units, or unit cells. These unit cells are packed against each other in three dimensions much as identical boxes are packed and stored in a warehouse. The unit cell may contain one or more than one molecule. Although the number of molecules per unit cell is always the same for all the unit cells of a single crystal, it may vary between different crystal forms of the same protein. The diagram shows in two dimensions several identical unit cells, each containing two objects packed against each other. The two objects within each unit cell are related by twofold symmetry to illustrate that each unit cell in a protein cr) stal can contain several molecules that are related by symmetry to each other. (The pattern is adapted from a Japanese stencil of unknown origin from the nineteenth century.)...

The number and complexity of unit processes and in turn unit operations comprising a water purification or wastewater treatment facility are functions of the legal and operational requirements of the treated water, the nature and degree of contamination of the incoming water (raw water to the plant), and the quantities of water to be processed. This means then, that water treatment facilities from a design and operational standpoints vary, but they do rely on overlapping and even identical unit processes. 

In addition to tlie applications cited above, tlie Poisson distribution can be used to obtain tlie reliability of a standby redundancy system, in which one unit is in tlie operating mode and n identical units are in standby mode. Unlike a... 

The effort required to specify the materials schedule for a new plant or to check a design very much depends on how much experience there is of similar or identical units in operation. Factors such as process conditions and raw material sources are taken into consideration before extrapolating the experience of another unit. 

Increased demand for softened MU, perhaps caused by an increase in plant capacity without increasing softener capacity installed, or the use of additional steam for processing so that the return of condensate is diminished. Here a larger plant is required. If the existing plant is not too old, a second identical unit usually is installed and the equipment is operated as duty/standby, with water-meter countdown crossover and immediate regeneration of the out-of-service unit. 

We can generalize this expression to write that for N identical units Z = -A . 

Summary. In summary, when modeling with the fugacity concept, all equilibria can be treated by Z values (one for each compartment) and all reaction, advection and transport processes can be treated by D values. The only other quantities requiring definition are compartment volumes and emission rates or initial concentrations. A major advantage is that since all D quantities are in equivalent units they can be compared directly and the dominant processes identified. By converting diverse processes such as volatilization, sediment deposition, fish uptake and stream flow into identical units, their relative importance can be established directly and easily. Further, algebraic manipulation... 

Polymers that are built from the repetition of identical "repeat units" are called "homopolymers" (from ancient Greek "d/iog — same).2 Linear homopolymer chains are obtained upon linking chemically identical units exclusively at both ends. However, repeat units are not always symmetrical in the chain direction. Depending on the orientation of the repeat unit, different microstructural... 

Due to either accidental or intentional events during the synthesis, branching points can occur along the chain (2D microstructure). Although this needs the presence at the branching point of, stricto sensu, a "non-identical" unit, these macromolecules remain classified as homopolymers. 

Unlike crystals that are packed with identical unit cells in 3D space, aperiodic crystals lack such units. So far, aperiodic crystals include not only quasiperiodic crystals, but also crystals in which incommensurable modulations or intergrowth structures (or composites) occur [14], That is to say, quasiperiodicity is only one of the aperiodicities. So what is quasiperiodicity Simply speaking, a structure is classified to be quasiperiodic if it is aperiodic and exhibits self-similarity upon inflation and deflation by tau (x = 1.618, the golden mean). By this, one recognizes the fact that objects with perfect fivefold symmetry can exist in the 3D space however, no 3D space groups are available to build or to interpret such structures. 

In this problem, the actual yield is the amount of product found by the scientist (45.2 g IF5) therefore, we need the theoretical yield to finish the problem. Since our actual yield has the units g IF5, our theoretical yield must have identical units. To determine the grams of IF5 from the moles of limiting reactant, we need the molar mass of IF5 [126.9 g I/mol 1 + 5 (19.00 g F/mol F) = 221.9 g nymol]. 

A All the units in the percent yield calculation, except %, must cancel. This requires that the yields in both the numerator and the denominator must have identical units including the same chemical species. 

In the first case, along the direction of the diagonal of the cubic cell, there is a sequence ABC of identical unit slabs ( minimal sandwiches ), each composed of two superimposed triangular nets of Zn and S atoms, respectively. The thickness of the slabs, which include the Zn and S atom nets, is 0.25 of the lattice period along the superimposition direction (that is along the cubic cell diagonal aj3). It is (0.25,3 X 541) pm = 234 pm. In the wurtzite structure there is a sequence BC of similar slabs formed by sandwiches of the same triangular nets of Zn and S atoms. Their thickness is —0.37 X c = 0.37 X 626.1pm = 232 pm). 

From now on we use only the GPF for the cyclic system and drop the subscript C. Since our system has m identical units, the sites will always be identical in the weak sense. There is always one intrinsic constant for the first site but, in general, we have more than one pair correlation, triplet correlation, etc. As in Section 7.1 we develop, for simplicity, the case of two states/= 2, but most of the results are quite general. 

It is common to find vapour pressures quoted in millimetres (mm) mercury (Hg) in older papers, although sometimes the identical unit Torr is cited instead. In both cases, the conversion to the SI unit, the Pascal (Pa), is simply ... 

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