Same plane

Figure 2-53. Tolane rotamers a) one with the two aromatic rings in the same plane, and b) one with the two rings perpendicular to each other.

The product of matrix operators is an operator. For example, rotation through 90°, followed by another rotation through 90° in the same direction and in the same plane, is the same as one rotation through 180°... 

The equivalence of the pairs of Cartesian coordinate displacements is a result of the fact that the displacement vectors are connected by the point group operations of the C2v group. In particular, reflection of Axr through the yz plane produces - Axr, and reflection of AyL through this same plane yields AyR. 

A regioselective aldol condensation described by Biichi succeeds for sterical reasons (G. Biichi, 1968). If one treats the diaidehyde given below with acid, both possible enols are probably formed in a reversible reaaion. Only compound A, however, is found as a product, since in B the interaction between the enol and ester groups which are in the same plane hinders the cyclization. BOchi used acid catalysis instead of the usual base catalysis. This is often advisable, when sterical hindrance may be important. It works, because the addition of a proton or a Lewis acid to a carbonyl oxygen acidifies the neighbouring CH-bonds. 

FIGURE 2 16 sp Hybridization (a) Electron configuration of carbon in its most stable state (b) Mixing the s orbital with two of the three p orbitals generates three sp hybrid orbitals and leaves one of the 2p orbitals untouched The axes of the three sp orbitals lie in the same plane and make angles of 120 with one another... 

FIGURE 4 9 tert Butyl cation (a) The positively charged carbon is sp hybridized Each methyl group IS attached to the positively charged carbon by a cr bond and these three bonds he in the same plane (b) The sp hybridized car bon has an empty 2p orbital the axis of which is perpen dicular to the plane of the carbon atoms... 

Although both stereoisomers yield 4 tert butylcyclohexene as the only alkene they do so at quite different rates The cis isomer reacts over 500 times faster than the trans The difference in reaction rate results from different degrees of rr bond develop ment in the E2 transition state Since rr overlap of p orbitals requires their axes to be parallel rr bond formation is best achieved when the four atoms of the H—C—C—X unit he in the same plane at the transition state The two conformations that permit this are termed syn coplanar and anti coplanar... 

Generally polymers involve bonding of the most substituted carbon of one monomeric unit to the least substituted carbon atom of the adjacent unit in a head-to-tail arrangement. Substituents appear on alternate carbon atoms. Tacticity refers to the configuration of substituents relative to the backbone axis. In an isotactic arrangement, substituents are on the same plane of the backbone axis that is, the configuration at each chiral center is identical. 

The disks are assumed to lie in the same plane. While this picture is implausible for bulk crystallization, it makes sense for crystals grown in ultrathin films, adjacent to surfaces, and in stretched samples. A similar mathematical formalism can be developed for spherical growth and the disk can be regarded as a cross section of this. 

Extensional flows occur when fluid deformation is the result of a stretching motion. Extensional viscosity is related to the stress required for the stretching. This stress is necessary to increase the normalized distance between two material entities in the same plane when the separation is s and the relative velocity is ds/dt. The deformation rate is the extensional strain rate, which is given by equation 13 (108) ... 

Compared with elbow fittings, bends with a centerhne radius of three or five nominal pipe diameters save the cost of joints and reduce pressure drop. Such bends are not suited for instaUation in a bank of pipes of unequal size when the bends are in the same plane as the bank. 

When both solids and gases pass through the distributor, such as in catalytic-cracldng units, a number of variations are or have been used, such as concentric rings in the same plane, with the annuli open (Fig. 17-9a) concentric rings in the form of a cone (Fig. 17-9b) grids of T bars or other structural shapes (Fig. 17-9c) flat metal perforated plates supported or reinforced with structural members (Fig. 17-9d) dished and perforated plates concave both upward and downward (Fig. 17-9e and f). The last two forms are generally more economical. 

The Coriolis veclor lies in the same plane as the velocity vector and is perpendicular to the rotation vector. If the rotation of the reference frame is anticlockwise, then the Coriolis acceleration is directed 90° clockwise from the velocity vector, and vice versa when the frame rotates clockwise. The Coriolis acceleration distorts the trajectory of the body as it moves rectilinearly in the rotating frame. 

This is similar to (a) except that now they are in the same plane and are not one below the other. Although heat dissipation would be slightly better than (b), this too is not a recommended configuration. 

To calculate the sectional modulus (or moment of resistance) of the four bus sections in parallel we have multiplied the sectional modulus of one bus by 4. This is a simple method when the busbars of each phase are in the same plane and equally spaced as in Figure 28.33(a) with no additional spacers between them to hold them together. 

Inlet and diseharge pressures are defined as the stagnation pressures at the inlet and diseharge, whieh are the sum of statie and veloeity pressures at the eorresponding points. Statie pressures should be measured at four stations in the same plane of the pipe as shown in the piping arrangements. Veloeity pressure, when less than 5% of the pressure rise, ean be eomputed by the formula... 

The CHA is shown in schematic cross-section in Fig. 2.5 [2.5]. Two hemispheres of radii ri (inner) and T2 (outer) are positioned concentrically. Potentials -Vi and -V2 are applied to the inner and outer hemispheres, respectively, with V2 greater than Vi. The source S and the focus E are in the same plane as the center of curvature, and Tq is the radius of the equipotential surface between the hemispheres. If electrons of energy E = eVo are injected at S along the equipotential surface, they will be focused at Eif ... 

Whereas the spot positions carry information about the size of the surface unit cell, the shapes and widths of the spots, i.e. the spot profiles, are influenced by the long range arrangement and order of the unit cells at the surface. If vertical displacements (steps, facets) of the surface unit cells are involved, the spot profiles change as a function of electron energy. If all surface unit cells are in the same plane (within the transfer width of the LEED optics), the spot profile is constant with energy. 

Step 3 Calculate the distance from the center of the flame to a position with coordinates x and z, representing personnel or equipment exposed to radiation from the flame. (Since the limiting case is the one in which the flame is blown over in the direction of the exposed personnel or equipment, they are in the same plane and a two-coordinate system is adequate for describing their relative locations). Note also that all wind directions must be considered, even in locations which have a prevailing wind direction. 

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