Alternate representations

Alternative representations of stream temperature and energy have been proposed. Perhaps the best known is the heat-content diagram, which represents each stream as an area on a graph (3) where the vertical scale is temperature, and the horizontal is heat capacity times flow rate. Sometimes this latter quantity is called capacity rate. The stream area, ie, capacity rate times temperature change, represents the enthalpy change of the stream. 

Figure 9.11 Alternative representations of the layer structures of (a) kaolinite, (b) pyrophyllite, and (c) talc. (After H. J. Emeleus and J. S. Anderson, 1960 and B. Mason and L. G. Berry, 1968.)...

Figure 9.13 Alternative representations of muscovite and chlorite (after B. Mason and 1.. G. Berry ...

Figure 22.4 Alternative representations of (a) infinite chains of vanadium atoms in VF5, (b) tetrameric structures of NbFs and TaFs, and (c) dimeric structure of MX5 (M = Mb, Ta X = Cl, Br).

Figure 22.5 Alternative representations of (a) the sheet structure of NbF4 and (b) the chain structure of MX4 (M = Nb, Ta X = Cl, Br, I) showing the displacement of the metal atoms which leads to diamagnetism.

Figure 23.8 Trinuclear. M-.M bonded species of Mo" and W. (a) (b) and (c) are alternative representations of the -MjOij unit (a) emphasizes its relationship to the edge-sharing octahedra of the Mj group in polymetallate ions (b) shows the (rrj-O) (rrr-Olj bridges and M-M bonds of its MiOs incomplete cubane" core and (c) emphasizes its triangular centre by viewing from the unoccupied comer of the cuboid (d) and (e) offer the same perspective as (c) but of (Mj02(02CR)6(H20)3] and (Mjirrj-X)(rti-OR)(OR)9] stmetures respectively.

An alternative representation is useful when data on mean molal heat capacities Cj (T) are available with 25 C as the reference temperature (T ). Then the equation... 

Alternative representations of benzene. The "circle" representation must be used carefully since it doesn t indicate the number of tt electrons in the ring. 

K r,f,r) = e r e " ° r) which neglects all quantum effects arising from the noncommutativity of the operators and v. In order to appreciate the nature of the approximation, let us consider the case where the energy potential v(r) = v + A r, with v and A constant quantities. Although the QMP for a particle subjected to a constant force is one of the few cases explicitly known [32], for our convenience we adopt the following exact alternative representation of the propagator for a particle moving in a linear potential [see Appendix A, eq. (A.8)]... 

Generally, a chemical plant is composed of several units with several streams and components. The set of material and energy balances constitutes a set of linear/nonlinear equations, which can be represented by an undirected graph. However, when the number of units and streams is large, the graphical representation becomes cumbersome. An alternative representation of the topological structure of the balances in a chemical process is achieved using the occurrence matrix. 

Fig. 7.10. Alternate representation of crystal structures of the a- and (3-forms of resorcinol, as viewed down the c-axis (001 plane). The figure is adapted from data given in Refs. [24] and [25].

Bicyclic systems such as trithiapentalene 61 or thia-aza-pentalenes 62 (Scheme 30) show significant charge separation. Alternative representations for the above systems include nonclassical structures 61b,c and 62c with S(IV). Density functional calculations showed that the stability and aromaticity of bis-... 

Considering now the simple structure of CsCl as an illustration, we see that the crystallographic description reported in Table 3.2 corresponds to the atom arrangement shown, with alternative representations, in Fig. 3.8. More generally, from... 

Carbodiphosphoranes of type 89 [209] have been known for many years. The bonding parameters and the reactivity indicate that the electronic situation in these compounds is reasonably described by the polar structure 89 with two electron pairs at the central carbon atom, although alternative representations (89" and 89 ") are also possible (Fig. 28) [210]. 

Fig. 1 also helps in writing down the four-dimensional sphere parametrizations corresponding to the alternative representations for Fock projection. 

As is well known, conventional hydrogenoid spherical orbitals are strictly linked to tetradimensional harmonics when the atomic orbitals for the tridimensional hydrogen atom are considered in momentum space. We have therefore studied an alternative representation, providing the Stark and Zeeman basis sets, related to the spherical one by orthogonal transformation, see eqs. (12) and (15). The latter can also be interpreted as suitable timber coefficients relating different tree structures of hyperspherical harmonics for R (Fig. 1). 

Many other methods, of course, take advantage of invariances with respect to orbital transformations to obtain alternative representations of wavefunctions, such as for example the commonly employed localization procedures for the doubly-occupied MOs from Hartree-Fock calculations [11-15]. We give here a brief account of procedures that particularly seek a valence bond representation of MO wavefunctions. 

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