Three dimensional diagram

Fig. 10.10 Three-dimensional diagram of steady states I and HI are domains of existence of single solution, II is a domain of existence of three solutions (two stable and one unstable). Boundary surfaces correspond to two stable solutions. Reprinted from Yarin et al. (2002) with permission...

The valence bonds of carbon have bxed directions and are equidistant in space, pointing from the center to the corners of a tetrahedron forming an angle of 109°. Thus, in propane, which we usually write as CH3-CH2 CH3, the carbon atoms are not connected in a straight line, but are actually as shown in the above three-dimensional diagram. 

The melting temperature of A is higher them that of B. Therefore, the melting temperature of a drops as the composition becomes richer in B. At specific temperatures on the diagram (see 1. 2.), a two-phase system appears, that of a Uquid plus that of a. Finally, as the temperature rises, the melt is homogenous and the solid, a, has melted. In the three-phase system, only the relationship between A, B and C can be illustrated on a two-dimensioncil drawing. A three-dimensional diagram would be required to show the effect of temperature as well. 

Other diagrams could be devised indicating the influence of pH, electrolyte concentration, etc., on the appearance of one or two phases. Three-dimensional diagrams might also be of great aid. 

Figure 24 shows the ternary phase diagram (solubility isotherm) of an unsolvated conglomerate that consists of physical mixtures of the two enantiomers that are capable of forming a racemic eutectic mixture. It corresponds to an isothermal (horizontal) cross section of the three-dimensional diagram shown in Fig. 21. Examples include A-acetyl-leucine in acetone, adrenaline in water, and methadone in water (each at 25°C) [141].

The activated complex must be described in terms of three intemuclear distances, i.e. RMi, Rm and RAC or in terms of two intemuclear distances and angle between the bonds. Thus, in order to plot energy against three parameters, a four dimensional diagram would be necessary. However, if only a linear activated complex (A-B-C ) is considered, only two distances are involved (because RAC = RMi + RhC) and a three dimensional diagram is adequate to describe the energy variation with intemuclear distances as shown in Fig. 2.5. 

Sketch a three-dimensional diagram of methanol. Hint Recall that the shape around an oxygen atom is bent. 

Figure 4,7 The h-pH-c three dimensional diagram for flotation of galena in the presence of xanthate (1 - floatability area 2-non floatability area)...

The present model has three parameters other than the residence time. These are k2, / , and ku. Figure 6.18 showed the division of the k2 P0 plane into different regions corresponding to unique, isola, and mushroom patterns for the case ku — 0. Mapping out the full k2-P0-ku would require a three-dimensional diagram, which would (i) be difficult to draw accurately and (ii) not be easy to read quantitative results from. Instead, Figs 6.20 and 6.21... 

In the three-dimensional diagram, the curve formed by the intersection of the two surfaces represents all of the equilibrium points of the two-phase system. Such a curve is obtained for each type of a two-phase equilibrium existing in a single-component system. At any triple point of the system three such curves meet at a point, giving the temperature and pressure of the triple point. 

These methods of standardization are suitable if interactions between the features have to be interpreted. One hundred and more cases or objects are shown as points in two-or three-dimensional diagrams. Similarities between objects can be demonstrated as clusters in a two- or three-feature space. 

Figure E9.1. Three-dimensional diagram of the bubble size variation as a function of the gas velocity and the operating pressure (Hf = 1 m calculated from Eq. (E9.1)).

Construction of a complete diagram which represents all these variables would require a four-dimensional space. However, if the pressure is assumed constant (customarily at 1 atm), the system can be represented by a three dimensional diagram with three independent variables, i.e., temperature and two composition variables. In plotting three dimensional diagrams, it is customary that the compositions are represented by triangular coordinates in a horizontal plane and the temperature in a vertical axis. 

In each case the probability density may be represented in spherical coordinates in the form of a three-dimensional diagram (see Fig. 2.2) in which the angles 0, ip correspond to the arguments of the function pb(0,ip), whilst the value of the function is plotted along the radius-vector r(0, ip). 

Draw a three-dimensional diagram, including any lone pairs of electrons. 

The Woodward-Hoffmann rules give a more satisfying description and we shall follow the routine outlined for cycloadditions. Note that for stage 3, we can use the three-dimensional diagram we have already made. 

Different from low molar mass organic samples, where single molecules are to be determined, for complex synthetic polymers the analytical task is the determination of a distributed property. The molecular heterogeneity of a certain complex polymer can be presented either in a three-dimensional diagram or a... 

The adsorbed amount of cyanide is proportional to the clay, humic acid, and iron content. It is difficult to tell which soil component is mainly responsible for cyanide adsorption because, in the case of examined samples, clay, humus, and Fe content increase simultaneously. More informative is the three-dimensional diagram, in which the amount of adsorbed cyanide versus clay number and humus content is plotted (Figure 3.9). 

Equilibria in the CaO-AljOj-SOj-HjO system were studied at 25 C by Jones (J20) and at 20°C by d Ans and Eick (D24). Eollowing Brown (B91), the latters results are shown on a three-dimensional diagram in which the... 

Here 6d, <5p, and <5h are the contributions to the solubility parameter from dispersive forces, dipole-dipole forces, and hydrogen bonding forces, respectively. Since the three forces can occur to varying degrees in different components and can be represented on a three dimensional diagram, this theory is termed the three-dimensional solubility parameter. Barton (1983, 1990) tabulates the contributions to the three dimensional solubility parameter for a variety of solvents and polymers. 

The equation of state (1.3) can be depicted in a three-dimensional diagram vith the three thermodynamic properties pressure p, temperature T, and specific volume v as axes. Two-dimensional projections with the third variable as a parameter are widely used to explain thermodynamic processes. 

Because of the complexity of Fig. 12.1, the detailed characteristics of binary VLB are usually depicted by two-dimensional graphs that display what is seen on various planes that cut the three-dimensional diagram. The three principal planes, each perpendicular to one of the coordinate axes, are illustrated in Fig. 12.1. Thus a vertical plane perpendicular to the temperature axis is outlined as ALBDEA. The lines on this plane represent a Pxy phase diagram at constant T, of which we have already seen examples in Figs. 10.1, 11.7, 11.9, and 11.11. If the lines from several such planes are projected on a single parallel plane, a diagram like Fig. 12.2 is obtained. It shows Pxy plots for three different temperatures. The one for represents the section of Fig. 12.1 indicated by ALBDEA. 

When N = 2, the phase mle becomes F = 4 — n. Since there must be at least one phase (n= 1), the maximum number of phase-mle variables which must be specified to fix the intensive state of the system is three namely, P, T, and one mole (or mass) fraction. All equilibrium states of the system can therefore be represented in three-dimensional P-T-composition space. Within this space, the states of pairs of phases coexisting at equilibrium (F=4 — 2 = 2) define surfaces. A schematic three-dimensional diagram illustrating these surfaces for VLE is shown in Fig. 10.1. 

It is always difficult to draw on paper a three-dimensional diagram, so generally the zig-zag arrangement of the carbon chain is simplified to a straight line, or what is called a projection formula. 

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