Diagonal diagrams

Figure Al.6.18. Liouville space lattice representation in one-to-one correspondence with the diagrams in figure A1.6.17. Interactions of the density matrix with the field from the left (right) is signified by a vertical (liorizontal) step. The advantage to the Liouville lattice representation is that populations are clearly identified as diagonal lattice points, while coherences are off-diagonal points. This allows innnediate identification of the processes subject to population decay processes (adapted from [37]).

Fig. 21. Single absorption equilibrium-stage diagram where the equiUbrium curve is for 8% SO2, 12.9% the diagonal lines represent the adiabatic temperature rise of the process gas within each converter pass the horizontal lines represent gas cooling between passes, where no appreciable conversion...

Three types of binary equilibrium cui ves are shown in Fig. 13-27. The y-x diagram is almost always plotted for the component that is the more volatile (denoted by the subscript 1) in the region where distillation is to take place. Cui ve A shows the most usual case, in which component 1 remains more volatile over the entire composition range. Cui ve B is typical of many systems (ethanol-water, for example) in which the component that is more volatile at lowvalues of X becomes less volatile than the other component at high values of X. The vapor and liquid compositions are identical for the homogeneous azeotrope where cui ve B crosses the 45° diagonal. A heterogeneous azeotrope is formed with two liquid phases by cui ve C,... 

An operating line can be located on the y-x diagram if (1) two points on the hne are known or (2) one point and the slope are known. The known points on an operating line are usually its intersection with the y-x diagonal and/or its intersection with another operating line. 

Figure 2.12. Two-dimensional (2D-)INADEQUATE diagram of 1-butanol (12) [ CDajzCO, 95% v/v, 25°C, 50 MHz, 128 scans, 128 experiments], (a) Contour plot with the AB systems of bonded C atoms on the horizontal axis (b) plots of the three AB systems of the molecule obtained from (a) (c) contour plot of the symmetrised INADEQUATE experiment showing the AB or AX systems of bonded C atoms in the HH COSY format (cross signals on the axes perpendicular to the diagonal)...

Let us now turn to the surfaces themselves to learn the kinds of kinetic information they contain. First observe that the potential energy surface of Fig. 5-2 is drawn to be symmetrical about the 45° diagonal. This is the type of surface to be expected for a symmetrical reaction like H -I- H2 = H2 -h H, in which the reactants and products are identical. The corresponding reaction coordinate diagram in Fig. 5-3, therefore, shows the reactants and products having the same stability (energy) and the transition state appearing at precisely the midpoint of the reaction coordinate. 

O Driscoll, et al. (11) have pointed out that the sign and magnitude of the vertical distance, V] - X], between the composition curve and diagonal on the cc diagram are measures of the direction and degree of drift of the instantaneous cc, (y)inst V]. It can be shown that 3 is an equivalent measure of drift. In fact, the two measures are related via... 

Figure 5.27 Schematic diagram of a COSY spectrum illustrating how intrinsically different cross-peaks are from diagonal peaks. The circles represent the cross-peaks in an AB coupled system. The diagonal peaks have dispersion shape, and the crosspeaks have an absorption shape though they alternate in sense. The diagonal peaks are centered at d 8,i) and (8 , 8 ) the cross-peaks appear at (8., 8 ) and (8 , 8 ).

The top and bottom operating lines intersect the diagonal at xd and xb respectively mark these points on the diagram. 

Thus, by knowing aAB from vapor-liquid equilibrium and by specifying xA, A can be calculated. Figure 4.3a also shows a typical vapor-liquid equilibrium pair, where the mole fraction of benzene in the liquid phase is 0.4 and that in the vapor phase is 0.62. A diagonal line across the x-y diagram represents equal vapor and liquid compositions. The phase equilibrium behavior shows a curve above the diagonal line. This indicates that benzene has a higher concentration in the vapor phase than toluene, that is,... 

Solution To determine the location of the azeotrope for a specified pressure, the liquid composition has to be varied and a bubble-point calculation performed at each liquid composition until a composition is identified, whereby X = y,-. Alternatively, the vapor composition could be varied and a dew-point calculation performed at each vapor composition. Either way, this requires iteration. Figure 4.5 shows the x—y diagram for the 2-propanol-water system. This was obtained by carrying out a bubble-point calculation at different values of the liquid composition. The point where the x—y plot crosses the diagonal line gives the azeotropic composition. A more direct search for the azeotropic composition can be carried out for such a binary system in a spreadsheet by varying T and x simultaneously and by solving the objective function (see Section 3.9) ... 

On an x-y diagram for Component i this is a straight line starting at the distillate composition with slope R/(R + 1) and which intersects the diagonal line at x,. ... 

FIGURE 8.3 Diagram of the miniature PHSS and respirometer system. The respirometer chamber (left) is equipped with three sensors, the PHSS and PNOS in the chamber stopper and the POS diagonal to the chamber bottom, and has a minimum volume of 2 ml. Tissue is held on a support just above the stir bar for adequate perfusion. The miniature PHSS (right) is approximately 5 mm long and 2 mm outside diameter. The anode is polished at the tip, the cathode is coiled around the insulated anode, the H2S-permeable membrane is cemented to the stainless steel sleeve, and the 30 pL electrolyte volume covers both anode and cathode (after [41]). 

Main sequence star The majority of stars, 92 per cent, that fall on the leading diagonal of the Herzprung-Russell diagram. 

Fig. 5.19. Evolutionary track in the HR diagram of an AGB model of total mass 0.6 Mq, initial composition (Y, Z) = (0.25, 0.001 Z /20). Heavy dots marked 2 to 11 indicate the start of a series of thermal pulses (see Fig. 5.20), which lead to excursions along the steep diagonal lines. Numbers along the horizontal and descending track indicate times in years relative to the moment when an ionized planetary nebula appears and (in parentheses) the mass of the envelope in units of Mq. R = 0.0285 indicates a line of constant radius (R in solar units) corresponding to the white-dwarf sequence. Shaded areas represent earlier evolutionary stages for stars with initial masses 3,5 and 7 Mq and the steep broken line marks the high-temperature boundary of the instability strip in which stars pulsate in their fundamental mode. The y-axis gives log L/Lq. Adapted from Iben and Renzini (1983).

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