Isothermal section

Sing (see Ref. 207 and earlier papers) developed a modification of the de Boer r-plot idea. The latter rests on the observation of a characteristic isotherm (Section XVII-9), that is, on the conclusion that the adsorption isotherm is independent of the adsorbent in the multilayer region. Sing recognized that there were differences for different adsorbents, and used an appropriate standard isotherm for each system, the standard isotherm being for a nonporous adsorbent of composition similar to that of the porous one being studied. He then defined a quantity = n/nx)s where nx is the amount adsorbed by the nonporous reference material at the selected P/P. The values are used to correct pore radii for multilayer adsorption in much the same manner as with de Boer. Lecloux and Pirard [208] have discussed further the use of standard isotherms. 

Eig. 3. Isothermal section at 23 3°C of the system MgO-MgCl2 H2 0 showing phases in equiHbrium with vapor phases in sealed containers (65). 

Fig. 26. Isothermal section of Al—Cu—Li system at 350 °C (adapted from ref. 43) a, isothermal section b, limits of soHd solubility for a-phase. SoHd lines...

It was shown some time ago that one can also use a similar thermodynamic approach to explain and/or predict the composition dependence of the potential of electrodes in ternary systems [22-25], This followed from the development of the analysis methodology for the determination of the stability windows of electrolyte phases in ternary systems [26]. In these cases, one uses isothermal sections of ternary phase diagrams, the so-called Gibbs triangles, upon which to plot compositions. In ternary systems, the Gibbs Phase Rule tells us... 

Of special interest are the isothermal sections at T4 and Ts, which are at temperatures above the critical temperature of component 1, but below the... 

Figure 2. Formation of ternary borides and phase equilibria within ternary boride systems of the type M-M-B or M-Y-B (M = metal, Y = honmetal). , complete isothermal section established B, part of a diagram only. Numbers in the lower part of each square correspond to the refs, to the ternary section. The number of ternary compounds observed is indicated in the right upper corner of each square.

No compound formation observed see isothermal section from Fig. 1, 6.7.2. 

The number of CrB representatives is highly increased by ternary compounds observed in the isothermal sections M -Mo-B and My-W-B just below the transformation T of the CrB-type high-T modifications of MoB and WB into low-T a-MoB or -WB. These ternary compounds therefore represent a stabilization of the high-T form of MoB and WB, respectively, toward lower T by statistical Mo(W)/metal substitution. Transition T (T ) range for WB from 2100°C (W rich) to 2180°C (B... 

Despite the occurrence of binary AIB2 borides (see also Fig. 2), no ternary representatives are known (Mn, Mo)B2 found from isothermal sections is a stabilized high-T phase by conversion to lower T by a statistical ( ) metal-metal substitution. Both MnB2 and M0B2 are high-T compounds stable above 1075°C and 1517°C respectively WB2 is claimed but is either metastable or impurity stabilized. Similar examples are observed with (W, Pd>2B5 (M02B5 type) as well as (Mo, Rh),, (B3 and (W, Ni), B3 (Mo,., 83 type). The phase relations in the B-rich section of the Mo(W)-B binaries, however, are not known precisely. 

Fig. 21 Three-dimensional representation of a ternary system of two enantiomers in a solvent, S. One of the faces of the prism (at left) corresponds to the binary diagram of D and L (here a conglomerate). Shaded area isothermal section representing the solubility diagram at temperature T0. (Reproduced with permission of the copyright owner, John Wiley and Sons, Inc., New York, from Ref. 141, p. 169.)...

Figure 4.17 Isothermal sections of the ternary phase diagram A-B-C shown in Figure 4.16 at (a) 650 °C and (b) 450 °C [17]. Here L denotes liq. Reprinted with permission of The American Ceramic Society, www.ceramics.org. Copyright [1984], All rights reserved.

For ternary systems with complex phase behaviour in the solid state it is more convenient to use only isothermal sections. This is shown for two temperatures for the ternary system Ti-Si-C in Figure 4.19 [10]. In this system several binary and ternary intermediate phases are stable, and the system is divided into several ternary sub-systems. Tielines for two-phase equilibria are also shown in the two isothermal sections. 

A simple example of a real ternary diagram is shown in Fig. 2.26, where the isothermal section, determined at 200°C, of the Al-Bi-Sb system is shown together with the relevant binary diagrams Al-Bi showing a miscibility gap in the liquid state and complete insolubility in the solid state, Bi-Sb with complete mutual... 

Figure 2.26. Isothermal section of the Al-Bi-Sh phase diagram at 200°C. In the triangle marked hy the asterisk, three phases (coincident with the two elements A1 and Bi together with the compound AlSh) are observed. In the other triangle two-phase alloys are formed. A few tie-lines are shown. The alloy marked hy , for instance, contains the compound AlSb together with a...

Figure 2.27. Isothermal section at 307°C of the Al-Zn-Si diagram. The boundary binary systems are shown. The isothermal section at 307°C is marked on the binary Al-Zn diagram. The corresponding single-phase (thick segment) and two-phase regions are indicated in the base edge of the triangle. By additions of Si (immiscible in the solid state in the other two elements) two- and three-phase fields are formed. ( ) = three-phase region. In the two-phase region on the left examples of tie-lines are presented.

The Al-Zn-Si is another example of simple ternary system its isothermal section at 307°C is shown in Fig. 2.27 together with its boundary binaries. Si, in the solid state, is practically insoluble in A1 or in Zn and in their binary solutions. In the... 

Figure 2.29. Isothermal sections of ternary phase diagrams (a) Al-Er-Mg system at 400°C, Saccone et al. (2002) and, (b) Al-Cu-Ti system at 540°C from Villars et al. (1995). A number of single-phase regions (dark grey) may be noticed, both extending from binary compounds and as ternary intermediate phases (r) in the Al-Er-Mg system and the four phases Tj t2 t3 and t4 in the Al-Cu-Ti system. The three-phase fields are marked by an asterisk, in the Al-Er-Mg system a few tie-lines are indicated in the two-phase fields.

As an example of more complex systems and descriptions, the Ni-Mg system is shown in Fig. 2.32 (adapted from Levinsky 1997). In (a) an isobaric section of the diagram is shown (a low pressure has been considered in order to have a certain extension of the gas phase which consists essentially of Mg vapour). In Fig 2.32(b) there is an isothermal section of the diagram at 700°C. Notice, for different values of pressure, the change in the sequence of phases stable at different compositions. A value of the pressure close to atmosphere is approached at the top of the figure. In Fig 2.32(c) the usual Tlx diagram is shown. This can be considered an isobaric phase diagram if pressure is relatively low but still higher than the sum of the equilibrium partial pressures of the components. 

Figure 2.31. Schematic representation of the P/T equilibria in a simple two-component system (forming continuous solid and liquid solutions). In (a) a perspective view of the P-T-X diagram is shown in (b) its projection on the P/T plane. Notice the two single-component systems represented, for instance, for the component B by the three lines SB/G (sublimation line of B representing the gas/so lid equilibrium), SB/LB (melting equilibrium of B) and the boiling line LB/G. The solid solution is indicated by a. Notice in (a) the isobaric and isothermal sections of the diagrams (compare with Fig. 2.1).

Figure 2.32. (a) Ni-Mg system. Isobaric section of the diagram at p = 103 Pa (from Levinsky 1997). (b) Ni-Mg system. Isothermal section of the diagram at 700°C (from Levinsky 1997). (c) Ni-Mg system. Tlx diagram at a pressure nearly constant and equal to the ambient. 

Figure 2.33. Ni-Co-O phase diagram (isothermal section at 1600 K). log p02 (oxygen partial pressure) is plotted against the molar fraction in the metallic alloy. The metallic alloy, (Ni, Co) solid solution is stable in (1) and the mixed oxide (Ni, Co)0 solid solution in (3). In the intermediate region (2) we have coexistence of alloy and oxide. For the value of the partial oxygen pressure corresponding to y, within the two-phase field, we will have the alloy of composition xt in equilibrium with an oxide containing the two metals in the ratio x2-...

Notice that in order to have a higher coherency among the different systems, we selected the temperatures of the different isothermal sections not on the basis of absolute values but as reduced values of the average melting temperatures (even if the reduction was made in a rather arbitrary way) defining, for a Me-X alloy ... 

Figure 4.42. Mo alloys multi-diagram. Notice that along the vertical axis, from Ta to Pt, the sequence of the isothermal sections Ta-W, W-Re, Re-Os, etc. is shown. In the same axis an approximate representation is also suggested for the other binary combinations (Ta-Re Ta-Os, Ta-Ir, etc., W-Os, W-Ir, etc.). A confirmation of the (even partial) validity of this representation is given by the phase sequences observed in the W-Re and Re-Os systems in comparison with that of the W-Os system. (For the symbols see Fig.4.41).

In order to compute the binding isotherm (Section 2.1) of any system, one must know all the microstates of the system. This cannot be done for even the smallest binding system. However, in order to understand the origin of cooperativity and the mechanism by which ligands cooperate, it is sufficient to consider simple models having only a few macrostates. This understanding will be helpful for the selection of methods to extract information from experimental data, and for the meaningful interpretation of this information. 

J The concept of counter-phases. When a stable compound penetrates from a binary into a ternary system, it may extend right across the system or exhibit only limited solubility for the third element. In the latter case, any characterisation also requires thermodynamic parameters to be available for the equivalent metastable compound in one of the other binaries. These are known as counter-phases. Figure 6.16 shows an isothermal section across the Fe-Mo-B system (Pan 1992) which involves such extensions for the binary borides. In the absence of any other guide-... 

Figure 9.15. Isothermal section in die temaiy system A-B-C diowing tie-lines in die various two-phase fields and their intersection to form a three-phase field,...

In a ternary isothermal section a similar procedure is used where an alloy is stepped such that its composition remains in a two-phase field. The three-phase field is now exactly defined by the composition of the phases in equilibrium and this also provides the limiting binary tie-lines which can used as start points for calculating the next two-phase equilibrium. 

In vertical sections through ternary and higher-order systems or isothermal sections for quaternary and higher systems, the position becomes more complex as tie-lines do not lie in the plane of the diagram. They therefore cannot be used to define the positions of phase boundaries and the procedures described above become inoperative. New concepts are required, such as viewing the diagram in a... 

Figure 10.2 Calinilated ternary isothermal sections for (a) Ni-Mo-Re at 1000°C (Saundeis 1997c), (b) Ca0-Mg0-AI203 at 2000 K (Hallstedt 1992) and (c) isotheimal section of Ti-Al-Nb at IOOO°C (Saundeis 1996a).

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