Tangent common

Next suppose AS , and AH , are both positive. In this case these two partially offset one another, and a plot of AGj resembling that shown in Fig. 8.2b may result. We are particularly interested in the two minima in this curve and the hump between them. A common tangent can always be drawn to two such minima so the above discussion shows that the minima at points P and Q in Fig. 8.2b each have the same values of AjUi and A/i2. Since AjUj is simply the difference between juj and its value for the pure component, the chemical potential for each component is seen to have the same value for both solution P and solution Q in Fig. 8.2b. 

Transition pressures were determined by equating enthalpies of different polymorphs via the common tangent construction. We found that ratile transforms to columbite structure at 11.8 GPa with a small volume change of 3%. These results agree well with experimental... 

Therefore this partial differential represents mathematically the tangent to any AG-concentration curve. In our case for a common tangent to AG, and AG, we must have not only a common slope... 

In Fig. 7.83 using the common tangent construction the equilibrium compositions of phase I-phase II at their boundary are found, from the points of contact, to be respectively Xg — A, and Xg = X2. 

Fig. 7.85 Common tangent method applied to AC" ref The value of the free...

Between oxygen pressures of 2-8 x 10" and 8 x 10 atm the outermost oxide layer will consist of the MO phase. Its exact surface composition can be predicted by using the common tangent in the same manner as described for MO. From Fig. 7.84 it is clear that, as the oxygen pressure in the atmosphere is reduced the composition of the surface oxide layer will vary in a predictable manner from Xq = 0-55 to Xq = 0-475. 

If two different states can exist permanently in contact, the points representing these states on the thermodynamic model have a common tangent plane. 

Again, the limiting position of the line joining points of the connodal curve and the direction of the common tangent to the connodal and spinodal curves at the plait point is given by ... 

Figure 4.2 Gibbs energy curves for the liquid and solid solution in the binary system Si-Ge at 1500 K. (a) A common tangent construction showing the compositions of the two phases in equilibrium, (b) Tangents at compositions that do not give two phases in equilibrium. Thermodynamic data are taken from reference [2],...

At an even lower temperature, T, a sample in equilibrium will consist of the crystalline phase h-Y2C>3(ss), MgO(ss) or a two-phase mixture of these (see Figure 4.8(f)). The compositions of the two phases in equilibrium are again given by the common tangent construction. 

Fig. 14.9 Total energy curves for seven structures of Si as a function of volume normalized to the experimental volume. The dashed line is the common tangent between the diamond and the p-tin phase.

Graphically, the conditions for thermodynamic equilibrium are equal to two points which have a common tangent. These points give the composition of a polymer-rich phase (I) and a solvent-rich phase (II) that can coexist in thermodynamic equilibrium. The summation of such points is also called the coexistence curve or binodal line. 

Note Binodal compositions are defined by pairs of points on the curve of Gibbs energy of mixing vs. composition that have common tangents, corresponding to compositions of equal chemical potentials of each of the two components in two phases. 

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