Triple intersections

Key features of Pourbaix diagrams are the points of intersection between the coexistence lines. In a simple diagram, three compounds of a dependent component can coexist at these points. Thus, if compounds i, j and k coexist at a point, 3 coexistence lines must radiate from the point, ij, jk and ik. The coordinates of potential triple intersection points can be determined by simultaneous solution of pairs of equations. For example the coordinates of the equilibrium point between sulphur, SOjj- and HS- are determined by solution of the equations... 

The results show the triple intersection is valid because x(1 1) x(1 3) and x(1 5) are higher than the remaining values of x-Thus coexistence lines 1,3, 1,5 and 3,5 do indeed radiate from pH = 7.618, pE = -U.36 5. Valid intersection points are stored together with values of i, j and k. 

Unfortunately, the activity-coefficient equations cannot conveniently be made explicit in terms of x, and the location of the constant activity curves on the triangular diagram is possible only by a lengthy series of interpolations. Location of the triple intersection points becomes an even more difficult trial-and-error procedure. While this can be done, for practical purposes use of the ternary activity-coefficient equations is ordinarily limited to cases where the solubility curve of the ternary liquid system is known. For such a situation, the calculations become relatively simple, since it is then merely necessary to compute activities of C along the solubility curve and to join equal values on opposite sides of the curve by the tie lines. 

A coincidence in time means that at least two special events occur at the same point in time, while the occurrence of two special events at the same value of concentrations represents a coincidence in value. For example, the maximum of a concentration dependence may coincide with the point at which this dependence intersects with another concentration dependence or the intersection between three concentration dependences may occur at the same moment in time providing a triple intersection. 

The triple intersection of Eig. 6.7 deserves special mentioning with respect to low-energy vibronic motion. It has been pointed out above that the surface topology near a degeneracy (conical or not) is not of central importance for the strength of the nonadiabatic coupling effects. Eor low-energy motion this shape does matter. 

The amorphous solids obtained by heating ionic polymers of aluminum phosphate are not hygroscopic to the degree that purely orthophosphate polymers are. They show little tendency to dissolve or absorb more than surface water from high humidity atmospheres. This is a result of the lattice of this solid, that is an infinite three-dimensional network of polyphosphates uniting at triple intersections of aluminum ion terminals. 

In the nonrelativistic case, at a given the quantity x was shown to be invariant under the hansformation in Eq. (16), for a = y = 0. This invariant, whose value depends on was used to systematically locate confluences, [18-21], intersection points at which two distinct branches of the conical intersection seam intersect. Here, we show that the scalar triple product, gij X is the invariant for q = 3. Since the g t, and h cannot be... 

To understand the conditions which control sublimation, it is necessary to study the solid - liquid - vapour equilibria. In Fig. 1,19, 1 (compare Fig. 1,10, 1) the curve T IF is the vapour pressure curve of the liquid (i.e., it represents the conditions of equilibrium, temperature and pressure, for a system of liquid and vapour), and TS is the vapour pressure curve of the solid (i.e., the conditions under which the vapour and solid are in equili-hrium). The two curves intersect at T at this point, known as the triple point, solid, liquid and vapour coexist. The curve TV represents the... 

Liquid helium-4 can exist in two different liquid phases liquid helium I, the normal liquid, and liquid helium II, the superfluid, since under certain conditions the latter fluid ac4s as if it had no viscosity. The phase transition between the two hquid phases is identified as the lambda line and where this transition intersects the vapor-pressure curve is designated as the lambda point. Thus, there is no triple point for this fluia as for other fluids. In fact, sohd helium can only exist under a pressure of 2.5 MPa or more. 

All five models for ethane show roughly the same information. The Wire model looks like a line formula in your chemistry textbook, except that all atoms, not just earbons, are found at the end of a line or at the intersection of lines. (The only exception occurs where three atoms lie on a line. Here, a Wire model will not show the exact position of the center atom.) The Wire model uses color to distinguish different atoms, and one, two and three lines to indicate single, double and triple bonds, respectively. 

The triple point is the location at which all three phases boundaries intersect. At the triple point (and only at the triple point), all three phases (solid, liquid, and gas) coexist in dynamic equilibrium. Below the triple point, the solid and gas phases are next-door neighbors, and the solid-to-gas phase transition occurs directly. 

For classes with fewer than four sites, the assertion is trivial. For chiral classes with four or more sites, there is at least one triple of sites which does not lie in a symmetry plane of the skeleton. For, if all sites lie in a common symmetry plane, molecules of the class with the ligands all different would possess planes of symmetry, i.e., the class would not be chiral. On the other hand, suppose that the sites do not lie all in a common mirror plane, but that nevertheless every triple of sites lies in a symmetry plane. It follows that every pair of sites lies on the intersection of two different symmetry planes, therefore on an axis of symmetry of the skeleton. But if more than four sites all lie pairwise on an axis of symmetry of a finite figure, they must all lie on a common axis, and the class is again achiral. For chiral classes, then, there is at least one triple of sites which does not lie on a plane of symmetry of the skeleton. Now consider a molecule in which the sites of this triple are occupied by ligands of three different kinds, the other sites by ligands different from these three, but identical with each other. Such a molecule is chiral, since the only improper operation which leaves the three different ligands invariant is a reflection in the plane of the triple, and this changes the rest of the molecule. The assertion follows immediately. 

The triple point of a substance is reached when the vapor pressure of the solid phase is equal to that of the liquid phase. If both solid and liquid are subjected to external pressure (which may be caused by capillary forces), their curves of vapor pressure versus temperature lie above those for uncompressed phases and intersect at a temperature different from the triple point. The melting point Tm observed at atmospheric pressure, as a rule, is very near to the triple point. Thus the freezing temperature Tmr of a drop of radius r should be different from Tm. 

An exceptional case of a very different type is provided by helium [15], for which the third law is valid despite the fact that He remains a liquid at 0 K. A phase diagram for helium is shown in Figure 11.5. In this case, in contrast to other substances, the solid-liquid equilibrium line at high pressures does not continue downward at low pressures until it meets the hquid-vapor pressure curve to intersect at a triple point. Rather, the sohd-hquid equilibrium line takes an unusual turn toward the horizontal as the temperature drops to near 2 K. This change is caused by a surprising... 

The dispersive (+ n, - m ) mode has already been seen clearly with the duMond diagrams, Figure 2.10. Here, the curves are no longer identical and the crystals must be displaced from the parallel position in order to get simultaneous diffraction. As the crystals are displaced, so the band of intersection moves up and down the curve. When the curves become very different, the K 1 and K 2 intensities are traced out separately. Then the peaks are resolved in the rocking curve, and if no better beam conditioner is available it is important in such cases to remove the K 2 component with a slit placed after the beam conditioner. A slit placed in front of the detector, with the detector driven at twice the angular speed of the specimen, also works very well. This is in effect a low resolution triple-axis measurement. 

Look back at the large phase diagram (Figure 7-1) and notice the intersection of the three lines at 0.01° and 6 X 10 atm. Only at this triple point can the solid, liquid, and vapor states of FljO all coexist. Now find the point at 374° C and 218 atm where the liquid/gas boundary terminates. This critical point is the highest temperature and highest pressure at which there is a difference between liquid and gas states. At either a temperature or a pressure over the critical point, only a single fluid state exists, and there is a smooth transition from a dense, liquid-like fluid to a tenuous, gas-like fluid. 

Invariant behavior occurs at the intersection of three univariant curves. This intersection defines a point at which three phases are in equilibrium. At these so called triple or invariant points, there are no degrees of freedom and both temperature and pressure assume fixed values. 

The phase diagram also illustrates why some substances which melt at normal pressure, will sublime at a lower pressure the line p = Pa intersects at Tg the locus OR of the points defining the solid-vapour equilibrium, i.e. at the pressure pj, the substance will sublime at the temperature T. Sometimes the opposite behaviour is observed, namely that a substance which sublimes at normal pressure will melt in a vacuum system under its own vapour pressure This is a non-equilibrium phenomenon and occurs if the substance is heated so rapidly that its vapour pressure rises above that of the triple point this happens quite frequently with aluminium bromide and with iodine. 

The point of intersection of I, R M is known as the triple point, TP. The resulting existence of the above three waves, causes a density discontinuity. The surface of this discontinuity, known as slipstream, S, represents a stream line for the flow relative to the intersection. Between this and the reflecting surface is the region of high pressure, known as Mach region here the pressure is approx twice that behind the incident wave. The top of this pressure region, the triple point, travels away from the reflected surface. As pressure and impulse appear to have their maximum values just above and below the triple point, respectively, the region of maximum blast effect is approximately that of the triple point... 

Oblique or transverse shocks lead to the formation of triple shock configurations, and to an increase in the temp and pressure behind the oblique shocks over that obtainable behind smooth shocks. This, in turn, makes for more favorable conditions for the initiation of further reaction. At triple wave intersections, conditions for reaction are particularly favorable. Thus the wave front becomes a complex three-dimensional cellular structure, quite different from the classical one-dimensional uniform wave front... 

Any given type of matter has a unique combination of pressure and temperature at the intersection of all three states. This pressure-temperature combination is called the triple point At the triple point, all three phases coexist. In the case of good old H2O, going to the triple point would produce boiling ice water. Take a moment to bask in the weirdness. 

As discussed for CO2 + n-alkane systems at carbon numbers n<24 the three-phase curve hhg ends a low temperature in a quadruple point s2l2lig. This is shown schematically in Figure 2.2-9a and b. In the quadrupel point three other three-phase curves terminate. The s2hh curve runs steeply to high pressure and ends in a critical endpoint where this curve intersects the critical curve. The s2l2g curve runs to the triple point of pure component B and the s/l/g curve runs to lower temperature and ends at low temperature in a second quadruple point s2silig (not shown). 

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