Phase diagrams lever rule

Systems that exhibit this effect include Sn/Bi, Sn/In, and Sn-Pb. This effect is illustrated in Figure 4 for the Sn/Ag/Bi system with graphs depicting the solid fraction as function of temperature and composition based on the phase diagram lever rule and nonequilibrium solidification. For example, considering the composition Sn-3.5Ag-7.5Bi, the last liquid... 

From 160°C to room temperature. The lead-rich phase becomes unstable when the phase boundary at 160°C is crossed. It breaks down into two solid phases, with compositions given by the ends of the tie line through point 4. On further cooling the composition of the two solid phases changes as shown by the arrows each dissolves less of the other. A phase reaction takes place. The proportion of each phase is given by the lever rule. The compositions of each are read directly from the diagram (the ends of the tie lines). 

Thus, one could expect to find a droplet morphology at those quench conditions at which the equilibrium minority phase volume fraction (determined by the lever rule from the phase diagram) is lower than the percolation threshold. However, the time interval after which a disperse coarsening occurs would depend strongly on the quench conditions (Fig. 40), because the volume fraction of the minority phase approaches the equilibrium value very slowly at the late times. 

If two gas mixtures R and S are combined, the resulting mixture composition lies on a line connecting the points R and S on the flammability diagram. The location of the final mixture on the straight line depends on the relative moles in the mixtures combined If mixture S has more moles, the final mixture point will lie closer to point S. This is identical to the lever rule used for phase diagrams. 

The temperature-composition diagram can be used to calculate the composition of the two-phase system according to the amount of each solvent present. For example, at temperature T, the composition of the most abundant phase, which consists of liquid A saturated with liquid B, is represented by the point a and the composition of the minor phase, consisting of liquid B saturated with liquid A, is represented by point a. The horizontal line connecting these two points is known as a tie line as it links two phases that are in equilibrium with each other. From this line the relative amounts of the two phases at equilibrium can be calculated, using the lever rule, under the conditions described by the diagram. The lever rule gets its name from a similar rule that is used to relate two masses on a lever with their distances from a pivot, i.e. ... 

If more of one of the liquids is added, the effect, according to the lever rule, is to shift the point of the pivot until balance is regained. Thus, at the given temperature the composition of the phases remains the same, i.e. each saturated with the other liquid, but the relative amount alters if more of liquid B is added, then in the lever diagram the pivot point will shift to the right and thus more of this phase will form at the expense of the other phase (which is mainly liquid A). This rule applies to all partially miscible liquids. 

Note that the resulting fractional amounts are in weight percent, because the abscissa axis of the phase diagram reports the fractional weights of the two components (similar application of baricentric coordinates to a molar plot of type 7.2 would have resulted in molar fractions of phases in the system). Applying the lever rule at the various T, we may quantitatively follow the crystallization behavior of the system (i.e., atT = 1350 °C, = 0.333 and Xl = 0.666 atT = 1300... 

Apply the Lever Rule to a two-phase field in a binary phase diagram. 

Equations (2.40) and (2.41) are the lever rule and can be used to determine the relative amounts of each phase in any two-phase region of a binary component phase diagram. For the example under consideration, the amount of liquid present mrns out to be... 

Fig. 1.8 Lever rule in phase diagrams. In Fig. 1.7, the composition X5 shows two phase mixtures, the compositions of which are Xi (M(O) phase) and Xj (MO phase). The mixing ratio of the two phases equals (x2 — X5)/(x5 — Xj), (b), which is...

Figure 7.9 P-xB diagram (b) for a binary two-phase system (a), showing the compositions of coexisting vapor Oeap) and liquid (xgq) phases for a particular vapor-pressure value (dotted line), and the connecting tie-line (heavy solid line) that connects vapor and liquid compositions at this pressure. Varying amounts (rcvap, nhq) of the two phases correspond to different positions along the tie-line, as determined by the lever rule (see text).

These results express the fact that any linear combination of conserved densities (a generalized moment density) is itself a conserved density in thermodynamics. We have shown, therefore, that if the free energy of the system depends only on K moment densities p,... pK, we can view these as the densities of K quasi-species of particles and can construct the phase diagram via the usual construction of tangencies and the lever rule. Formally this has reduced the problem to finite dimensionality, although this is trivial... 

A comprehension of Figure4.3 has value because a similar phase diagram could be drawn for a natural gas of fixed composition between the quadruple points (Qi and QaJ. The same phase transitions and boundaries would qualitatively occur, with the artificial constraint that all hydrocarbon phases be of the same composition as the original gas. A second useful outcome of binary phase diagrams like Figure 4.3 is the use of the lever rule (Koretsky, 2004, p. 367) at constant temperature to determine relative phase amounts note that the lever rule can be applied for quantitatively correct phase diagrams. 

This process will continue till the temperature ultimately reaches the eutectic line. According to the phase diagram in Fig. 16-(a) below eutectic line temperature, assuming at 700° C solid state will consist two phases a and 3 whose proportionality will follow the lever rule that is, ab/ac = (3/a. 

Since the point M lies in the two-phase region of the triangular diagram, the term mixture applies only on a scale larger than the size of the droplets formed. The droplet dispersion formed by agitation has sufficient interfacial area (see Section I.C) for equilibrium to be reached quickly, so that point M represents the mean of the extract composition (point E) and the raffinate composition (point R) which are connected by the appropriate tie-line. A further application of the inverse lever rule permits calculation of the relative amounts of extract and raffinate. In this example, the material balance based on 1 kg of feed is summarized as follows ... 

The binodal branches do not coincide with the phase diagram axes. This means that the biopolymers are limitedly cosoluble. For instance, on mixing a protein solution A and a polysaccharide solution B a mixture of composition C can be obtained. This mixed solution spontaneously breaks down into two liquid phases, phase D and phase E. Phase D is rich in protein and E is rich in polysaccharide. These two liquid phases form a water-in-water (WIW) emulsion. Hie phase volume ratio is estimated by the inverse lever rule. The phase D/phase E volume ratio equals the ratio of the tieline segments EC/CD. Point F represents the phase separation threshold, that is, the minimal critical concentration of biopolymers required for phase separation to occur. 

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