Phase curve

FIG. 13-27 Typical binary eqiiilihriiim curves. Curve A, system with normal volatility. Curve B, system with homogeneous azeotrope (one liquid phase). Curve C, system with heterogeneous azeotrope (two liquid phases in eqiiilih-riiim with one vapor phase). 

To find values of gain and phase at a different frequeneies along a -20dB/decade gain slope and its associated phase curve ... 

The phase curve passes through —90° at a = Wn- Its shape depends upon ( and is obtained from the standard curves given in Figure 6.11. 

Flow parameter (Norton Co.) = F = FP Concentration of solute in liquid, lb mol solute/lb mol solute free solvent (or stream) Concentration of solute in liquid, in equiUbri-um -with the gas, lb mol solute/lb mol solvent Concentration of solute in liquid, mole fraction, or mol fraction of more volatile component in liquid phase Curve fit coefficients for C2, Table 9-32 Curve fit coefficients for Cg, Table 9-32 Concentration of solute in liquid in equilibrium -with gas, mol fracdon Concentradon of solute in gas, lb mol solute/lb mol solute free (solvent) (stream) Capacity parameter (Norton)... 

Fig.4.7. Photolysis of cetene in the layer adsorbed on a ZnO film (curve 1) and in the gas phase (curve 2) at room temperature.

Figure 3 Effect of seeding and inhibitors on aggregation reaction. The lag phase (curve c) is characteristic of reactions in which formation of nuclei for polymerization is an unfavorable process. Addition of preformed nuclei or seeds" (curve a) abolishes the lag phase. Inhibitors may affect the formation of nuclei and influence eitherthe lag phase, the extension of the nuclei changing the growth phase, or both (curve d). The inhibitor example (curve d) acts more strongly at nuclei formation than on the slope or plateau level of the growth phase.

Fig. 13 Energy profiles for C C, Cl in DMF. Curve a potential energy in the gas phase. Curve b potential energy in the solvent (D p = 62 meV). Curve c variation of the solvation free energy. Curve d solvent reorganization energy.

Does Surface Precipitation occur at Concentrations lower than those calculated from the Solubility Product As the theory of solid solutions (see Appendix 6.2) explains, the solubility of a constituent is greatly reduced when it becomes a minor constituent of a solid solution phase (curve b in Fig. 6.10).Thus, a solid species, e.g., M(OH)2 can precipitate at lower pH values in the presence of a hydrous oxide (as a solid solvent), than in its absence. 

G. GENERAL TRANSFER FUNCTIONS IN SERIES. The historical reason for the widespread use of Bode plots is that, before the use of computers, they made it possible to handle complex processes fairly easily. A complex transfer function can be broken down into its simple elements leads, lags, gains, deadtimes, etc. Then each of these is plotted on the same Bode plots. Finally the total complex transfer function is obtained by adding the individual log modulus curves and the individual phase curves at each value of frequency. 

Fig. 9.14 Flooding point diagram of centrifugal extractors (see Fig. 9.9). Curve 1 marks the lower limit, when the rotor is filled with heavy phase curve 2 marks an upper hmit, when the rotor is filled with light phase curve 3 marks the limit of the total throughput of both phases.

Fig. 12. Simulation of chemostat wiAout (curves 1) and with (e = 0.2, thus, E = 2.9 if M = 0) dispersed organic phase (curves 2 D = 0.2 h )...

Fig. 19. Partial pressure of Te2(g) along the CdTe(s) three-phase curve, upper left, and along the HgTe(s) three-phase curve plotted on a log scale versus 1000/T Uppermost line labeled P° is the vapor pressure of Te(l). Short line segments are for liquids with labeled atom fraction . Experimental points for CdTe are from Brebrick (1971b), those for HgTe are from Su et al. (1981) and Brebrick and Strauss (1965).

Fig. 21. Partial pressure of Cd along the three-phase curve for CdTe(s). Vapor pressure of Cd(l) is line labeled P d. Squares are from Lorenz (1962), circles from Brebrick (1971b), and triangles from de Nobel (1959).

Figures 24-26 show the partial pressures of Hg, Te2, and Cd along the three-phase curves for various solid solutions, Hgj xCdxTe(s). (We used the letter u in Parts II and IV to denote the composition of the solid solution and x for the atom fractions of the components in the liquid. In these figures and...

Fig. 24. Partial pressure of Hg along the three-phase curves for various solid solutions. The labels are the value of x in the formula, Hg, xCd,Te(s). Experimental points from Schwartz el al. (1981) and Tung et al. (1981b, 1982). 

Fig. 25. Partial pressure of Te2 along the three-phase curves for various solid solutions. Same credit source as for Fig. 24.

Fig. 40. Partial pressure of mercury in atmospheres as a function of 103 times the reciprocal absolute temperature along the three-phase curves for the liquids, (Hg . zCd2)yTei The value of z is shown near the bottom, -rich leg of each curve. The circles mark the pressure and temperature where =. ...

Fig. 42. Partial pressure of mercury in atm along the three-phase curve for the liquid (Hg0 7Cd0 3)JTe1 y is shown as a solid curve. The uppermost line gives the vapor pressure of pure mercury. Each cross along the three-phase curve marks the pressure and temperature where is equal to the value given near that cross. The dashed lines are calculated results for the liquids (Hg0 7Cd0.3),Te, = 0.4,0.5,0.6, and 0.7, for temperatures above the liquidus temperature. The solid symbols are experimental values (Steininger, 1976). Solid circles are for = 0.50 diamonds are for = 0.40.

Figure 42 shows the calculated values of PHf along the three-phase curve for the liquid (Hg0.7Cd0 as a solid curve. The numbers near the crosses on this curve give the corresponding value of y. The dashed curves are calculated for liquids (Hg0 Cdo Tej for = 0.40, 0.50, 0.60, and 0.70 at... 

Some Rate Constants and Activation Energies. When we apply Equation 8 to the initial slopes of the corrected liquid-phase curves in... 

Often the essentials of phase diagrams in P,7,x-space are represented in a P,7-projection. In this type of diagrams only non-variant (F=0) and monovariant (F=l) equilibria can be represented. Since pressure and temperature of phases in equilibrium are equal, a four-phase equilibrium is now represented by one point and a three-phase equilibrium with one curve. Also the critical curve and the azeotropic curve are projected as a curve on the P, 7-plane. A four-phase point is the point of intersection of four three-phase curves. The point of intersection of a three-phase curve and a critical curve is a so-called critical endpoint. In this intersection point both the three-phase curve and the critical curve terminate. 

In Figure 2.2-3 the curves Ig are the vapour pressure curves of the pure components which end in a critical point l=g. The curves l=g, h=g and h=g are vapour-liquid critical curves and the curves h=h are curves on which two liquid phases become critical. The points of intersection of a critical curve with a three-phase curve hhg is a critical endpoint. Distinction can be made between upper critical endpoints (UCEP) and lower critical endpoints (LCEP). The UCEP is highest temperature of a three-phase curve, the LCEP is the lowest temperature of a three-phase curve. The point of intersection of the hhg curve with a l/=g curve is a critical endpoint in which the li liquid phase and the vapour phase are critical in the presence of a non-critical l2 phase (h+(h=g)) and the point of intersection of the hhg curve with a h=h curve is a critical endpoint in which the two liquid phases h and // are critical in the presence of a non-critical vapour phase (h=h)+g)-... 

Type V fluid phase behaviour shows at temperatures close to 7C-A a three-phase curve hhg which ends at low temperature in a LCEP (h=h)+g and at high temperature in a UCEP (h+h) g The critical curve shows two branches. The branch h=g runs from the critical point of pure component A to the UCEP. The second branch starts in the LCEP and ends in the critical point of pure component B. This branch of the critical curve is at low temperature h=h in nature and at high temperature its character is changed into h=g- The h=h curve is a critical curve which represents lower critical solution temperatures. In Figure 2.2-7 four isothermal P c-sections are shown. 

In type VI phase behaviour a three-phase curve l2hg is found with an LCEP and an UCEP. Both critical endpoints are of the type (l2=li)+g and are connected by a l2=h critical curve which shows a pressure maximum. For this type of phase behaviour at constant pressure closed loop isobaric regions of l2+li equilibria are found with a lower critical solution temperature and an upper critical solution temperature. 

In Figure 2.2-8 the critical endpoint temperatures for the family 0f CO2 + n-alkanes systems are plotted as a function of the carbon number n. If in a particular binary system the three-phase curve hhg is followed to low temperature then at a certain temperature a solid phase is formed (solid n-alkane or solid C02 at low carbon numbers). This occurs at one unique temperature because we now have four phases in equilibrium in a binary system, so according to the phase rule F= 0. Below this so-called quadruple point temperature the hhg curve is metastable. 

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). 

When the supercritical fluid has a relatively high solubility in the molten heavy component, the S-L-V curve can have a negative dP/dT slope [64]. The second type of three-phase S-L-V curve shows a temperature minimum [65], In the third type, where the S-L-V curve has a positive dP/dT slope, the supercritical fluid is only slightly soluble in the molten heavy component, and therefore the increase of hydrostatic pressure will raise the melting temperature and a new type of three-phase curve with a temperature minimum and maximum may occur [66]. 

Finally, use the free-energy density vs. composition curves and work the tangent-to-curve construction in reverse. Using the result that Aga = —12 x 107Jm-3, the corresponding tangent to the a-phase curve will be at about 33 at. % B. 

The diagrams in Fig. 1 lb can be obtained by the so-called frequency-sweep method, where the lateral position and the distance Zc are fixed, while the frequency is varied around (O0. The Zc value corresponds to a given set-point ratio of the amplitude in contact to the free amplitude, rsp=Asp/Af. Depending on the tip-sample interaction, both the amplitude and the phase curve shifts in a certain direction. When the overall force is repulsive, the resonance frequency moves to higher values and results in a positive phase shift A(p=90 °-(p>0, where the phase shift of 90 ° corresponds to the free cantilever oscillations at ks=0 in Eq. 12. When the force is attractive the resonance frequency decreases compared to the free cantilever and Acp becomes negative. The situation in Fig. lib corre-... 

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