The temperature-concentration diagram

So fai- we have considered scaling as function of n. q iir the excluded volume limit for fixed temperature. We may extend the approach to include temperature variations close toT = O, assuming 

As has bexm discussed in Chap. 7. in the region t 1, implying, 5e 1, the two-pammeter hypothesis is falid. Thus t should occur only in the combination 

Note that this is completely consistent with our RG result (8,57). The crossover from the dilute to the seniidilute region takes phme at a concentration c = c n, t), where the chains just fill the volume  

This equation also can be read as defining a critical chain length n = n of concentration crossover  

Comparison of eqs. (9.26), (9.12) shoves that is nothing but the blob size Tig, so that we find the temperature dependence of 

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The work of Daoud and Jarmink [DJ76] on the temperature-concentration diagram proceeds by translating results established in the theory of phase transitions to the polymer problem. They also consider the region T < S. For T > 0 their work is equivalent to the argument as given here and discusses an extensive set of physical observables. 

When we warm up the solution at fixed concentration C in the domain IV of the temperature-concentration diagram, the square radius 3X2 and the overlap ratio increase. The chain is then subject to two contradictory effects first the solvent quality increases, second the screening of the interaction becomes stronger. Let us first recall that the average square distance 3X2 varies in the following manner (see Chapter 12, Section 3.3.2)... 

The temperature-concentration diagram in Figure 9.27 illustrates the mixing of a saturated solution A and unsaturated solution B to give a mixture with a composition and temperature represented by a point somewhere along line AB, determined by the mixture rule (section 4.4). A supersaturated mixture will be produced if the relative flowrates result in the mixture point lying in the sector below the solubility curve. For example, if A and B are mixed at equal mass flowrates, a supersaturated mixture Mi would be produced (distance AMi = BMi). For the mixture to enter the unsaturated zone, the B A ratio would have to exceed about 7 2 in the case illustrated. The unsaturated mixture M2, for example, is the result of an 8 1 B A ratio (AM2 = 8BM2). 

Two liquid phases always occur in the case of strong positive deviation from Raoult s law. The LLE behavior as a function of temperature only depends on the temperature dependence of the activity coefficients. The possible temperature dependencies for binary systems at constant pressure are shown in Figure 5.67 in the form of the temperature-concentration-diagrams, the scxalled binodal curves. 

Figure 9.1 Typical arrangements for the cooling and evaporative crystallizations and path of the crystallization in the temperature-concentration diagram.

The micrographs in Fig. 7.88 show clearly how from a knowledge of the AG -concentration diagrams it is possible to select the exact reaction conditions for the production of tailor-made outermost surface phase layers of the most desired composition and thus of the optimum physical and chemical properties for a given system. In addition it shows that according to thermodynamics, there can be predictable differences in the composition of the same outermost phase layer prepared at the same conditions of temperature but under slightly different vapour pressures. 

The maximum temperature will occur if there are no heat losses (adiabatic process). As no heat or material is removed, the problem can be solved graphically in the enthalpy-concentration diagram (Figure 3.3). The mixing operation is represented on the diagram... 

Gels are obtained for concentrations shown in the temperature-concentration phase diagram (Figure 1). Electron spin resonance (ESR) shows (10) that for a given temperature only a fraction (p) of the initial steroid concentration is transferred from the solution to the gel network. The picture of this gel is thus of a supersaturation gel there is a dynamic equilibrium between free molecules in solution and aggregated steroid molecules included in the long objects which constitute the gel network. The free steroid molecules concentration at a temperature where the gel state is stable is (1-p), while C p is the steroid concentration within the solid-iike gel aggregates. 

The solids, at any rate the polymerised forms, are regarded as solid solutions of Pa and P/3 in varying proportions. Temperature-concentration diagrams similar to those representing a two-eomponent system have been constructed for these pseudo-components. 

The phase diagram of a ternary system in which the three species do not form solid solutions with each other and the constituent binary systems form eutectics, is shown in Figure 4(b). The temperatures A B and Tc correspond to the melting points of A, B, and C, respectively. The vertical faces of the prism represent the temperature-concentration behavior of the three binaries. Note that the behavior of each binary system is that shown in Figure 2d. The solidus lines are not shown for the sake of clarity. Points E g, E j. are the eutectic points of the three binary... 

Figure 11.5 The temperature-concentration phase diagram for aqueous j crystallin (MW 20 000) systems (pH = 7, /= 0.24 mol kg" ) , cloud point measurements , concentration measurements of separated phases A = critical point.

Fig. 4.19 Temperature-concentration diagram for a binary mixture as well as the temperature and concentration profiles in the vapour and the condensate. Indices 0 cold wall, I interface, G core flow of vapour (G Gas), a boiling and dew point lines b condensate and vapour boundary layer c temperature profile d concentration profile...

Liquid-vapour equilibria for Cl -COCl have been calculated on the assumption that the system conforms to ideality [541]. In the region of low dichlorine concentration (<2.5%), the system has been shown, by analysis of the gas and liquid phases, to behave consistently with these calculations. The temperature-composition diagrams are illustrated in Fig. 6.13 at 101.3 and 152.0 kPa [541]. 

The curved boundary lines on which the isotherms of Fig. 16.8 terminate represent conditions of temperature and concentration under which solid phases form. These are various solid hydrates of sodium hydroxide. The enthalpies of all single-phase solutions lie above this boundary line. The enthalpy-concentration diagram can also be extended to include solid phases. 

The figure-that follows for the ethanol + water sy.s-tem is an unusual one in that it shows both vapor-liquid equilibrium and the enthalpy concentration diagrams on a single plot. This is done as follows. The lower collection of heavy lines give the enthalpy concentration data for the liquid at various temperatures and the upper collection of lines is the enthalpy-concentration data for the vapor, each at two pressures, 0.1013 and 1 013 bar. (There are also enthalpy-concentration lines for several other temperatures.) The middle collection of lines connect the equilibrium compositions of liquid and vapor. For example, at a pressure of 1.013 bar, a saturated-vapor containing 71 wt % ethanol with an enthalpy of 1535 kJ/kg is in equilibrium with a liquid containing 29 wt % ethanol with an enthalpy of 315 kJ/kg at a temperature of 85°C. Note also that the azeotropes that form in the ethanol -f water system are indicated at each pressure. 

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