Critical Phenomena in Binary Systems

Introduction 233. 2. Critical Point for Pure Substances 234. 3. Critical Point for 

Binary Mixtures 236. 4. Conformal Solutions 239. 5. Second Order Terms and 

Critical Phenomena in the Vaporization of Binary jVIixtures 242. 6. Explicit 

Form of the Second Order Terms - Discussion 247. 7. Experimental Evidence 253. 

In this chapter we shall study in some detail the critical phenomena in binary systems and especially the relation between critical phenomena and intermolecular forces. In 2-3 we summarize the basic tbermod3naamic relations we need in the subsequent treatment. In 4-7 we consider critical vaporization phenomena while in 8 we study critical solution phenomena. 

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Figure 2 Critical phenomena in binary systems where gas-liquid and liquid-liquid equilibria interfere schematic representation for symbols see Section 1). a, b, and c, p T) projections of the phase diagram d, p x) isotherm for T = const. = Ti of binary systems corresponding to type 2b or 2c...

Figure 3 Critical phenomena in binary systems with positive homogeneous (a artd b) and heterogeneous (c to e and f to h) azeotropes (schematic representation see text)...

Figure 4 Interference of crystallization and gas-liquid criticed phenomena in binary systems schematic representation for symbols see Section 1 Qi = LGSiSn Qa = LLGSi). a. Type found for HaO + NaCl b, type found for HaO + SiOa c, crystallization behaviour in systems of type 2b such as proposed in the literature ...

The occurrence of critical phenomena in binary systems (critical vaporiiationor critical solution temperatures) will be studied in Ch. XII where the thermodynamic conditions for phase separation will be considered in detail. Here we shall summarize some basic relations to which we shall refer in many chapters of this book. 

Problems concerning the conditions of stability of homogenous systems for critical phases in ternary systems are very similar to those for the gas-liquid phenomena in binary systems, because of two independent variables at constant temperature and pressure. The conditions for stability are (82G/dnl)T P 2 3>0 and (820, given by Equations (5.146) and (5.147), respectively. Inspection of the condition equivalent to Equation (5.148) given by Equation (5.152) shows that (82(j>/dnl)TP> 2i 3 and, therefore, it is the condition expressed by Equation (5.147) or (5.150) that determines the boundary between stable... 

Over the last 10 years or so, a great deal of work has been devoted to the study of critical phenomena in binary micellar solutions and multicomponent microemulsions systems [19]. The aim of these investigations in surfactant solutions was to point out differences if they existed between these critical points and the liquid-gas critical points of a pure compound. The main questions to be considered were (1) Why did the observed critical exponents not always follow the universal behavior predicted by the renormalization group theory of critical phenomena and (2) Was the order of magnitude of the critical amplitudes comparable to that found in mixtures of small molecules The systems presented in this chapter exhibit several lines of critical points. Among them, one involves inverse microemulsions and another, sponge phases. The origin of these phase separations and their critical behavior are discussed next. 

Critical phenomena in binary fluid mixtures are substantially analogous to those in one-component fluids when proper allowance is made for the increase in the number of independent thermodynamic variables by one. Thus any extensive property of a one-component system ie,g. the energy U) can be expressed as a function of three variables, for example the entropy 5, the volume V, and the amount of substance but the binary mixture requires four (e.g. S, V, and 2). For most purposes the size of the system is irrelevant and one reduces the number of variables by one by using molar quantities Um, Sxa, Vm, for the binary mixture three independent variables (e.g. 5m, Im, and a composition variable such as the mole fraction x of the second component) then sufiice. 

The ultrasonic absorption in relation to the transitions and critical phenomena in microemulsions has been studied by Lang et al. (72). The ultrasonic absorption is very sensitive to the concentration fluctuations which occur near the critical temperature or composition in binary liquids. Similar absorption maxima were also expected as the composition of the systems was varied in the vicinity of composition where water-in-oil microemulsions convert into the oil-in-water microemulsions. However, the most puzzling feature of these data is probably the very continuous change of the relaxation parameters with composition even in the range where W/0 microemulsions turn into 0/W microemulsions. 

Diepen GAM, Scheffer FEC. (a) On critical phenomena of saturated solutions in binary systems. J Am Chem Soc 1948 70 4081 (b) The solubility of naphthalene in supercritical ethylene. J Am Chem Soc 1948 70 4085. 

Systems of type la (without critical phenomena in solid saturated solutions) (Figure 1.14). This is the simplest type of binary system. Most of the divariant (L-G, L-Sa, L-Sb, G-Sa, G-Sb) and monovariant equilibria (L-G-Sa, L-G-Sb, L = G) in the binary system (A-B) are the monovariant and nonvariant, respectively, with equilibria of one-component subsystems (A and B), spreading into two-component region of composition. Only the phase equilibria with two solid phases (invariant eutectic equihbrium (L-G-Sa-Sb) monovariant equilibria (L-Sa-Sb, G-Sa-Sb) and divariant equi-lihrium (Sa-Sb)) appear in the binary mixture as a result of an interaction of phase equilibria that extend fi om one-component subsystems. 

So far, the extension of the Hory-Huggins theory has enabled the modeling of several hitherto unexplainable anomalous phenomena, like uncommon molecular weight dependencies of second osmotic virial coefficients, the existence of multiple critical points for binary systems, or the odd swelling behavior of cellulose in water. 

Summary The classical treatment of the physicochemical behavior of polymers is presented in such a way that the chapter will meet the requirements of a beginner in the study of polymeric systems in solution. This chapter is an introduction to the classical conformational and thermodynamic analysis of polymeric solutions where the different theories that describe these behaviors of polymers are analyzed. Owing to the importance of the basic knowledge of the solution properties of polymers, the description of the conformational and thermodynamic behavior of polymers is presented in a classical way. The basic concepts like theta condition, excluded volume, good and poor solvents, critical phenomena, concentration regime, cosolvent effect of polymers in binary solvents, preferential adsorption are analyzed in an intelligible way. The thermodynamic theory of association equilibria which is capable to describe quantitatively the preferential adsorption of polymers by polar binary solvents is also analyzed. 

One of the most interesting aspects of energy transport is the excitation percolation transition (, and its similarity (10) to magnetic phase transitions and other critical phenomena (, 8). In its simplest form the problem is one of connectivity. In a binary system, made only of hosts and donors, the question is can the excitation travel from one side of the material to the other The implicit assumption is that there are excitation-transfer-bonds only between two donors that are "close enough", where "close enough" has a practical aspect (e.g. defined by the excitation transfer probability or time). Obviously, if there is a succession of excitation-bonds from one edge of the material to the other, one has "percolation", i.e. a connected chain of donors forming an excitation conduit. We note that the excitation-bonds seldom correspond to real chemical bonds rather more often they correspond to van-der-Walls type bonds and most often they correspond to a dipole-dipole or equivalent quantum-mechanical interaction. 

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