Equilibrium crystallization, definition

In Chapter III, surface free energy and surface stress were treated as equivalent, and both were discussed in terms of the energy to form unit additional surface. It is now desirable to consider an independent, more mechanical definition of surface stress. If a surface is cut by a plane normal to it, then, in order that the atoms on either side of the cut remain in equilibrium, it will be necessary to apply some external force to them. The total such force per unit length is the surface stress, and half the sum of the two surface stresses along mutually perpendicular cuts is equal to the surface tension. (Similarly, one-third of the sum of the three principal stresses in the body of a liquid is equal to its hydrostatic pressure.) In the case of a liquid or isotropic solid the two surface stresses are equal, but for a nonisotropic solid or crystal, this will not be true. In such a case the partial surface stresses or stretching tensions may be denoted as Ti and T2-... 

The growth of crystals—or more generally the solidification of a sohd from a fluid phase—is definitely not an equilibrium problem. Why, therefore, should we discuss here equihbrium thermodynamics, instead of treating directly, for example the coagulation of two atoms and then simply following the growth of the cluster by adding more particles with time ... 

Figure 5. A schematic representation of superposed steady-state reservoirs of constant volumes Vi (fractional crystallization is omitted in this schema). At steady-state, Vi/xi=V2/x2=..., where x is the residence time. This is analogous to the law of radioactive equilibrium between nuclides 1 and 2 Ni/Ti=N2/T2=...A further interest of this simple model is to show that residence times by definition depend on the volume of the reservoirs.

For this method, either a weighed amount of the solute (or a definite amount of the solvent) is placed in a suitable vessel. While agitating the system at constant temperature, known amounts of the solvent (or the solute) are added gradually until the solubility limit is reached. Appropriate checks must be carried out to ensure that the system is very close to equilibrium when the content or temperature of the system is recorded. In this method of temperature variation, attention is usually focused on the last small crystal. The equilibrium temperature is taken as the mean of the two temperatures at which the crystal either slowly grows or slowly dissolves. This procedure may also be carried out at the microscale by examining a small volume of the system under a hot-stage microscope. 

One could go on with examples such as the use of a shirt rather than sand reduce the silt content of drinking water or the use of a net to separate fish from their native waters. Rather than that perhaps we should rely on the definition of a chemical equilibrium and its presence or absence. Chemical equilibria are dynamic with only the illusion of static state. Acetic acid dissociates in water to acetate-ion and hydrated hydrogen ion. At any instant, however, there is an acid molecule formed by recombination of acid anion and a proton cation while another acid molecule dissociates. The equilibrium constant is based on a dynamic process. Ordinary filtration is not an equilibrium process nor is it the case of crystals plucked from under a microscope into a waiting vial. 

The conformations of the furanose ring in 250 nucleoside and nucleotide structures were analysed by Bartenev et al. (1987). These authors made the assumption, referred to above, that intermolecular interactions have a random effect on the structure in the crystal, and that the probability JVg of a structure crystallizing in a non-ground-state conformation is the same as the probability of it arising in thermal equilibrium at ambient temperature T in solution (6). (A difficulty arises immediately with the definition of the temperature, because structural parameters for molecules in crystals are... 

For crystal growth at constant rate, if the crystal composition can respond to interface melt composition through surface equilibrium, steady state may be reached (Smith et ah, 1956). At steady state, (dCldt) = 0 by definition. Hence,... 

The definition of crystal is itself a developing concept, as demonstrated by the ongoing discussions [5, 6]. Most ot the theoretical background proposed in this chapter is valid for a perfect crystal, i.e., an infinite mathematical object with an idealized crystal structure ideal crystal) in thermodynamic equilibrium at a given presstrre P and temperature T. In textbooks, only the gas phase thermodynamics is usually discussed in detail, whereas little attention is paid to the solid state. A full thermodynamic treatment of solids is beyond the scope of this chapter and the reader is referred to specific books on the subject, for example [7]. 

Schlenk was the one who first took triphenylmethyl-type radicals to the monomeric extreme and thus produced the final evidence for the existence of free radicals. The first example in this direction was phenylbis(biphenylyl)-methyl (11), which was isolated as white crystals from operations carried out in the apparatus described by Schmidlin. " Upon dissolution of 11 in benzene, a red color developed, and cryoscopic studies revealed that the monomeric phenylbis(biphenylyl)methyl constituted 80% of the equilibrium mixture. Trisbiphenylylmethyl (12) was even more extreme it formed black crystals and was a 100% monomeric free radical in an almost black solution. Finally, Schlenk et al. established the connection between the conducting solutions of triphenylhalomethanes and the free radical triphenylmethyl by showing that the cathodic reduction of triphenylbromomethane in liquid SO2 gave rise to triphenylmethyl. These findings were considered the definitive evidence for the free radical hypothesis, and Schlenck was nominated for the Nobel Prize in 1918 and several times afterwards for this achievement, amongst others (Table 2). 

The Stassfurt deposits have been the subject of elaborate investigations by J. H. van t Hoff and his school.16 In 1849, J. Usiglio 17 studied the deposition of salts when sea.water is cone, by evaporation, and examined the residues analytically. He found that calcium carbonate was first eliminated, then calcium sulphate, then sodium chloride, and the more soluble salts accumulated in the mother liquid. This method of investigation does not allow sufficient time for the various salts to attain a state of equilibrium, and it therefore follows that the natural evaporation of brines probably furnishes somewhat different results. Moreover, it is difficult, if not impossible, to identify the several substances which separate from the mother liquid formed during the later stages of the evaporation. J. H. van t Hoff followed the synthetic method in his study of this subject. He started from simple soln. like those of sodium and potassium chlorides, under definite conditions of temp., and gradually added the pertinent constituents until the subject became so complicated that the crystallization of the constituents from concentrating sea water was reduced to a special case of a far more comprehensive work. 

A mixture of ammonia and hydrogen sulphide does not unite if the press, is less than a certain critical value, which depends on the temp. If the press, is at or above this value crystals of ammonium hydrosulphide are formed if the press, be increased,more crystals will form and if the press, be reduced, crystals will decompose. When the two gases are present in eq. amounts this press, is called the dissociation pressure of the solid. In the present case, if the vapour phase has the same composition as the solid with which it is in equilibrium, the system is univariant, and there is a definite dissociation press, for each temp. F. Isambert found the dissociation press, of ammonium hydrosulphide, in mm. of mercury, increases rapidly with a rise of tomp. ... 

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