Ideal solutions thermodynamic properties

When the ideal solution is used as the reference for real solutions, thermodynamic properties are designated by (RL) for Raoulfs law reference. This reference is often used in solutions in which all solutes are liquids at the temperature of interest, especially when the compositions of components are varied over a considerable range. In this case, for every component, we write Eq. (3) as... 

All the theories described above are based on the ideal solution thermodynamics, on the one hand, and on a rather heuristic molecular treatment of micelles as a phase particle, on the other hand. Despite of their obvious successes in predicting micellar solution properties, these theories have some essential drawbacks. The number of adjusting parameters at the evaluation of different contributions to the free energy is too high, as well as the number of oversimplifications, which have been used in order to estimate these parameters. For example, the micellar core is considered as a very small fluid phase droplet surrounded by a second fluid phase and the free energy of micelle surface is estimated as the interfacial tension between these two fluid phases. In order to elucidate this problem Eriksson et al, [24] attempted to... 

Osmotic pressure is one of four closely related properties of solutions that are collectively known as colligative properties. In all four, a difference in the behavior of the solution and the pure solvent is related to the thermodynamic activity of the solvent in the solution. In ideal solutions the activity equals the mole fraction, and the mole fractions of the solvent (subscript 1) and the solute (subscript 2) add up to unity in two-component systems. Therefore the colligative properties can easily be related to the mole fraction of the solute in an ideal solution. The following review of the other three colligative properties indicates the similarity which underlies the analysis of all the colligative properties ... 

All other thermodynamic properties for an ideal solution foUow from this equation. In particular, differentiation with respect to temperature and pressure, followed by appHcation of equations for partial properties analogous to equations 62 and 63, leads to equations 191 and 192 ... 

If M represents the molar value of any extensive thermodynamic property, an excess property is defined as the difference between the actual property value of a solution and the value it would have as an ideal solution at the same temperature, pressure, and composition. Thus,... 

P rtl IMol r Properties. The properties of individual components in a mixture or solution play an important role in solution thermodynamics. These properties, which represent molar derivatives of such extensive quantities as Gibbs free energy and entropy, are called partial molar properties. For example, in a Hquid mixture of ethanol and water, the partial molar volume of ethanol and the partial molar volume of water have values that are, in general, quite different from the volumes of pure ethanol and pure water at the same temperature and pressure (21). If the mixture is an ideal solution, the partial molar volume of a component in solution is the same as the molar volume of the pure material at the same temperature and pressure. 

Denotes excess thermodynamic property Denotes value for an ideal solution Denotes value for an ideal gas Denotes liquid phase... 

This equation is the basis for development of expressions for all other thermodynamic properties of an ideal solution. Equations (4-60) and (4-61), apphed to an ideal solution with replaced by Gj, can be written... 

The residual Gibbs energy and the fugacity coefficient are useful where experimental PVT data can be adequately correlated by equations of state. Indeed, if convenient treatment or all fluids by means of equations of state were possible, the thermodynamic-property relations already presented would suffice. However, liquid solutions are often more easily dealt with through properties that measure their deviations from ideal solution behavior, not from ideal gas behavior. Thus, the mathematical formahsm of excess properties is analogous to that of the residual properties. 

As an example of how the approximate thermodynamic-property equations are handled in the inner loop, consider the calculation of K values. The approximate models for nearly ideal hquid solutions are the following empirical Clausius-Clapeyron form of the K value in terms of a base or reference component, b, and the definition of the relative volatility, Ot. 

The excess molar thermodynamic function Z is defined as the difference in the property Zm for a real mixture and that for an ideal solution. That is,... 

The reason is that classical thermodynamics tells us nothing about the atomic or molecular state of a system. We use thermodynamic results to infer molecular properties, but the evidence is circumstantial. For example, we can infer why a (hydrocarbon + alkanol) mixture shows large positive deviations from ideal solution behavior, in terms of the breaking of hydrogen bonds during mixing, but our description cannot be backed up by thermodynamic equations that involve molecular parameters. 

Chapters 7 to 9 apply the thermodynamic relationships to mixtures, to phase equilibria, and to chemical equilibrium. In Chapter 7, both nonelectrolyte and electrolyte solutions are described, including the properties of ideal mixtures. The Debye-Hiickel theory is developed and applied to the electrolyte solutions. Thermal properties and osmotic pressure are also described. In Chapter 8, the principles of phase equilibria of pure substances and of mixtures are presented. The phase rule, Clapeyron equation, and phase diagrams are used extensively in the description of representative systems. Chapter 9 uses thermodynamics to describe chemical equilibrium. The equilibrium constant and its relationship to pressure, temperature, and activity is developed, as are the basic equations that apply to electrochemical cells. Examples are given that demonstrate the use of thermodynamics in predicting equilibrium conditions and cell voltages. 

Polymer solutions always exhibit large deviations from Raoult s law, though at extreme dilutions they do approach ideality. Generally however, deviation from ideal behaviour is too great to make Raoult s law of any use for describing the thermodynamic properties of polymer solutions. 

The first term on the right is the formula for the chemical potential of component a at density pa = na/V in an ideal gas, as would be the case if interactions between molecules were negligible, fee is Boltzmann s constant, and V is the volume of the solution. The other parameters in that ideal contribution are properties of the isolated molecule of type a, and depend on the thermodynamic state only through T. Specifically, V/A is the translational contribution to the partition function of single a molecule at temperature T in a volume V... 

Figure 3.3 Thermodynamic properties of an arbitrary ideal solution A-B at 1000 K. (a) The Gibbs energy, enthalpy and entropy, (b) The entropy of mixing and the partial entropy of mixing of component A. (c) The Gibbs energy of mixing and the partial Gibbs energy of mixing of component A.

As in the nonelectrolyte case, the problem of representing the thermodynamic properties of electrolyte solutions is best regarded as that of finding a suitable expression for the non-ideal part of the chemical potential, or the excess Gibbs energy, as a function of composition, temperature, dielectric constant and any other relevant variables. 

By using a thermodynamic model based on the formation of self-associates, as proposed by Singh and Sommer (1992), Akinlade and Awe (2006) studied the composition dependence of the bulk and surface properties of some liquid alloys (Tl-Ga at 700°C, Cd-Zn at 627°C). Positive deviations of the mixing properties from ideal solution behaviour were explained and the degree of phase separation was computed both for bulk alloys and for the surface. 

If Equation (14.2), Equation (14.6), or Equation (14.7) is used to define an ideal solution of two components, values for the changes in thermodynamic properties resulting from the formation of such a solution follow directly. 

TABLE 14.1. Thermodynamic Properties of Two Ideal Solutions (A and A ) of Different Mole Fractions, A,- and A/, Prepared from the Same Components... 

As we saw in section 3.8.1, Raoult s law describes the properties of an ideal solution, in which the thermodynamic activity of the component is numerically equivalent to its molar concentration ... 

The notion of an ideal behavior also is defined here for those cases in which Pi is constant over a range of solution compositions, while variations with solution composition are said to characterize nonideal behavior. In the present studies values of purification factors are affected by the kinetics of the process. Accordingly, these quantities may not be true thermodynamic properties. 

Swollen tensile and compression techniques avoid both of these problems since equilibrium swelling is not required, and the method is based on interfacial bond release and plasticization rather than solution thermodynamics. The technique relies upon the approach to ideal rubberlike behavior which results when lightly crosslinked polymers are swelled. At small to moderate elongations, the stress-strain properties of rubbers... 

In one patented process (19) a m-p-xylene fraction is produced from xylene mixtures by distillation, and is subsequently cooled to about —70° F. to produce p-xylene crystals, which are removed in high purity by filtering or centrifuging. The yield of p-xylene is limited by eutectic formation with m-xylene. As the mixture behaves as an ideal solution, the yield and temperature level can be calculated from the thermodynamic properties of xylenes, which were reported by Kravchenko (12). 

To understand why a solute lowers the vapor pressure, we need to look at the thermodynamic properties of the solution. We saw in Section 8.2, specifically Eq. 1, that at equilibrium, and in the absence of any solute, the molar free energy of the vapor is equal to that of the pure solvent. We now need to consider the molar free energies of the solvent and the vapor when a solute is present. We shall consider only nonvolatile solutes, which do not appear in the vapor phase, and limit our considerations to ideal solutions. 

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