Ideal solution laws

The entropy of mixing of very similar substances, i.e. the ideal solution law, can be derived from the simplest of statistical considerations. It too is a limiting law, of which the most nearly perfect example is the entropy of mixing of two isotopic species. 

The ideal solution law, Henry s Law, also enters into the establishment of performance of ideal and non-ideal solutions. 

Deviations from idealized behaviours (e.g. ideal gas laws, ideal solution laws, Trouton s or Hildebrand s rule, etc,). 

The activities of each solvent are shown in columns five and six of Tables I and II and are plotted in Figure 1 for 2-propanol. The standard state for the activities of each solvent was taken as the state for the pure component at the same temperature and pressure as that of the mixture. Assuming the vapor mixture obeys the ideal solution law, that is 0 2 = Wn + the activities were cal-... 

Critical Tables (7) give values of vapor pressure of 5.0 and 7.5% NaCl solutions over the range of 0° to 110° C. From these data the BPE for a 7.0% solution (50% recovery) at 1 atm. is readily calculated to be 2.34° F. From the ideal solution law (which should apply well to water in dilute solutions) and the Clausius-Clapeyron equation we get... 

The vaporization of solvent molecules from the pure liquid solvent described above should not differ from its vaporization from an infinitely dilute solution of some solute(s) in it, since the vast majority of solvent molecules have other solvent molecules in their surroundings in both cases. As the solute concentration increases in the dilute solution range, it is expected that Raoulf s law will be obeyed, that is, the vapour pressure of the solvent will be proportional to its mole fraction in the solution. If this is indeed the case, the solution is an ideal solution. At appreciable concentrations of the solute this will no longer be the case, due to solute-solute interactions and modified solute-solvent ones. The vapour pressure as well as other thermodynamic functions of the solvent and, of course, of the solute will no longer obey ideal solution laws. The consideration of these effects is beyond the scope of this book. 

D) Experimental results are frequently reported in terms of deviation functions. The usefulness of these functions arises from the fact many properties of various systems obey approximate laws. Thus, we speak of the deviations from ideal gas behavior or deviations from the ideal solution laws. The advantage of such deviation functions is that their values are usually much smaller than the whole value, and consequently greater accuracy can be obtained with simpler calculations, either graphically or algebraically. As an example, the molar volume of a mixture of liquids is approximately additive in the mole fractions, so that we may write c... 

This set may be solved in principle to yield P and Xj, each as a function of r, with the knowledge of a value of P and Xj for fixed values of r. The solution is easily obtained on the assumption that the ideal solution laws are applicable and that either the ideal gas equation is followed in the case of a gas phase or the volumes are independent of the pressure. Then, for an ideal liquid solution, Equation (14.27) becomes... 

Amagat s law of additive volumes holds for all pressures, which means that the ideal-solution law holds for the gaseous mixture, but not necessarily for the pure gases per se, and therefore the fugacity (/) is given by... 

The ideal-solution law is a more general statement of Raoult s law in terms of fugacity. In solution of highly alike substances. 

The ideal-solution law. Equation (4.314), reduces to Raoult s law. Equation (4.313), under the simplifying conditions ... 

Expressed in fugacity, the ideal-solution law is not restricted to low pressure, as it is in Raoult s law. 

A solution is ideal if it satisfies the ideal-solution law. No real solution is rigorously ideal, but solutions of similar substances approach ideal-solution behavior as the similarity increases. Solutions of xylene isomers, for example, deviate from ideal-solution law by about 1% at the maximiun. Close members of the same homologous series are often assumed to be ideal. It is not unusual to calculate mixtures of paraffin hydrocarbons with the ideal-solution equation. Ideal-solution law is the basis for ideal K values often used in industry. However, ideal-solution law is of great value in another way, and that is to provide a basis for introducing a correction factor, known as the activity coefficient. 

An ideal mixtnre is one in which the ideal-solution law is followed in all phases. For the liquid phase in an ideal mixtnre, by Eqnation (4.314), the fugacity of a component i is given by... 

Example 4.8. The propylene-l-butene system at moderate pressures might be expected to obey the ideal solution laws. Use (4-92) to compute K-values at 100°F (310.93°K) over a pressure range of approximately 60 psia (413.69 kPa) to 200 psia (1.379 MPa). Compare the results to Raoult s law K-values and to experimental data of Goff, Farrington, and Sage."... 

One of the reasons for the failure of the ideal solution law is the assumption that a large polymeric solute molecule is interchangeable with the smaller solvent molecule. The law also neglects intermolec-ular forces since the heat of mixing (AH x) assumed to be zero. The Flory-Huggins theory attempted to remedy these shortcomings in the ideal solution law. > ... 

These two kinds of ideality permeate discussions of liquid and solid solution properties, and are formalized by two ideal solution laws - Raoulf s law and Henry s law. 

The term in square brackets in Eq. (138) expresses the variation of activity coefficient of the neutral species with concentration. Thus, in addition to the statistical contribution to diffusion, expressed by the familiar gradient-in-concentration term, there is a chemical driving force due to the variation of free energy with composition and hence position. The term in square brackets is known as the thermodynamic enhancement factor and was identified by Darken [1948]. The diffusion coefficient A is known as the chanical diffusion coefficient, and its use is appropriate whenever diffusion takes place in an appreciable concentration gradient and when ideal solution laws cannot be applied to the solute. The concept was extended by C. 

In brief, there is no purely thermodynamic criterion which leads us to prefer one definition of the mole fraction to any other. The ideal solution laws may be obeyed if this choice is made in one way but not in another, and the most appropriate choice for this purpose is determined by molecular considerations. 

The question may be asked whether Henry s law and Raoult s law continue to be compatible with this equation when the small changes in total pressure are allowed for. This involves the pressure dependence of Ki in equation (7 49). This point will be discussed from a rather different standpoint in the next chapter where it will be shown that the thermodynamic consistency of the ideal solution laws implies constancy of the partial molar volumes. 

According to the theory developed by Debye, Hfickel and others, the departure of electrolytes from the ideal solution laws may be... 

Enthalpy change in reaction under conditions where the species obey the perfect gas laws or the ideal solution laws An integration constant having the dimensions of an enthalpy change... 

Without loss of generality, let us choose the model of the new phase as a Hne (strictly stoichiometric) intermediate phase with composition C = Ci = 0.5, and exclude the elastic contributions to the Gibbs energy. The parent phase in the vicinity of the phase transition points will be described by the ideal solution law and will be also denoted as the a-phase. In fact, the model of regular solution would be more reasonable since the existence of intermediate phases usually correlates with negative mixing energy. Yet, for simpHcity, below we restrict ourselves to the... 

A plot of T against (1//) will give a straight line of slope, -x/A, and the intercept at 1/f) = 0 will be T°. From the slope of the line, the purity of the sample is determined. The physical assumptions for the calorimetric determination of purity are that the ideal solution laws are obeyed and that the impurity is liquid-soluble and solid-insoluble. The ideal solution law is applicable for small amounts of impurity but solid solutions can occur when the major component has a small enthalpy of fusion or is very similar chemically to the substance. For the case of solid solution formation, a treatment has been given by Mastrangelo and Domte [55-mas/dor] which can be used to obtain the freezing temperatue for zero impurity. 

Most mixtures do not obey Raoult s law or the corrected Raoult s law given by Eqs. (3-8) and (3-10). The deviations from the ideal solution laws can be due to the vapor phase, the liquid phase, or both. These deviations are both chemical and physical in nature. The most important factors involved in these deviations are believed to be (1) the fact that the molecules have volume and (2) the fact that the molecules exert forces on each other that may be attractions, or repulsions, or actual chemical effects. 

Deviations from ideal solution laws are more important for the liquid phase than those for the vapor phase because they are encountered even at low pressures, and in general their magnitudes are greater. The densities of the liquids are such that the volume of the molecules and the forces between them are always significant. Deviations for the liquid phase are of at least two main types (1) Those due to the fact that the vapor does not obey the perfect-gas law. Thus, if one tries to define the fugacity of a component in the liquid phase as mol fraction times the vapor pressure of the pure component, this can be satisfactory only when the vapor at a pressure equal to the vapor pressure is essentially a perfect gas. (2) The deviations that are due to special phenomena associated with the liquid phase such as associations or chemical combinations. 

Solution Deviations. The corrections so far considered have been limited to those associated with the fact that the vapor does not obey the perfect-gas law. A large number of mixtures, in fact most of them, do not obey the ideal solution laws even at very low pressure, and the deviations cannot be predicted by the use of gas-phase fugacity corrections. The deviations are the residt of the forces between the molecules in the liquid phase, and these forces can Be ery Targely diie... 

The ideal solution laws such as Raoult s law and Raoult s law corrected for gas law deviation are applicable to binary or multicomponent systems. They treat each component independently of any other component present t.e., the relationship between the mol fraction in the vapor and in the liquid for a given component depends only on the temperature and total pressure. In many cases, these simplified rules are not applicable, and there is interreaction between the various components. It would be particularly desirable to have a satisfactory theoretical approach to the problem of multicomponent vapor-liquid equilibria since the experimental determination for this... 

We see that most of the statements made about Raoult s law apply to Henry s law. It is normally shown as an ideal solution law, yi = 100, with the pure species vapor pressme replaced by the Henry s law constant. Table 8.4 also shows the equation for x in addition to the working equation for y, because Henry s law is most often used to estimate the... 

Thus, we see that Henry s law fits into our computational scheme for VLE, as an ideal solution law, with the choice of. liquid phase Often it is applied in examples like the air-water example in Chapter 3, in which one species in the gas (e.g., water) exists as a vapor, while one or more other species in the gas (e.g., nitrogen and oxygen) are present as gases above their critical temperatures. Purists would describe that as part VLE (for the water) and part gas-Hquid equilibrium (for the nitrogen and oxygen). The gaseous phase would be called a gas, not a vapor, but we see that this is a matter of arbitrary definitions. 

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