Solutions nonideal

An ideal solution is an exception rather than the rule. Real solutions are, in general, nonideal. Any solution in which the activity of a component is not equal to its mole fraction is called non-ideal. The extent of the nonideality of a solution, i.e., the extent of its deviation from 

The first term on the left hand side in the above equation is equal to zero because Xi is independent of temperature. Hence 

The ideal solubility of naphthalene in any liquid solvent at 20 C should thus be 0.266 mole fraction the experimental values in solvents which are chemically similar to the solute, and which might be expected to form approximately ideal solutions, viz., benzene and toluene, are 0.241 and 0.224 mole fraction,respectively. In hydroxylic solvents, which form nonideal solutions, the values are quite different, for example, 0.018 in methanol and 0.0456 in acetic acid. 

For an ideal solution the heat of solution is equal to the heat of fusion of the solid at the given temperature and pressure, and consequently for a given solute the former should be independent of the nature of the solvent. That this should be the case can be readily shown by means of a procedure exactly analogous to that employed in 34h in connection with the heat of solution of gases. 

Deviations from Ideal Behavior.—Theoretical considerations show that if a mixture of tw6 liquids is to behave ideally, the two types of molecules must be similar. The environment of any molecule in the solution, and hence the force acting upon it, is then not appreciably different from that existing in the pure liquid. It is to be anticipated, therefore, that under these conditions the partial vapor pressure (or fugacity) of each constituent, which is a measure of its tendency to escape from the solution, will be directly proportional to the number of molecules of the constituent in the liquid phase. This is equivalent to stating that a mixture of two liquids consisting of similar molecules would be expected to obey Raoult s law. Such is actually the case, for the relatively few liquid systems which are known to behave ideally, or to approximate to ideal behavior, consist of similar molecules, e.g., ethylene bromide and propylene bromide, n-hexane and n-heptane, r -butyl chloride and bromide, ethyl bromide and iodide, and benzene and toluene. 

If the constituents of a mixture differ appreciably in nature, deviations from ideal behavior are to be expected and are, in fact, observed. These deviations are most frequently positive in nature, so that the actual par- 

For a nonideal solution exhibiting positive deviations fi/ft for each component is greater than its mole fraction N -. It is an experimental fact that as the temperature is increased most liquid solutions tend toward ideal behavior. This means that for a system of given composition for which the deviations from Raoult s law are positive, the ratio / // usually decreases with increasing temperature. According to equation (34.4), therefore, which holds for solution of all types, the numerator Jff — tti must be negative thus Hi is greater than The total heat content of the solution, con- 

In actuality, Raoult s law is only an approximation for real systems. Although it is a good approximation for many solvents, for which Xa 851 if the solution is dilute, it is often a very poor approximation for solutes, for which Xa 0.5. In the limit as Xa — 0, though, there is still a linear relationship 

We can equate the chemical potential of the solute to the chemical potential of the vapor in equilibrium with it. Assume the vapor is an ideal gas  

Comparing Equations 26 and 27 to Equation 8 implies a new standard state (to be denoted as the Henry s Law standard state or HL s.s.) whose chemical potential is related to that for the gaseous standard state by 

Both Equations 29 and 30 are valid in the region where the limit of Equation 22 holds, but at higher concentrations of A, Equation 29 fails to hold. However, Equation 30 is a special case of Equation 8 and as such it defines the activity p for any value of XA. Thus, 

It is then convenient to define an activity coefficient y A such that9 

Any deviations from assumptions (1) to (4) will constitute a deviation from ideality — an ideal solution is a rare occurrence — and several more realistic types of solution can be identified  

Irregular solutions, in which both A and A/ deviate from their ideal 

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A solution which obeys Raoult s law over the full range of compositions is called an ideal solution (see Example 7.1). Equation (8.22) describes the relationship between activity and mole fraction for ideal solutions. In the case of nonideal solutions, the nonideality may be taken into account by introducing an activity coefficient as a factor of proportionality into Eq. (8.22). 

Thus, usiag these techniques and a nonideal solution model that is capable of predictiag multiple Hquid phases, it is possible to produce phase diagrams comparable to those of Eigure 15. These predictions are not, however, always quantitatively accurate (2,6,8,91,100). 

Since the infinite dilution values D°g and Dba. re generally unequal, even a thermodynamically ideal solution hke Ya = Ys = 1 will exhibit concentration dependence of the diffusivity. In addition, nonideal solutions require a thermodynamic correction factor to retain the true driving force for molecular diffusion, or the gradient of the chemical potential rather than the composition gradient. That correction factor is ... 

Density and Specific Gravity For binary or pseudobinary mixtures of hquids or gases or a solution of a solid or gas in a solvent, the density is a funcrion of the composition at a given temperature and pressure. Specific gravity is the ratio of the density of a noncompress-ible substance to the density of water at the same physical conditions. For nonideal solutions, empirical calibration will give the relationship between density and composition. Several types of measuring devices are described below. 

The separation of components by liquid-liquid extraction depends primarily on the thermodynamic equilibrium partition of those components between the two liquid phases. Knowledge of these partition relationships is essential for selecting the ratio or extraction solvent to feed that enters an extraction process and for evaluating the mass-transfer rates or theoretical stage efficiencies achieved in process equipment. Since two liquid phases that are immiscible are used, the thermodynamic equilibrium involves considerable evaluation of nonideal solutions. In the simplest case a feed solvent F contains a solute that is to be transferred into an extraction solvent S. 

Here Q is the solute concentration and R the gas constant. This is in fact obeyed over a rather wide range of concentrations, almost up to solute mole fractions of 0.61, with an error of only 25 percent. This is remarkable, since the van t Hoff equation is rigorous only in the infinitely dilute limit. Even in the case of highly nonideal solutions, for example a solution with a ratios of 1.5 and e ratios of 4, the van t Hoff equation is still obeyed quite well for concentrations up to about 6 mole percent. It appears from these results that the van t Hoff approximation is much more sensitive to the nonideality of the solutions, and not that sensitive... 

Let us now focus attention on the common case where all three binaries exhibit positive deviations from Raoult s law, i.e., afj- > 0 for all ij pairs. If Tc for the 1-3 binary is far below room temperature, then that binary is only moderately nonideal and a13 is small. We must now choose a gas which forms a highly nonideal solution with one of the liquid components (say, component 3) while it forms with the other component (component 1) a solution which is only modestly nonideal. In that event,... 

A hypothetical solution that obeys Raoult s law exactly at all concentrations is called an ideal solution. In an ideal solution, the interactions between solute and solvent molecules are the same as the interactions between solvent molecules in the pure state and between solute molecules in the pure state. Consequently, the solute molecules mingle freely with the solvent molecules. That is, in an ideal solution, the enthalpy of solution is zero. Solutes that form nearly ideal solutions are often similar in composition and structure to the solvent molecules. For instance, methylbenzene (toluene), C6H5CH, forms nearly ideal solutions with benzene, C6H6. Real solutions do not obey Raoult s law at all concentrations but the lower the solute concentration, the more closely they resemble ideal solutions. Raoult s law is another example of a limiting law (Section 4.4), which in this case becomes increasingly valid as the concentration of the solute approaches zero. A solution that does not obey Raoult s law at a particular solute concentration is called a nonideal solution. Real solutions are approximately ideal at solute concentrations below about 0.1 M for nonelectrolyte solutions and 0.01 M for electrolyte solutions. The greater departure from ideality in electrolyte solutions arises from the interactions between ions, which occur over a long distance and hence have a pronounced effect. Unless stated otherwise, we shall assume that all the solutions that we meet are ideal. 

For nonideal solutions, the thermodynamic equilibrium constant, as given by Equation (7.29), is fundamental and Ei mettc should be reconciled to it even though the exponents in Equation (7.28) may be different than the stoichiometric coefficients. As a practical matter, the equilibrium composition of nonideal solutions is usually found by running reactions to completion rather than by thermodynamic calculations, but they can also be predicted using generalized correlations. 

Of great importance for the development of solution theory was the work of Gilbert N. Lewis, who introduced the concept of activity in thermodynamics (1907) and in this way greatly eased the analysis of phenomena in nonideal solutions. Substantial information on solution structure was also gathered when the conductivity and activity coefficients (Section 7.3) were analyzed as functions of solution concentration. 

The behaviour of most metallurgically important solutions could be described by certain simple laws. These laws and several other pertinent aspects of solution behaviour are described in this section. The laws of Raoult, Henry and Sievert are presented first. Next, certain parameters such as activity, activity coefficient, chemical potential, and relative partial and integral molar free energies, which are essential for thermodynamic detailing of solution behaviour, are defined. This is followed by a discussion on the Gibbs-Duhem equation and ideal and nonideal solutions. The special case of nonideal solutions, termed as a regular solution, is then presented wherein the concept of excess thermodynamic functions has been used. 

A particular type of nonideal solution is the regular solution which is characterized by a nonzero enthalpy of mixing but an ideal entropy of mixing. Thus, for a regular solution,... 

Fig. 10 Dependence of vapor pressure of a solution containing a volatile solute, illustrated for (A) an ideal solution and (B) a nonideal solution and shown as a function of mole fraction of the solute. Individual vapor pressure curves are shown for the solvent (0) the solute ( ), and for the sum of these (X).

Whenever the solute and solvent exhibit significant degrees of mutual attraction, deviations from the simple relationships will be observed. The properties of these nonideal solutions must be determined by the balance of attractive and disruptive forces. When a definite attraction can exist between the solute and solvent, the vapor pressure of each component is normally decreased. The overall vapor pressure of the system will then exhibit significant deviations from linearity in its concentration dependence, as is illustrated in Fig. 10B. 

Before closing this section we note that even in nonideal solutions we can use the standard state of Equation 16 for the solute. Since Equation 16 only holds for ideal solutions, one generalizes to obtain48... 

For a nonideal solution, CPi is replaced by the partial molar heat capacity, CPl, but such information may not be available. 

To overcome the problem of non-ideality the work be carried out at the Q temperature because in nonideal solutions the apparent Molecular weight is a linear function of concentration at temperatures near Q and the slope depending primarily on the second virial coefficient. 

The thermodynamic development above has been strictly limited to the case of ideal gases and mixtures of ideal gases. As pressure increases, corrections for vapor nonideality become increasingly important. They cannot be neglected at elevated pressures (particularly in the critical region). Similar corrections are necessary in the condensed phase for solutions which show marked departures from Raoult s or Henry s laws which are the common ideal reference solutions of choice. For nonideal solutions, in both gas and condensed phases, there is no longer any direct... 

An important attribute of Equation 5.16 is that the pressure exerted on both phases, Ptot, is common to both isotopomers. The important difference between Equations 5.16 and 5.9 is that the isotopic vapor pressure difference (P/ — P) does not enter the last two terms of Equation 5.16 as it does in Equation 5.9. Also isotope effects on the second virial coefficient AB/B = (B — B)/B and the condensed phase molar volume AV/V are significantly smaller than those on AP/P ln(P7P). Consequently the corrections in Equation 5.16 are considerably smaller than those in Equations 5.9 and 5.10, and can sooner be neglected. Thus to good approximation ln(a") is a direct measure of the logarithmic partition function ratio ln(Qv Q7QvQcO> provided the pressure is not too high, and assuming ideality for the condensed phase isotopomer solution. For nonideal solutions a modification to Equation 5.16 is necessary. 

The chemical potential of the solute in a nonideal solution is given by... 

V - V is here the volume change of the solute as it is transferred from pure solvent to a nonideal solution, all at a pressure p. 

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