Ideal Real Solutions

Up to this point, we have considered only ideal solutions. If the solute does not behave as an inert particle (i.e., if it interacts with other solutes including its own kind), the energy of this interaction must be included in the internal energy of the system and the (A. 16) and those that follow must be defined in terms of activities. 

The activity coefficient /, is a measure of the nonideal behavior of species i. For a solution of species i in a noninteracting solvent, it is defined as 

The activity coefficient can have a value smaller or greater than one. 

These interactions exist in real systems for which the activity has the physical meaning of the effective concentration. Thus, only for dilute real solutions does A = a. In mixtures, the activity coefficient is usually, but not always, less than one and is affected by all the species in the multicomponent mixture. 

When the system contains two (or more) phases the components of the system interact with these phases in one of two ways. The first is that the component exists in both phases, in which case at equilibrium the chemical potential of that component must be equal in both phases, which follows from the Gibbs equation (A.20). Because the number of moles transferred from phase 1 equals the number of moles received by phase 2(A i = — A 2) 

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Raoult s law does not apply over the entire concentration range in a non-ideal, real solution. However, when one component is in a large enough excess to be considered a solvent, Raoult s law may be expressed as ... 

In analogy to the gas, the reference state is for the ideally dilute solution at c, although at the real solution may be far from ideal. (Teclmically, since this has now been extended to non-volatile solutes, it is defined at... 

Ideal Adsorbed Solution Theory. Perhaps the most successful approach to the prediction of multicomponent equiUbria from single-component isotherm data is ideal adsorbed solution theory (14). In essence, the theory is based on the assumption that the adsorbed phase is thermodynamically ideal in the sense that the equiUbrium pressure for each component is simply the product of its mole fraction in the adsorbed phase and the equihbrium pressure for the pure component at the same spreadingpressure. The theoretical basis for this assumption and the details of the calculations required to predict the mixture isotherm are given in standard texts on adsorption (7) as well as in the original paper (14). Whereas the theory has been shown to work well for several systems, notably for mixtures of hydrocarbons on carbon adsorbents, there are a number of systems which do not obey this model. Azeotrope formation and selectivity reversal, which are observed quite commonly in real systems, ate not consistent with an ideal adsorbed... 

Solubility Properties. Fats and oils are characterized by virtually complete lack of miscibility with water. However, they are miscible in all proportions with many nonpolar organic solvents. Tme solubiHty depends on the thermal properties of the solute and solvent and the relative attractive forces between like and unlike molecules. Ideal solubiHties can be calculated from thermal properties. Most real solutions of fats and oils in organic solvents show positive deviation from ideaHty, particularly at higher concentrations. Determination of solubiHties of components of fat and oil mixtures is critical when designing separations of mixtures by fractional crystallization. 

The ideal gas is a useful model of the behavior of gases and serves as a standard to which real gas behavior can be compared. This is formalized by the introduction of residual properties. Another useful model is the ideal solution, which sei ves as a standard to which real solution behavior can be compared. This is formalized by introduction of excess propei ties. 

Thus the formation of an ideal solution from its components is always a spontaneous process. Real solutions are described in terms of the difference in the molar Gibbs free energy of their formation and that of the corresponding ideal solution, thus ... 

For ideal solutions, the activity coefficient will be unity, but for real solutions, 7r i will differ from unity, and, in fact, can be used as a measure of the nonideality of the solution. But we have seen earlier that real solutions approach ideal solution behavior in dilute solution. That is, the behavior of the solvent in a solution approaches Raoult s law as. vi — 1, and we can write for the solvent... 

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. 

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... 

Activity ax is termed the rational activity and coefficient yx is the rational activity coefficient This activity is not directly given by the ratio of the fugacities, as it is for gases, but appears nonetheless to be the best means from a thermodynamic point of view for description of the behaviour of real solutions. The rational activity corresponds to the mole fraction for ideal solutions (hence the subscript x). Both ax and yx are dimensionless numbers. 

The behaviour of real solutions approaches that of ideal solutions at high dilution. The molar conductivity at limiting dilution, denoted A0, is... 

To deal with the properties of real solutions, most workers determine the magnitude of a pure number that can be multiplied by the mole fraction to bring the equation of state back to an ideal form ... 

Most real solutions cannot be described in the ideal solution approximation and it is convenient to describe the behaviour of real systems in terms of deviations from the ideal behaviour. Molar excess functions are defined as... 

The entropy of mixing of many real solutions will deviate considerably from the ideal entropy of mixing. However, accurate data are available only in a few cases. The simplest model to account for a non-ideal entropy of mixing is the quasi-regular model, where the excess Gibbs energy of mixing is expressed as... 

It is generally observed that as the temperature increases, real solutions tend to become more ideal and r can be interpreted as the temperature at which a regular solution becomes ideal. To give a physically meaningful representation of a system r should be a positive quantity and larger than the temperature of investigation. The activity coefficient of component A for various values of Q AB is shown as a function of temperature for t = 3000 K and xA = xB = 0.5 in Figure 9.3. The model approaches the ideal model as T - t. 

The solid curve in Figure 16.1 shows the activity of the solvent in a solution as a function of the mole fraction of solvent. If the solution were ideal, Equations (14.6) and (16.1) would both be applicable over the whole range of mole fractions. Then, fli =Xi, which is a relationship indicated by the broken line in Figure 16.1. Also, because Equation (16.1) approaches Equation (14.6) in the limit as Xj 1 for the real solution, the solid curve approaches the ideal line asymptotically as Xj 1. 

Thus, the ideal solution is a reference for the solvent in a real solution, and the activity coefficient of the solvent measures the deviation from ideality. 

Deviations from ideality in real solutions have been discussed in some detail to provide an experimental and theoretical basis for precise calculations of changes in the Gibbs function for transformations involving solutions. We shall continue our discussions of the principles of chemical thermodynamics with a consideration of some typical calculations of changes in Gibbs function in real solutions. 

K being a constant which is usually determined experimentally during cell calibration. Lj is the heat of evaporation of the solvent, the density of the solution, and c the polymer concentration. Finally, because the given deviation is valid only for ideal solutions but only real solutions can be studied in practice, the above equation is developed in a power law series with respect to c ... 

The ideal solution is a model fluid which serves as a standard to which real solution behavior can be compared. Equation 151, which characterizes the... 

Since real solutions tend toward ideality as the solute concentration decreases,... 

After reaching Equation (20) we abandoned a completely general discussion of osmotic pressure in favor of the simpler assumption of ideality. The ideal result, Equation (25), applies to real solutions in the limit of infinite dilution. The objective of this section is to examine the extension of Equation (20) to nonideal solutions or, more practically, to solutions with concentrations that are greater than infinitely dilute. 

For real solutions, the partial pressures PA, PB and total vapor pressure P deviate from the idealized limit (7.45a-c), with deviations of either sign. The following is an illustrative diagram for positive deviations (P > Fideal), showing the partial pressures (solid lines) and total pressure (heavy solid line) exceeding ideal values (dotted lines) ... 

As defined by (7.49a, b), a Henry s law solution is a more general and useful approximation than an ideal solution as defined by (7.45) or (7.47), but each of these approximations is often inadequate for real solutions at concentrations of chemical interest. 

In general, activity is merely an alternative way to express chemical potential. The general objective is to express fil in a form that emulates the ideal gas expression (6.55), but with the actual vapor pressure PL of component i (rather than that assumed from Dalton s law). The trick will be to choose a standard-state divisor in (6.55) that makes this expression valid for the components of a real solution. 

For ideal solutions, these two conventions become equivalent (P° = Pj = A Henry)5 but for real solutions the distinction between solvent (Pa) and solute (Pg) standard states must be kept in mind. 

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