Definition of the ideal solution

The ideal gas law is an important example of a limiting law. As the pressure approaches zero, the behavior of any real gas approaches that of the ideal gas as a limit. Thus all real gases behave ideally at zero pressure, and f or practical purposes they are ideal at low finite pressures. From this generalization of experimental behavior, the ideal gas is defined as one that behaves ideally at any pressure. 

We arrive at a similar limiting law from observing the behavior of solutions. For simplicity, we consider a solution composed of a volatile solvent and one or more involatile solutes, and examine the equilibrium between the solution and the vapor. If a pure liquid is placed in a container that is initially evacuated, the liquid evaporates until the space above the liquid is filled with vapor. The temperature of the system is kept constant. At equilibrium, the pressure established in the vapor is the vapor pressure of the pure liquid (Fig. 13.1a). If an involatile material is dissolved in the liquid, the equilibrium vapor pressure p over the solution is observed to be less than over the pure liquid (Fig. 13.1b). 

Since the solute is involatile, the vapor consists of pure solvent. As more involatile material is added, the pressure in the vapor phase decreases. A schematic plot of the vapor pressure of the solvent against the mole fraction of the involatile solute in the solution, %2 is shown by the solid line in Fig. 13.2. At X2 = 0, p = p° as X2 increases, p decreases. The important feature of Fig. 13.2 is that the vapor pressure of the dilute solution (X2 near zero) approaches the dashed line connecting p° and zero. Depending on the particular combination of solvent and solute, the experimental vapor-pressure curve at higher concentrations of solute may fall below the dashed line, as in Fig. 13.2, or above it, or even lie exactly on it. However, for all solutions the experimental curve is tangent to the dashed line at X2 = 0, and approaches the dashed line very closely as the solution becomes more and more dilute. The equation of the ideal line (the dashed line) is 

If X is the mole fraction of solvent in the solution, then x X2 = h and the equation 

Raoult s law is another example of a limiting law. Real solutions follow Raoult s law more closely as the solution becomes more dilute. The ideal solution is defined as one that follows Raoult s law over the entire range of concentrations. The vapor pressure of the solvent over an ideal solution of an involatile solute is shown in Fig. 13.3. All real solutions behave ideally as the concentration of the solutes approaches zero. 

A solution will be said to be ideal if the chemical potential of every component is a linear function of the logarithm of its mole fraction according to the relation 

The advantage of using (8 14) as the definition of ideality, rather than Baoult s or Henry s laws, is that the present chapter is brought more closely into relation with Chapter 3 on gaseous mixtures. In fact an equation of the same form as (8 14) has already been used in 3 9 for the definition of the ideal gaseous solution. All mixtures which are called ideal—and also the special type of gaseous mixture which is called perfect—show the same dependence of the chemical potentials on the composition. 

A solution is ideal only if (8 14) applies to every component in a given region of composition. However, it is not necessary that (8 14) shall apply over the whole range of composition. A solution may approximate to ideal behaviour in one region and not in others as shown in the last section, there are grounds for expecting that a solution will approach closer and closer to ideality the more dilute it becomes in all but one of its components. 

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Because thermodynamics describes macroscopic behaviors, we need a macroscopic definition of the ideal solution in addition to the microscopic description given above. We define an ideal solution as one that, for each of its components, at all T and P and over the entire range of concentrations,... 

Begin with the equilibrium relationship given by Eq. (1-8) and the definition of the ideal solution K value given by Eq. (1-10) and obtain the following formulas for the Kb method for nonideal solutions... 

Often we know or can easily compute values of the fugacity for the pure component at the mixture T and at some convenient pressure P then it is natural to base the definition of the ideal solution on this known pure-component fugacity. To do so, we choose x" = 1 then the slope of the ideal-solution straight line is... 

The activities of the various components 1,2,3. .. of an ideal solution are, according to the definition of an ideal solution, equal to their mole fractions Ni, N2,. . . . The activity, for present purposes, may be taken as the ratio of the partial pressure Pi of the constituent in the solution to the vapor pressure P of the pure constituent i in the liquid state at the same temperature. Although few solutions conform even approximately to ideal behavior at all concentrations, it may be shown that the activity of the solvent must converge to its mole fraction Ni as the concentration of the solute(s) is made sufficiently small. According to the most elementary considerations, at sufficiently high dilutions the activity 2 of the solute must become proportional to its mole fraction, provided merely that it does not dissociate in solution. In other words, the escaping tendency of the solute must be proportional to the number of solute particles present in the solution, if the solution is sufficiently dilute. This assertion is equally plausible for monomeric and polymeric solutes, although the... 

This is in accordance with the definition of an ideal solution given in Section 3.2. 

The relation between Raoult s law and the definition of an ideal solution given by Equation (8.57) is obtained by a study of Equation (10.35) or (10.40). If a solution is ideal, then A/i must be zero and the right-hand side of both equations must be zero. If we write Pyt in both equations as Pt, the partial pressure of the component, and Pyj in Equation (10.40) as P[, then the logarithmic term becomes lnfP P ), which is zero when Raoult s law, given in the form Pl = P[xl, is obeyed. We then see that to define an ideal solution in terms of Raoult s law and still be consistent with Equation (10.57) requires that the experimental measurements be made at the same total pressure and that the vapor behaves as an ideal gas. 

For solutions comprised of species of equal molecular volume in which all molecular interactions are the same, one can show by the methods of statistical thermodynamics that the lowest possible value of the entropy is given by an equation analogous to Eq. (10.7). Thus we complete the definition of an ideal solution by specifying that its entropy be given by the equation ... 

The use of the foregoing definition of an ideal solution implies certain properties of such a solution. The variation of the fugacity / of a pure liquid i with temperature, at constant pressure and composition, is given by equation (29.22), viz.. 

The rigid requirement of the ideal solution that every component obey Raoult s law over the entire range of composition is relaxed in the definition of the ideal dilute solution. To arrive at the laws governing dilute solutions, we must examine the experimental behavior of these solutions. The vapor-pressure curves for three systems are described below. 

A typical definition of an ideal solution is one that obeys Raoult s Law. This definition is asymptotically correct in dilute solution. The osmotic pressure can then be expressed as ... 

The excess functions used here are in excess of those which apply to a particular kind of ideal solution viz. that (essentially) given by Raoult s law. This choice of Ideality is arbitrary and for some situations a different definition of ideal solution may be more suitable. Further, choosing (essentially) Raoult s law as our definition of an ideal solution, we are naturally led to Che use of mole fraction x as our choice of composition variable. That is not necessarily the best choice and there are several cases (notably, polymer solutions and solutions of electrolytes) where other measures of composition are much more convenient. 

If, as shown above, for ideal gas mixtures the fugacity of one species in the mixture is equal to its partial pressure, then we would like to extend that simple idea to nonideal gas mixtures, and to solutions of liquids and solids. We can, using the definition of an ideal solution. An ideal solution is like an ideal gas in the following respects ... 

The definition of the fugacity of a species in solution is parallel to the definition of the pure-species fugacity. An equation analogous to the ideal gas expression, Eq. (4-73), is written for species i in a flmd mixture ... 

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. 

Another definition of an ideal polarized electrode is based on the practical form of this electrode. At an ideal polarized electrode either no exchange of charged particles takes place between the electrode and the solution or—if thermodynamically feasible—exchange occurs very slowly as a result of the large activation energy. 

According to the definition of the A-B bond dissociation enthalpy, reactants and products in reaction 5.1 must be in the gas phase under standard conditions. That is to say that those species are in the ideal gas phase, implying that inter-molecular interactions do not exist. DH (A-B) refers, therefore, to the isolated molecule AB, and it does not contain any contribution from intermolecular forces. Though this is obviously the correct way of defining the energetics of any bond, there are many literature examples where bond dissociation enthalpies have been reported in solution. 

Crystals Obtained by Acid Addition. Figure 4 shows the effect of initial solution composition on the impurity content of crystals obtained by acid addition. Clearly, this corresponds to the definition of an ideal system as presented above. These data show the order followed in impurity incorporation in the L-Ile crystals is L-Val > L-Leu > L-a-ABA, although there is only one data point on a-amino butyric acid. Also, the value of purification factors for all impurities is less than one. This means that purification by crystallization was indeed occurring. 

This relationship constitutes the basic definition of the activity. If the solution behaves ideally, a, =x, and Equation (18) define Raoult s law. Those four solution properties that we know as the colligative properties are all based on Equation (12) in each, solvent in solution is in equilibrium with pure solvent in another phase and has the same chemical potential in both phases. This can be solvent vapor in equilibrium with solvent in solution (as in vapor pressure lowering and boiling point elevation) or solvent in solution in equilibrium with pure, solid solvent (as in freezing point depression). Equation (12) also applies to osmotic equilibrium as shown in Figure 3.2. 

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