Ideal solutions

The ideal solution is the vector of the best values achievable for each criterion. In other words, if // = max fi, then the ideal solution = fu fi Because 

In an ideal solution the components obey Raoult s law. The activity equals the mole fraction  

The partial and integral molar free energies of mixing are, therefore, given by AG, d = R TlnXA AG, d = R TlnXB AGM,id = RT(XA lnXA + XB lnXB) 

The equality of activity and mole fraction in an ideal solution has interesting consequences. These consequences, in fact, are the characteristics of an ideal solution, and are presented in the following. 

At constant temperature and composition, the variation of free energy with pressure is related to the volume of the system as 

Expressing this equation in terms of relative partial molar quantities, the following relationships are obtained  

Our master, Leonardo da Vinci, once said, Think of the end before the beginning. In other words, all inventors should always think not only how to solve a problem but how to develop the ideal solution or a concept for the ideal engineering system in a given design situation. 

The Ideal Solution is a concept of the ideal engineering system in a given design situation, that is, a concept for a system providing all required useful functions, creating no harmful functions, and not using any resources. 

Design a small chamber for testing various metals in acid. 

Testing requires providing direct contact between the acid and the metal. Our initial thought is that the metal specimen should be surrounded by acid—submerged in a chamber filled with acid. In fact, our inventive challenge was so formulated that this understanding of the problem is directly implied. However, if we analyze the available resources, we will realize that we have available only the metal to be tested and some acid. If the chamber 

In this situation, the chamber should be designed using the metal available for testing. In this way, no other materials will be used and the inventive problem will be solved utilizing only the available resources. We will have the ideal solution. 

the forces between like and unlike molecules are the same and 

As in the case of the ideal gas, an ideal solution is also a hypothetical fluid. Here, however, intermolecular forces are present, but we assume that those between unlike molecules are the same with those between like molecules. 

If an ideal liquid solution is in equilibrium with the vapor above it that braves as an ideal gas, the relationship between the compositions of the two phases is described analytically by Raoult s law  

P partial pressure of component i, i.e. the part of the total pressure due to component /  

mole fraction of component i in the liquid and vapor phases respectively  

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If the parameters were to become increasingly correlated, the confidence ellipses would approach a 45 line and it would become impossible to determine a unique set of parameters. As discussed by Fabrics and Renon (1975), strong correlation is common for nearly ideal solutions whenever the two adjustable parameters represent energy differences. 

Raoult s law When a solute is dissolved in a solvent, the vapour pressure of the latter is lowered proportionally to the mole fraction of solute present. Since the lowering of vapour pressure causes an elevation of the boiling point and a depression of the freezing point, Raoult s law also applies and leads to the conclusion that the elevation of boiling point or depression of freezing point is proportional to the weight of the solute and inversely proportional to its molecular weight. Raoult s law is strictly only applicable to ideal solutions since it assumes that there is no chemical interaction between the solute and solvent molecules. 

A fairly simple treatment, due to Guggenheim [80], is useful for the case of ideal or nearly ideal solutions. An abbreviated derivation begins with the free energy of a species... 

Figure III-9u shows some data for fairly ideal solutions [81] where the solid lines 2, 3, and 6 show the attempt to fit the data with Eq. III-53 line 4 by taking ff as a purely empirical constant and line 5, by the use of the Hildebrand-Scott equation [81]. As a further example of solution behavior, Fig. III-9b shows some data on fused-salt mixtures [83] the dotted lines show the fit to Eq. III-SS.

The data in Table III-2 have been determined for the surface tension of isooctane-benzene solutions at 30°C. Calculate Ff, F, F, and F for various concentrations and plot these quantities versus the mole fraction of the solution. Assume ideal solutions. 

Condensed phases of systems of category 1 may exhibit essentially ideal solution behavior, very nonideal behavior, or nearly complete immiscibility. An illustration of some of the complexities of behavior is given in Fig. IV-20, as described in the legend. 

If an ideal solution is formed, then the actual molecular A is just Aav (and Aex = 0). The same result obtains if the components are completely immiscible as illustrated in Fig. IV-21 for a mixture of arachidic acid and a merocyanine dye [116]. These systems are usually distinguished through the mosaic structure seen in microscopic evaluation. 

Various functional forms for / have been proposed either as a result of empirical observation or in terms of specific models. A particularly important example of the latter is that known as the Langmuir adsorption equation [2]. By analogy with the derivation for gas adsorption (see Section XVII-3), the Langmuir model assumes the surface to consist of adsorption sites, each having an area a. All adsorbed species interact only with a site and not with each other, and adsorption is thus limited to a monolayer. Related lattice models reduce to the Langmuir model under these assumptions [3,4]. In the case of adsorption from solution, however, it seems more plausible to consider an alternative phrasing of the model. Adsorption is still limited to a monolayer, but this layer is now regarded as an ideal two-dimensional solution of equal-size solute and solvent molecules of area a. Thus lateral interactions, absent in the site picture, cancel out in the ideal solution however, in the first version is a properly of the solid lattice, while in the second it is a properly of the adsorbed species. Both models attribute differences in adsorption behavior entirely to differences in adsorbate-solid interactions. Both present adsorption as a competition between solute and solvent. 

Alternative approaches treat the adsorbed layer as an ideal solution or in terms of a Polanyi potential model (see Refs. 12-14 and Section XVII-7) a related approach has been presented by Myers and Sircar [15]. Adsorption rates have been modeled as diffusion controlled [16,17]. 

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. 

As the time separation 11 -, s approaches w tlie second tenn in this correlation vanishes and the remaining tenn is the equilibrium density-density correlation fomuila for an ideal solution. The second possibility is to consider a non-equilibrium initial state, c(r, t) = (r). The averaged solution is [26]... 

The entropy of a solution is itself a composite quantity comprising (i) a part depending only on tire amount of solvent and solute species, and independent from what tliey are, and (ii) a part characteristic of tire actual species (A, B,. ..) involved (equal to zero for ideal solutions). These two parts have been denoted respectively cratic and unitary by Gurney [55]. At extreme dilution, (ii) becomes more or less negligible, and only tire cratic tenn remains, whose contribution to tire free energy of mixing is... 

At the outset it will be profitable to deal with an ideal solution possessing the following properties (i) there is no heat effect when the components are mixed (ii) there is no change in volume when the solution is formed from its components (iii) the vapour pressure of each component is equal to the vapour pressure of the pure substances multiplied by its mol fraction in the solution. The last-named property is merely an expression of Raoult s law, the vapour pressure of a substance is pro-... 

Let us consider a mixture forming an ideal solution, that is, an ideal liquid pair. Applying Raoult s law to the two volatile components A and B, we have ... 

Solutions in water are designated as aqueous, and the concentration of the solution is expressed in terms of the number of moles of solvent associated with 1 mol of the solute. If no concentration is indicated, the solution is assumed to be dilute. The standard state for a solute in aqueous solution is taken as the hypothetical ideal solution of unit molality (indicated as std. state or ss). In this state... 

Boiling points versus composition diagram for a near-ideal solution, showing the progress of a distillation. 

There is a parallel between the composition of a copolymer produced from a certain feed and the composition of a vapor in equilibrium with a two-component liquid mixture. The following example illustrates this parallel when the liquid mixture is an ideal solution and the vapor is an ideal gas. 

An ideal gas obeys Dalton s law that is, the total pressure is the sum of the partial pressures of the components. An ideal solution obeys Raoult s law that is, the partial pressure of the ith component in a solution is equal to the mole fraction of that component in the solution times the vapor pressure of pure component i. Use these relationships to relate the mole fraction of component 1 in the equilibrium vapor to its mole fraction in a two-component solution and relate the result to the ideal case of the copolymer composition equation. 

We define Fj to be the mole fraction of component 1 in the vapor phase and fi to be its mole fraction in the liquid solution. Here pj and p2 are the vapor pressures of components 1 and 2 in equihbrium with an ideal solution and Pi° and p2° are the vapor pressures of the two pure liquids. By Dalton s law, Plot Pi P2 Pi/Ptot these are ideal gases and p is propor-... 

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). 

Comparing Eqs. (8.29) and (8.30) also leads to the conclusion expressed by Eq. (8.22) aj = Xj. Again we emphasize that this result applies only to ideal solutions, but the statistical approach gives us additional insights into the molecular properties associated with ideality in solutions ... 

Since the 0 s are fractions, the logarithms in Eq. (8.38) are less than unity and AGj is negative for all concentrations. In the case of athermal mixtures entropy considerations alone are sufficient to account for polymer-solvent miscibility at all concentrations. Exactly the same is true for ideal solutions. As a matter of fact, it is possible to regard the expressions for AS and AGj for ideal solutions as special cases of Eqs. (8.37) and (8.38) for the situation where n happens to equal unity. The following example compares values for ASj for ideal and Flory-Huggins solutions to examine quantitatively the effect of variations in n on the entropy of mixing. 

Evaluate ASj for ideal solutions and for athermal solutions of polymers having n values of 50, 100, and 500 by solving Eqs. (8.28) and (8.38) at regular intervals of mole fraction. Compare these calculated quantities by preparing a suitable plot of the results. 

We express the calculated entropies of mixing in units of R. For ideal solutions the values of are evaluated directly from Eq. (8.28) ... 

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 ... 

Vapor pressure lowering. Equation (8.20) shows that for any component in a binary liquid solution aj = Pj/Pi°. For an ideal solution, this becomes... 

One way to describe this situation is to say that the colligative properties provide a method for counting the number of solute molecules in a solution. In these ideal solutions this is done without regard to the chemical identity of the species. Therefore if the solute consists of several different components which we index i, then nj = S nj j is the number of moles counted. Of course, the total mass of solute in this case is given by mj = Sjnj jMj j, so the molecular weight obtained for such a mixture is given by... 

Precisely the same substitutions and approximations that we wrote in Eq. (8.71) can be applied again to In aj. Thus for the case of dilute, ideal solutions, Eq. (8.78) becomes... 

Calculate Apj, ATf, ATj, and n for solutions which are 1% by weight in benzene of solutes for which M = 10 and 10 . Assume that these solutions are adequately described by dilute ideal solution expressions. Consult a handbook for the physical properties of benzene. Comment on the significance of the results with respect to the feasibility of these various methods for the determination of M for solutes of high and low molecular weight. 

Equation (8.85) can be solved for Xj and the latter substituted into Eq. (8.83) to give an expression for H which applies to dilute-but not necessarily ideal—solutions ... 

At equilibrium, a component of a gas in contact with a liquid has identical fugacities in both the gas and liquid phase. For ideal solutions Raoult s law applies ... 

Mixtures. A number of mixtures of the hehum-group elements have been studied and their physical properties are found to show Httle deviation from ideal solution models. Data for mixtures of the hehum-group elements with each other and with other low molecular weight materials are available (68). A similar collection of gas—soHd data is also available (69). 

For ideal solutions (7 = 1) of a binary mixture, the equation simplifies to the following, which appHes whether the separation is by distillation or by any other technique. 

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

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 ... 

The summabihty relation (eq. 122) appHed to the special case of an ideal solution is written as equation 194 ... 

Equations for the mixture properties of an ideal solution foUow immediately. 

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