Ideal solutions, mixing

The first two terms describe mechanical mixing of endmembers A and B, the third is the ideal solution mixing term, and the last term is the regular solution contribution, in which solid solution composition. (If a> is not so independent, the solution is not regular). The term (oNJ n in Eq. (1.39) is also sometimes called the excess Gibbs free energy of mixing, or AG (excess). 

For the ideal solution, mixing is a spontaneous process that is entropically driven. 

Since the entropy of mixing is always positive, and the heat of mixing is zero for an ideal solution, mixing in all proportions always occurs spontaneously. 

Therefore, the molar Gibbs enthalpy of mixing of an ideal solution is zero at all temperatures and all pressures. This means that the components of an ideal solution mix without the need for heat... 

Therefore the compounds of the ideal solution mix without any change in volume. This means that the volume of one mole of the ideal solution is the sum, weighted per molar fraction, of molar volumes of pure compounds. This result is valid at all temperatures and all pressures. 

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 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-... 

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. 

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

All three quantities are for the same T, P, and physical state. Eq. (4-126) defines a partial molar property change of mixing, and Eq. (4-125) is the summability relation for these properties. Each of Eqs. (4-93) through (4-96) is an expression for an ideal solution property, and each may be combined with the defining equation for an excess property (Eq. [4-99]), yielding ... 

At the same time it is recognized that the pairs of substances which, on mixing, are most likely to obey Raoult s law are those whose particles are most nearly alike and therefore interchangeable. Obviously no species of particles is likely to fulfill this condition better than the isotopes of an element. Among the isotopes of any element the only difference between the various particles is, of course, a nuclear difference among the isotopes of a heavy element the mass difference is trivial and the various species of particles are interchangeable. Whether the element is in its liquid or solid form, the isotopes of a heavy element form an ideal solution. Before discussing this problem we shall first consider the solution of a solid solute in a liquid solvent. 

The contribution that (46) makes to the free energy of mixing is — kT In Wc and it will be noticed that, if the right-hand side of (47) is multiplied by kT, it becomes identical with (45), which is the total change in the free energy, when an ideal solution is formed from its components. 

We will see later that this same equation applies to the mixing of liquids or solids when ideal solutions form. 

This result is nearly equal to 4.87 J K hmoT. the value that would be calculated for the entropy of mixing to form an ideal solution. We will show in Chapter 7 that the equation to calculate AmixSm for the ideal mixing process is the same as the one to calculate the entropy of mixing of two ideal gases. That is. AmixSm = -R. Vj ln.Yj. 

The reason is that classical thermodynamics tells us nothing about the atomic or molecular state of a system. We use thermodynamic results to infer molecular properties, but the evidence is circumstantial. For example, we can infer why a (hydrocarbon + alkanol) mixture shows large positive deviations from ideal solution behavior, in terms of the breaking of hydrogen bonds during mixing, but our description cannot be backed up by thermodynamic equations that involve molecular parameters. 

Ideal solution behavior over extended ranges in both composition and temperature requires that the following conditions be fulfilled (i) the entropy of mixing must be given by ... 

The Gibbs energy of mixing of an ideal solution is negative due to the positive entropy of mixing obtained by differentiation of Ald.xGm with respect to temperature ... 

Figure 3.3 Thermodynamic properties of an arbitrary ideal solution A-B at 1000 K. (a) The Gibbs energy, enthalpy and entropy, (b) The entropy of mixing and the partial entropy of mixing of component A. (c) The Gibbs energy of mixing and the partial Gibbs energy of mixing of component A.

The simplest model beyond the ideal solution model is the regular solution model, first introduced by Hildebrant [9]. Here A mix, S m is assumed to be ideal, while A inix m is not. The molar excess Gibbs energy of mixing, which contains only a single free parameter, is then... 

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