Entropy mixing example

When equilibrium has been established after removing the barrier, the two liquids together occupy a volume of about 1000 mL. Formation of a homogeneous solution has increased disorder, or randomness, because the molecules of each substance are now mixed and distributed in a volume twice as large as that which they occupied before mixing. The amoxmt of disorder in the system is given by a thermod5mamic quantity called entropy. This example illustrates our second basic principle Processes in which the disorder (entropy) of the systan increases tend to occur spontaneously. 

The thermodynamic excess functions differ from the thermodynamic functions of mixing only for quantities which involve the entropy. For example, the excess enthalpy A is identical with the enthalpy of mixing given by (1.6.6). Furthermore the excess volume v is identical with the volume of mixing given by (1.6.7). The excess entropy (in terms of activity coefficients) is given by (cf. 1.6.5)... 

For example, the expansion of a gas requires the release of a pm holding a piston in place or the opening of a stopcock, while a chemical reaction can be initiated by mixing the reactants or by adding a catalyst. One often finds statements that at equilibrium in an isolated system (constant U, V, n), the entropy is maximized . Wliat does this mean ... 

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. 

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. 

Figure 8.1 The entropy of mixing (in units of R) as a function of mole fraction solute for ideal mixing and for the Flory-Huggins lattice model with n = 50, 100, and 500. Values are calculated in Example 8.1.

Remember that the hump which causes the instability with respect to phase separation arises from an unfavorable AH considerations of configurational entropy alone favor mixing. Since AS is multiplied by T in the evaluation of AGj, we anticipate that as the temperature increases, curves like that shown in Fig. 8.2b will gradually smooth out and eventually pass over to the form shown in Fig. 8.2a. The temperature at which the wiggles in the curve finally vanish will be a critical temperature for this particular phase separation. We shall presently turn to the Flory-Huggins theory for some mathematical descriptions of this critical point. The following example reminds us of a similar problem encountered elsewhere in physical chemistry. 

Most of the assumptions are based on idealized models, indicating the limitations of the mathematical methods employed and the quantity and type of experimental data available. For example, the details of the combinatorial entropy of a binary mixture may be well understood, but modeling requires, in large measure, uniformity so the statistical relationships can be determined. This uniformity is manifested in mixing rules and a minimum number of adjustable parameters so as to avoid problems related to the mathematics, eg, local minima and multiple solutions. 

One practical example of demixing that might be attributed to a difference in crystallizability is the incompatibility in blends of polymers with different stereochemical compositions. The stereochemical isomers contain both chemical and geometrical similarities, but differ in the tendency of close packing. In this case, both the mixing energy B and the additional mixing entropy due to structural asymmetry between two kinds of monomers are small. However, the stereochemical differences between two polymers will result in a difference in the value of Ep. Under this consideration, most experimental observations on the compatibility of polymer blends with different stereochemical compositions [89-99] are tractable. For more details, we refer the reader to Ref. [86]. 

The excess molar entropy of mixing is the real entropy of mixing minus the ideal entropy of mixing. Using a binary A-B solution as an example, Ae xSm s... 

Exchange of unimers between two different types of block copolymer micelles has often been referred to as hybridization. This situation is more complex than for the case described above because thermodynamic parameters now come into play in addition to the kinetic ones. A typical example of such hybridization is related to the mixing of micelles formed by two different copolymers of the same chemical nature but with different composition and/or length for the constituent blocks. Tuzar et al. [41] studied the mixing of PS-PMAA micelles with different sizes in water-dioxane mixtures by sedimentation velocity measurements. These authors concluded that the different chains were mixing with time, the driving force being to reach the maximum entropy. 

The inner structure of polyelectrolyte multilayer films has been studied by neutron and X-ray reflectivity experiments by intercalating deuterated PSS into a nondeut-erated PSS/PAH assembly [94, 99]. An important lesson from these experiments is that polyelectrolytes in PEMs do not present well-defined layers but are rather interpenetrated or fussy systems. As a consequence, polyelectrolyte chains deposited in an adsorption step are intertwined with those deposited in the three or four previous adsorption cycles. When polyelectrolyte mobility is increased by immersion in NaCl 0.8 M, the interpenetration increases with time as the system evolves towards a fully mixed state in order to maximize its entropy ]100]. From the point of view of redox PEMs, polyelectrolyte interpenetration is advantageous in the sense that two layers of a redox polyelectrolyte can be in electrochemical contact even if they are separated by one or more layers of an electroinactive poly ion. For example, electrical connectivity between a layer of a redox polymer and the electrode is maintained even when separated by up to 2.5 insulating bUayers [67, 101-103]. 

An example of such behavior is met with in aqueous ethanol (EtOH) at, say, 75°C. At Xboh< 0.24, the heat of mixing is negative (heat is evolved on mixing the components), but at higher ethanol contents it is positive (heat is absorbed and the mixture cools). In the equimolar mixture AH = 220 but AGab = 960 J mol, due to a negative entropy of mixing, but since... 

Diffusion in general, not only in the case of thin films, is a thermodynamically irreversible self-driven process. It is best defined in simple terms, such as the tendency of two gases to mix when separated by a porous partition. It drives toward an equilibrium maximum-entropy state of a system. It does so by eliminating concentration gradients of, for example, impurity atoms or vacancies in a solid or between physically connected thin films. In the case of two gases separated by a porous partition, it leads eventually to perfect mixing of the two. 

Perfluoropolymers have an extremely low degree of intermolecular inter-achon. Thus, the entropy of mixing is the dominant driving force for solubihty. Fluids that have even weak enthalpic interachons with themselves are poorer solvents for perfluoropolymers. For example, aromahe and other unsaturated perfluorocarbons are poorer solvents than their saturated counterparts. Using... 

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