Mixing process entropy change

FIGURE 2.4. A spontaneous process of racemi-zation is performed along a different route in three steps. First, the system of 2N identical particles is deassimilated into two groups, each of which contains N particles. One group of d particles is then transformed into / particles. Finally, the two distinguishable components d and / are mixed. The entropy changes only in the deassimilation process in the first step. 

The entropy changes ASa and ASB can be calculated from equation (2.69), which applies to the isothermal reversible expansion of ideal gas, since AS is independent of the path and the same result is obtained for the expansion during the spontaneous mixing process as during the controlled reversible expansion. Equation (2.69) gives... 

Further understanding of the kinetic of template polymerization needs consideration of the process entropy. Applying a well known lattice model, it is easy to see that entropy changes, AS, in free polymerization and the template polymerization, differs considerably. According to the principles of statistical thermodynamics, the entropy of mixing is given by the equation ... 

The entropy of a solution is increased by the mixing of solvents, and it is decreased by interactions among the solvent molecules or interactions of solutes with the solvent. The mixing of two miscible liquids is a thermodynamically favorable process because it increases the number of positions available to the molecules. The entropy change on going from the unmixed liquids to the mixed state can be calculated from the expression... 

The addition of a diluent to a crystalline polymer depresses its melting point, as is shown schematically in Figure 2.43. The upper sketch again shows the standard reference case. In the lower sketch, solvent molecules are available to mix with the polymer chains once they separate from the crystalline lattice. The final state is now a polymer solution, instead of a molten polymer. This additional disordering greatly increases the entropy change for the process and therefore decreases the melting point, frequently to the extent of 40-50 °C. 

The mixing of two gaseous substances, or of two non-polar liquids, are further examples of entropy-driven processes. These involve negligible enthalpy changes (no strong chemical bonds are formed or broken) but the increased randomness and disorder in the system lead to a positive entropy change. 

Next, the probability function Ptj for the maximum and minimum values of 1(0, R) is discussed mathematically. The self-entropy H(C) in Eq. (2.38) is decided only by the fraction of each component in the feed, and the value does not change through the mixing process. Then, the maximum and minimum values of the mutual information entropy are determined by the value of the conditional entropy H(C/R). Since the range of the variable j is fixed as l[Pg.70]

Separation and mixing. For a separation process, the changes in enthalpy and entropy are generally positive and negative, respectively. So their direction factor becomes negative and its absolute value is rather large, because the enthalpy change is relatively small. For example, when 250 moles of aqueous solution of 40 mol% methanol is distilled and separated into aqueous... 

Here Sq is the actual uptake of heat and Sq is the additional heat that would have been absorbed from region II for a reversible change, and hence it is a positive quantity. In an actual change of state, the entropy increment dS contains the quantity of entropy Sq /T. This may be, for example, due to a mixing process or a chemical reaction within region I. The relation of Camot-Clausius gives the change of the entropy of a closed system... 

Entropies, enthalpies, and volume changes arising from the mixing process may be readily deduced from Eqs. (3.13.3)-(3.13.7), and are left as an exercise (Exercise 3.13.3). 

It may be noted, incidentally, that the practical or virtual entropy also does not include the entropy of mixing of different isotopic forms of a given molecular speciee. This quantity is virtually unchanged in a chemical reaction, and so the entropy change of the process is unaffected by its complete omission. 

The Hansen characterization is usually considered a sphere, even although it is really a modified spheroid. The constant 4 in Equation 10.4 modifies the spheroid to a sphere. The cohesion energy parameters of those liquids where affinities are highest are located within the sphere. The center of the sphere has the values of the bpp, and parameters, taken as characteristic for the solute. This may not be quite true because of entropy effects. The radius of the sphere, R, reflects the condition where the free energy change, AG, for the process being considered is zero. This is discussed here in terms of the mixing process, for which reason the superscript M is used. The entropy effects, and enthalpy effects, ATP, are in balance. Thus, on the sphere surface... 

We shall now show that mixing is not an irreversible process, and the entropy change, in the process depicted in figure H.l, is not due to the mixing process, but to expansion. Therefore, the reference to the quantity (H.10) or (H.12) as entropy of mixing is inappropriate and should be avoided. 

---