Mixing molar entropy

The partial molar entropy, enthalpy or volume of mixing can be derived from eq. (3.21) and are given by the relations... 

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... 

The molar entropy of mixing for the ideal solution follows as... 

Figure 9.4 The molar entropy (a) and enthalpy (b) of mixing of a quasi-chemical solution A-B at 1000 K for selected values of yAB withzA = zB =2. The distribution of pairs AA,BB and AB for the same conditions is given in (c).

Figure 9.6 (a) Molar entropy of mixing of ideal polymer solutions for r = 10, 100 and 1000 plotted as a function of the mole fraction of polymer compared with the entropy of mixing of two atoms of similar size, r = 1. (b) Activity of the two components for the same conditions. 

The ideal molar entropy of mixing is then given by... 

The relative partial molar enthalpies of the species are obtained by using Eqs. (70) and (75) in Eq. (41). When the interaction coefficients linear functions of T as assumed here, these enthalpies can be written down directly from Eq. (70) since the partial derivatives defining them in Eq. (41) are all taken at constant values for the species mole fractions. Since the concept of excess quantities measures a quantity for a solution relative to its value in an ideal solution, all nonzero enthalpy quantities are excess. The total enthalpy of mixing is then the same as the excess enthalpy of mixing and a relative partial molar enthalpy is the same as the excess relative partial molar enthalpy. Therefore for brevity the adjective excess is not used here in connection with enthalpy quantities. By definition the relation between the relative partial molar entropy of species j, Sj, and the excess relative partial molar entropy sj is... 

The excess molar enthalpy hV is simply the heat of mixing at constant pressure related to 1 mole of solution.) And from the excess molar enthalpy and the excess chemical potential, we can obtain the excess molar entropy of the system from the following equation ... 

As already pointed out, Yu is 1 if a compound forms an ideal solution. In this rather rare case, the term RTkiyu, which we denote as partial molar excess free energy of compound i in solution t, Gpe, is 0. This means that the difference between the chemical potential of the compound in solution and its chemical potential in the reference state is only due to the different concentration of the compound i in the two states. The term R In xtf=S 1 expresses the partial molar entropy of ideal mixing (a purely statistical term) when diluting the compound from its pure liquid (xiL =1) into a solvent that consists of otherwise like molecules. 

Once the species present in a solution have been chosen and the values of the various equilibrium constants have been determined to give the best fit to the experimental data, other thermodynamic quantities can be evaluated by use of the usual relations. Thus, the excess molar Gibbs energies can be calculated when the values of the excess chemical potentials have been determined. The molar change of enthalpy on mixing and excess molar entropy can be calculated by the appropriate differentiation of the excess Gibbs energy with respect to temperature. These functions depend upon the temperature dependence of the equilibrium constants. 

The same form of equation can be used to calculate the standard molar entropy S (iso) of the isomer group. The entropy of formation of the isomer group is equal to the mole-fraction-weighted entropy of formation plus the entropy of mixing the isomers. 

Generally, the chemical potential of a constituent substance i in a mixture consists of a unitary part, which is inherent to the pure substance i and independent of its concentration, and a communal part, which depends on the concentration of constituent i [Ref. 3.]. The communal part of the chemical potential of a constituent i in a mixture arises from the entropy of mixing of i For an ideal mixture the molar entropy of mixing of i, s,M, is given from Eq. 3.51 by = -j ln x, and hence the communal part of the chemical potential is expressed by p 4 = -TsM = RT nx, at constant temperature, where x, is the molar fraction of... 

The molar entropy of mixing of the ideal solution is thus not equal to zero. This is due to the fact that the mixing of pure components proceeds spontaneously and is connected with rearrangement of the melt and thus with an increase in entropy. 

If 1 mole of exchange sites is being considered, then S is equal to Avogadro s number and the gas constant, R, can replace kS to give the molar entropy of mixing ... 

Experimental values for some of the partial molar quantities can be obtained from laboratory measurements on mixtures. In particular, mixture density measurements can be used to obtain partial molar volumes, and heat-of-mixing data yield information on partial molar enthalpies. Both of these measurements are considered here. In Chapter 10 phase equilibrium measurements that provide information on the partial molar Gibbs energy of a component in a mixture are discussed. Once the partial molar enthalpy and partial molar Gibbs energy are known at the anie temperature, the partial molar entropy can be computed from the relation S-, = (G — H-,)/T. 

Fig. 10.1. Hypothetical processes in which two ideal gases A and B, originally separated by a partition, mix to fonn an ideal gaseous solution with a change in molar entropy equaling...

Since any impure crystal has at least the entropy of mixing at the absolute zero, its entropy cannot be zero such a substance does not follow the third law of thermodynamics. Some substances that are chemically pure do not fulfill the requirement that the crystal be perfectly ordered at the absolute zero of temperature. Carbon monoxide, CO, and nitric oxide, NO, are classic examples. In the crystals of CO and NO, some molecules are oriented differently than others. In a perfect crystal of CO, all the molecules should be lined up with the oxygen pointing north and the carbon pointing south, for example. In the actual crystal, the two ends of the molecule are oriented randomly it is as if two kinds of carbon monoxide were mixed, half and half. The molar entropy of mixing would be... 

Construct a graph of the molar entropy of mixing AS IR vs. volume fraction for a polymer solution where the polymer degree of polymerization Zj = 10 and 1000. You should do this by plotting AS IR values starting... 

For an isothermal-isobaric addition of a small amount of component i to a mixture, the reversible heat effect is given by the change in partial molar entropy on mixing. 

---