Entropy of solution

More fundamental treatments of polymer solubihty go back to the lattice theory developed independentiy and almost simultaneously by Flory (13) and Huggins (14) in 1942. By imagining the solvent molecules and polymer chain segments to be distributed on a lattice, they statistically evaluated the entropy of solution. The enthalpy of solution was characterized by the Flory-Huggins interaction parameter, which is related to solubihty parameters by equation 5. For high molecular weight polymers in monomeric solvents, the Flory-Huggins solubihty criterion is X A 0.5. 

Heat of Precipitation. Entropy of Solution and Partial Molal Entropy. The Unitary Part of the Entropy. Equilibrium in Proton Transfers. Equilibrium in Any Process. The Unitary Part of a Free Energy Change. The Conventional Standard Free Energy Change. Proton Transfers Involving a Solvent Molecule. The Conventional Standard Free Energy of Solution. The Disparity of a Solution. The E.M.F. of Galvanic Cells. 

Entropy of Solution and Partial Molal Entropy. If the heat of solution of a solute is known, and the free energy of solution is known at some low concentration, then the entropy of solution AS at the same concentration can at once be found from the relation... 

Consider, for example, the saturated solution of a sparingly soluble crystal. Let AII>a, and AS, t denote the heat of solution and the entropy of solution when a few additional pairs are taken into the saturated solution. The condition for equilibrium between the solid and the solution. is, of course, that there shall be no change in the free energy in this process a saturated solution is one for which AF is zero. Hence we may write at once... 

To obtain the entropy of solution we have only to divide the heat of solution by the temperature. 

Let us apply this to the values for silver chloride at 25° given above. Dividing 15,740 by the temperature, we obtain for the entropy of solution of crystalline AgCl in its saturated solution the value... 

In electrochemistry it is customary to multiply each of those quantities by Avogadro s constant and, when a few additional ions enter the already saturated solution, to speak of the entropy of solution per mole. Let the entropy of one mole of the crystalline solid be denoted by Scr and let Si and S2 denote, respectively, the entropy of the solution before, and after, the entry of the additional solute, both expressed in calories per mole. The total initial entropy is obviously (S + Si) and the final entropy is St. The difference between the final and the initial entropy is by definition AS,at. 

Since the saturated solutions of AgT and AgCl are both very dilute, it is of interest to examine their partial molal entropies, to see whether we can make a comparison between the values of the unitary terms. As mentioned above, the heat of precipitation of silver iodide was found by calorimetric measurement to be 1.16 electron-volts per ion pair, or 26,710 cal/mole. Dividing this by the temperature, we find for the entropy of solution of the crystal in the saturated solution the value... 

Conventional Partial Molal Ionic Entropies. Correlation between Ionic Entropy and Viscosity. Conventional Partial Molal Entropy of (H30)+ and (OH)-. The Conventional and the Unitary Entropy of Solution. Solutes in Aaueous Solution. Solutes in Methanol Solution. 

The Conventional and the Unitary Entropy of Solution. In Sec. 55 we discussed the free energy of solution, by considering the quantity (AF — 2RT In x). Again taking a uni-univalent solute, let us now fix attention on the quantity... 

The left-hand side of (165) or (166) gives the unitary part of the entropy of solution. In electrochemistry, however, it is the left-hand side of (167) which is the conventional entropy of solution at infinite dilution usually denoted by A[Pg.179]

With the usual 1000 grams of solvent as the b.q.s. we have in aqueous solution M = 55.51 thus 2R In M is equal to 16 e.u. To obtain the unitary part of the entropy of solution of a uni-univalent crystal in water at any temperature, we have to subtract 16.0 e.u. from the conventional... 

The molar entropy of solution of a crystal—the AS0 of (168)—may be regarded as the difference between (1) the molar entropy of the crystal, and (2) the partial molal entropy of the solute in a solution of a certain Concentration. The question arises In a solution of what concentration Now we notice that (168) may be written in the form... 

Solutes in Aqueous Solution. As mentioned in See. 88, when we say that we expect to find a correlation between the /1-coefficients of viscosity of various species of ions, and their entropy of solution, this refers only to the unitary part of the entropy, the part associated with the ionic co-sphere. We are inclined to adopt the view that a negative //-coefficient for a pair of ions should be accompanied by a positive increment in entropy, while a positive //-coefficient should be accompanied by a decrease in entropy. The values of AS0, the conventional entropy of solution, to be found in the literature, do not, give a direct answer to this question, since they contain the cratic term, which in water at room temperature amounts to 16 e.u. This must be subtracted. 

Table 27 contains data for some uni-univalent solutes for which both the entropy of solution at 25°C and the viscosity //-coefficient in aqueous solution at 18 or 25° are known. In column 3 from the entropy of solution 16.0 e.u. have been subtracted for the cratic term. 

Solutes in Methanol Solution. In Table 23 we have seen that for four solutes in methanol the viscosity //-coefficients are positive. This is the case even for KC1 and KBr, for which the coefficients are negative in aqueous solution. In Sec. 88 it was pointed out that it would be of interest to see whether this inversion is likewise accompanied by a change in sign for the ionic entropy. Although no accurate values for the entropy of solution of salts in methanol arc available, reliable estimates have been made for KC1, KBr, and NaCl.1 Since the /1-coefficients of KC1 and KBr have been determined both in methanol and in water, all the required data are available for these two solutes. The values of A/S" given in Table 29 have been taken from Table 34 in Chapter 12, where the method of derivation is explained. The cratie term included in each of these values is 14 cal/deg, as already mentioned in Sec. 90. 

Table 29. Entropies of Solution and Viscosities in Methanol and in Water...

The Number of Dipoles per Unit Volume. The Entropy Change Accompanying Proton Transfers. The Equilibrium between a Solid and Its Saturated Solution. Examples of Values of L and AF°. The Change of Solubility with Temperature. Uni-divalent and Other Solutes. Lithium Carbonate in Aqueous Solution. H2COj in Aqueous Solution. Comparison between HjCOj and Li2C03 in Aqueous Solution. Heats of Solution and the Conventional Free Energies and Entropies of Solution. 

Turning next to AgBr, we see from Table 33 that the value of L increases from 0.931 electron-volt at 15° to 0.935 at 35°, a difference of 0.004. Dividing by 20, we find that the average value of dL/dT in the neighborhood of 25° is 2 X 10 1 electron-volt/deg. Multiplying by 23,060, we find this is equivalent to 4.6 cal/mole. It follows that the value of the conventional entropy of solution AS0 in the neighborhood of 25°C is approximately... 

Heats of Solution and the Conventional Free Energies and Entropies of Solution. Table 34 gives for various common salts the observed values of the heat of solution, and the conventional free energy... 

The heat of solution tends at extreme dilution to the value +4207 cal/mole. Calculate the conventional free energy of solution at 25°C and the conventional entropy of solution. 

As seen from Tables 23 and 21 the ion pair (K+ + Cl") increases the viscosity of methanol but diminishes that of water. We recall that the values for the entropy of solution in Table 29 show a parallel trend in the galvanic cells of Sec. 112 placed back to back, this difference in ionic entropy between aqueous and methanol solutions would alone be sufficient to give rise to an e.m.f. We must ask whether this e.m.f. would be in the same direction, or in the direction opposite to the e.m.f. that would result from a use of (199). 

Electrostriotion, 188, 190-192 Energy levels, 34, 151-152 Entropy, of crystals, 95, 180, 211, 267 partial molal (see Partial molal entropy) of solution (see Solution)... 

On the other hand, upon closer examination even the copper-gold solid solutions evince serious discrepancies with the quasichemical theory. There is a composition range where the entropy of solution is larger than that for random mixing (see Fig. 1) where... 

Fig. 1 The entropy of solution of face-centered cubic solid solutions of Cu-Au at 427°C, from reference 41.

Other ordering systems show striking discrepancies with the predictions of the quasi-chemical theories. Cu-Pt,67 Co-Pt,38 and Pb-Tl36 are binaries the solid solutions of which exhibit a positive partial excess free energy for one of their components, as well as positive excess entropies of solution. Co-Pt goes even further in deviating from theory in that it has a positive enthalpy of solution,... 

It is simplest to consider these factors as they are reflected in the entropy of the solution, because it is easy to subtract from the measured entropy of solution the configurational contribution. For the latter, one may use the ideal entropy of mixing, — In, since the correction arising from usual deviation of a solution (not a superlattice) from randomness is usually less than — 0.1 cal/deg-g atom. (In special cases, where the degree of short-range order is known from x-ray diffuse scattering, one may adequately calculate this correction from quasi-chemical theory.) Consequently, the excess entropy of solution, AS6, is a convenient measure of the sum of the nonconfigurational factors in the solution. 

Four of the solid solutions of Table III have excess entropies of solution which include contributions from magnetic disordering in both the alloy and in one or both of the pure components. These contributions can be quite large, and since there is no assurance... 

Bismuth, excess entropy of solution of noble metals in liquid bismuth, 133 Block polymers, 181 Bond energies in the halogens, 61 Boron fluoride as initiator in polymerization, 156... 

Lead, excess entropy of solution of noble metals in, 133 Lead-thalium, solid solution, 126 Lead-tin, system, energy of solution, 143 solution, enthalpy of formation, 143 Lead-zinc, alloy (Pb8Zn2), calculation of thermodynamic quantities, 136 Legendre expansion in total ground state wave function of helium, 294 Lennard-Jones 6-12 potential, in analy-... 

In addition to chemical reactions, the isokinetic relationship can be applied to various physical processes accompanied by enthalpy change. Correlations of this kind were found between enthalpies and entropies of solution (20, 83-92), vaporization (86, 91), sublimation (93, 94), desorption (95), and diffusion (96, 97) and between the two parameters characterizing the temperature dependence of thermochromic transitions (98). A kind of isokinetic relationship was claimed even for enthalpy and entropy of pure substances when relative values referred to those at 298° K are used (99). Enthalpies and entropies of intermolecular interaction were correlated for solutions, pure liquids, and crystals (6). Quite generally, for any temperature-dependent physical quantity, the activation parameters can be computed in a formal way, and correlations between them have been observed for dielectric absorption (100) and resistance of semiconductors (101-105) or fluidity (40, 106). On the other hand, the isokinetic relationship seems to hold in reactions of widely different kinds, starting from elementary processes in the gas phase (107) and including recombination reactions in the solid phase (108), polymerization reactions (109), and inorganic complex formation (110-112), up to such biochemical reactions as denaturation of proteins (113) and even such biological processes as hemolysis of erythrocytes (114). 

---