Entropy unitary

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... 

We have already pointed out that, in general, wAB will vary with temperature. In this respect Wab resembles the quantities D, L, Y, and J, all of which are sensitive to temperature. When, for any process, we differentiate (60) with respect to the temperature, we obtain the change in entropy AS for the process. Now at, all temperatures the unitary term in (60) is independent of the composition of the solution and obviously, if we differentiate it with respect to the temperature, the quantity so obtained will necessarily be independent of the composition of the solution, and so will provide a unitary term in AS. We must write then... 

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. 

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... 

We can now answer the question under discussion. Since 29.1 e.u. is considerably smaller than the total difference (116.8 — 75.8) = 41.0 e.u., mentioned above, we conclude that the unitary term for the ion pair (Ag+ + I") is greater than for (Ag+ + Cl"). The contribution from the Ag+ ion and its co-sphere must be, within the experimental error, the same in the two cases. Finally then, we reach the conclusion that in water at 25°C the entropy associated with the iodide ion I" and its co-sphere is greater than that associated with the ion Cl" and its co-sphere in fact, the excess lies in the neighborhood of (41 — 29) = 12 e.u. 

The heat of solution of silver bromide in water at 25°C is 20,150 cal/mole. Taking the value of the entropy and the solubility of the crystalline solid from Tables 44 and 33, find by the method of Secs. 48 and 49 the difference between the unitary part of the partial inolal entropy of the bromide ion Br and that of the iodide ion I-. 

We notice that, although there is only one solute species on the left-hand side, there are two on the right-hand side. The process is therefore accompanied by an increase in entropy, and the AF of the process will contain a term — T AScrat,c Let us first discuss the values of the unitary terms to do this we may carry out the process in a different manner. Choosing two distant water molecules, we transfer a proton from one to the other. According to Table 12, at 25°C the work required amounts... 

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. 

Suppose that we arbitrarily set the partial molal entropy of the K+ ion equal to zero. This means that we assign to the Cl- ion the whole of the partial molal entropy of the ion pair (K+ + Cl-) that is to say, we assign to the Cl- ion, not only the unitary term for the Cl- ion, but also the cratic term for both ions, and also the unitary term for the potassium ion. 

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... 

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 30. The Unitary Partial Molal Entropy in Aqueous Solution...

The conclusions are evidently relevant to the amount of entropy lost by ions in methanol solution—see Table 29. If, however, the expression (170) is used for an atomic ion, we know that it is applicable only for values of R that are large compared with the ionic radius—that is to say, it will give quantitative results only when applied to the solvent dipoles in the outer parts of the co-sphere. The extent to which it applies also to the dipoles in the inner parts of the co-sphere must depend on the degree to which the behavior of these molecules simulates that of the more distant molecules. This can be determined only by experiment. In Table 29 we have seen that for the ion pair (K+ + Br ) and for the ion pair (K+ + Cl-) in methanol the unitary part of ASa amounts to a loss of 26.8 e.u. and 30.5 e.u., respectively, in contrast to the values for the same ions in aqueous solution, where the loss of entropy in the outer parts of the co-sphere is more than counterbalanced by a gain in entropy that has been attributed to the disorder produced by the ionic field. 

In both processes the ions are supposed to be introduced into solutions where the interionic forces are negligible. When in (191) the COJ ion is formed, the solvent in the co-sphere of this ion loses a certain amount of entropy. Likewise, in (192) when the CO, ion is formed, the solvent in the co-sphere of the ion loses precisely the same amount of entropy. At the same time, the amount of entropy associated with the thermal energy of the COJ" ion in aqueous solution is, of course, the same in (192) as in (191). In the process (192) we shall be concerned with the unitary term for the two Li+ ions in contrast to (191) where two protons are added to two II20 molecules. 

Turning next to the unitary part of AS0, this is given in Table 36 under the heading — N(dL/dT). It was pointed out in Secs. 90 and 106 that, to obtain the unitary part of AS0 in aqueous solution, one must subtract 16.0 e.u. for a uni-univalent solute, and 24.0 e.u. for a uni-divalent solute. In methanol solution the corresponding quantities are 14.0 and 21.0 e.u. In Table 36 it will be seen that, except for the first two solutes KBr and KC1, the values are all negative, in both solvents. It will be recalled that for KBr and KC1 the B-coefficients in viscosity are negative, and we associate the positive values for the unitary part of the entropy, shown in Table 29, with the creation of disorder in the ionic co-spheres. In every solvent the dielectric constant decreases with rise of temperature and this leads us to expect that L will increase. For KBr and KC1 in methanol solution, we see from Table 36 that dL/dT has indeed a large positive value. On the other hand, when these crystals dissolve in water, these electrostatic considerations appear to be completely overbalanced by other factors. 

This result holds equally well, of course, when R happens to be the operator representing the entropy of an ensemble. Both Tr Wx In Wx and Tr WN In WN are invariant under unitary transformations, and so have no time dependence arising from the Schrodinger equation. This implies a paradox with the second law of thermodynamics in that apparently no increase in entropy can occur in an equilibrium isolated system. This paradox has been resolved by observing that no real laboratory system can in fact be conceived in which the hamiltonian is truly independent of time the uncertainty principle allows virtual fluctuations of the hamiltonian with time at all boundaries that are used to define the configuration and isolate the system, and it is easy to prove that such fluctuations necessarily increase the entropy.30... 

RMT). K systems are most strongly mixing classical systems with a positive Kolmogorov entropy. The conjecture turned out valid also for less chaotic (ergodic) systems without time-reversal invariance leading to the Gaussian unitary ensemble (GUE). 

Cantor and SchimmeP provide a lucid description of the thermodynamics of the hydrophobic effect, and they stress the importance of considering both the unitary and cratic contributions to the partial molal entropy of solute-solvent interactions. Briefly, the partial molal entropy (5a) is the sum of the unitary contribution (5a ) which takes into account the characteristics of solute A and its interactions with water) and the cratic term (-R In Ca, where R is the universal gas constant and ( a is the mole fraction of component A) which is a statistical term resulting from the mixing of component A with solvent molecules. The unitary change in entropy 5a ... 

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... 

Table 6 shows thermodynamic parameters of the binding processes. In spite of the polymeric catalyst and the low molecular weight analogues the similar correla-tionship between AH and ASU is given, where unitary entropy changes, ASU, as... 

The first reason that led Latora and Baranger to evaluate the time evolution of the Gibbs entropy by means of a bunch of trajectories moving in a phase space divided into many small cells is the following In the Hamiltonian case the density equation must obey the Liouville theorem, namely it is a unitary transformation, which maintains the Gibbs entropy constant. However, this difficulty can be bypassed without abandoning the density picture. In line with the advocates of decoherence theory, we modify the density equation in such a way as to mimic the influence of external, extremely weak fluctuations [141]. It has to be pointed out that from this point of view, there is no essential difference with the case where these fluctuations correspond to a modified form of quantum mechanics [115]. 

Tegmark M. How unitary cosmology generalizes thermodynamics and solves the inflationary entropy problem. Phys. Rev. D. 2012 85 123517, 19 pages. DOI 10.1103 / PhysRevD. 85.123517... 

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