Dilution entropy

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... 

The one-pot synthesis of 9 described above appears to afford only modest yields of azacrowns. One might wonder why any crown at all would be formed under non-high dilution conditions intended to yield only open-chained material. Vogtle suggests that this can be explained in terms of template, steric and entropy effects . These factors are of doubtless significance, but it is interesting to note that in the synthesis of poly-azamacrocycles, Richman and Atkins found that there was no significant template effect observed. The question of the template effect in Ihe syntheses of 9 has recently been addressed by Kulstad and Malmsten They conclude that the formation of 9 is assisted by the presence of alkali metal cations. 

The process described above is usually called osmosis and this usually imphes a flow of fluid in one direction or the other. If the permeating species, usually called the solvent, flows from the pure compartment to the mixture compartment then it is called osmosis pure and simple. This seems the natural process since the solvent dilutes the solution and this involves an increase in entropy and/or a decrease in free energy, so the resultant flow is spontaneous and the system tends to equihbrium. However, the starting conditions may be such that the difference of pressure... 

Consider 1 mole of a completely dissociated uni-univalent solute in aqueous solution at extreme dilution at 25°C, each positive and each negative ion having a diameter equal to 3.0 angstroms. Find from (19), in calories per mole per degree, what would be the total amount of entropy lost by the solvent in the fields of all these ions, if (19) could correctly be used for a sphere as small as 3 angstroms. 

Review of Solutions in General. In the discussion of these various examples we have noticed at extreme dilution the prevalence of the term — In Xb, or alternatively — In yB. The origin of this common factor in many different types of solutions can be shown, as we might suspect, to be of a fundamental nature. For this purpose let us make the familiar comparison between a dilute solution and a gas. Since the nineteenth century it has been recognized that the behavior of any solute in extremely dilute solution is, in some ways, similar to that of a gas at low pressure. Now when a vessel of volume v contains n particles of a perfect gas at a lixed temperature, the value of the entropy depends on the number of particles per unit volume, n/v. In fact, when an additional number of particles is introduced into the vessel, the increment in the entropy, per particle added, is of the form... 

That is to say, however many different factors contribute to the constant term that is independent of the concentration, the other term that depends on the concentration should, at extreme dilution, be of the form — In yB. Since we identify1 the entropy with A (In Wtk + hi Wcf), (58) is the form that we have obtained in each of the examples examined above. 

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... 

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]

Let us now consider the same charged sphere immersed in various liquids with widely different values of n. By diluting water with dioxane at constant temperature, we can reduce n from 3.3 X 1022 toward zero. Clearly when n, the number of dipoles per unit volume, approaches zero, the total entropy lost per unit volume must approach zero. From this point of view the expression (170) is seen to have a somewhat paradoxical appearance, since e, which, according to Table 32, is roughly proportional to n, occurs in the denominator. This means that, as the number of dipoles per cubic centimeter decreases, the total amount of entropy lost progressively increases. The reason for this is that, when... 

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. 

Variational principle on the correlation problem, 213, 318, 320, 326 Vibrational electronic modes, 12 Vibrational entropy in dilute solutions, 133... 

Part (a) is the driving force for the adsorption. If only (a) were present, adsorbed chains would lie flat on the surface. Parts (b) and (c) are the opposing forces (b) accounts for the entropy loss of a bond on the surface as compared to the solution, (c) represents the separation into a concentrated surface phase and a dilute solution. Part (d) arises from polymer-polymer, solvent-solvent and polymer-solvent interactions, which usually favour accumulation of segments. At equili-... 

The thermodynamic linear expansion factor has been related to Flory or thermodynamic interaction parameter, %, and the entropy of dilution parameter, Xs, through the Flory-Fox [10] equations. 

Hence the theoretical configurational entropy of mixing AaSm cannot be compared in an unambiguous manner with the experimentally accessible quantity ASm- It should be noted that the various difficulties encountered, aside from those precipitated by the character of dilute polymer solutions, are not peculiar to polymer solutions but are about equally significant in the theory of solutions of simple molecules as well. 

The chemical potential difference —ju may be resolved into its heat and entropy components in either of two ways the partial molar heat of dilution may be measured directly by calorimetric methods and the entropy of dilution calculated from the relationship A i = (AHi —AFi)/T where AFi=/xi —/x or the temperature coefficient of the activity (hence the temperature coefficient of the chemical potential) may be determined, and from it the heat and entropy of dilution can be calculated using the standard relationships... 

Fig. 113.—Comparison of observed entropies of dilution (points and solid lines with results calculated for ASi according to Eq. (28) (broken line). Data for polydimethyl-siloxane, M =3850, in benzene, A (Newing ), obtained from measured activities and calorimetric heats of dilution. Entropies for polystyrene (Bawn et in methyl ethyl ketone,, and in toluene, O, were calculated from the temperature coefficient of the activity. The smoothed results for benzene solutions of rubber, represented by the solid curve without points, were obtained similarly.

Gee and Orr have pointed out that the deviations from theory of the heat of dilution and of the entropy of dilution are to some extent mutually compensating. Hence the theoretical expression for the free energy affords a considerably better working approximation than either Eq. (29) for the heat of dilution or Eq. (28) for the configurational entropy of dilution. One must not overlook the fact that, in spite of its shortcomings, the theory as given here is a vast improvement over classical ideal solution theory in applications to polymer solutions. 

The heat and entropy of dilution may be derived by differentiation, but the resulting expressions are unwieldy. It is preferable to undertake the evaluation of F2, or of 2, at different temperatures and then to deduce the primary entropy and heat of dilution parameters and Ki by means of the equations given above (see below). 

Entropy of dilution parameters xj/i are calculable, according to Eq. (7), from the slopes of the lines in Fig. 122. Values obtained in this manner are 0.65 and 1.055 for the polyisobutylene and the polystyrene systems, respectively. These are considerably higher than the values... 

If we now calculate Cm from Eq. (7), the results of the foregoing analysis yield numerical values for the entropy of dilution parameters ypi in the various solvents. From the 0 s obtained simultaneously, the heat of dilution parameter Ki — 0 pi/T may be computed. To recapitulate, the value of in conjunction with gives at once Cm i(1--0/T). Acceptance of the value of Cm given by Eq. (7) as numerically correct makes possible the evaluation of the total thermodynamic interaction i(l —0/7"), which is equal to ( i—/ci). If the temperature coefficient is known, this quantity may be resolved into its entropy and energy components. 

It will be observed that entropies of dilution (as indicated by i) are highly variable from one system to another. This is contrary to the theory developed from consideration of lattice arrangements, according to which pi should be approximately 1/2 and nearly independent of the system. For polystyrene in methyl ethyl ketone, the entropy of dilution is nearly zero i.e., this solvent is a poor one not because of an adverse energy of interaction but because of the low entropy. First neighbor interactions apparently contribute to the entropy as well as to the energy, a point which was emphasized in Chapter XII. It will be noted also that cyclic solvents almost without exception are more favorable from the standpoint of the entropy than acyclic ones. 

ASiy ASt Corresponding partial molar entropies of dilution. 

Parameter characterizing the entropy of dilution of polymer with solvent. 

Frank, H. S. Evans, M. W. (1945). Entropy in binary liquid mixtures partial molal entropy in dilute solutions structure and thermodynamics in aqueous electrolytes. Journal of Chemical Physics, 13, 507-32. 

The net retention volume and the specific retention volume, defined in Table 1.1, are important parameters for determining physicochemical constants from gas chromatographic data [9,10,32]. The free energy, enthalpy, and. entropy of nixing or solution, and the infinite dilution solute activity coefficients can be determined from retention measurements. Measurements are usually made at infinite dilution (Henry s law region) in which the value of the activity coefficient (also the gas-liquid partition coefficient) can be assumed to have a constant value. At infinite dilution the solute molecules are not sufficiently close to exert any mutual attractions, and the environment of each may be considered to consist entirely of solvent molecules. The activity... 

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