Entropy of dilution

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. 

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. 

Becaus-e of the similarity in the relations for osmotic pressure in dilute solutions and the equation for an ideal gas, van t Hoff proposed his bombardment theory in which osmotic pressure is considered in terms of collisions of solute molecules oil the semipeniieable membrane. This theoiy has a number of objections and has now been discarded. Other theories have also been put forward involving solvent bombardment on the semipermeable membrane, and vapor pressure effects. For example, osmotic pressure has been considered as the negative pressure which must be applied to the solvent to reduce its vapor pressure to that of the solution. It is, however, more profitable to interpret osmotic pressures using thermodynamic relations, such as the entropy of dilution,... 

Entropies of dilution Significantly lower than theory... 

Polymer solutions, even at very low concentrations, show considerable deviations from this law, largely due to their abnormally high entropy of dilution. It has been calculated on this basis7 that the osmotic pressure-concentration equation should be of the continued series form ... 

The entropy of dilution of a component of an ideal gas or liquid increases with the logarithm of the volume available for dilution. Specifically, the entropy gain of n moles of a component due to dilution is... 

The value for solute i in a given phase depends on two factors the intrinsic thermodynamic affinity of the solute to the phase and the dilution of the solute. The latter affects p( through entropy (i.e., the entropy of dilution). In analytical separations, solute concentrations are generally... 

Unusual Properties of PDMS. Some of the unusual physical properties exhibited by PDMS are summarized in List I. Atypically low values are exhibited for the characteristic pressure (a corrected internal pressure, which is much used in the study of liquids) (37), the bulk viscosity i, and the temperature coeflScient of y (4). Also, entropies of dilution and excess volumes on mixing PDMS with solvents are much lower than can be accounted for by the Flory equation of state theory (37). Finally, as has already been mentioned, PDMS has a surprisingly high permeability. 

Characteristic pressure, unusually low Bulk viscosity (i ), unusually low Temperature coeflScient of i, unusually low Entropies of dilution, significantly lower than predicted by theory Excess volumes on mixing, significantly lower than predicted by theory... 

In this respect, and to focus attention simply on PMA, the body of data presented by Liquori and coll, seems fairly conclusive. It should also be recalled that it had previously bear concluded from light scattering studies that the heat and entropy of dilution of PMA are negative 16). And this evidence leads Silberberg, Eliassaf and Katchalsky to assiune that there is a "particularly marked build-up of structure (of the solvent) in the solutions of PMA in other words hydrophobic bonding would greatly impose on the physico-chemical properties of this polyacid in dilute aqueous solution. In the system considered, "specific solvent effects thus appear to be extremely relevant. 

Vesnaver, G., Kranjc, Z., Pohar, C., and Skerjanc, J. Free enthalpies, enthalpies, and entropies of dilution of aqueous-solutions of alkaline-earth poly(styrenesulfonates) at different temperatures. Journal of Physical Chemistry, 1987, 91, No. 14, p. 3845-3848. 

Determine for the polymer-solvent system, (a) the temperature at which theta conditions are attained, (b) the entropy of dilution parameter 1/) and (c) the heat of dilution parameter k at 27°C. [Specific volume of polymer = 0.96 cm /g molar volume of cyclohexane at 27°C = 108.7 cm /mol.]... 

It will be observed even for the limited data in Table 3.1 that entropies of dilution (as indicated by ip) are highly variable from one polymer-solvent system to another and from one solvent to another for the same polymer depending on the geometrical character of the solvent. This is contrary to the theory developed from consideration of lattice arrangements according to which Ip should be approximately and nearly independent of the system. It may be noted that theories of polymer solutions fail to take into account the specific geometrical character of the solvent in relation to the polymer segment. This is a serious deficiency which must be borne in mind in applications of these theories. 

The corresponding standard heat and entropy can be obtained using the following equation, which includes a contribution from the entropy of dilution. 

Finally, we note that not all aqueous polymer solutions exhibit the foregoing behaviour. Thus, for example, poly(acrylic acid) and poly(acry-lamide) at low pH display positive values for the enthalpy and entropy of dilution parameters (Silberberg et ai, 1957). Day and Robb (1981) have confirmed calorimetrically that the heat of dilution of poly(acrylamide) in water is endothermic, althou, in contrast, they found that the substituted polyacrylamides poly(N-methyl acrylamide) and poly(N,N-dimethyl acrylamide) displayed negative heats of dilution. It was proposed that the endothermic character of the poly(acrylamide) solutions was a consequence of the energy required to separate the associated amide dipoles. 

---