Entropy of dilution parameter

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. 

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. 

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.]... 

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. 

The macroscopic thermodynamic approach Croucher and Hair (1978) have followed Eichinger (1970) in considering the temperature dependence of x in macroscopic thermodynamic terms. In the classical Flory-Huggins theory, the enthalpy and entropy of dilution parameters Ki and are constants related to the d-temperature by 0 = (ici/( i)r and to Xi by (i—Xi) = l i —ki. These relationships lead immediately to... 

Patterson (1968) has also derived expressions for the enthalpy and entropy of dilution parameters... 

Since the excess chemical potential is a fi-ee energy then % must also be a free energy, i.e. it has both an entropy and an enthalpy component. Flory defined enthalpy and entropy of dilution parameters, Ki and respectively, such that the partial molar enthalpy of dilution was... 

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). 

Parameter characterizing the entropy of dilution of polymer with solvent. 

Flory (1953) has defined enthalpy (>ci) and entropy (i i) of dilution parameters whereby... 

Fortunately, there is an independent, universal method that in principle allows the type of steric stabilization to be ascertained. Determination of the entropy and enthalpy (kj) of dilution parameters for the stabilizing... 

Flory and Krigbaum defined an enthalpy (Kj) parameter and an entropy of dilution ( /i) parameter such that the thermodynamic functions used to describe these long-range effects are given in terms of the excess partial molar quantities... 

The parameter will vary with PS molecular weight and with the free energy of dilution (Flory, 1953), kT jj2 — Q/T)v, where is the entropy of dilution, 9 is the Flory theta temperature, and V2 is the volume fraction of solute. 

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... 

Recently, the Pitzer equation has been applied to model weak electrolyte systems by Beutier and Renon ( ) and Edwards, et al. (10). Beutier and Renon used a simplified Pitzer equation for the ion-ion interaction contribution, applied Debye-McAulay s electrostatic theory (Harned and Owen, (14)) for the ion-molecule interaction contribution, and adoptee) Margules type terms for molecule-molecule interactions between the same molecular solutes. Edwards, et al. applied the Pitzer equation directly, without defining any new terms, for all interactions (ion-ion, ion-molecule, and molecule-molecule) while neglecting all ternary parameters. Bromley s (1) ideas on additivity of interaction parameters of individual ions and correlation between individual ion and partial molar entropy of ions at infinite dilution were adopted in both studies. In addition, they both neglected contributions from interactions among ions of the same sign. 

This assumption is based on the fact that the polymer-solvent interaction parameter [see Eq. (8)] of the tributyrin-cellulose tributyrate system, as evaluated from melting-point depressions, is nearly zero at about 100° C [Mandelkern and Flory (160)]. It does not follow, however, that the system is athermal, for the parameter generally involves an entropy contribution. Furthermore, the heat and entropy parts of this parameter vary with the concentration in a complicated way, especially in polar systems [see, for example, Takenaka (243) Zimm (22) Kurata (154)]. Thus it is extremely hazardous to predict dilute solution properties from concentrated solution properties such as the melting-point depression, at least on a highly quantitative level as in the present problem. 

The thermodynamic parameters 1/) and K introduced above, pertaining to polymer-solvent interactions in dilute solutions, may be determined from thermodynamic studies of dilute solutions of the polymer, e.g., from osmotic pressure or turbidity measurements at different temperatures. These parameters may also be determined, at least in principle, from viscosity measurements on polymer solutions (see Frictional Propcitics of Polymers). The parameter ij, which is a measure of the entropy of mixing, appears to be related to the spatial or geometrical character of the solvent. For those solvents having cyclic structures, which are relatively compact and symmetrical (e.g., benzene, toluene, and cyclohexane), xp has relatively higher values than for the less symmetrical acyclic solvents capable of assuming a number of different configurations. Cyclic solvents are thus more favorable... 

The separation of these cumulative effects is not an easy task, but is necessary for the determination of thermodynamic parameters, such as chemical bond strengths. Measuring very dilute water solutions at 3.9 °C, where the thermal expansion coefficient of water vanishes (or at slightly lower temperatures in more concentrated aqueous solutions, such as buffer solutions) can be used to separate the so-called structural volume changes from the thermal effects due to radiationless deactivation.253,254 In this way, it is also possible to determine the entropy changes concomitant with the production or decay of relatively short-lived species (e.g. triplet states), a unique possibility offered by these techniques.254 255... 

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