Dilution parameters

Figure 3. Numerical integration of Equation E2 showing net oxidant [03-N0 ] yields from an initial [NMHCJq = SOOppb. A. Variation with the product reactivity parameter Q here a=)9=l and S=l. B. Variation with the dilution parameter S here a=)9=l, Q=l. When S is small, a nearly stoichiometric yield of oxidant=1000ppb is achieved at higher values of S, oxidant and reactants are diluted with less oxidant accumulation.

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 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. 

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.]... 

Assuming that the volume of one monomer unit is the same as that of a cyclohexane molecule, estimate the theta temperature 9 and the entropic dilution parameter ip. 

Calculate from the data an estimate of the theta temperature Q and the entropic dilution parameter f/. [Polymer density = 0.92 g/cm molar volume of ethylbenzene = 122 cm /mol]... 

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

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. 

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... 

The dilution parameters are readily determined by a range of experimental measurements, such as the temperature dependence of the second virial coefficient. Alternatively, if the 0-temperature is measured separately, determination of the intramolecular expansion factor a, e.g. by viscometry, allows the values of and k, to be calculated using the Flory relationship, as... 

The signs of the dilution parameters for different types of steric stabilization... 

Of course, if k i is negative and ip, is positive, the value of 0 is negative and so it is unlikely to be known independently. In these circumstances, the measurement of a for a range of molecular weights will suffice to determine both dilution parameters. 

Ottewill and Walker (1968) have used the relationship between the interaction parameter Xi and the entropy and enthalpy dilution parameters tj/i and ic, respectively), viz i Xi =(< i to write equation (12.10) as... 

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... 

The magnitude of the effect of an impurity depends upon its amount and its nature. Fisher and Scesney have made detailed calculations for a soluble model (a mobile-electron ferromagnet), utilizing a dilution parameter X which is approximately Tg - r° /T where T is the critical temperature of the system with the impurity while 7g is that of the pure system. They find that appreciable values of X are required in order to observe a renormalization effect. For example, they find as a useful approximation for the observed jS deter-minedt by a log-log fit of experimental data in the range of t from 10 to 10 ... 

Dilute solutions were treated in the mean-field fiumework by Flory (Krig-baum and Flory 1952, Flory 1953). The concept is that although the individual polymer molecules are widely separated fi-om each other in dilute solution and exclude other polymer molecules fi-om the space pervaded by the molecule (the excluded volume principle) within the space the average chain unit concentration is uniform and the previously determined relations can be applied with little modification. The major innovation for dilute solutions is the introduction of dilution parameters. The first term on the right-hand side of equation (5.2.8) is identical to the expression obtained for an ideal solution. Consequently the second term on the right-hand side is identified as an excess contribution. Using the relation of equation (5.2.7) we can write down the excess chemical potential as... 

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 implications of the dilution parameter are most relevant when estimating individual doses from exposure to final products. The selected dilution parameters, pertaining to mass-specific clearance levels for those nuclides that are both gamma emitters and dominantly recovered in the final product (e.g. Co) are the following ... 

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