Energy free, of mixing

For the spontaneous merger of two phases to occur, the following condition must prevail  

Ssoiute The unit of AH, AS, and AGmix is Joules per Kelvin.11 The 0 temperature is that temperature where AHmix = 0 and polymer dissolution exhibiting ideal behavior is instigated by AS only. 

By itself, ASmix is incapable of predicting spontaneity and randomness this is demonstrated in crystallization and helix formation that anomalously result in a high degree of order (-AS), but are nevertheless spontaneous processes more significantly driven by a loss of latent heat ( — AH). 

For some constant-temperature, physical processes, e.g., phase separation and sedimentation, there is a corresponding d(AGmix) for every dp  

Contemporary polymer theory considers segments of the primary structure to be the statistical unit comparable in size to that of solvent molecules. The large number of segments in polymers and the small scale of AGmix, A Hmix, and A.Smix allow their thermodynamics to be preferably described statistically (Smith, 1982), thereby permitting the following equations  

If we consider now a relation of the type represented in (10.1.1) and (10.1.2), we may write the following expression for the maximum amount of work recoverable from mixing under ideal conditions when a mole of each feed stream (/i for concentrate, /2 for sea water) is mixed with the other to produce a product stream (j = p) dumped into the sea  

Generally the reverse osmosis concentrate is at a high pressure (less than that of the entering high-pressure feed brine) this pressure energy is recovered to a large extent in energy recovery devices. The above expression assumes that the concentrate pressure has been reduced to atmospheric. 

Another way to carry out mixing of two saline solutions would be what is practiced in the concept known as osmotic power plant. Suppose you have a saline water (1) at a pressure (Pi) less than its osmotic pressure ( rj. Let the aqueous solution (2) on the other side of the reverse osmosis membrane (having a lower osmotic pressure 1C2 Tti) be present at atmospheric pressure, Pg. If the osmotic pressure of this solution 2 is less than the pressure of the saline water 1 (i.e. 2 P ), then water from solution 2 will go to solution 1 and will dilute it However, this water will increase the volume of the saline solution present at pressure Pi. Therefore one could use this extra energy through an appropriate hydraulic device/arrangement (Loeb, 1976). 

Dilute solutions of solutes/macrosolutes are ffequendy encountered in practice. Examples are a dilute fermentation broth containing small volatile molecules such as alcohol, small nonvolatile molecules such as amino acids, large protein molecules that are secreted externally or those that are obtained in a cell lysate from a cell culture. In these cases, recovery of these molecules in a substantially pure condition is the separation goal. Fractionation of isotopes where the desired isotope is present in a very dilute solution is another problem of the same type (separation of D2O from H2O separation of jgojopgg 

There are considerable differences in the separation problems discussed above. In the case of purification of proteins produced by fermentation or cell culture, one needs to concentrate and then purify to obtain as pure a protein as possible that is now substantially free from 

Having calculated the entropy and enthalpy contributions to mixing, these can now be combined to give the expression for the free eneigy of mixing, AG = ABP — TAS as 

It is more useful to express Equation 8.30 in terms of the chemical potentials of the pure solvent (Pi), by differentiating the expression with respect to the number of solvent molecules, to obtain the partial molar Gibbs free energy of dilution (after multiplying by Avogadro s number). 

This could also be carried out for the polymer (N2), but as it makes no difference which one is taken (both having started from AG ), Equation 8.31 is more convenient to use. Although this expression is not strictly vahd for the dilute solution regime, it can be converted into a structure that is extremely informative about deviations from ideal solution behavior encoimtered when measuring the molar mass by techniques such as osmotic pressure. If the logarithmic term is expanded using a Taylor series, 

This can be modified by remembering that r = (VJV and (jtj = CjVj, where Vj is the partial specific volume of the polymer. This can be related to the polymer molecular weight through Vj = so that (t )2/r) = CjVi/Mj and, finally. 

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We may also define a free energy of mixing [240]. The alternative (and equally acceptable) definition of G given in Eq. 111-87 is... 

Fig. IV-22. Excess free energy of mixing of condensed films of octadecanol-docosyl sulfate at 25°C, at various film pressures. Top curve t = 5 dyn/cm bottom curve ir = 50 dyn/cm intermediate curves at 5-dyn/cm intervals. The curves are uncorrected for the mixing term at low film pressure. (From Ref. 246.)...

The molar Helmholtz free energy of mixing (appropriate at constant volume) for such a synnnetrical system of molecules of equal size, usually called a simple mixture , is written as a fiinction of the mole fraction v of the component B... 

Figure A2.5.15. The molar Gibbs free energy of mixing versus mole fraetionxfor a simple mixture at several temperatures. Beeause of the synuuetry of equation (A2.5.15) the tangent lines indieating two-phase equilibrium are horizontal. The dashed and dotted eiirves have the same signifieanee as in previous figures.

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... 

Solubility Parameter. CompatibiHty between hydrocarbon resins and other components in an appHcation can be estimated by the Hildebrand solubiHty parameter (2). In order for materials to be mutually soluble, the free energy of mixing must be negative (3). The solubiHty of a hydrocarbon resin with other polymers or components in a system can be approximated by the similarities in the solubiHty parameters of the resin and the other materials. Tme solubiHty parameters are only available for simple compounds and solvents. However, parameters for more complex materials can be approximated by relative solubiHty comparisons with substances of known solubiHty parameter. 

Basic Thermodynamics. Equilibrium-phase behavior of mixtures is governed by the free energy of mixing and how this quantity, consisting of enthalpic... 

One proposed approach (75) to modeling the phase behavior for hydrogen bonding pairs uses the following expression for the free energy of mixing (eq. 7). 

Practical Solubility Concepts. Solution theory can provide a convenient, effective framework for solvent selection and blend formulation (3). When a solute dissolves in a solvent, a change in free energy occurs as a result of solvent—solute interactions. The change in free energy of mixing must be negative for dissolution to occur. In equation 1,... 

Fig. 25. Relationship between the measured interfacial strength and the (negative) Gibbs free energy of mixing, (-AG )o5, for glass beads treated with various silane coupling agents embedded in a PVB matrix. Error bars correspond to 95% mean confidence intervals. Redrawn from ref. [165].

Most polymer pairs are thermodynamically incompatible, in the sense that their free energy of mixing is positive. This does not mean that there is absolutely no interdiffusion at all at the interface between them adjacent to the interface limited interdiffusion occurs, which can be seen as an increasing of the low surface entropy implied by a smooth surface [30-33]. This nanoscale roughening of an interface can increase the adhesion between the polymers. 

In a thermodynamic sense, the compatibility of polymers is similar to the dissolving solute in a solvent. The thermodynamic standard of solubility is the free energy of mixing Ga. If AGm < 0, then two components are soluble to each other. According to the definition ... 

The free energy of mixing Umix for the fee Cu-Zn alloys is shown as a function of concentration in Fig. 1. It is obtained from the usual formula... 

Figure 1. The free energies of mixing of fee disordered alloys. The filled eireles eonneeted with a solid line are the energies ealeulated with the LSMS. The erosses eonneeted with a dotted line are the energies calculated with the CPA-LSMS without the Conlomb energy, while the open circles connected with dotted lines include the Conlomb contribution. The plusses connect with a dashed-dotted line are the energies calculated with the SCF-KKR-CPA without the Coulomb energy, while the squares connected with dashed-dotted lines include the Coulomb contribution.

Considering an (incompressible) polymer mixture with volume fraction (j)A = (j) of A-monomers and volume fraction (j)B = 1 - (j) of B monomers, the mean-field expression for the excess-free energy of mixing is given by the well-known Flory-Huggins expression " ... 

The contribution that (46) makes to the free energy of mixing is — kT In Wc and it will be noticed that, if the right-hand side of (47) is multiplied by kT, it becomes identical with (45), which is the total change in the free energy, when an ideal solution is formed from its components. 

To obtain AmixGm, the molar Gibbs free energy of mixing, we divide equation (7.5) by n — J2 n> t0 obtain... 

Here the first term is the free energy of pure crystalline lead, the second term an assumed linear dependence on composition, and the third term the free energy of mixing lead and thallium in the atomic positions 0, J J, 0, J and J, J, 0 of the unit cube. Curves representing these expressions are shown in Fig. 7. The value of a... 

Solubility occurs where the free energy of mixing, AG , is negative. This value is related to the enthalpy of mixing, AH, and the entropy of mixing, AiSjjj, by the Gibbs equation ... 

Substituting back into equation (5.3), Raoult s law shows that the free energy of mixing (or dilution) may be given by ... 

The symbol Xs stands for the interaction energy per solvent molecule divided by kT. Combining equations (5.7) and (5.8) gives the Flory-Huggins equation for the free energy of mixing of a polymer solution ... 

Lohse et al. have summarized the results of recent work in this area [21]. The focus of the work is obtaining the interaction parameter x of the Hory-Huggins-Stavermann equation for the free energy of mixing per unit volume for a polymer blend. For two polymers to be miscible, the interaction parameter has to be very small, of the order of 0.01. The interaction density coefficient X = ( y/y)R7 , a more relevant term, is directly measured by SANS using random phase approximation study. It may be related to the square of the Hildebrand solubility parameter (d) difference which is an established criterion for polymer-polymer miscibility ... 

If the configurational entropy A>Sm is assumed to represent the total entropy change LSm on mixing, the free energy of mixing is simply obtained by combining Eqs. (10) and (20). That is,... 

If xi contains an entropy contribution, the form of the chemical potential is unaltered but its resolution into entropy and heat contributions must be carried out according to operations like those applied above to the free energy of mixing. 

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