Jacobson-Stockmayer equation

There is, of course, an equilibrium between the monomers and polymer in the lipase-catalyzed polycondensation of dialkyl esters and glycols. In the lipase CC- or MM-catalyzed polymerization of dimethyl succinate and 1,6-hexanediol in toluene, adsorption of methanol by molecular sieves or elimination of methanol by nitrogen bubbling shifted the thermodynamic equilibrium (101). When dicar-boxy lie acid dialkyl esters and a , >-alkylene glycols were used as monomers, cyclic oligomers were formed from any monomer combinations examined (102). The yield of the cyclics depended on the monomer structure, initial concentration of the monomers, and reaction temperature. The ring-chain equilibrium was observed and the molar distribution of the cyclic species obeyed the Jacobson-Stockmayer equation. 

Keywords Cyclic polymers Jacobson-Stockmayer equation Living radical polymerization Ring-closure... 

This equation cannot actually apply to a polyelectrolyte, but can be used to estimate entropy changes. The two charged segments join together through electrostatic attraction if and only if these come within a distance (say a) less than the Debye-Huckel screening length. The volume (Vj) of closest approach will be Vj X An 13. This leads to Jacobson-Stockmayer equation written for the present system as [20]... 

Values for this function are listed by Jacobson and Stockmayer. The same equations apply to a polymer prepared from a single monomer... 

The parameter P in these equations was described by Jacobson and Stockmayer as the probability that a chain would be in an appropriate configuration for ring formation (equation ). Jacobson and Stockmayer actually set equal to (vxb ) which... 

One problem of Jacobson and Stockmayer s interpretation is observed in the cases of very small but unstrained rings, where much higher concentrations of rings were formed than were predicted. Flory and Semiyen ( ) explained this deviation by suggesting that not only did two termini have to meet within a volume Vg in order to establish a bond, they also had to approach each other from a specified direction. This direction was specified by a solid angle fraction 6a)/4iT. They explained that this term should appear in the entropy expression for process 1 in the inverse form, i.e. 4Tr/6o). In process 2, if the chains were sufficiently long, there would be no correlation between the probability for two termini of a molecule to meet within Vg and to approach within the solid angle 6a). In this case, the term 6a)/4ir would be valid for inclusion into the entropy term for process 2 and, hence, when the entropies for processes 1 and 2 were summed, these terms would cancel (and the equation for AS(3) from Jacobson and Stockmayer s theory should be valid). However, for short chains, the probability of approach... 

According to Jacobson and Stockmayer (JS), the cyclization probability is related to the mean separation of the reaction sites, and the equilibrium concentration of each cyclic oligomer is predicted to decrease according to Equation 7.1 ... 

Jacobson and Stockmayer [15] showed the fraction of chains and rings on the temperature-concentration phase plane, and found very interesting phenomena that are analogous to Bose-Einstein condensation. When the parameter B exceeds a certain critical value, 100% rings are formed below a critical concentration of polymers. In fact, when p = 1, we have the coupled equations... 

The Jacobson and Stockmayer theory [25] gives a simple theoretical expression for the molar cyclisation equilibrium constants for macrocyclics in ring-chain equilibrates. Thus Equation (6) can be used to calculate values provided the corresponding values are known. Conversely, values can be obtained by simply measuring values (shown to be equal to molar concentrations). 

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