Isomeric partition functions

The geometry optimizations will end with a set of m isomeric structures. Their relative concentrations can be expressed as their mole fractions, w,-, using the isomeric partition functions q. In terms of and the ground-state energy changes AT/q, the mole fractions are given [195-197] ... 

The examples cited above are only two of the many possible cases of H-bond isomerization. Because of the low kinetic barriers separating these species, equilibration of H-bonded isomer populations to limiting thermodynamic values is generally expected to be much faster than for covalent isomers. Methods of quantum statistical thermodynamics can be used to calculate partition functions and equilibrium population distributions for H-bonded isomers,41 just as in the parallel case for covalent isomers and conformers. 

N 069 " Rotational Isomeric State Treatment of the Cystine Residue. Configuration Partition Function and... 

If we want to know the equilibrium constant for the isomerization A B, we need again K = NB/NA. But we have NA and NB summed up over all the energy states in the partition functions,... 

It is easy to list the various chemical and biological events influenced by flexibility, but unfortunately efforts to quantitate this structural attribute have been few. Mann analyzed the conformation of alkanes by modifying the number of gauche arrangements with Pitzer s steric partition function. Luisiranked alkanes on a. scale of conformational rigidity based on three-states rotational isomerism. Unfortunately, these schemes are designed for acyclic hydrocarbons and have no inherent capability to be adapted to heteroatomic molecules. 

For dissociation and isomerization of molecules, the activated complex is usually obtained by fixing the value of the reaction coordinate representing a linear combination of the bond lengths. If neither molecules nor the activated complex possess internal rotations, it may be put with high accuracy that Fj t = E f Since F — F, the ratio F /F is in fact only that of vibrational partition functions. If the temperatures are high enough (kT hc i, hwf), then Fy b = kT/hcOj, and... 

In Sect. 2.1 we carried out a first estimate of the fraction of trans- and gauche-states in polyethylene. In this estimate, independence of the rotational isomeric states of different bonds was implicitly assumed. We now may check for the modification introduced by the pentane effect, as the Ising-model also provides us with equations for the fractions of the different pairs of conformational states. We rewrite the partition function... 

Because of the third assumption, these theories are called Rotational Isomeric State (RIS) models. Adopting these approximations, the partition function can be considerably simplified and the statistical mechanics of chain molecules can be mapped onto the well-known ID Potts model with K states. [20]... 

The problem can be further simplified taking Into account only a discrete number of rotational states (which are chosen in general to be coincident with conformational potential energy minima), in the calculation of the partition function. This is the rotational isomeric model, first proposed by Volkenstein and principally de-velopped by Flory and his school. 

The shape of the curve may be explained in terms of the previous considerations as arising from an intrinsic torsional potential and the interaction between the nonbonded methyl groups.There are three minima of the e-nergy, corresponding to T,G, G conformations, the last two being energetically equivalent. For the evaluation of the partition function these are chosen as rotational isomeric states, and their statistical weights are ... 

The conformational partition function, being the sum of all the terms appearing in the table, may be written for n-pentane, in the rotational isomeric approximation,... 

The configuration partition function, representing the sum of all such factors, one for each conformation of the chain as represented by the scheme of rotational Isomeric states, is... 

The HLB numbers decrease with increasing chain length, e.g., from 13.25 for sodium decane 1-sulfonate to 9.45 for the C18 homolog [72]. Typical HLB numbers for positional isomers range from 12.3 for sodium dodecane 1-sulfonate to 13.2 for the more hydrophilic 6 isomer [73]. The HLB numbers of alkanesulfonates are less influenced by the isomeric position of the functional group and by substituents than the cM values [68]. HLB numbers can be correlated with partition coefficients for the distribution of a surfactant between the aqueous and oily phases, which emphasizes that the partition coefficient is dependent on the carbon number [68]. 

The transport and partitioning of an organic compound in the environment is a function of the physical and chemical properties of that compound and the site-specific characteristics of the environment (e.g., percentage soil organic matter). Based on the environmental correlations with physical properties (Lyman et al. 1982), the physical and chemical properties of the three isomeric cresols are sufficiently similar to indicate that similar transport and partitioning processes will be... 

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