Symmetry, numbers

Equation XVI-21 provides for the general case of a molecule having n independent ways of rotation and a moment of inertia 7 that, for an asymmetric molecule, is the (geometric) mean of the principal moments. The quantity a is the symmetry number, or the number of indistinguishable positions into which the molecule can be turned by rotations. The rotational energy and entropy are [66,67]... 

The factor of 2 in the denominator of the H2 molecule s rotational partition function is the "symmetry number" that must be inserted because of the identity of the two H nuclei. 

Innumerable reactions occur in acid catalyzed hydrocarbon conversion processes. These reactions can be classified into a limited number of reaction families such as (de)-protonation, alkyl shift, P-scission,... Within such a reaction family, the rate coefficient is assumed to depend on the type, n or m cfr. Eq. (1), of the carbenium ions involved as reactant and/or product, secondary or tertiary. The only other structural feature of the reactive moiety which needs to be accounted for is the symmetry number. The ratio of the symmetry number of the... 

SO that the total number of collisions in the AB mixture is 200jtd u the same as in the volume containing twice the amount of A and no B. Had we forgotten to take the symmetry numbers into account, then the calculated number of collisions in the volume with Pa = 20 would have been too high. 

In effect, the division by two is the result of the molecular symmetry, as specified by the character table for the group 0. In general it is useful to define a symmetry number a (= 2 in this case), as shown below. The well-known example of the importance of nuclear spin is that of ortho- arid para-hydrogen (see Section 10.9.5). 

We admit some cowardice. Most of our earlier estimates of entropy invoked symmetry numbers and/or were studies of bond cleavage reactions dominated by translational effects. We hesitate to compare isomers with the same carbon or heavy atom skeleton when effects of a few kJ mol-1 are crucial. 

E. Taskinen and K. Nummelin, J. Org. Chem., 50, 4833 (1985). These authors performed the isomer equilibration at several temperatures and so could use the experimentally derived equilibrium constant to derive the enthalpy of rearrangement. There was no need for assuming the entropy of isomerization is 0 or just determined by symmetry number corrections. 

The question of the choice of the appropriate symmetry number for con-formationally labile molecules is a subtle one and is at best an approximation. It is the opinion of this author that, if a = 18 for the n-alkanes is accepted as correct, then for the sake of consistency the a-values for all of the cycloalkanes have to be set at 2n, as shown by the following example. Consider the hypothetical reaction (36) for which a statistical correction is clearly... 

For a different view on the symmetry number of cyclopentane see Kabo and Andreevskii (1973) who prefer ct = 1 for this ring. 

Thermochemical data from the compilation of Stull et at., 1969. Entropy values are based on a 1 M standard state. The asterisk denotes symmetry-corrected quantities. Symmetry numbers were chosen as follows 18 for the n-alkanes, cis-3-hexene, dibuthyl sulphide, diethyl ether, and diethyl amine 2n for the cycloalkanes and 2 for all of the remaining ring compounds 3 for the alkanols, alkanethiols and alkyl amines 9 for the methyl alkyl sulphides... 

Avogadro s number and aRi is the symmetry number of the xr-meric cyclic oligomer, which represents a statistical correction for the number of equivalent bonds that can undergo the reverse ring-opening reaction. For Gaussian chains (57) takes the form (58) which is identical to (56), apart from the pres-... 

From Equation 4.79, it is then recognized that the isotope effect is given by a symmetry number factor and terms which depend only on the normal mode vibrational frequencies. There are no terms in the equality that depend explicitly on atomic and molecular masses or on moments of inertia. 

Here H is the Hamiltonian function for one N atomic molecule (i) and s is its symmetry number. One might have expected this result immediately from the Kirkwood formulation for the classical canonical partition function. H is a function of the 3N Cartesian momenta and the 3N Cartesian coordinates of molecule i. 

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