Symmetry number ratio

We next show that (Q/Q ). depenis only on the symmetry number ratio and a mass factor lndependent of chemical composition. Write the Hamiltonian of the molecule in Cartesian coordinates. Then the classical partition function 

ACS Symposium Series American Chemical Society Washington, DC, 1975. 

Thus the reduced partition function ratio, (s/s )f, Is just the chemical Isotope fractionation factor of the chemical species against the gaseous atom. With the convention prime Is the light Isotope It Is easy to prove that ln(s/s )f Is always positive. This follows from the fact that u. u.. 

We return now to Eq. (10) and examine the G(u) function. It is a monotonlc positive function. Its value Is 1/2 as u goes to Infinity and It approaches zero as u goes to zero. At small u It has the value u/12. For the case of small u, Eq. (11) becomes 

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It should be emphasized that the symmetry number ratio (s2/si) is entered as a multiplier of (s2/si)f so that the symmetry number factors which lead to no isotopic enrichment in themselves are left out. To obtain the complete isotope effect one has to multiply the above expression by symmetry number ratios so that the symmetry number ratios in front of the f expressions are removed. So the symmetry number factor in Equations 4.143 and 4.144 is given by... 

A small correction for rotational symmetry number ratios has been applied to observed values to obtain the... 

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... 

How does one determine the symmetry number As illustrated in the section above it is equal to the number of rotations that take the molecule into itself. Another and very attractive method is based on the use of group theory. Students who have taken a course in inorganic chemistry have been introduced to group theory. If the reader is uncomfortable with this topic the next few paragraphs can be skipped, especially since this method of finding molecular symmetry numbers need not to be used for finding the ratios of symmetry numbers, Si/s2, required to understand isotopomer fractionation. 

Equation 4.117 makes complete sense. One of the first things one learns in dealing with phase space integrals is to be careful and not over-count the phase space volume as has already been repeatedly pointed out. In quantum mechanics equivalent particles are indistinguishable. The factor n ni is exactly the number of indistinguishable permutations, while A accounts for multiple minima in the BO surface. It is proper that this factor be included in the symmetry number. Since the BO potential energy surface is independent of isotopic substitution it follows that A is also independent of isotope substitution and cannot affect the isotopic partition function ratio. From Equation 4.116 it follows... 

Isotope effects on equilibria have been formulated earlier in this chapter in terms of ratios of (s2/si)f values, referred to as reduced isotopic partition function ratios. From Equation 4.80, we recognize that the true value of the isotope effect is found by multiplying the ratio of reduced isotopic partition function ratios by ratios of s2/si values. Using Equation 4.116 one now knows how to calculate s2/si from ratios of factorials. Note well that symmetry numbers only enter when a molecule contains two or more identical atoms. Also note that at high temperature (s2/si)f approaches unity so that the high temperature equilibrium constant is the symmetry number factor. 

The symmetry number factors are derived from the reduced isotopic partition function ratio of the RHt species. 

If one substitutes Equations4.123a into 4.124, one immediately finds the interesting result that the [H]/[D] ratio in the isotopomers of RHt-nDn is exactly equal to the ratio [H]/[D] in the gas phase atomic species. We have thus demonstrated that proper consideration of symmetry numbers leads to the result that the RHt species have [H]/[D] ratios exactly the same as those for the atomic species in the high temperature limit. QED. 

It has been previously noted that the first quantum correction to the classical high temperature limit for an isotope effect on an equilibrium constant is interesting. Each vibrational frequency makes a contribution c[>(u) to RPFR and this contribution can be expanded in powers of u with the first non-vanishing term proportional to u2/24, the so called first quantum correction. Similarly, for rates one introduces the first quantum correction for the reduced partition function ratios, includes the Wigner correction for k /k2 and makes use of relations like Equation 4.103 for small x and small y, to find a value for the rate constant isotope effect (omitting the noninteresting symmetry number term)... 

The ratio of symmetry numbers s s° in equation 11.40 merely represents the relative probabilities of forming symmetrical and unsymmetrical molecules, and ni and nf are the masses of exchanging molecules (the translational contribution to the partition function ratio is at all T equal to the power ratio of the inverse molecular weight). Denoting as AX, the vibrational frequency shift from isotopically heavy to light molecules (i.e., AX, = X° — X ) and assuming AX, to be intrinsically positive, equation 11.40 can be transated into... 

The theory of IEs was formulated by Bigeleisen and Mayer.9 The IE on the acid-base reaction of Equation (1) is defined as the ratio of its acidity constant KA to the acidity constant of the isotopic reaction, Equation (2). The ratio KJ KA is then the equilibrium constant XEIE for the exchange reaction of Equation (3). That equilibrium constant may be expressed in terms of the partition function Q of each of the species, as given in Equation (4), which ignores symmetry numbers. 

The number of special coordinates, or dimensionality of a problem, can be reduced using three basic strategies symmetry, aspect ratio and series resistances. 

The quotient of factorials in equation (10) is also equal to the ratio of symmetry numbers of and XLm. 

The energy spectrum of the nucleus according to the semi-empirical shell model [23] appears not at zero ratio, but in two disjoint parts at ratios 0.22 and 0.18. This shift relates to the appearance of the symmetric arrangement at ratio 1.04 rather than 1. An Aufbau procedure based on this result fits the 8-period table derived from the number spiral, but like the observed periodic table, at ratio r, the shell-model result also has hidden symmetry. At ratio zero, the inferred energy spectrum not only fits the 8-period table but also... 

The main effect is already taken into account if symmetry numbers are included in the densities of states. The symmetry number is a correction to the density of states that allows for the fact that indistinguishable atoms occupy symmetry-related positions and these atoms have to obey the constraints of the Pauli principle (i.e. the wave function must have a definite symmetry with respect to any permutation), whereas the classical density of states contains no such constraint. The density of states is reduced by a factor that is equal to the dimension of the rotational subgroup of the molecule. When a molecule is distorted, its symmetry is reduced, and so its symmetry number changes by a proportion that is equivalent to the number of indistinguishable ways in which the distortion may be produced. For example, the rotational subgroup of the methane molecule is T, whose dimension is 12, whereas the rotational subgroup of a distorted molecule in which one bond is stretched is C3, whose dimension is 3. The ratio of these symmetry numbers, 4, is the number of ways in which the distortion can occur, i.e. the reaction path degeneracy. 

The symmetry numbers also allow for the degeneracy of possible return paths from the transition state, which has caused problems in the past [34,32]. The only corrections necessary to the use of symmetry numbers are the additional possibility that the reactant or the transition state may have a number of optical isomers. In this case Gold has shown that the rate coefficient needs to be corrected by a factor /m equal to the ratio of the number of optical isomers in the transition state to that for the reactant [31]. 

The elementary kinetic constant k is therefore the product of a term calculated from a difference of intrinsic free enthalpies 12 and a ratio of symmetry numbers, which we will call single-events number ne — 0 ob/[Pg.276]

Both techniques converge to the same result, but the symmetry number method, though less intuitive, is easier to handle and we will limit our discussion to this technique. According to Benson (1976), the symmetry number cr of a molecule affects its rotational entropy by a factor —i ln(cr). Consequently, the statistical contribution to the stability constant of equilibrium (79) is given by the ratio of the symmetry numbers of the reactants and products (Eq. (82)). 

We can then evaluate the ratio of partition functions by a Monte Carlo calculation, drawing random samples uniformly from configuration space (this inefficient scheme will be improved on later) and nonuniformly in the Fourier coefficient space according to P(a). We then simply multiply the average over many Monte Carlo samples by which is an analytic function, to obtain Q Ki. This result is then divided by <7, the symmetry number of the molecule, to finally obtain an approximation to the properly sym-... 

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