Rotational symmetry number

S ince no torsional conformers are expected for each of these rather rigid isomers, the contributions to entropy are only caused by the distinct molecular rotational degrees of freedom of each isomer considered as a rigid unit which can be evaluated by symmetry considerations. The symmetry contribution to entropy is -R In a, where a is the symmetry number i.e., the number of undistinguishable positions adopted by the isomer (considered rigid) by simple rotations. Symmetry numbers are o s 2 for the Q isomer and a = 6 for the Dj one. 

Table 10.1 Rotational symmetry numbers for molecular point groups...

Consider the molecular rotational partition function for the CO molecule, a linear diatomic molecule. The moment of inertia of CO is / = 1.4498 x 10-46 kg-m2, and its rotational symmetry number is a = 1. Thus, evaluating Eq. 8.65 at T = 300 K, we find the rotational partition function to be... 

The rotational symmetry number for CH3 is er =6. The ground electronic state has a degeneracy go = 2. The lowest-lying excited electronic state is 9.117 x 10-19 J above the ground state, and it also has a degeneracy of 2. Although the excited electronic level makes a negligible contribution to the thermodynamic quantities below, we include the calculations to show quantitatively how small these contributions are. With the mass of the CH3 molecule m = 2.497 x 10-26 kg, we have all of the physical constants needed to calculate the thermodynamic quantities of interest. 

Evaluate the translational, rotational, and vibrational contributions to the entropy for 0.1 moles of the A127C135 molecule at 900°C and a pressure of 1 mBar. Assume a bond length of 2.13 A, vibrational frequency rotational symmetry number a = 1. 

Perform all of the tasks in the previous problem, but for the NH3 molecule. You will need the following physical constants associated with NH3 vibrational frequencies co = 3506, 1022, 3577, 3577, 1691, 1691 cm-1, moments of interia = 4.414 x 10-47 kg-m2, 2.809 x 10-47 kg tn2, 2.809 x 10 47 kg-m2, rotational symmetry number a = 3. Consider just one excited electronic energy state lying 46,205 cm-1 above the ground electronic state. Both the ground and first excited electronic states have degeneracies of 1. Experimental data for NH3 can be found in thermotables. csv. 

Here, v,- are the vibrational energy levels, T is the temperature, m the molecular weight, p the pressure, ABC the product of the moments of inertia, a the rotational symmetry number and p the number of ways in which a given conformer or isomer may be formed. 

Rotational symmetry numbers for some simple molecules. (From Dannenfelser et al., 1996)... 

Calculation of partition functions requires spectroscopic quantities for the rotational and vibrational partition functions. The quantities required are moments of inertia, rotational symmetry numbers and fundamental vibration frequencies for all normal modes of vibration. The translational terms require the mass of the molecule. All terms depend on temperature. Calculation of partition functions is routine for species for which a detailed spectroscopic analysis has been made. 

In these equations, P ( ) is the particular propoerty of interest, G(R) the group contribtuions for the alkyl substitent, and 0)2 (F)2, etc., the specific group values. One must bear in mind that intrinsic entropy (S ) is given by the group additivity (32) and that S = S - Rln o/n where o is the rotational symmetry number and n is the number of optical isomers of the particular compound or radical. 

The results by the latter two methods have been found repeatedly to give standard free energy changes agreeing to +0.1-0.2 kcal or better, and are favoured if discrepancies exist. The results of part (ii) of Table 1 provide a number of examples of excellent agreement of values at 300 and 600 K. The agreement indicates that simple gas phase proton-transfer equilibria involve little or no entropy effects (other than for rotational symmetry numbers). 

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

The vibrational analysis is followed by a standard, classical statistical thermodynamic analysis at 298.18 K (25°C) and latm pressure. (For details, see McQuarrie (2000)). Computed quantities include the principal axes and moments of inertia, the rotational symmetry number and symmetry classification, and the translational, rotational, vibrational, and total enthalpy and entropy, respectively. Both the temperature and pressure can be altered from standard conditions and/or scanned across a requested range of values. The total zero-point energy at 0 K is given by summed over all real frequencies (converted to kcalmoP see O Eq. 10.36). 

Rotational Symmetry Number is 2 The Molecule is an Asymmetric Top Translational Enthalpy 0.889 kcal/mol... 

Explaining symmetry numbers is not easy. Just consider the following examples asymmetric molecules have = 1 sample rotational symmetry numbers are 3 for NH3, 2 for H2O, 4 for ethylene, 6 for ethane, 12 for benzene and methane. Molecules with the same symmetry have the same symmetry number, so methane and carbon tetrachloride both have cr = 12. 

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