Partition rotation

Hydrogen is a molecule of very low mass and the moment of inertia is low, so separation between consecutive rotational energy levels is large and there is only partition rotational excitation among the hydrogen molecules and the extent of this excitation varies with temperature. 

Since translational and internal energy (of rotation and vibration) are independent, the partition function for the gas can be written... 

Thus the kinetic and statistical mechanical derivations may be brought into identity by means of a specific series of assumptions, including the assumption that the internal partition functions are the same for the two states (see Ref. 12). As discussed in Section XVI-4A, this last is almost certainly not the case because as a minimum effect some loss of rotational degrees of freedom should occur on adsorption. 

Unlike the situation embodied in seetion A2.4.1. in whieh the theory was developed in an essentially isotropie maimer, the presenee of an eleetrode introduees an essentially non-isotropie element into the equations. Negleetmg rotational-dependent interaetions, we see that the overall partition fiinotion ean be written... 

Mullin A S, Park J, Chou J Z, Flynn G W and Weston R E Jr 1993 Some rotations like it hot seleotive energy partitioning in the state resolved dynamios of oollisions between COj and highly vibrationally exoited pyrazine Chem. Phys. 175 53-70... 

For translational, rotational and vibrational motion the partition function Ccin be calculated using standard results obtained by solving the Schrodinger equation ... 

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. 

Constant in rotational partition function of gases Constant relating wave number and moment of inertia Z = constant relating wave number and energy per mole... 

The total partition function may be approximated to the product of the partition function for each contribution to the heat capacity, that from the translational energy for atomic species, and translation plus rotation plus vibration for the diatomic and more complex species. Defining the partition function, PF, tlrrough the equation... 

The classical value is attained by most molecules at temperatures above 300 K for die translation and rotation components, but for some molecules, those which have high heats of formation from die constituent atoms such as H2, die classical value for die vibrational component is only reached above room temperature. Consideration of the vibrational partition function for a diatomic gas leads to the relation... 

The special case where only rotators are present, Np = 0, is of particular interest for the analysis of molecular crystals and will be studied below. Here we note that in the other limit, where only spherical particles are present, Vf = 0, and where only symmetrical box elongations are considered with boxes of side length S, the corresponding measure in the partition function (X Qxp[—/3Ep S, r )], involving the random variable S, can be simplified considerably, resulting in the effective Hamiltonian... 

Here Zint is the intramolecular partition function accounting for rotations and vibrations. However, in equilibrium, the chemical potential in the gas phase is equal to that in the adsorbate, fi, so that we can write the desorption rate in (I) as... 

Consider a nucleus that can partition between two magnetically nonequivalent sites. Examples would be protons or carbon atoms involved in cis-trans isomerization, rotation about the carbon—nitrogen atom in amides, proton exchange between solute and solvent or between two conjugate acid-base pairs, or molecular complex formation. In the NMR context the nucleus is said to undergo chemical exchange between the sites. Chemical exchange is a relaxation mechanism, because it is a means by which the nucleus in one site (state) is enabled to leave that state. 

Here /, are the three moments of inertia. The symmetry index a is the order of the rotational subgroup in the molecular point group (i.e. the number of proper symmetry operations), for H2O it is 2, for NH3 it is 3, for benzene it is 12 etc. The rotational partition function requires only information about the atomic masses and positions (eq. (12.14)), i.e. the molecular geometry. 

The above treatment has made some assumptions, such as harmonic frequencies and sufficiently small energy spacing between the rotational levels. If a more elaborate treatment is required, the summation for the partition functions must be carried out explicitly. Many molecules also have internal rotations with quite small barriers, hi the above they are assumed to be described by simple harmonic vibrations, which may be a poor approximation. Calculating the energy levels for a hindered rotor is somewhat complicated, and is rarely done. If the barrier is very low, the motion may be treated as a free rotor, in which case it contributes a constant factor of RT to the enthalpy and R/2 to the entropy. 

C) The error in AE" /AEq is 0.1 kcal/mol. Corrections from vibrations, rotations and translation are clearly necessary. Explicit calculation of the partition functions for anharmonic vibrations and internal rotations may be considered. However, at this point other factors also become important for the activation energy. These include for example ... 

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