Hydrogen rotational partition function

Isotope (H (deuterium), discovered by Urey et al. (1932), is usually denoted by symbol D. The large relative mass difference between H and D induces significant fractionation ascribable to equilibrium, kinetic, and diffusional effects. The main difference in the calculation of equilibrium isotopic fractionation effects in hydrogen molecules with respect to oxygen arises from the fact that the rotational partition function of hydrogen is nonclassical. Rotational contributions to the isotopic fractionation do not cancel out at high T, as in the classical approximation, and must be accounted for in the estimates of the partition function ratio /. 

In a different approach, Bottinga (1969a) evaluated the nonclassical rotational partition function for water vapor, molecular hydrogen, and methane through the asymptotic expansion of Strip and Kirkwood (1951) ... 

The vaporization of a pure liquid or the reverse process, the condensation of the liquid, provides an interesting test of this delayed equilibration hypothesis. Thus as a hydrogen-bonded molecule, which vibrates in the liquid, separates from the surface it frees itself from the potential energy restrictions which prevented rotation. However, in so far as the evaporating molecule has insufficient collisions with neighbors to equilibrate to the free rotational partition function, fgy of the gas, it will retain substantially the partition function, fb of the condensed phase even in the activated complex. Consequently for the condensation process the usual... 

Nuclear Spin Effects on Rotation. There is an interesting effect on the rotational partition function, even for the hydrogen molecule, due to nuclear spin statistics. The Fermi postulate mandates that the overall wavefunction (including all sources of spin) be antisymmetric to all two-particle interchanges. A simple molecule like (1H1)2, made of two electrons (S = 1/2) and two protons (spin 7=1/2), will have two kinds of molecule ... 

Problem Calculate the rotational partition function of (i) hydrogen gas, (ii) iodine chloride gas, at 300 K. 

The quantum-mechanical interpretation of the appearance of the symmetry number in the rotational partition function has its basis in the symmetry of the total wave function of the molecule. We will consider only the case of the hydrogen molecule. 

A major advance in the theory of primary hydrogen isotope effects came when the approximation was made that substitution of deuterium for hydrogen does not greatly affect the classical properties of the molecule, such as the mass or moments of inertia and consequently neither the translational nor rotational partition functions . This left only the quantum mechanical vibrational partition function as a source of the isotope effect. Writing the deuterium isotope effect in terms of the complete vibrational partition function, equation (6) is obtained, where Ut = hvJkT, Vi is the frequency of the /th vibrational mode and N is the number of atoms in the molecule. The products and summations are... 

For light rotors it may be possible to observe sufficient torsional transitions for the internal rotation partition function to be determined by direction summation (p. 271). If only a few low-lying torsional levels are observed, they may be fitted to a suitable potential from which higher torsional energy levels can be computed. Thermodynamic functions of hydrogen peroxide calculated by use of this procedure give a value of 5 (298.15 K) in excellent agreement with the calorimetric value. ... 

L, J, Dq and Bq are constants describing the hydrogen molecules. J is the Einstein temperature of the gas molecule, Dq its dissociation energy (e.g. the ground state energy of the gas molecule relative to the atoms at rest) and the rotational constant of the X2 molecule. Most of the translational rotational partition function is given by LT /2, The values of the constants L, J and M necessary to calculate gS are listed in Table I (see Ref. 14 for more details). ... 

It is possible that two or three of these degrees of freedom internal to the molecule may be better described as rotations than as vibrations. Such would be the case, for example, with the H2 molecule. For molecules containing only one atom other than hydrogen atoms - e.g. CIH, CH4, NH4 - we obtain better results when we consider that a movement is indeed a high-temperature rotation, but also a low-temperature vibration. There would be a rather sharp transition within a certain temperature range. In the case of a rotation, a rotational partition function term replaces a vibrational term in equation [1.30] and the corresponding terms in relations [1.30] and [1.32]. 

The Product Rule. Within the harmonic approximation, an important relation between the isotopic sensitivities of the classical translational and rotational partition functions and the vibration frequencies of a molecule is provided by the Product Rule [14,23-25]. For isotopic substitution of a single hydrogen the rule may be stated as follows ... 

Carry out the summation of the rotational partition function separately for the ortho and para forms of hydrogen gas. Compare with the result obtained with the integral approximation using a symmetry number of 2 and a nuclear spin degeneracy factor of 4. 

A better approximation to the rotational partition function could be obtained by explicit summation, but we wiU use this approximation. The translational partition function of atomic hydrogen is... 

The LMR detection of polyatomic radicals with heavy (not hydrogen) atoms is hindered for several reasons (1) the population of the lower state is low, it decreases with rotational partition function (2) Zeeman splitting is small, it decreases both with moments of inertia and rotational quantum numbers of the radical hence LMR spectra are observable only at low magnetic fields because of a weak coupling of S with N. Moreover, the spectra are often unresolved because of a large number of components. 

Another implication of the importance of hindered rotations in the transition state is the effect of deuteration on k. Because the rotational partition function is dominated by the moment of inertia of the rotating moiety, and because Fig. 5 shows that these involve hydrogen atoms, the... 

Each of the partition functions is now regarded as a product of independent translational, rotational, and vibrational partition functions—the implication being that vibration-rotation interaction is negligible, and it is then assumed that the rotations are classical and the vibrations harmonic. If the structure of each molecular species is known, the moments of inertia can be calculated, and if necessary— as may well be the case for hydrogen isotopes—a correction can be applied to account for the fact that the rotational partition function has not reached its classical value. If complete vibrational analjrses of all the molecules are also available, the vibrational partition functions can be set up, and an approximate correction for neglect of anharmonicity can also be made. Having done all this, we can calculate the isotope effect. 

Because the partition function ratio / is defined in such a way that the classical rotational and translational contributions are canceled, equations 11.40, 11.41 and 11.43 must be modified by introducing the ratio of the deviations from classical rotational behavior of heavy and light hydrogen molecules. For small values of [Pg.779]

The internal rotations around the skeletal bonds of PE are hindered due to interaction between the neighboring hydrogens. Since second and higher orders of interactions are not negligible, the internal rotations are interdependent. The statistics of such interdependent rotations is developed and applied to obtain the configurational partition function and the mean-square end-to-end... 

The partition function ratio / for the isotopic hydrogen molecules, including a correction for nonclassical rotation, is known. For convenience a function / is defined... 

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