Nuclear-spin partition function

The partition function for a molecule is formed of the partition functions for individual types of energy increments (motions), i.e. from the translational, rotational, internal rotational (free rotation, hindered rotation), vibrational, electronic and nuclear spin partition functions... 

The nuclear spin partition function is given by (2i + 1) where i is the nuclear spin quantum number, since the energy of nuclear orientations is very small compared with kT Thus, for the hydrogen atom ( H), / = 1/2 and for the chlorine atom i = 3/2 giving nuclear spin contributions of 2 and 4, respectively, to the partition function. However, it is conventional to omit these factors from calculated entropies. 

Nuclear Spin Contribution. The nuclear spin partition function is the product of the nuclear spin multiplicity 2ij -H 1) for all the atoms in the molecule, where /V is the nuclear spin of the yth atom. Since, apart from processes involving molecular hydrogen and its isotopes (and the transmutation of the elements), nuclear spins are conserved, this contribution is conventionally omitted from the total entropy leaving the practical or virtual entropy (these adjectives are frequently omitted also). 

If the nucleus has an odd mass number, the overall wave function is anti-symmetrical with regard to the nuclei it is the product of all the translational, rotational, vibrational, electronic and nuclear wave functions. With all the translational, vibrational and electronic wave functions being symmetrical, we only have to consider the rotational and nuclear functions, one of which must be symmetrical and the other anti-symmetrical or vice versa. The g(g-l)/2 wave functions with anti-symmetrical nuclear spin must have corresponding symmetrical rotational wave functions, i.e. with even values of j the g(g+l)/2 wave functions with symmetrical nuclear spin must have corresponding rotational wave functions, i.e. with odd values of j. The combined nuclear-rotational partition function will therefore be ... 

We now consider these terms separately. As far as nuclear energy is concerned, it may be taken equal to zero. The nuclear partition function then becomes fn = Qna, where Qub is the nuclear spin statistical weight. Since the allowed rotational levels are dependent upon the nuclear spin wave functions, as we have seen in section 14c, it is convenient to combine the nuclear spin statistical weight with the rotational partition function, and write... 

There are in principle also energy levels associated with nuclear spins. In the absence of an external magnetic field, these are degenerate and consequently contribute a constant term to the partition function. As nuclear spins do not change during chemical reactions, we will ignore this contribution. 

In Eq. (44), gei(T ) is the ratio of transition state and reactant electronic partition functions [31] and the rotational degeneracy factor = (2ji + l)(2/2 + 1) for heteronuclear diatomics, and will also include nuclear spin considerations in the case of homonuclear diatomics. 

Proceeding in the spirit above it seems reasonable to inquire why s is equal to the number of equivalent rotations, rather than to the total number of symmetry operations for the molecule of interest. Rotational partition functions of the diatomic molecule were discussed immediately above. It was pointed out that symmetry requirements mandate that homonuclear diatomics occupy rotational states with either even or odd values of the rotational quantum number J depending on the nuclear spin quantum number I. Heteronuclear diatomics populate both even and odd J states. Similar behaviors are expected for polyatomic molecules but the analysis of polyatomic rotational wave functions is far more complex than it is for diatomics. Moreover the spacing between polyatomic rotational energy levels is small compared to kT and classical analysis is appropriate. These factors appreciated there is little motivation to study the quantum rules applying to individual rotational states of polyatomic molecules. 

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 ... 

Finally, nuclear spin also produces an additional constant factor in the partition function a nucleus with spin quantum number s contributes a factor 2s + 1 to... 

In order to illustrate the consequences of equation (70), it will be assumed that the partition functions for the reactants and the complex can be expressed as products of the appropriate numbers of translational, rotational and vibrational partition functions. For simplicity we shall also neglect factors associated with nuclear spin and electronic excitation. If = total number of atoms in a molecule of species i and = 0 for nonlinear molecules, 1 for linear molecules, and 3 for monatomic molecules, then the correct numbers of the various kinds of degrees of freedom are obtained in equation (70) by letting... 

At all reasonable temperatures the rotational levels of a molecule containing more than two atoms, like those of diatomic molecules, are occupied sufficiently for the behavior to be virtually classical in character. Assuming that the molecule can be represented as a rigid rotator, the rotational partition function, excluding the nuclear spin factor, for a nonlinear molecule is given by... 

Nuclear spin contribution to rotational heat capacity should be taken into account. The Cf orth0) and Cj para) are obtained from the corresponding partition functions... 

Webnucleo s Nuclear Data Tool allows an internet user to explore some of the key properties of nuclei that govern how the species behave in nuclear reactions. These are the atomic number Z, the neutron number N, the mass number A, the mass excess, the ground-state spin, and the nuclear partition function. Students can use the Nuclear Data Tool to explore these quantities and then to study questions such as what reactions among nuclei are energetically possible or even to simply look up needed quantities for their homework. 

The ortho-para ratio is determined by the statistical calculation of the availability of states given by the partition function. Calculated ortho-para equilibria for H2, D2, and T2 are shown in Figure 6 at the temperature range from 0 to 300K. When the two nuclear spins are parallel, the resultant nuclear spin quantum number is 1 (i.e. 1/2 -E 1 /2) and the state is threefold degenerate. When the two spins are opposed, however, the resultant nuclear spin is zero and the state is nondegenerate. Therefore para-H2 has the lower energy and this state is favored at lower temperatures. The equilibrium concentration of H2 approaches three parts ortho to one part para at room temperature. As the nuclear spin quantum number of the deuteron is 1 rather than 1/2 for the proton (see Table 1), the D2 system is described by Bose-Einstein... 

The statistical weight factors gi depend on the vibrational state v = (nivi, ti2V2,the rotational state with the rotational quantum number J, the projection K onto the symmetry axis in the case of a symmetric top, and furthermore on the nuclear spins / of the A nuclei. The partition function... 

Internal Partition Function for Monatomic Gases.—For the present purpose, for the internal energy of a monatomic gas only the nuclear spin and electronic states need be considered. On the assumption that these energies are additive, the partition function can again be factored ... 

Internal Partition Functions for Polyatomic Molecules.— The internal partition function for a polyatomic molecule comprises contributions from nuclear spin and electronic levels, and from rotational and vibrational degrees of freedom. On the assumption that the corresponding energies are additive and independent, these contributions can be factored, and the corresponding contributions to the thermodynamic functions are additive. 

Nuclear partition function and correction of symmetry due to nuclear spin... 

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