Diatomic molecules inertia, moments

E10.6 For the diatomic molecule Na2, 5 = 230.476 J-K-1-mol" at T= 300 K, and 256.876 J-K-,-mol-1 at T= 600 K. Assume the rigid rotator and harmonic oscillator approximations and calculate u, the fundamental vibrational frequency and r, the interatomic separation between the atoms in the molecule. For a diatomic molecule, the moment of inertia is given by l pr2, where p is the reduced mass given by... 

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 microwave experiment studies rotational structure at a given vibrational level. The spectra are analyzed in terms of rotational models of various symmetries. The vibration of a diatomic molecule is, for instance, approximated by a Morse potential and the rotational frequencies are related to a molecular moment of inertia. For a rigid classical diatomic molecule the moment of inertia I = nr2 and the equilibrium bond length may be calculated from the known reduced mass and the measured moment, assuming zero centrifugal distortion. 

Each level, with its unit usually given in cm is designated by the quantum number, /, which is likewise related to the total angular momentum, while B is the rotational constant. For a diatomic molecule the moment of inertia h can be calculated from the nuclear masses m, mj and the equilibrium distance r ... 

The allowable energy levels for free rotation of a rigid diatomic molecule with moment of inertia I are given by... 

The above equations show something else about the rotational spectrum its relationship to B, which in turn is related to the moment of inertia. For diatomic molecules, the moment of inertia I is defined simply as ftr, where is the reduced mass and r is the internuclear separation of the two atoms. Rotational spectroscopy is therefore useful in calculating the sizes of diatomic molecules. This is a general capability of rotational spectroscopy, but it is most easily illustrated for diatomic molecules. 

Here the total moment of inertia I of the molecule takes the place of iRe2 in the diatomic molecule case... 

As in diatomic molecules the structure of greatest importance is the equilibrium structure, but one rotational constant can give, at most, only one structural parameter. In a non-linear but planar molecule the out-of-plane principal moment of inertia 4 is related to the other two by... 

For diatomic molecules, B0 is the rotational constant to use with equation (10.125), while Be applies to equation (10.124). They are related by Bq = Be 2 - The moment of inertia 70(kg-m2) is related to 50(cm ) through the relationship /0 = h/ 8 x 10 27r22 oc), with h and c expressed in SI units. For polyatomic molecules, /a, /b, and Iq are the moments of inertia to use with Table 10.4 where the rigid rotator approximation is assumed. For diatomic molecules, /0 is used with Table 10.4 to calculate values to which we add the anharmonicity and nonrigid rotator corrections. 

The quantity I (= /jiR ) is the moment of inertia for the diatomic molecule with the intemuclear distance fixed at Re and Be is known as the rotational constant (see Section 5.4). 

Consider the simple diatomic molecule CO with a carbon atom at one end of the bond and an oxygen atom at the other. The moment of inertia is a measure of how heavy each atom is and the length of the bond between them. The moment of inertia carries important information regarding the structure of the molecule and, more importantly, is very useful in identifying a molecule. The energy separation between the allowed end-over-end rotations of a diatomic, Ej, is given by ... 

The moment of inertia for diatomic molecules is fairly easy to understand but for other linear molecules such as OCS or non-linear polyatomic molecules it is more complicated. The general expression for the moment of inertia is given by ... 

A rotating diatomic molecule consists of masses mi and m2 circling the centre of mass at distances 7 1 and r2 respectively. The moment of inertia is / = mir + m2r. By definition, the centre of mass is located such that m r = m2r2, and hence... 

A single particle of (reduced) mass p in an orbit of radius r = rq + r2 (= interatomic distance) therefore has the same moment of inertia as the diatomic molecule. The classical energy for such a particle is E = p2/2m and the angular momentum L = pr. In terms of the moment of inertia I = mr2, it follows that L2 = 2mEr2 = 2EI. The length of arc that corresponds to particle motion is s = rep, where ip is the angle of rotation. The Schrodinger equation is1... 

Kratzer and Loomis as well as Haas (1921) also discussed the isotope effect on the rotational energy levels of a diatomic molecule resulting from the isotope effect on the moment of inertia, which for a diatomic molecule, again depends on the reduced mass. They noted that isotope effects should be seen in pure rotational spectra, as well as in vibrational spectra with rotational fine structure, and in electronic spectra with fine structure. They pointed out the lack of experimental data then available for making comparison. 

All together one would obtain an effeetive moment of inertia tensor which includes the rotational g tensor again. This correction is normally ignored for polyatomic molecules, but allows to estimate the rotational g factor of diatomic molecules from field-free rotation-vibration spectra [5,10,11]. 

We see from (4.104) that, although the vibrational quantum number is not changing, the frequency of a pure-rotational transition depends on the vibrational quantum number of the molecule undergoing the transition. (Recall that vibration changes the effective moment of inertia, and thus affects the rotational energies.) For a collection of diatomic molecules at temperature T, the relative populations of the energy levels are given by the Boltzmann distribution law the ratio of the number of molecules with vibrational quantum number v to the number with vibrational quantum number zero is... 

For any homonudear diatomic molecule whose nuclei have nonzero spin, it should, in principle, be possible to isolate a modified form with only even-numbered rotational levels populated. (For /=0, half the rotational levels do not exist.) However, only for H2 and D2 has this been achieved. The small moments of inertia of these light molecules give a relatively large spacing between 7 = 0 and 7= 1 rotational levels, so that we can get nearly all the molecules into the 7 = 0 level at a temperature above the freezing point of the substance. 

For a symmetric top, the selection rules are such that we can determine only B0 [see (5.85)]. Knowledge of Ib°, the moment of inertia about a principal axis perpendicular to the symmetry axis, is not sufficient to determine the molecular structure, except for a diatomic molecule. To get added information, the microwave spectra of isotopically substituted spe-... 

As we saw in Chapter 1, the importance of numbers in chemistry derives from the fact that experimental measurement of a particular chemical or physical property will always yield a numerical value to which we attach some significance. This might involve direct measurement of an intrinsic property of an atom or molecule, such as ionization energy or conductivity, but, more frequently, we find it necessary to use theory to relate the measured property to other properties of the system. For example, the rotational constant, B, for the diatomic molecule CO can be obtained directly from a measurement of the separation of adjacent rotational lines in the infrared spectrum. Theory provides the link between the measured rotational constant and the moment of inertia, I, of the molecule by the formula ... 

The most direct application of particle-on-a-sphere result is to the rotational motion of diatomic molecules in a gas. As with vibrations (see Section 3.2), the real situation looks a little more complicated, but can be solved in a similar way. A molecule actually rotates about its centre of mass the coordinates 8 and can be used to define its direction in space. If we replace the mass in Schrodinger s equation by the reduced mass given by eqn 3.22, and let r be the bond length, then the moment of inertia is... 

The rigid-rotor Hamiltonian for a diatomic molecule with the moment of inertia I = pf o is... 

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