Diatomic molecule reduced mass

Vibrationally excited diatomic molecules will only emit if they are polar, and most of the available results are for reactions which produce diatomic hydrides. Because of their unusually small reduced mass, these molecules have high frequency and very anharmonic vibrations, and their rotational levels are widely spaced. Consequently, their spectra can be resolved more easily than those of nonhydrides, where there are many more individual lines in the vibration-rotation spectrum. Furthermore, the molecular dynamics of these reactions are particularly interesting because of the special kinematic features that arise when an H atom is involved in a reactive collision and because these... 

The hamionic oscillator of two masses is a model of a vibrating diatomic molecule. We ask the question, What would the vibrational frequency be for H2 if it were a hamionic oscillator The reduced mass of the hydrogen molecule is... 

The Hamiltonian in this problem contains only the kinetic energy of rotation no potential energy is present because the molecule is undergoing unhindered "free rotation". The angles 0 and (j) describe the orientation of the diatomic molecule s axis relative to a laboratory-fixed coordinate system, and p is the reduced mass of the diatomic molecule p=mim2/(mi+m2). 

D is the chemical energy of dissociation which cair be obtained from thermodynamic data, aird is the reduced mass of the diatomic molecule... 

We now need to investigate the quantum-mechanical treatment of vibrational motion. Consider then a diatomic molecule with reduced mass /c- His time-independent Schrodinger equation is... 

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... 

The isotopic difference between the mean squares of the displacements in equation (7) can be calculated if the carbon-hydrogen oscillator is treated as a diatomic molecule. It is easily shown that for constant potential the mean square of the displacement from the equilibrium position of the harmonic oscillator will be inversely proportional to the square root of the reduced mass, /x, and hence... 

The force constant that is associated with the stretching vibration of a bond is often taken as a measure of the strength of the bond, although it is more correctly a measure of the curvature of the potential energy function around the minimum (Figure 2.1) that is, the rigidity of the bond. For a diatomic molecule, the frequency of vibration v is determined by the force constant k and the reduced mass /x = + m2), where m and m2 are the masses of... 

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... 

Figure 1.1 The diatomic molecule of masses m] and m2. The transformation to the relative coordinate r, r = r -r2 is useful when the potential depends on r only. The reduced mass, p is, as usual, p = m1m2/(m1 + m2).

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. 

It should be noted in passing that Mulliken also examined the isotope effect on the quadratic terms in the equations for the band heads. These ratios should theoretically show an isotope effect proportional to the reduced masses of the diatomic molecules (rather than the square root of the reduced masses). While Mulliken concludes that these ratios also confirm that the molecule is BO rather than BN, the four experimental ratios show a fairly large scatter so that the case for identifying the molecule is not as strong as that from the experimental a and b ratios. He also measured some of the rotational lines in the spectra of BO and considered the measured and theoretical isotope effects. Here one experimental isotope ratio checks the theoretically calculated ratio quite well, but for the other two the result was unsatisfactory. However, Mulliken judged the error to be within the experimental uncertainty. 

By the introduction of the (x, y) coordinate system, one has reduced the problem to the motion of a particle of mass (i in a two-dimensional rectilinear space (x, y). Thus, the problem of the collision between an atom and a diatomic molecule in a collinear geometry has been converted into a problem of a single particle on the potential energy surface expressed in terms of the coordinates x and y rather than the coordinates rAB and rBc The coordinates x and y which transform the kinetic energy to diagonal form in such way that the kinetic energy contains only one (effective) mass are referred to as mass scaled Jacobi coordinates. 

This study is one of the earliest attempts to calculate equilibrium fractionation factors using measured vibrational spectra and simple reduced-mass calculations for diatomic molecules. For the sake of consistency I have converted reported single-molecule partition function ratios to units. 

The vibrational frequency of a diatomic molecule (a one-dimensional system) is proportional to the square root of force constant (the second derivative of the energy with respect to the interatomic distance) divided by the reduced mass (which depends on the masses of the two atoms). 

In treating the vibrational and rotational motion of a diatomic molecule having reduced mass (i, equilibrium bond length re and harmonic force constant k, we are faced with the following radial Schrodinger equation ... 

In order to make the example concrete rather than abstract leta consider the vibration of a diatomic iodine molecule. First define the parameters of diatomic iodine the equilibrium separation in Angstroms = 10-n> m, the force constant, k, in N nr1, and the reduced mass in ng. 

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... 

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