Molecules reduced mass

Molecule Reduced mass (atomic units) Dissociation energy kcals Zero-point energy kcals Inter- nuclear distance A Moment of inertia gmcm2 X io40 Force constant dynes cm l X io 8 Vibrational frequency cm l... 

In this equation, D is the difference between the origin of energies and the minimum value of the curve. De is the residual energy, which is the vibration energy at absolute zero hV(/2, a-q is the inter-atom distance for the minimum energy, which is the equilibrium distance of the molecule. Constant a depends on the light speed c molecule reduced mass p. (p is related to the atomic masses by the equations in [10.5]) Plank s constant h and value x as defined by [10.5]. Constant a is written ... 

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... 

Here, u is the displacement of the /ith molecule from its equilibrium position and M the reduced mass of each molecular site. Second, the electron is described within the frame of the tight-binding approximation, where it is assumed that the effect of the potential at a given site of the one-dimensional chain is limited to its nearest neighbors. In that case, the energy dispersion of the electron is given by... 

The method gives VJF, where F is a reduced mass of the rotating group and the rest of the molecule. F is not always accurately known. 

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... 

Here p is the radius of the effective cross-section, (v) is the average velocity of colliding particles, and p is their reduced mass. When rotational relaxation of heavy molecules in a solution of light particles is considered, the above criterion is well satisfied. In the opposite case the situation is quite different. Even if the relaxation is induced by collisions of similar particles (as in a one-component system), the fraction of molecules which remain adiabatically isolated from the heat reservoir is fairly large. For such molecules energy relaxation is much slower than that of angular momentum, i.e. xe/xj > 1. 

According to theory (22), the rate constant for reaction between an ion and a molecule can be expressed in terms of the polarizability of the neutral species and of the reduced mass of the reacting pair... 

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 same is true for the distribution of relative velocities, provided one replaces the mass with the reduced mass fi (defined in Eq. 55) of the two molecules ... 

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... 

In the foregoing, U is the interaction potential, M is the reduced mass of the colliding system, ftk and ftk are respectively the momentum of the projectile before and after the collision, ig and in are respectively the wavefunctions of the atom (or molecule) in the ground and nth excited states, and the volume element dt includes the atomic electron and the projectile. Since U for charged-particle impact may be represented by a sum of coulombic terms in most cases, Eq. (4.11) can be written as (Bethe, 1930 Inokuti, 1971)... 

In Eqs. (5.1) and (5.2), m is the reduced mass of the colliding system, V is the interaction potential at ion-molecule separation r, 6 is the angle between the direction of r and the center-of-mass velocity, and the dot indicates differentiation with respect to time. Integration of (5.1) just gives the angular momentum L, which is conserved in the collision. Substitution in (5.2) gives... 

Here /(R) and pflX) denote the shift and generalized momentum for the molecular vibration of the low frequency a>9 and reduced mass m, at the Rth site of the adsorbate lattice bi+(K) and K) are creation and annihilation operators for the collectivized mode of the adsorbate that is characterized by the squared frequency /2(K) = ml + d>, a,(K)/m , with O / iat(K) representing the Fourier component of the force constant function /jat(R). Shifts i//(R) for all molecules are assumed to be oriented in the same arbitrary direction specified by the unit vector e they are related to the corresponding normal coordinates, ue (K), and secondary quantization operators ... 

Consider an arbitrary two-dimensional Bravais lattice, with its sites R occupied by adsorbed molecules and molecular vibrations representing two modes, of a high and low frequency. Frequencies (ohh reduced masses mh/y vibrational coordinates w/,/(R), and momenta pA /(R) are accordingly labeled by subscripts h and / referring to the high-frequency and the low-frequency vibration. The most general form of the Hamiltonian appears as140... 

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