Kinetic energy, classical rotational translational

The first term on the right is the translational kinetic energy of the molecule as a whole this simply adds a constant to the total energy, and we shall omit this term. The second and third terms are ihe rotational and vibrational kinetic energies of the molecule. The final term is the energy of interaction between rotation and vibration. To get the classical-mechanical Hamiltonian function, we add the potential energy V to (5.2), where U is a function of the relative positions of the nuclei. 

Figure 11. Rotational energy , dependence of the cross section for O + HCl and O + DCl reactions at E uk = 10 kcal/mol. Compared are the results of the classical trajectory calculations ( ) [47], kinematic mass model results which only include the energetic effects of the reactant rotation (o). kinematic mass model results which include rotational effects on the distribution of collisions with the barrier in addition to the rotational energetic effects ( ), and the kinematic mass model results for rotationally unexcited reactants 0 = 0) and the translational kinetic energy increased by the amoimt of (A) [62].

This equation describes the classic p-V-T behavior of an ideal gas. It reveals a well-known fact that gas pressnre is due to the translational kinetic energy of the molecules. The internal degrees of freedom that are distributed between rotational and vibrational motion do not contribute to gas pressure. The equation of state can be written in more conventional form if Boltzmann s constant k is written as the gas constant R divided by Avogadro s number Navo, and the molar volume of the gas is identified as n = Navo(V/N). Hence, pv = RT. 

We recall from classical mechanics that an expression for rotational motion results from the corresponding one for translational motion by replacing mass by moment of inertia, momentum by angular momentum, and velocity by angular velocity. Therefore, the middle part of the above formula for kinetic energy represents an analog of... 

Simulations may be classified as static and dynamic. In a static simulation no explicit account is taken of thermal motions in the system, which is therefore treated as if it were at a temperature of absolute zero. A molecular dynamics simulation, on the other hand, requires the specification of a temperature, which defines the kinetic energy to be distributed between the available degrees of freedom. By solving a set of classical equations of motion, such thermally induced reorientation phenomena as the vibrations, rotations, and translations of the system may be described. The two approaches have their advantages, which become clear in the following sections. 

In a way, one may interpret this potential as the sum of the classical Coulomb potential and the orbital (or quantum) kinetic moment energy or repulsive nature in the light of above considerations, see Figure 3.5. In fact, the kinetic energy of the total electronic motion in Hydrogen (and hydrogenic atoms) consists of two terms one responsible for translation and other for spherical rotation (the so-called orbital motion). 

For rigid molecules, the computed Cp needs to be augmented by the kinetic energy contributions for total translations and rotations, e.g., 3/2/ -f 3/2/ = 3/ for non-linear molecules. Calculation of the liquid Cp for flexible molecules is more complicated since the classical treatment of vibrations is improper an approximate, general solution is to compute only the contribution to Cp from the fluctuations in the intermolecular energy and to estimate the intramolecular contribution from the experimental or a quantum mechanical value of Cp of the ideal gas, less / , to remove the gas-phase PV contribution. For the NVT ensemble, the density, a, and k are not available since the volume is fixed however, Cv can be computed from the fluctuation in the potential energy. The pressure can also be calculated from equation (20)... 

Because of their importance to nucleation kinetics, there have been a number of attempts to calculate free energies of formation of clusters theoretically. The most important approaches for the current discussion are harmonic models, " Monte Carlo studies, and molecular dynamics calcula-tions. In the harmonic model the cluster is assumed to be composed of constituent atoms with harmonic intermolecular forces. The most recent calculations, which use the harmonic model, have taken the geometries of the clusters to be those determined by the minimum in the two-body additive Lennard-Jones potential surface. The oscillator frequencies have been obtained by diagonalizing the Lennard-Jones force constant matrix. In the harmonic model the translational and rotational modes of the clusters are treated classically, and the vibrational modes are treated quantum mechanically. The harmonic models work best at low temjjeratures where anharmonic-ity effects are least important and the system is dominated by a single structure. 

The logical way to begin the mathematical treatment of the vibration and rotation of a molecule is to set up the classical expressions for the kinetic and potential energies of the molecule in terms of the coordinates of the atoms, and then to use these expressions to obtain the wave equa-lion for vibration, rotation, and translation. Following this, it should bo proved that when the proper coordinate system is used, the complete wave equation can be approximately separated into three equations, one for translation, one for rotation, and one for vibration. Unfortunately, this procedure is not a very simple one and utilizes more quantum-mechanical technique than is required for the discussion of the vibrational ( ((nation itself. Consequently, the actual carrying out of the separation ill be deferred until Chap. 11, and only a summary of the results thus (ibi allied will be presented at this point. The reader who prefers to follow the more logical order may turn to Chap. 11 before continuing with the present sections. 

C -C bond, 8 degrees of freedom are purely kinetic (3 translations, 3 rotations and 2 internal rotations) and 31 degrees of freedom are vibrational. Therefore, alanine has, by classical theory, an energy of (4 + 31) kT per molecule in the gas phase due to translations, rotations and vibrations, namely over and above its equilibrium potential energy V (r ). According to classical sta-... 

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