Degrees of freedom translational

However, we know that a non-linear moleeule has three rotational and three translational degrees of freedom, all of whieh ean be assigned to symmetry speeies (Seetion 4.3.1). These are indieated in Table 6.5 and subtraeted from the total number of degrees of freedom to give the total number of vibrational degrees of freedom. 

When experimental data are not available, methods of estimation based on statistical mechanics are employed (7,19). Classical kinetic theory suggests a contribution to CP of S R for each translational degree of freedom in the molecule, a contribution of S R for each axis of rotation, and of R for each vibrational degree of freedom. A cmde estimate of CP for small molecules can be obtained which neglects vibrational degrees of freedom ... 

In order to provide a more general description of ternary mixtures of oil, water, and surfactant, we introduce an extended model in which the degrees of freedom of the amphiphiles, contrary to the basic model, are explicitly taken into account. Because of the amphiphilic nature of the surfactant particles, in addition to the translational degrees of freedom, leading to the scalar OP, also the orientational degrees of freedom are important. These orientational degrees of freedom lead to another OP which has the form of the vector field. 

The assignment of (hr) - 5) vibrational modes for a linear molecule and (hr) - 6) vibrational modes for a nonlinear molecule comes from a consideration of the number of degrees of freedom in the molecule. It requires hr) coordinates to completely specify the position of all t) atoms in the molecule, and each coordinate results in a degree of freedom. Three coordinates (x, y, and z) specify the movement of the center of mass of the molecule in space. They set the translational degrees of freedom, since translational motion is associated with movement of the molecule as a whole. Two internal coordinates (angles) are required to specify the orientation of the axis of a linear molecule during rotation, while three angles are required for a nonlinear... 

This relaxation proceeds without energy exchange between rotational and translational degrees of freedom and is supposed to be the same in EFA as in exact theory f = With this assumption we obtain a result identical to the ELIOS approximation [190] ... 

Excess energy in Reaction (75) would not be readily accommodated in the translational degrees of freedom available. 

In the liquid state, the molecules are still free to move in three dimensions but stiU have to be confined in a container in the same manner as the gaseous state if we expect to be able to measure them. However, there are important differences. Since the molecules in the liquid state have had energy removed from them in order to get them to condense, the translational degrees of freedom are found to be restricted. This is due to the fact that the molecules are much closer together and can interact with one another. It is this interaction that gives the Uquid state its unique properties. Thus, the molecules of a liquid are not free to flow in any of the three directions, but are bound by intermolecular forces. These forces depend upon the electronic structure of the molecule. In the case of water, which has two electrons on the ojQ gen atom which do not participate in the bonding structure, the molecule has an electronic moment, i.e.- is a "dipole". 

Therefore, as we change the state of matter, the translational degrees of freedom in liquids become severely restrieted in relation to those of the gciseous state. And, the vibrational and rotational degrees of freedom appear to be somewhat restricted, even though many of the liquid vibrational and rotational states have been found to be quite similar to those of the gaseous state. 

It should thus be clear that as we change the state of matter, the translational degrees of freedom present in gases beeome restiieted in liquids and disappear in solids. For gaseous moleeules, both vibrational and rotational degrees of fireedom are present while those of the liquid state are modified to the point where only vibrational states ean be said to truly free states. The same eannot be said for molectiles in the solid state. In the solid. 

In order to dissolve ionic solutes so readily, water molecules must solvate the ions as they enter solution. Consequently, water molecules lose their translational degrees of freedom as a result of their association with specific ions. It is possible to estimate the number of water molecules in clusters of the type (H20) using mass spectrometry (Kebarle, 1977). 

Even in J-type shock models, it is not appropriate to use thermal rate coefficients because the internal degrees of freedom will cool rapidly (via radiation) in the low density medium, whereas the translational degree of freedom will cool much more slowly. Appropriate rate coefficients are then those in which only translation is strongly excited such rate coefficients can be considerably lower than thermal rates for systems in which vibrational energy is the most efficient at inducing reaction. 

These activated complexes differ from ordinary molecules in that in addition to the three normal translational degrees of freedom, they have a fourth degree of translational freedom corresponding to movement along the reaction coordinate. This degree of freedom replaces one vibrational degree of freedom that would otherwise be observed. 

Coherent states for the translational degrees of freedom (in d dimensions) are Gaussians located at a point (q, p) in phase space. For convenience we choose them as... 

The first case has already been considered section 2.0 the second case leads to a strong classical spin-orbit coupling, which is reflected in a Hamiltonian nature of the classical combined dynamics. In both situations the procedure is to find a suitable approximate Hamiltonian Hq( ) that propagates coherent states exactly along appropriate classical spin-orbit trajectories (x(l,),p(t),n(l,)). (For problems with only translational degrees of freedom this has been suggested in (Heller, 1975) and proven in (Combescure and Robert, 1997).) Then one treats the full Hamiltonian as a perturbation of the approximate one and calculates the full time evolution in quantum mechanical perturbation theory (via the Dyson series), i.e., one iterates the Duhamel formula... 

The internal energy is, as indicated above, connected to the number of degrees of freedom of the molecule that is the number of squared terms in the Hamiltonian function or the number of independent coordinates needed to describe the motion of the system. Each degree of freedom contributes jRT to the molar internal energy in the classical limit, e.g. at sufficiently high temperatures. A monoatomic gas has three translational degrees of freedom and hence, as shown above, Um =3/2RT andCy m =3/2R. 

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