Statistical thermodynamics translational energy

The thermophysical properties necessary for the growth of tetrahedral bonded films could be estimated with a thermal statistical model. These properties include the thermodynamic sensible properties, such as chemical potential /t, Gibbs free energy G, enthalpy H, heat capacity Cp, and entropy S. Such a model could use statistical thermodynamic expressions allowing for translational, rotational, and vibrational motions of the atom. 

This covers most hypersonic flow regimes with high thermodynamic nonequilibrium. The statistical uncertainties of the rate constants were below 5% for all cases run. It is found that under reentry flow conditions most of the endothermicity is derived from the N2-O relative translational energy and the NO molecules are formed vibrationally and rotationally very excited. Also, the QCT thermal rate constants showed good agreement with the available experimental data. ... 

To simplify notation for these two terms let 2f0[G3(MP2)] s E0 and G3MP2 Enthalpy = //29g. The thermal correction to the enthalpy (TCH), converting energy at 0 K to enthalpy at 298, (H29% -E0 = -78.430772-(—78.4347736) = 0.0040016 h) is a composite of two classical statistical thermodynamic enthalpy changes for translation and rotation, and a quantum harmonic oscillator term for the vibrational energy. 

The equilibrium constants Kf are not measurable and we must resort to statistical thermodynamics to estimate these values theoretically. The partition function (Q) is a quantity with no simple physical significance but it may be substituted for concentrations in the calculation of equilibrium constants (Eqns. 4 and 5) [5], (It is assumed that there is no isotopic substitution in B.) Partition functions may be expressed as the product of contributions to the total energy from translational, rotational and vibrational motion (Eqn. 6). 

We diall assume that on a collision between two butadiene molecules the probability of reaction has a constant value a when the rdative translational energy along the line of centres exceeds a definite energy e and is otherwise zero. We call the energy of activation and a the "probability factor . The formula for is then (Fowler and Guggenheim, Statistical thermodynamics , Cambridge University... 

From elementary statistical thermodynamics we know that the equilibrium constant can be written in terms of the partition functions of the individual molecules taking part in a reaction. These quantities represent the sum over all energy states in the system—translational, rotational, vibrational, and electronic. The probability that a molecule will be in a particular energy state, f ,-, is given by the Boltzmann law,... 

We have seen that statistical thermodynamics gives the same translational (that is, internal) energies and pressures that we find from other, phenomenological perspectives. But values for A and G depend on the entropy of our gaseous sample. It remains to be seen how (or rather, if ) statistical thermodynamics predictions for S agree with phenomenological values of entropy. 

The vibrational analysis is followed by a standard, classical statistical thermodynamic analysis at 298.18 K (25°C) and latm pressure. (For details, see McQuarrie (2000)). Computed quantities include the principal axes and moments of inertia, the rotational symmetry number and symmetry classification, and the translational, rotational, vibrational, and total enthalpy and entropy, respectively. Both the temperature and pressure can be altered from standard conditions and/or scanned across a requested range of values. The total zero-point energy at 0 K is given by summed over all real frequencies (converted to kcalmoP see O Eq. 10.36). 

The partition function contribution, PFC, in equation (2) includes contributions from statistical thermodynamics that consist of the conformational population contribution (POP), the torsional contribution (TOR), and the contributions from translational rotation and a PV term required to convert the energy into enthalpy (T/R) ... 

There is a factor F in this expression that is related to the various forms of molecule energy (translation, rotation, vibration). This can be calculated through statistical thermodynamics. Thus it is a priori possible to calculate the speed, hence the second name of this theory. We will see how to calculate this speed in Chapters 10 and 11. 

An argon atom has only three degrees of freedom translational motion in the X, y, and z coordinates. If the translational energy of the argon atom is 4.14 X 10 J, what is a possible translational quantum state for the atom How much energy is required to promote the atom one quantum level in each dimension The energy differences between each translational quantum level is so small at these levels (i.e. a continuum) that in statistical thermodynamics it is approximated as infinitesimal. 

In the previous examples we only considered electronic energy changes and approximated the entropy as all or nothing. In essence, we assumed that gas-phase species have 100% of their standard state entropy and surface species possess no entropy at all. These assumptions can certainly be improved and in order to construct thermodynamically consistent microkinetic models this is not just optional, but absolutely necessary. Entropy and enthalpy corrections for surface species can be calculated using statistical thermodynamics from knowledge of the vibrational frequencies, and the translational and rotational degrees of freedom (DOF). In contrast to gas-phase molecules, adsorbates cannot freely rotate and move across the surface, but the translational and rotational DOF are frustrated within the potential energy well imposed by the surface. In the harmonic limit the frustrated translational and rotational DOF can conveniently be described as vibrational modes, which in turn means that any surface adsorbate will have iN vibrational DOFs that are all treated equally. 

Calculation of Thermodynamic Properties We note that the translational contributions to the thermodynamic properties depend on the mass or molecular weight of the molecule, the rotational contributions on the moments of inertia, the vibrational contributions on the fundamental vibrational frequencies, and the electronic contributions on the energies and statistical weight factors for the electronic states. With the aid of this information, as summarized in Tables 10.1 to 10.3 for a number of molecules, and the thermodynamic relationships summarized in Table 10.4, we can calculate a... 

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