Molecular partition function calculation

Here /i j3 is the chemical potential of the ideal gas at the standard pressure. It will be seen subsequently that qi for an ideal gas depends linearly on the volume V, so fif is a function only of the temperature. It does of course depend on the distribution of energy levels of the ideal gas molecules. The form of Equation 4.59 for the chemical potential of an ideal gas component is the same as that previously derived from thermodynamics (Equation 4.47). The present approach shows how to calculate m through the evaluation of the molecular partition function. Furthermore, the... 

The partition function ratios needed for the calculation of the isotope effect on the equilibrium constant K will be calculated, as before, in the harmonic-oscillator-rigid-rotor approximation for both reactants and transition states. One obtains in terms of molecular partition functions q... 

At the actual reaction temperature, the molecular partition function for all intermediates are calculated from the properties of each intermediate. 

We see that the partition function of a molecule is the product of the contributions of the translational, rotational, vibrational, and electronic partition functions, which we can calculate separately, as discussed next. We will see in Section 8.5 that any thermodynamic quantity of interest can be derived from the molecular partition function. Thus it is important to be able to evaluate q. 

It is also important to keep in mind the independent (state) variables that were specified in deriving q. That is, the partition function was derived with the number of molecules N, the volume of the system V, and the temperature T specified as the independent variables. Thus, when taking the derivative with respect to temperature, as will be needed later, it is good to keep in mind that q = q(N, V, T). The partition function for the entire system of identical molecules, with independent variables N, V, and T, is denoted by a capital Q. If the molecules are indistinguishable, as would normally be the case when calculating thermochemical properties for a given species, then the system partition function is related to the molecular partition function by... 

The partition function provides the bridge to calculating thermodynamic quantities of interest. Using the molecular partition function and formulas derived in this section, we will be able to calculate the internal energy E, the heat capacity Cp, and the entropy S of a gas from fundamental properties of the molecule, such as its mass, moments of inertia, and vibrational frequencies. Thus, if thermodynamic data are lacking for a species of interest, we usually know, or can estimate, these molecular constants, and we can calculate reasonably accurate thermodynamic quantities. In Section 8.6 we illustrate the practical application of the formulas derived here with a numerical example of the thermodynamic properties for the species CH3. 

All calculations will be done for the standard pressure of 1 bar and, unless otherwise noted, at T = 298.15 K for one mole of gas. Table 8.1 lists the calculated molecular partition function, thermal energy (energy in excess of the ground-state energy), heat capacity, and entropy. The individual contributions from translation, rotation, each of the six vibrational modes, and from the first excited electronic energy level are included. 

The translational contribution to the molecular partition function, which is calculated using Eq. 8.59, clearly makes the largest contribution. (In obtaining this value, we also made use of the ideal gas law to calculate the volume V = 0.02479 m3 of a mole of gas at this temperature and pressure.) The rotational partition function is evaluated via Eq. 8.67, and the vibrational partition function for each mode is found via Eq. 8.71. Only the very... 

The information needed to evaluate the molecular partition functions qb (13.63), may in principle be obtained from experimental spectroscopic measurements or theoretical calculations on each molecule i. Each type of energy contribution to qt (translational, rotational, vibrational, electronic) in principle requires associated quantum energy levels... 

The thermodynamic functions of the gaseous lanthanide trihalides have been calculated using standard statistical thermodynamic methods which relate the functions Cp, S, and H to the molecular partition function Q (Lewis et al., 1961) ... 

For the surface, we calculate the Helmholtz free energy from Eq. (45) of Chapter 5 A = —RT In Q. We assume that surface molecules are distinguishable (by their position) and noninteracting, so that the system partition function is a product of N molecular partition functions. However, because we are not interested in which of the N out of a total of M surface sites are occupied, we must include a degeneracy factor of M /N M — A) . The energy of a molecule on the surface is taken as zero. 

Another application of intermediate coupling calculations has been to use the calculated results to reevaluate dissociation energies derived using the third-law method and mass- spectral data. Balasubramanian and Pitzer have shown how this can be accomplished in their calculations on Sn2 and Pb2 (90). This method requires the molecular partition function, which can be written... 

It is interesting to see how the statistical treatment of equilibrium sys-tem may be applied to the calculation of kinetic data, at least to the same accuracy as was obtained when the assumption of a Maxwellian distribution was employed. Let us assume that we have a mixed gas of hard sphere molecules A and B, capable of forming a weakly bound complex AB. Let us further assume that the molecules A and B possess no internal energy. We can then write the molecular partition functions for A, B, and AB ... 

Hence, in the light of our both accounts of causality, the molecular dynamics model represents causal processes or chains of events. But is the derivation of a molecule s structure by a molecular dynamics simulation a causal explanation Here the answer is no. The molecular dynamics model alone is not used to explain a causal story elucidating the time evolution of the molecule s conformations. It is used to find the equilibrium conformation situation that comes about a theoretically infinite time interval. The calculation of a molecule s trajectory is only the first step in deriving any observable structural property of this molecule. After a molecular dynamics search we have to screen its trajectory for the energetic minima. We apply the Boltzmann distribution principle to infer the most probable conformation of this molecule.17 It is not a causal principle at work here. This principle is derived from thermodynamics, and hence is statistical. For example, to derive the expression for the Boltzmann distribution, one crucial step is to determine the number of possible realizations there are for each specific distribution of items over a number of energy levels. There is no existing explanation for something like the molecular partition function for a system in thermodynamic equilibrium solely by means of causal processes or causal stories based on considerations on closest possible worlds. 

Table 5.1 lists relationships needed to calculate the various terms of the molecular partition function. Further, the activity for any species j... 

The molecular partition functions can be used to calculate the equilibrium constant for the reaction between F and H2. For this gas-phase reaction... 

Equation (29.75) is an important link between quantum mechanics and chemistry. Knowing the energy levels of the molecules, we can calculate the molecular partition functions. Then we use Eq. (29.75) to obtain the equilibrium constant for the chemical reaction. 

The values of v and can be calculated if we write the equilibrium constant in terms of molecular partition functions per unit volume, qjV. [See Eq. (29.75).] Then... 

In the molecular-motion contribution, the molecular partition function is the product of translational, rotational, vibrational, and electronic partition functions. If the molecule in solution is assumed to have the entire volume of the solution available to it, the ratio of gas-phase and solution-phase translational partition functions equals one. Likewise, the electronic partition function ratio will be one. It is unclear what one should use for the rotational partition function in solution, but if this is assumed to have the same form as that in the gas phase, the rotational partition function ratio (which involves the moments of inertia) will be very close to one, since structural changes from gas to solution are slight. Significant contributions to the vibrational partition function are made only by the low-frequency vibrational normal modes, and these modes sometimes show substantial changes in frequency on going from the gas phase to solution. If a vibrational calculation is done in the gas phase and in solutitm, one can calculate AG°oiv m, but the most common procedure is to omit it, assuming that its contribution is negligible. 

The molecular partition functions, Q, can be related to molecular properties of reactants and products. The partition function expresses the probability of encountering a molecule, so that the ratio of partition functions for the products versus the reactants of a chemical reaction expresses the relative probability of encountering products versus reactants and, therefore, the equilibrium constant. The partition function can be written as a product of independent factors at the level of various approximations, each of which is related to the molecular mass, the principal moments of inertia, the normal vibration frequency, and the electronic energy levels, respectively. When the ratio of isotopic partition function is calculated, the electronic part of the partition function cancels, at the level of the Born-Oppenheimer approximation, an approximation stating that the motion of nuclei in ordinary molecular vibrations is slow relative to the motions of electrons. 

The pyrolysis mechanism of PPE and its derivatives given in Scheme 7.2 consist of bimolecular and unimolecular reactions. Applying transition state theory, we calculate the rate constants for the hydrogen abstraction reactions using Eq. (7.19) and the rate constants for reactions 1 and 3-5 using Eq. (7.21). The Wigner correction (Eq. (7.20)) is utilized to approximate quantum effects and the molecular partition functions are defined through Eqs (7.14), (7.15), (7.17), (7.18), and (7.28). 

At this juncture, the MAIN routine is given the temperature at which it is required to calculate the rate, whereupon it proceeds to calculate the molecular partition function for this temperature with PFNCTN. Hence,... 

A comparison of Eqs. (30) and (33) indicates that the rate constant k = coK. The equilibrium constant for production of activated complexes is related to molecular partition functions (Q), calculated using statistical mechanics ... 

Although we have derived this equation assuming that the reaction coordinate for atom transfer corresponds to translational motion, the same expression is obtained if the reaction coordinate is assumed to be vibrational motion. According to Eq. (37), the reaction rate constant may be calculated using the relevant molecular partition functions, known from statistical mechanics, remembering that Oahb clude the translational motion... 

Here Q is the zero point energy-corrected molecular partition function per unit volume for each species, k-Q the Boltzmann s constant, T the absolute temperature, h the Planck constant, and Eo=V] + Y eq, where sq is the zero point energy of the transition state minus the sum of the zero point energies of the reactants. Details for the calculation of... 

In this equation k is called the transmission coefficient and is taken to be equal to unity in simple transition-state theory calculations, but is greater than imity when tunneling is important (see below), c° is the inverse of the reference volume assumed in calculating the translational partition function (see below), m is the molecularity of the reaction (ie, m = 1 for unimolecular, 2 for bimolecular, and so on), is Boltzmann s constant (1.380658 x 10 J molecule K ), h is Planck s constant (6.6260755 x 10 J s), Eq (commonly referred to as the reaction barrier) is the energy difference between the transition structure and the reactants (in their respective equiUbriiun geometries), Qj is the molecular partition function of the transition state, and Qi is the molecular partition function of reactant i. 

Statistical thermod5rnatnics enables us to express the entropy as a function of the canonical partition function Zc (relation [A2.39], see Appendix 2). This partition function is expressed by relation [A2.36], on the basis of the molecular partition functions. These molecular partition functions are expressed, in relation [A2.21], by the partition functions of translation, vibration and rotation. These are calculated on the basis of the molecule mass and relation [A2.26] for a perfect gas, the vibration frequencies (relation [A2.30]) of its bonds and of its moments of inertia (expression [A2.29]). These data are determined by stud5dng the spectra of the molecules - particularly the absorption spectra in the iirffared. Hence, at least for simple molecules, we are able to calculate an absolute value for the entropy - i.e. with no frame of reference, and in particular without the aid of Planck s hypothesis. 

In Appendix 2 (section A2.5), we recapped the relations that exist between the equilibrium constant for different types of reaction, the molecular partition functions and the difference between the fundamental energies at the temperature of 0 K of the substances involved in the reaction. These partition functions and this energy difference can be calculated on the basis of the vibration frequencies of the molecules, their moments of inertia and their masses. Those data can be accessed on the basis of the absorption spectra, essentially in the infrared. Thus, it is possible to calculate the equilibrium constants a priori. 

By applying the definition [A2.20], we can calculate the contributions of each of the motions of the molecule to the molecular partition function. 

Thus, the equilibrium constant in the solid phase can be calculated on the basis of the molecular partition functions of vibration - i.e. the vibration frequencies of the molecules. Those values also enable us to calculate the residual energy of these molecules, and hence, in the case of perfect solutions, to determine the exponential term in relation [A2.70]. 

Quantum mechanics helps determine the energies of molecules and the molecular partition functions, as we will see in Chapter 6, can be calculated from the characteristics of molecules (mass, moments of inertia, vibrational frequencies, etc.). The aim of this section is to determine the value of the canonical partition functions, since this is useful to link them to molecular partition functions. 

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