Partition functions, calculation formulae

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. 

At high temperatures (/S -r 0) the centroid (3.53) collapses to a point so that the centroid partition function (3.52) becomes a classical one (3.49b), and the velocity (3.63) should approach the classical value Uci- In particular, it can be directly shown [Voth et al. 1989b] that the centroid approximation provides the correct Wigner formula (2.11) for a parabolic barrier at T > T, if one uses the classical velocity factor u i. A. direct calculation of Ax for a parabolic barrier at T > Tc gives... 

Schrodinger equation. When the molecule is too large and difficult for quantum mechanical calculations, or the molecule interacts with many other molecules or an external field, we turn to the methods of molecular mechanics with empirical force fields. We compute and obtain numerical values of the partition functions, instead of precise formulas. The computation of thermodynamic properties proceeds by using a number of techniques, of which the most prominent are the molecular dynamics and the Monte Carlo methods. 

In order to calculate thermodynamic activation parameters, we need to know how to evaluate the translational, rotational, and vibrational parts of the partition functions. This can be accomplished by means of the standard formulas of statistical mechanics (see, for example, Dole, 1954). 

Typically the experiment with a transactinoid element lasts days or even weeks, and results in a single experimental value of the adsorption constant. The only possibility to obtain an estimate of the adsorption enthalpy based on such a result is to calculate the entropy change from the first principles [4] and substitute it into Eq. 5.7. The values of Aa( sS are calculated from the formulae of statistical mechanics for the particular model of the adsorbed state. The evaluation starts with the partition function of single molecule qm and with the molar partition function Z to calculate the absolute molar entropy from the general equation ... 

In this work 2 was a sphere of radius R and the nucleus was placed at the center of the sphere. This reduced the problem to that of the radial function only. In 1911, H. Weyl solved some vibrational problems [3], which now may be interpreted as describing the structure of the highly excited part of the spectrum of a free particle in a bounded region 2 with Dirichlet boundary conditions. Weyl s famous asymptotic formulae for the density of states in a region of large volume, that depends on the volume but not on the form of the region 2 (see e.g. Sect. VI.4. in [4], or Sect. XIII.15 in [5]), are usually used in physical chemistry when the partition function is calculated for translational motion of an ideal gas. Nowadays the next term in this asymptotic expression is usually studied in the theory of chaos (see e.g. Sect. 7.2 of [6]). 

First of all, we examine the partition function Z — an important function in thermodynamics and statistics, and calculate the free energy of the system according to the formula... 

The Gibbs free energies are calculated using standard thermodynamic tables which are easily usable by machine since they give the data in the form of polynomial coefficients. The data are sometimes limited to 6000 K and it is therefore necessary to make extrapolations or to carry out calculations of partition functions from spectroscopic data In the latter case, which is certainly more reliable, one can determine standard thermodynamic functions with the aid of the classical formulae of statistical thermodynamics. The results may then be fitted to polynomials so that they match the tabulated data . Furthermore the calculation of partition functions is necessary for spectroscopic diagnostics and for the calculations of reaction rate parameters. 

In the gas phase, it is usually sufhcient to calculate the partition functions and associated thermal corrections to the enthalpy and entropy using the standard textbook formulae [31] for an ideal gas under the harmonic oscillator-rigid rotor approximation, provided one then makes explicit corrections for low-frequency torsional modes. These modes can be treated instead as one-dimensional hindered internal rotations using the torsional eigenvalue summation procedure described in Ref. [32]. Rate and equilibrium constants can then be obtained from the following standard textbook formulae [31] ... 

The internal partition function for molecules having inversion may be factored, to a good approximation, into overall rotational and vibrational partition functions. Although inversion tunnelling results in a splitting of rotational energy levels, the statistical weights are such that the classical formulae for rotational contributions to thermodynamic functions may be used. The appropriate symmetry number depends on the procedure used to calculate the vibrational partition function. 

In the previous section we obtained a general formula for the translational partition function. In this section we obtain formulas for the other factors in the molecular partition function for dilute gases and carry out example calculations of partition functions. 

Calculate the rotational partition function of Br2 at 25.0 K using the formula in Problem 25.31. Compare your result with that obtained from Eq. (25.4-13). 

The thermodynamic functions of a dilute gas can be calculated from the molecular partition function of the gas. The necessary formulas are based on the postulates of statistical mechanics and on the definition of the statistical entropy... 

Now that we see that we can combine partition functions for all the quantized energy systems into a total partition function, we can think of other ways to use the quantized energy formulas. There is a curious history for this approach. We can see above that gvib is an important part of the total partition function and yet for many years low-resolution infrared spectra blurred many of the 3N — 6 vibrational modes of molecules typically larger than benzene. Thus the equations for quantum thermodynamics were known before 1940 but could only be applied to cases of small molecules in the gas phase using experimental vibrational frequencies. Since about 1985, quantum chemistry programs have included the calculation of vibrational frequencies with some correction factors that now make it possible to write down the full partition function by including theoretical... 

The calculation of the rate constant by the method of activated complex using formula (4.76) needs the knowledge of o, F, and F. The partition functions of the reactants F can be calculated through the molecular parameters (vibrational frequencies, rotational constants), which are available from the corresponding reference books. As for Eg and the statistical sum F, the calculations of PES are necessary. However, note that the calculation of F is not very sensitive to the specific features of PES. Therefore, to choose the structure of the activated complex, as a rule, approximate approaches are sufficient. 

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