The molecular partition function

Then the values of H and G follow immediately from the definitions  

In a certain sense we have solved the problem of obtaining thermodynamic functions from the properties of molecules. These functions have been related to Q, which, by its definition, is related to the energy levels of the system, which are in turn related to the energy levels of the molecules in the system. To make these expressions useful, we must express the partition function in terms of the energies of the molecules. 

Consider the quantum state of the system that has the energy This energy is composed of the sum of the energies of the molecules 2, , plus any interaction energy W between the molecules  

For the present we assume that the particles do not interact (ideal gas) and set W = 0. Each energy corresponds to one of the allowed quantum states of the molecule. Because the energy has the form given by Eq. (29.28), it is possible to write the partition function as a product of partition functions of the individual molecules q. The final form is, for 

The sum in Eq. (29.30) is over all the quantum states of the molecule, so q is the molecular partition function. If quantum states have the same energy, they are said to be degenerate degeneracy = gi. The terms in the partition function can be grouped according to the energy level. The states yield g equal terms in the partition function. The expression in Eq. (29.30) can be written as 

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The denominator in this expression is the molecular partition function ... 

The numbers iVj and N- are only equal if there are no degeneracies. The sum in the denominator runs over all available molecular energy levels and it is called the molecular partition function. It is a quantity of immense importance in statistical thermodynamics, and it is given the special symbol q (sometimes z). We have... 

The q terms are the molecular partition functions of the superscript species. For the transition state,, the vibration along which the reaction takes place is omitted in the partition function, q. ... 

Equilibrium constants in the model were evaluated from the partition functions of the intermediates, assuming a uniformity of sites. The molecular partition function... 

The authors recognize that the symbol q has previously been used for thermodynamic heat. In using the letter q to symbolize the molecular partition function, usual practice is being followed. This usage should not give rise to confusion. 

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 formulae given in Table 4.1 for the molecular partition functions enable us to write the partition function ratio qheavy/qiight or q2/qi where, by the usual convention, the subscript 2 refers to the heavy isotopomer and 1 refers to the light isotopomer if heavy and light are appropriate designations. Then, a ratio of such partition function ratios enables one to evaluate the isotope effect on a gas phase equilibrium constant, as pointed out above. Before continuing, it is appropriate to... 

The second product is over the 3N—6(3N—5) normal mode frequencies of the ideal gas harmonic molecule to which Equation 4.78 applies. Thus the product over vibrations Equation 4.90 is indeed the quantum mechanical contribution to the molecular partition function for the ideal gas. 

This result indicates that (s2/si)f (compare Equations4.78 and 4.93) is just the quantum effect on the molecular partition functions of the normal mode vibrations. This result has now been derived without the explicit use of the Teller-Redlich product rule. 

For practical purposes the rules for diatomic molecules concerning even and odd J reduce to the statement that for homonuclear diatomic molecules the molecular partition function must be divided by two (s = 2), while for heteronuclear diatomic molecules no division is necessary (s = 1). The idea of the symmetry number, s,... 

The equilibrium constants are expressed in terms of the molecular partition functions... 

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

The equilibrium constant for each step is alculated from the molecular partition functions. 

Some important systems, which certainly do not fulfill the assumptions of harmonic transition state theory are gas phase reactions. In the gas phase, there are zero-modes such as translation and rotation, and these lead to totally different configuration integrals than those obtained from a normal mode analysis. For these species one can in a simple manner modify the terms going into the HTST rate by incorporating the molecular partition functions [3,119]. 

We have thus reduced the problem from finding the ensemble partition function Q to finding the molecular partition function q. In order to make further progress, we assume that the molecular energy e can be expressed as a separable sum of electronic, translational, rotational, and vibrational terms, i.e.,... 

The summation that appears in Eqs. 8.45, 8.46, and 8.49, is important enough that it is given a special name, the molecular partition function, denoted by q,... 

Thus the terms in the molecular partition function summation become... 

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... 

We note that the molecular partition function q that we have derived is for the special case of a collection of N indistinguishable molecules of one particular species. Mixtures of gases will have a different partition function, and the general case is not treated (and will not be needed) here. 

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. 

Here we use the label i to denote a molecular energy level, which may denote at once the specific translational (t), rotational (r), vibrational (u), and electronic (e) energy level of the molecule. From Eq. 8.46 and the definition of the molecular partition function q,... 

Using the previously derived expressions for q, we can now obtain expressions for each of the entropy terms. Equation 8.59 gives the molecular partition function for three-dimensional translational motion of a gas. Substituting this qtrans into Eq. 8.102, we obtain... 

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... 

For a temperature of 298.15 K, a pressure of 1 bar, and 1 mole of H2S, prepare a table of (1) the entropy (J/mol K), and separately the contributions from translation, rotation, each vibrational mode, and from electronically excited levels (2) specific heat at constant volume Cv (J/mol/K), and the separate contributions from each of the types of motions listed in (1) (3) the thermal internal energy E - Eo, and the separate contributions from each type of motion as before (4) the value of the molecular partition function q, and the separate contributions from each of the types of motions listed above (5) the specific heat at constant pressure (J/mol/K) (6) the thermal contribution to the enthalpy H-Ho (J/mol). 

We can now utilize some of the statistical mechanics relationships derived in Chapter 8 to find expressions for the free energy and the equilibrium constant in term of the molecular partition functions. From the definition of the free energy (Eq. 9.1) the expression for the enthalpy of an ideal gas (Eq. 8.121), and recalling that Ho = Eq (for an ideal gas), we obtain... 

Hard-Sphere Collision Limit It is interesting to evaluate the behavior of Eq. 10.9 when both reactants A and B are atomic species. In this situation the only degrees-of-freedom contribution to the molecular partition functions are from translational motion, evaluated via Eq. 8.59. The atomic species partition functions have no vibration, rotational, or (for the sake of simplicity) electronic contributions. 

Let qs be the molecular partition function for an adsorbed species. Consider the adsorption of N molecules on some portion of the surface containing a total of M possible adsorption sites. The system partition function Qs of the collection of N adsorbed species is... 

The numerator on the right-hand side of Eq. 11.100 is just the molecular partition function for the 2D gas (i.e., qxy), and the denominator is the product of the x-direction and y-direction partition functions for the bound motion in the potential well surrounding a surface site in the immobile-species case. 

The molecular partition function allows evaluation of the associated chemical potential as... 

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... 

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