Molecular partition functions

When the molecular partition function q is defined as (D.19)3, each energy level eo, i, S2. corresponding to each state o, i, is assumed to be 

Assuming = llhgT, let us consider the case where the temperature approaches absolute zero degrees. We set the ground state o of energy as T 0. Since q is divergent if eo 0 as T 0, we have the conditions 

On the other hand as T oo, the combination of the energy states must be infinite  

In this particular case, if the energy levels are distributed uniformly, we have 

If we apply the free particle solution in a one-dimensional domain (0, L), which will be shown in (D.47) to obtain the ground state energy e = iJi = the 

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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... 

Instead of formulating the reaction rate expression in terms of molecular partition functions, it is often convenient to employ an approach utilizing pseudo thermodynamic functions. From equation 4.3.29, the second-order rate constant is given by... 

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... 

Note the subscript C to indicate classical (or high temperature). In Equation 4.81 the p s are momenta and the q s the associated coordinates (not to be confused with q s previously used to symbolize molecular partition functions). In Cartesian coordinates dpjdqj = dpxidpyidpzidxidy1dzi with xi, yi, zi, the coordinates of atom... 

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 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... 

When treating polyatomics it is convenient to define an average molecular partition function, In = (lnQ)/N, for an assembly of N molecules. In the dilute vapor (ideal gas) this introduces no difficulty. There is no intermolecular interaction and In = (In Q)/N = ln(q) exactly (q is the microcanonical partition function). In the condensed phase, however, the Q s are no longer strictly factorable. Be that as it may, continuing, and assuming In = (In Q)/N, we are led to an approximate result which is superficially the same as Equation 5.10,... 

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.,... 

In a polyatomic molecule witli many vibrations, we simplify the vibrational partition function much as the original molecular partition function was simplified we assume that the total vibrational energy can be expressed as a sum of individual energies associated with each mode, in which case, for a non-linear molecule, we have... 

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... 

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