In Terms of the Molecular Partition Function

In the case of a pure ideal gas of volume V and temperature T, because there are no interactions among the molecules, the expression for the partition function is greatly simplified for it can be expressed in terms of the partition functions of the individual molecules, q  

The lower case symbols refer to molecules, is the degeneracy of molecular energy level ej, while the capital symbols 0, and ), refer to the whole gas. 

The development of Eq. 16.8.1 is given, among others, by Denbigh and by Reed and Gubbins. The basic assumption involved in its development is that each molecule is in a different q.s., and it is valid at temperatures above about 20 K. The kind of molecular statistics under these conditions is referred to as Maxwell-Boltzmann. At lower temperatures, the possibility of several molecules being in one quantum state must be considered, and Bose-Einstein or Fermi-Dirac statistics must be used (Reed and Gubbins). 

The term N arises from the fact that the particles are identical and free to move throughout the volume V. 

1n the case of an ideal gas mixture containing JVj molecules of type 1 and N2 of type 2  

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

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... 

The denominator in this equation has been given a special name, partition function, often symbolized by Z, which is derived from the German Zustandsumme (sum over states). The successive terms in the partition function describe the partition of the configurations among the respectives states available. One can express the thermodynamic state functions of an ideal gas in terms of the molecular partition function Z as follows ... 

The concentration of the reactants is determined by the quasi-equilibrium between reactants and transition state, for which the equilibrium constant is given by the ratio of the concentration of the transition state [ABC] and the reactants. Employing statistical considerations, the equUibrium constant can also be expressed in terms of the molecular partition functions [77] ... 

Because the quantity K is regarded as an equilibrium constant, you can express it in terms of the molecular partition functions developed in Chapter 13 (see Equation (13.18)) ... 

Now that we have the expression for the chemical potential in terms of the molecular partition function, we can write Eq. (26.1-27) in the form... 

The formulas for the thermodynamic functions in the previous section apply to any kind of system. They can be applied to a dilute gas by using Eq. (27.1-27) to express the canonical partition function in terms of the molecular partition function. 

Show that the formulas in Eq. (27.2-8) lead to the same formulas for the thermodynamic functions of a dilute gas in terms of the molecular partition function as in Section 26.1. 

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. ... 

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]. 

The term is the molecular partition function of the component /, the partition function expressed in relation to a molecule, so ... 

In the harmonic oscillator-rigid rotor approximation polyatomic molecules obey the same separation of their energy into four independent terms as in Eq. (25.4-5). In this approximation the molecular partition function of a polyatomic substance factors into the same four factors as in Eq. (25.4-6). The translational partition function is given by... 

The presence of potential energy in real fluids does not allow the simplification of the partition function in terms of the molecular ones, as is the case with the ideal gas. 

There are three approaches that may be used in deriving mathematical expressions for an adsorption isotherm. The first utilizes kinetic expressions for the rates of adsorption and desorption. At equilibrium these two rates must be equal. A second approach involves the use of statistical thermodynamics to obtain a pseudo equilibrium constant for the process in terms of the partition functions of vacant sites, adsorbed molecules, and gas phase molecules. A third approach using classical thermodynamics is also possible. Because it provides a useful physical picture of the molecular processes involved, we will adopt the kinetic approach in our derivations. 

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.,... 

Equation 11.81 is thus an expression for the Langmuir adsorption equilibrium constant in terms of the surface and gas molecular partition functions, qs and qg, respectively. 

For the vibrational term qivib, a classical high-T continuum approximation is seldom valid, and evaluation of the discrete sum over states is therefore required over the quantum vibrational distribution. (As pointed out in Sidebar 5.13, accurate treatment of molecular vibrations is crucial for accurate assessment of entropic contributions to AGrxn.) A simple quantum mechanical model of molecular vibrations is provided by the harmonic oscillator approximation for each of the 3N — 6 normal modes of vibration of a nonlinear polyatomic molecule of N atoms (cf. Sidebar 3.8). In this case, the quantum partition function can be evaluated analytically as... 

A precise treatment of the free energy change of protonation, expressed in terms of the partition functions and the various energy components, can be found in a discussion of the basic strengths of nitrogen heterocycles by Chalvet, O., Daudel, R., and Peradejordi, F., J. Chim. Phys. 59, 709 (1962). In their treatment, as in the present case, possible effects due to relative differences in molecular partition functions are neglected. 

In terms of the characterization of the thermodynamic behavior, the phase diagram and the variation of response functions - for example specific heat and the membrane-water partition coefficients seen in crossing the phase diagram - are important indicators of the molecular interactions [49],... 

Where e and e are the energy levels in an ideal system and an additional term only dependent of the whole properties of the phase, respectively. The molecular partition function becomes ... 

The treatment of units is here the same as that employed in connection with the translational partition function in the problem in 16e m, h and h are in c.g.s. units, and if P is in atm., R in the In R term should conveniently be in cc.-atm. deg. mole The actual weight m of the molecule may be replaced by M/N, where M is the ordinary molecular weight. Making these substitutions and converting the logarithms, it is found that... 

It has also been demonstrated by several authors [2] that a successful fit of the experimental data requires not only the assumption that k < 1, but also considering another important physical factor. This is the energetic surface heterogeneity of the actual gas/ solid interfaces. In terms of localized adsorption, this means a variation in the value of e on various surface sites. The accompanying variations in the molecular partition function ji are of lesser importance and are usually neglected. 

Thus the internal energy of a molecular system may be expressed directly in terms of the partition function z. If the energy levels of the system are known, z, and hence U, can be evaluated. Thus a thermodynamic function can be calculated from a knowledge of molecular properties. 

It is also possible to express the thermodynamic properties in terms of what is called a canonical partition function, Z. This is related to z by Z = zN for an ideal solid and by Z — zN/N for a perfect gas, where z is the molecular partition function. 

Much has been written about the relative merits of standard free energies, enthalpies and entropies as fundamental properties to elucidate chemical processes (see, for example, Taft, 1956 Leffler and Grunwald, 1963 Hepler, 1963 Larsen and Hepler, 1969 Wells, 1968 Exner, 1964a, b Hammett, 1970 Bell, 1973). In our opinion this question can only be answered in terms of the use to which the data will be put. Since AG°, AH° and AS° at room temperature all contain kinetic energy (partition function) terms, none of these properties corresponds exactly to the potential energy. Physical organic chemists are not put off much by this fact since they are usually more concerned with how properties change in response to systematic variation of molecular structure or solvent than they arc in particular properties of individual compounds. 

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