Equilibrium constant calculation from partition functions

In statistical mechanics the properties of a system in equilibrium are calculated from the partition function, which depending on the choice for the ensemble considered involves a sum over different states of the system. In the very popular canonical ensemble, that implies a constant number of particles N, volume V, and temperature T conditions, the quasiclassical partition function Q is... 

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] ... 

Equation 18.60 shows that statistical thermodynamics can calculate temperature-dependent equilibrium constants from partition functions. Because the partition functions themselves are ultimately determined from the energy levels of the chemical species, we see once again how a knowledge of energy levels—obtained from spectroscopy—helps us make thermodynamic predictions about chemical reactions. 

Thus, given sufEcient detailed knowledge of the internal energy levels of the molecules participating in a reaction, we can calculate the relevant partition functions, and then the equilibrium constant from Eq. (67). This approach is applicable in general Determine the partition function, then estimate the chemical potentials of the reacting species, and the equilibrium constant can be determined. A few examples will illustrate this approach. 

For t vo systems in chemical equilibrium we can calculate the equilibrium constant from the ratio of partition functions by requiring the chemical potentials of the t vo systems to be equal. 

Table 10.4 lists the rate parameters for the elementary steps of the CO + NO reaction in the limit of zero coverage. Parameters such as those listed in Tab. 10.4 form the highly desirable input for modeling overall reaction mechanisms. In addition, elementary rate parameters can be compared to calculations on the basis of the theories outlined in Chapters 3 and 6. In this way the kinetic parameters of elementary reaction steps provide, through spectroscopy and computational chemistry, a link between the intramolecular properties of adsorbed reactants and their reactivity Statistical thermodynamics furnishes the theoretical framework to describe how equilibrium constants and reaction rate constants depend on the partition functions of vibration and rotation. Thus, spectroscopy studies of adsorbed reactants and intermediates provide the input for computing equilibrium constants, while calculations on the transition states of reaction pathways, starting from structurally, electronically and vibrationally well-characterized ground states, enable the prediction of kinetic parameters. 

Isotope effects on equilibria have been formulated earlier in this chapter in terms of ratios of (s2/si)f values, referred to as reduced isotopic partition function ratios. From Equation 4.80, we recognize that the true value of the isotope effect is found by multiplying the ratio of reduced isotopic partition function ratios by ratios of s2/si values. Using Equation 4.116 one now knows how to calculate s2/si from ratios of factorials. Note well that symmetry numbers only enter when a molecule contains two or more identical atoms. Also note that at high temperature (s2/si)f approaches unity so that the high temperature equilibrium constant is the symmetry number factor. 

Pak and Chang have previously developed a partition function for liquid water applying the modified theory of significant liquid structure proposed by Chang et al. There, it is assumed that Ice-I-like, Ice-III-like, and gas-like molecules exist in liquid water and the molecules like Ice I and Ice III, both of which are oscillating torsionally, are in thermodynamic equilibrium. The equilibrium constant has been taken equal to the ratio of the partition functions of the two species. Various thermodynamic properties and the surface tension of liquid water from the partition function were successfully calculated. 

The equilibrium constants Kf are not measurable and we must resort to statistical thermodynamics to estimate these values theoretically. The partition function (Q) is a quantity with no simple physical significance but it may be substituted for concentrations in the calculation of equilibrium constants (Eqns. 4 and 5) [5], (It is assumed that there is no isotopic substitution in B.) Partition functions may be expressed as the product of contributions to the total energy from translational, rotational and vibrational motion (Eqn. 6). 

D24.3 The Eyring equation (eqn 24.53) results from activated complex theory, which is an attempt to account for the rate constants of bimolecular reactions of the form A + B iC -vPin terms of the formation of an activated complex. In the formulation of the theory, it is assumed that the activated complex and the reactants are in equilibrium, and the concentration of activated complex is calculated in terms of an equilibrium constant, which in turn is calculated from the partition functions of the reactants and a postulated form of the activated complex. It is further supposed that one normal mode of the activated complex, the one corresponding to displaconent along the reaction coordinate, has a very low force constant and displacement along this normal mode leads to products provided that the complex enters a certain configuration of its atoms, which is known as the transition stale. The derivation of the equilibrium constant from the partition functions leads to eqn 24.51 and in turn to eqn 24.53, the Eyring equation. See Section 24.4 for a more complete discussion of a complicated subject. 

The heat capacity C , thermodynamic functions S , -(G -Hj.)/T, H"- Hj., and the equilibrium constant K for the formation of PH as an ideal gas from the elements have been calculated for a standard-state pressure of 1 atm and tabulated for 298.15 K and between 0 and 6000 K at 100 K intervals [2, 4]. The JANAF data with T = 298 K are based on early spectroscopic data [7, 8] and semiempirical estimates for Dq and the electronically excited states [3]. In the third edition of the JANAF Tables [1], the old values were only converted to Joule units and to a standard pressure of 0.1 MPa. In the Russian compilation [4], values for a reference temperature T = 0 K are given that are based on a partition function established by using spectroscopic data for the X a A, A rij states [9,10] and theoretical data for the b and c 11 states [3, 11]. Selected values from [4] are ... 

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