Partition function translational motion

This equation contains the factor r, the rate of decomposition of the transition state which depends on the effective mass m, and k, neither of which is simple to estimate. However, the statistical mechanical expression for also contains m and 8 in the one-dimensional translational partition function corresponding to translation along the reaction coordinate. Combining the factor r above with the one-dimensional translational partition function for motion across the barrier, one obtains (see references at the end of the chapter) the simple result... 

The following derivation is modified from that of Fowler and Guggenheim [10,11]. The adsorbed molecules are considered to differ from gaseous ones in that their potential energy and local partition function (see Section XVI-4A) have been modified and that, instead of possessing normal translational motion, they are confined to localized sites without any interactions between adjacent molecules but with an adsorption energy Q. 

For translational, rotational and vibrational motion the partition function Ccin be calculated using standard results obtained by solving the Schrodinger equation ... 

We assign the 1 /N term to the translational partition function because all gases have translational motion, so that... 

We now have equations for the partition functions for the ideal gas and equations for relating the partition functions to the thermodynamic properties. We are ready to derive the equations for calculating the thermodynamic properties from the molecular parameters. As an example, let us calculate Um - t/o.m for the translational motion of the ideal gas. We start with... 

Partition functions are very important in estimating equilibrium constants and rate constants in elementary reaction steps. Therefore, we shall take a closer look at the partition functions of atoms and molecules. Motion, or translation, is the only degree of freedom that atoms have. Molecules also possess internal degrees of freedom, namely vibration and rotation. 

The function g is the partition function for the transition state, and Qr is the product of the partition functions for the reactant molecules. The partition function essentially counts the number of ways that thermal energy can be stored in the various modes (translation, rotation, vibration, etc.) of a system of molecules, and is directly related to the number of quantum states available at each energy. This is related to the freedom of motion in the various modes. From equations 6.5-7 and -16, we see that the entropy change is related to the ratio of the partition functions ... 

For translation motion, the partition functions of each degree of freedom is given by... 

In Equation 4.71 the individual qvib s have been specified qVib(v ) to indicate that these partition functions depend on the normal mode frequencies. It is interesting to note that the partition function for translation, which is usually considered in terms of the problem of the particle in a three dimensional rectangular box, is, itself a product of three partition functions one for motion in the x dimension, one for y, etc. 

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... 

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). 

The Boltzmann distribution of the populations of a collection of molecules at some temperature T was discussed in Section 8.3.2. This distribution, given by Eq. 8.46 or 8.88, was expressed in terms of the quantum mechanical energy levels and the partition function for a particular type of motion, for instance, translational, vibrational, or rotational motion. It is useful to express such population distributions in other forms, particularly to obtain an expression for the distribution of velocities. The velocity distribution function basically determines the (translational) energy available for overcoming a reaction barrier. It also determines the frequency of collisions, which directly contributes to the rate constant k. 

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. 

Note that the ratio qs/qg appears in the expression for Kp (Eq. 11.81). The gas-phase partition function could be written as a product of the contributions from translational motion times the contributions from all other internal degrees of freedom,... 

Note that the 1 /N term is assigned to the translational partition function, since all gases have translational motion, but only molecular gases have rotational and vibrational degrees of freedom. The electronic partition function is usually equal to one unless unpaired electrons are present in the atom or molecule. 

We insert Eq. (5.2) into Eq. (5.1), and note that the factors in front of the integral are related to the partition function associated with the relative translational motion of the reactants, see Eq. (A. 14) ... 

The partition functions can be factorized into contributions corresponding to the various forms of motion when they are uncoupled (see Appendix A.l), and it is advantageous to rewrite the expression for the rate constant in terms of partition function ratios for the translational, rotational, vibrational, and electronic motion ... 

We place the dividing surface between reactants and products at a critical separation rc and calculate the rate constant according to Eq. (6.8) (assuming, as in Section 4.1.2, that the atoms react with a probability of one when r = rc). In the relevant partition function for the activated complex, there are both translational degrees of freedom for the center-of-mass motion and rotational degrees of freedom... 

The harmonic approximation reduces to assuming the PES to be a hyperparaboloid in the vicinity of each of the local minima of the molecular potential energy. Under this assumption the thermodynamical quantities (and some other properties) can be obtained in the close form. Indeed, for the ideal gas of polyatomic molecules the partition function Q is a product of the partition functions corresponding to the translational, rotational, and vibrational motions of the nuclei and to that describing electronic degrees of freedom of an individual molecule ... 

The quantized energy ej can be of electronic, vibrational, rotational or translational type, readily calculated from the quantum laws of motion. In a macrosystem the sum over all the quantum states for the complete set of molecules, the sum over states defines the canonical partition function ... 

To obtain a more quantitative idea of these changes, let us consider individual partition functions for single-component, translational, rotational, and vibrational motions. They are given by (Secs. IX.8, 9, 10) ... 

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