Calculation of the translational partition function

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. 

The energy levels of a molecule confined to a cubic box of side l are given, as indicated in Section 9.1, by 

However, the translational levels are so close together we can replace the summation by an integral over nx 

An identical contribution arises from motion in the y and z-directions and 

The translational partition functions of gases are very large which tells us that there are many translational energy levels accessible to the molecules at normal temperatures. We can use the partition function to calculate the contribution of translational motion to the molar thermodynamic properties employing the relations of Table 9.3. 

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This definition is tantamount to replacing the volume V in the calculation of the translational partition function with the molar volume... 

The ratio of the translational partition functions is virtually 1 (because the masses nearly cancel explicit calculation gives 0.999). The same is true of the vibrational partition functions. Although the moments of inertia cancel in the rotational partition functions, the two homonuclear species each have er = 2, so... 

The objective of a simulation is to generate particle motion by using appropriate algorithms and to obtain an adequate distribution function, and thus, the macroscopic properties. In thermod5mamics, we use this method to calculate the energy of interaction of a collection of molecules, the part of configuration of the translational partition function and the radial distribution function. 

This is the translational partition function for any particle of mass m, moving over a line of length I, in one dimension. Please note that this result is exactly the same as that we calculated from quantum mechanics for a particle in a one-dimensional box. For a particle moving over an area A on a surface, the partition function of translation is... 

Hence, we conclude that the translational partition function of a particle depends on its mass, the temperature, the dimensionality as 3vell as the dimensions of the space in vhich it moves. As a result, translational partition functions may be large numbers. The translational partition function is conveniently calculated per volume, which is the quantity used, for example, when the equilibrium conditions are determined, as we shall see later. The partition function can conveniently be written as... 

The translational partition function is a function of both temperature and volume. However, none of the other partition functions have a volume dependence. It is thus convenient to eliminate the volume dependence of 5trans by agreeing to report values that use exclusively some volume that has been agreed upon by convention. The choices of the numerical value of V and its associated units define a standard state (or, more accurately, they contribute to an overall definition that may be considerably more detailed, as described further below). The most typical standard state used in theoretical calculations of entropies of translation is the volume occupied by one mole of ideal gas at 298 K and 1 atm pressure, namely, y° = 24.5 L. 

Problem Calculate the translational partition function for 1 mole of oxygen at 1 atm. pressure at 25 C, assuming the gas to behave ideally. 

Calculate the translational partition function and entropy of one mole of xenon at 1 atm pressure and 298 K. The relative molecular mass of xenon is 131,30. 

The volume of the bo. used to calculate the translational partition function for the activated complex was taken as I dm . True or False ... 

The vibrational and rotational components can be calculated from the harmonic oscillator and rigid rotor models, for example, whose expressions can be found in many textbooks of statistical thermodynamics [20]. If a more sophisticated correction is needed, vibrational anharmonic corrections and the hindered rotor are also valid models to be considered. The translational component can be calculated from the respective partition function or approximated, for example, by 3I2RT, the value found for an ideal monoatomic gas. 

In this equation k is called the transmission coefficient and is taken to be equal to unity in simple transition-state theory calculations, but is greater than imity when tunneling is important (see below), c° is the inverse of the reference volume assumed in calculating the translational partition function (see below), m is the molecularity of the reaction (ie, m = 1 for unimolecular, 2 for bimolecular, and so on), is Boltzmann s constant (1.380658 x 10 J molecule K ), h is Planck s constant (6.6260755 x 10 J s), Eq (commonly referred to as the reaction barrier) is the energy difference between the transition structure and the reactants (in their respective equiUbriiun geometries), Qj is the molecular partition function of the transition state, and Qi is the molecular partition function of reactant i. 

Here tSEehc, tSZPVE, Er and A5 are the diiferences in electronic energy, zero-point vibrational energy, thermal energy and entropy between the products and reactants, respectively. To calculate these thermodynamic parameters, the basic thermochemical quantity is the partition function q(F,7), which is the corresponding quantity of the total partition function Q in volume V. The total partition function Q is the product of partition functions for translation, rotation, vibration and electronic degrees of freedom. 

Statistical thermod5rnatnics enables us to express the entropy as a function of the canonical partition function Zc (relation [A2.39], see Appendix 2). This partition function is expressed by relation [A2.36], on the basis of the molecular partition functions. These molecular partition functions are expressed, in relation [A2.21], by the partition functions of translation, vibration and rotation. These are calculated on the basis of the molecule mass and relation [A2.26] for a perfect gas, the vibration frequencies (relation [A2.30]) of its bonds and of its moments of inertia (expression [A2.29]). These data are determined by stud5dng the spectra of the molecules - particularly the absorption spectra in the iirffared. Hence, at least for simple molecules, we are able to calculate an absolute value for the entropy - i.e. with no frame of reference, and in particular without the aid of Planck s hypothesis. 

Equation 17.54 is a useful conclusion. The (translational) partition function, originally defined as an infinite sum of negative exponentials of the energy levels, is equal to an expression in terms of the mass of the gas particles, the absolute temperature, the system volume, and some fundamental universal constants. This expression lets us calculate explicit values for q, which can then be used to determine values for energy, entropy, heat capacity, and so on. These calculated values—determined from a statistical rather than a phenomenological perspective—can then be compared to experimental values. We will thus get the first chance to see how well a statistical approach to thermodynamics compares with experiment. 

Thus, we can calculate the configuration part of the canonical partition function of translation, which is ... 

In the previous section we obtained a general formula for the translational partition function. In this section we obtain formulas for the other factors in the molecular partition function for dilute gases and carry out example calculations of partition functions. 

To calculate the width of a zone that is located at the top of the barrier, we assume that the activated particle should not have any translational degree of freedom in the direction of the movement (with such a translation, the particle would lose its activated character). Thus, the value of the corresponding partition function is 1 and then we have ... 

C) The error in AE" /AEq is 0.1 kcal/mol. Corrections from vibrations, rotations and translation are clearly necessary. Explicit calculation of the partition functions for anharmonic vibrations and internal rotations may be considered. However, at this point other factors also become important for the activation energy. These include for example ... 

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. 

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... 

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