Partition Function of a Diatomic Molecule

The energy and entropy are calculated from the partition function of a classical diatomic molecule. The internal entropy of a molecule depends logarithmically on its effective volume. The effective length of a molecule depends reciprocally on the force constant of the bond. 

A diatomic molecule has two atoms and hence six degrees of freedom. It is convenient to separate three degrees of freedom corresponding to the center of mass. The other degrees of freedom are then relative coordinates. In the present case the number of internal molecular degrees of freedom is 

The same holds for the internal momenta. Earlier in (4.1) the center of mass coordinates and momenta was defined. The center of mass coordinates are those of the vector 

By substitution it can be found that reduced mass and momentum are given by 

The degrees of freedom corresponding to center of mass motion and relative motion do not interact, thus p,f. can be written as 

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Finally, the rotational partition function of a diatomic molecule follows from the quantum mechanical energy level scheme ... 

Partition functions of a diatomic molecule (per degree of freedom)... 

This expression may be used for the vibrational partition function of a diatomic molecule at all temperatures the only approximation involved is that the oscillations are supposed to be harmonic in nature. The anhar-monicity correction must be made for precision calculations, but its effect is not large. The only property of the molecule required for the evaluation of Q, by equation (16.30) is the vibration frequency, which can be obtained from a study of its spectrum. The values of this frequency for a number of diatomic molecules are given in Table IX. ... 

Since the vibrational levels are nondegenerate, the vibrational partition function of a diatomic molecule is... 

It is important to note in Figure 1 that both curves show a decrease with temperature, and it should be possible to fit B smoothly onto A by multiplying by a suitable scale factor, possibly as shown by the dashed line. To explain the data shown in Figure 1 the temperature dependence of fs/fg is needed. The rotational partition function for a diatomic molecule that is free is... 

In order to use the procedure described above to calculate the partition function of a given molecule as a function of T and V, one must know the appropriate I and v values for the various degrees of freedom of the molecule. A great deal of this type of data is tabulated for diatomic molecules in [11] and for polyatomic molecules in [12]. Updated information may be found in [13], [14], and [15]. 

The rotational and vibrational factors in the partition function of the diatomic molecule A are given by equations (16.24) and (16.30), respectively. These are independent of the pressure (or volume) and require no adjustment or correction to the standard state. 

Expressions for the equation of state, de Broglie thermal wavelength, and rotational and vibrational partition functions for a diatomic molecule, provided by (28-87) and (28-89), as well as multiplication by Navo, allow one to determine the chemical potential on a molar basis. The final result for p, is consistent with its definition from classical thermodynamics ... 

Rgure 25.4 A Graphical Representation of the Rotational Partition Function (Drawn for CO at 298.15 K). This figure is analogous to Figure 25.2 for the translational partition function. The area under the bar graph is equal to the rotational partition function for a diatomic molecule, and the area under the curve is equal to the Integral approximation to the rotational partition function. 

Here m is the mass of the molecule. The rotational and vibrational partition functions for a diatomic molecule are [130]... 

The classical value is attained by most molecules at temperatures above 300 K for die translation and rotation components, but for some molecules, those which have high heats of formation from die constituent atoms such as H2, die classical value for die vibrational component is only reached above room temperature. Consideration of the vibrational partition function for a diatomic gas leads to the relation... 

The rotational microwave spectrum of a diatomic molecule has absorption lines (expressed as reciprocal wavenumbers cm ) at 20, 40, 60, 80 and 100 cm . Calculate the rotational partition function at 100 K from its fundamental definition, using kT/h= 69.5 cm" at 100 K. 

The adsorption of diatomic or dimeric molecules on a suitable cold crystalline surface can be quite realistically considered in terms of the dimer model in which dimers are represented by rigid rods which occupy the bonds (and associated terminal sites) of a plane lattice to the exclusion of other dimers. The partition function of a planar lattice of AT sites filled with jV dimers can be calculated exactly.7 Now if a single dimer is removed from the lattice, one is left with two monomers or holes which may separate. The equilibrium correlation between the two monomers, however, is appreciable. As in the case of Ising models, the correlation functions for particular directions of monomer-monomer separation can be expressed exactly in terms of a Toeplitz determinant.8 Although the structure of the basic generating functions is more complex than Eq. (12), the corresponding determinant for one direction has been reduced to an equally simple form.9 One discovers that the correlations decay asymptotically only as 1 /r1/2. 

We noted in Section 8.2 that only half the values of j are allowed for homonuclear diatomics or symmetric linear polyatomic molecules (only the even-y states or only the odd- y states, depending on the nuclear symmetries of the atoms). The evaluation of qmt would be the same as above, except that only half of the j s contribute. The result of the integration is exactly half the value in Eq. 8.64. Thus a general formula for the rotational partition function for a linear molecule is... 

This is the correct expression for the rotational partition function of a heteronuclear diatomic molecule. For a homonuclear diatomic molecule, however, it must be taken into account that the total wave function must be either symmetric or antisymmetric under the interchange of the two identical nuclei symmetric if the nuclei have integral spins or antisymmetric if they have half-integral spins. The effect on Qrot is that it should be replaced by Qrot/u, where a is a symmetry number that represents the number of indistinguishable orientations that the molecule can have (i.e., the number of ways the molecule can be rotated into itself ). Thus, Qrot in Eq. (A.19) should be replaced by Qrot/u, where a = 1 for a heteronuclear diatomic molecule and a = 2... 

Derive the value of the universal constant a in the expression Qr aa IT for the rotational partition function of any diatomic (or any linear) molecule I is the moment of inertia in e.g.s. units and T is the absolute temperature. Calculate the rotational partition function of carbon dioxide (a linear symmetrical molecule) at 25 C. 

After he had succeeded In enriching deuterium by the Raleigh distillation of liquid hydrogen, Urey undertook both theoretical and experimental investigations of the differences in the chemistry of protium and deuterium compounds. On the theoretical side Urey and Rittenberg (17) utilized the methods developed for the calculation of the partition function and the free energy of a diatomic molecule from spectroscopic data. For an ideal gas... 

In the molecular orbital method [8] the bonding is described in terms of linear combinations of atomic orbitals and the localised description of the chemical bond is lost, but as Mulliken [9] showed, it is possible to partition the electron density and thereby get an estimate of the spatial distribution and bonding character of an orbital. The procedure can be illustrated by the simple case of a diatomic molecule with one basis function per atom containing N electrons in a molecular orbital. Let the molecular orbital be written as ... 

The partition function of a system plays a central role in statistical thermodynamics. The concept was first introduced by Boltzmann, who gave it the German name Zustandssumme, i.e., a sum over states. The partition function is an important tool because it enables the calculation of the energy and entropy of a molecule, as well as its equilibrium. Rate constants of reactions in which the molecule is involved can even be predicted. The only input for calculating the partition function is the molecule s set of characteristic energies, ,-, as determined by spectroscopic measurements or by a quantum mechanical calculation. In the next section the entropy and energy of an ideal monoatomic gas and a diatomic molecule is computed. 

The general expressions for the energy and the entropy of a diatomic molecule are given. At high frequencies (these are often important) use the quantum mechanical expressions. The temperature dependent enthalpy and entropy depend on the frequencies of the system. The equilibrium constant for dissociation is related to the product and quotient of partition functions. Usually the partition function due to the rotational degrees of motion is much larger than that of the vibrational motion. 

Since the vibrational energy is a sum of 3n - 5 or 3n - 6 terms and since the quantum numbers for each vibration are independent of each other, the vibrational partition function of a polyatomic substance is a product of factors, and each factor is analogous to that of a diatomic molecule ... 

We represent the vibration of a diatomic molecule by a harmonic oscillator with mass II (the reduced mass of the molecule) and frequency v. The classical vibrational partition function for a harmonic oscillator is... 

First, we inquire under what condition is the steric factor unity, meaning that there are no steric requirements. We already know from Section 3.2 tiiat this condition is met when the two colliding particles are structureless. The partition function for the reactants is then Q= Q Q, one translational partition function (for a 3D motion) for each reactant. The point of no return has been identified in Section 3.2.7 the two particles are at a distance d apart and the barrier height is Eq. The transition state is here a diatomic. It has six degrees of freedom. Three are the motion of its center of mass. Of the other three, one is the vibrational motion, which is the reaction coordinate. Therefore, it should not be counted as an internal coordinate of the transition state. The other degrees of freedom are the two planes of rotation of a diatomic molecule. Therefore = 2x2r-the transition theory result for the reaction rate constant, in the absence of any steric effect, is... 

The expression for can be generalized by including the rotational energy and the rotational partition function in Eq. (3.5.6). The rotational energies of a diatomic molecule, given for the rigid rotator by Eq. (3.3.19), are... 

For calculations of rotational partition functions, the moment of inertia (eg, the molecule structure) should be known. In case of a diatomic molecule, the rotation partition... 

Under most circumstances the equations given in Table 10.4 accurately calculate the thermodynamic properties of the ideal gas. The most serious approximations involve the replacement of the summation with an integral [equations (10.94) and (10.95)] in calculating the partition function for the rigid rotator, and the approximation that the rotational and vibrational partition functions for a gas can be represented by those for a rigid rotator and harmonic oscillator. In general, the errors introduced by these approximations are most serious for the diatomic molecule." Fortunately, it is for the diatomic molecule that corrections are most easily calculated. It is also for these molecules that spectroscopic information is often available to make the corrections for anharmonicity and nonrigid rotator effects. We will summarize the relationships... 

Note that a diatomic molecule in the gas phase has only one vibration, but as soon as it adsorbs on the surface it acquires several more modes, some of which may have quite low frequencies. The total partition function of vibration then becomes the product of the individual partition functions ... 

Berlin showed that a diatomic molecule can be partitioned into a binding region and an antibinding region, as shown in Figure 6.1. These regions are separated by two surfaces of revolution given by the function B ... 

Consider the molecular rotational partition function for the CO molecule, a linear diatomic molecule. The moment of inertia of CO is / = 1.4498 x 10-46 kg-m2, and its rotational symmetry number is a = 1. Thus, evaluating Eq. 8.65 at T = 300 K, we find the rotational partition function to be... 

For a diatomic molecule the allowed rotational states will depend on whether or not the two nuclei are identical. At temperatures at which the energy difference of adjacent rotational states is small compared to kT we can write the approximate partition function for N molecules ... 

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