Function, partition

Once the partition function is known, the average energy and entropy per molecule, as well as the chemical potential can be directly calculated. These relations together with expressions for the partition function are summarized in Table 4.1. 

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. 

however, possible to replace the phase integral of Gibbs by a sum-over-statos or partition function, which in quantum statistics plays the same role for the calculation of thermodynamic properties that Gibbs phase integral plays in classical statistics. The partition function Q is defined as 

The individual products then represent the relative proba- 

The Gibbs phase integral may be employed in place of the partition function without serious error whenever the temperature of the system is such that the energy kT is much larger than the difference in energy of two adjacent quantum states. 

For many purposes it is possible to separate the total energy of a molecule into almost independent terms. We may thus write  

By following the Boltzmann formulation for the distribution function of a set of independent molecules among various energy states we have 

The exponential sum appearing in expression [A2.20] is found quite frequently in statistical thermodynamics. This sum is also called the partition function of 

The system s full partition function includes terms related to the i different types of energies nuclear, electronic, molecular vibrations, molecule rotations, as well as translation and interactions between the different types of molecules. 

To simplify this, we agree that for a molecule these different types of energies are independent (this is not quite trae, especially as the interdependence of the vibration and rotation energies of a molecule are known). We can therefore write the overall energy of a molecule like the sum of the different types of energy contributions  

We can show the partial partition functions related to the different types of energies  

The overall partial partition function can therefore be written as the product of the different types of partial partition functions  

We saw in Chapter 13 that the energy levels of the vib-rotor result from both vibrational and rotational levels. There we ignored the fact that the gas molecules also have translational energies and that within the stmcture of the molecule there are electronic and even nuclear energy levels. It is possible to combine the separate partition fimctions into one total partition function. We can illustrate this by considering only the combined levels of the vib-rotor in a single partition function 

Thus we see that we can easily write a total partition function as a product of each separate 

Now that we see that we can combine partition functions for all the quantized energy systems into a total partition function, we can think of other ways to use the quantized energy formulas. There is a curious history for this approach. We can see above that gvib is an important part of the total partition function and yet for many years low-resolution infrared spectra blurred many of the 3N — 6 vibrational modes of molecules typically larger than benzene. Thus the equations for quantum thermodynamics were known before 1940 but could only be applied to cases of small molecules in the gas phase using experimental vibrational frequencies. Since about 1985, quantum chemistry programs have included the calculation of vibrational frequencies with some correction factors that now make it possible to write down the full partition function by including theoretical 

In this section, classical and quantum methods of evaluating state densities are described to sketch the background theory. 

At equilibrium at a temperature T, the fraction of molecules occupying the ith quantum state is given by the Boltzmann distribution law 

It will be convenient to measure the E f relative to the molecular zero points and to account for any change of zero point energy separately. 

The partition function may be factorised into separate translational, vibrational and rotational terms. The translational and rotational levels are closely spaced compared with kT and the summation may be replaced with an integration without significant error. The resulting quantities 

The vibrational partition function for a quantised harmonic oscillator is 

We have found that the number of molecules in a specified state is given by — Q eilkr (Section 9.3). 

However, we shall continue to write the partition function in its simple form remembering that it represents a sum over states rather than energy levels. 

The internal energy of a system can be expressed in terms of the partition function. We note 

An ensemble is a collection of systems each of which contains N particles, occupies a volume V, and possess energy E. Each system represents one of the possible microscopic states and each is represented by a distribution of points in phase space. A phase space is defined by 3n coordinates and 3n momenta for a dynamic system consisting of n particles. There are three types of ensembles  

Microcanonical Ensemble The microcanonical ensemble is the assembly of all states in which the total energy E, the number of molecules N, and the total volume V are all fixed. It is a closed and isolated system. In a microcanonical ensemble, there is no fluctuation of any of the three variables N, E, and V. 

Canonical Ensemble In the canonical ensemble, all states have fixed V (volume) and N (the number of molecules), but the energy E fluctuates. The ensemble could be considered as a closed system in contact with a heat bath that has infinite heat capacity. 

Grand Canonical Ensemble In a grand canonical ensemble, not only E fluctuates but also N. The grand canonical ensemble is an open isothermal system. Both heat (energy) and mass (particles) can be transported across the walls of the system. 

The grand ensemble partition function is usually expressed as 

The constant is exactly analogous to any other equilibrium constant, and hence should be related to AG, Aff and A5, the standard free energy, enthalpy and entropy changes, respectively, accompanying the formation of the transition state from the reactants by means of the familiar thermodynamic relationships. In this manner, eq. (6.13) can 

The vibrational partition function of a harmonic oscillator has equally spaced energy levels (eq. (1.5)). Representing the separation between the energy levels by e, the vibrational partition function can be written as a geometric progression  

The linear, rigid rotor has energy levels given by 

In general, at room temperature, the difference between the rotational energy levels is much smaller than k T. Under these conditions, many rotational levels are occupied and it is reasonable to replace the summation by the integral 

The H2 molecule is one of the rare exceptions to the conditions formulated above, because its small mass, hence the small moment of inertia, leads to relatively large energetic separations between the energy levels. Given the H-H bond length /jjjj=0.741 A, we have / = 4.5X 10 kg m and AEj = 3 kJ mol , whereas at room temperature, iRT = 3.7 kJ mol-i. 

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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. 

Since translational and internal energy (of rotation and vibration) are independent, the partition function for the gas can be written... 

Of these, A are indistinguishable since the molecules are not labeled, and the complete partition function for N molecules becomes... 

It is now necessary to examine the partition function in more detail. The energy states for translation are assumed to be given by the quantum-mechanical picture of a particle in a box. For a one-dimensional box of length a. 

Thus the kinetic and statistical mechanical derivations may be brought into identity by means of a specific series of assumptions, including the assumption that the internal partition functions are the same for the two states (see Ref. 12). As discussed in Section XVI-4A, this last is almost certainly not the case because as a minimum effect some loss of rotational degrees of freedom should occur on adsorption. 

The grand canonical ensemble is a set of systems each with the same volume V, the same temperature T and the same chemical potential p (or if there is more than one substance present, the same set of p. s). This corresponds to a set of systems separated by diathennic and penneable walls and allowed to equilibrate. In classical thennodynamics, the appropriate fimction for fixed p, V, and Tis the productpV(see equation (A2.1.3 7)1 and statistical mechanics relates pV directly to the grand canonical partition function... 

Since other sets of constraints can be used, there are other ensembles and other partition functions, but these tliree are the most important. 

Once the partition function is evaluated, the contributions of the internal motion to thennodynamics can be evaluated. depends only on T, and has no effect on the pressure. Its effect on the heat capacity can be... 

The T-P partition ftmction can also be written in temis of the canonical partition function Qj as ... 

Thennodynamics of ideal quantum gases is typically obtained using a grand canonical ensemble. In principle this can also be done using a canonical ensemble partition function, Q =. exp(-p E ). For the photon and... 

Kaufman B 1949 Orystal statistics II. Partition function evaluated by Spinor analysis Phys. Rev. 65 1232... 

There is an inunediate coimection to the collision theory of bimolecular reactions. Introducing internal partition functions excluding the (separable) degrees of freedom for overall translation. 

Finally, the generalization of the partition function q m transition state theory (equation (A3.4.96)) is given by... 

These equations lead to fomis for the thermal rate constants that are perfectly similar to transition state theory, although the computations of the partition functions are different in detail. As described in figrne A3.4.7 various levels of the theory can be derived by successive approximations in this general state-selected fomr of the transition state theory in the framework of the statistical adiabatic chaimel model. We refer to the literature cited in the diagram for details. 

This is connnonly known as the transition state theory approximation to the rate constant. Note that all one needs to do to evaluate (A3.11.187) is to detennine the partition function of the reagents and transition state, which is a problem in statistical mechanics rather than dynamics. This makes transition state theory a very usefiil approach for many applications. However, what is left out are two potentially important effects, tiiimelling and barrier recrossing, bodi of which lead to CRTs that differ from the sum of step frmctions assumed in (A3.11.1831. 

All molecules in the second and subsequent layers are assumed to behave similarly to a liquid, in particular to have the same partition fimction. This is assumed to be different to the partition function (A2.2) of molecules adsorbed into the first layer. 

Otlier expressions for tire diffusion coefficient are based on tire concept of free volume [57], i.e. tire amount of volume in tire sample tliat is not occupied by tire polymer molecules. Computer simulations have also been used to quantify tire mobility of small molecules in polymers [58]. In a first approach, tire partition functions of tire ground... 

In the present study we try to obtain the isotherm equation in the form of a sum of the three terms Langmuir s, Henry s and multilayer adsorption, because it is the most convenient and is easily physically interpreted but, using more a realistic assumption. Namely, we take the partition functions as in the case of the isotherm of d Arcy and Watt [20], but assume that the value of V for the multilayer adsorption appearing in the (5) is equal to the sum of the number of adsorbed water molecules on the Langmuir s and Henry s sites ... 

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