Partition function potentials

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. 

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... 

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. 

The canonical ensemble is the name given to an ensemble for constant temperature, number of particles and volume. For our purposes Jf can be considered the same as the total energy, (p r ), which equals the sum of the kinetic energy (jT(p )) of the system, which depends upon the momenta of the particles, and the potential energy (T (r )), which depends upon tlie positions. The factor N arises from the indistinguishability of the particles and the factor is required to ensure that the partition function is equal to the quantum mechanical result for a particle in a box. A short discussion of some of the key results of statistical mechanics is provided in Appendix 6.1 and further details can be found in standard textbooks. 

The thermodynamic properties that we have considered so far, such as the internal energy, the pressure and the heat capacity are collectively known as the mechanical properties and can be routinely obtained from a Monte Carlo or molecular dynamics simulation. Other thermodynamic properties are difficult to determine accurately without resorting to special techniques. These are the so-called entropic or thermal properties the free energy, the chemical potential and the entropy itself. The difference between the mechanical emd thermal properties is that the mechanical properties are related to the derivative of the partition function whereas the thermal properties are directly related to the partition function itself. To illustrate the difference between these two classes of properties, let us consider the internal energy, U, and the Fielmholtz free energy, A. These are related to the partition function by ... 

In an ideal gas there are no interactions between the particles and so the potential ener function, 1 ), equals zero. exp(- f (r )/fcBT) is therefore equal to 1 for every gas partic in the system. The integral of 1 over the coordinates of each atom is equal to the volume, ai so for N ideal gas particles the configurational integral is given by (V = volume). T1 leads to the following result for the canonical partition function of an ideal gas ... 

If hvQ is small compared with kT, the partition function becomes kT/hvQ. The function kT jh which pre-multiplies the collision number in the uansition state theoty of the bimolecular collision reaction can therefore be described as resulting from vibration of frequency vq along the transition bond between the A and B atoms, and measures the time between each potential n ansition from reactants to product which will only occur provided that die activation energy, AEq is available. 

Free energy calculations rely on the following thermodynamic perturbation theory [6-8]. Consider a system A described by the energy function = 17 + T. 17 = 17 (r ) is the potential energy, which depends on the coordinates = (Fi, r, , r ), and T is the kinetic energy, which (in a Cartesian coordinate system) depends on the velocities v. For concreteness, the system could be made up of a biomolecule in solution. We limit ourselves (mostly) to a classical mechanical description for simplicity and reasons of space. In the canonical thermodynamic ensemble (constant N, volume V, temperature T), the classical partition function Z is proportional to the configurational integral Q, which in a Cartesian coordinate system is... 

Strictly speaking, the concept of itself makes no sense for a potential like the Eckart one, unless one artificially introduces Zo as the partition function of a bound initial state, which is not described by this potential. That is to say, it is reasonable to consider the combination kZo, not k alone. 

The classical bath sees the quantum particle potential as averaged over the characteristic time, which - if we recall that in conventional units it equals hjk T- vanishes in the classical limit h- Q. The quasienergy partition function for the classical bath now simply turns into an ordinary integral in configuration space. 

Two situations are considered which differ in the number of constraints imposed. In the first one the shear strain in x and y directions is fixed, infinitesimal, reversible transformations are governed by the thermodynamic potential [see Eq. (9)], and X is the relevant partition function [see Eq. (52)]. Here the shear stress is computed as a function of the registry... 

The structure of the chapter is as follows. First, we start with a brief introduction of the important theoretical developments and relevant interesting experimental observations. In Sec. 2 we present fundamental relations of the liquid-state replica methodology. These include the definitions of the partition function and averaged grand thermodynamic potential, the fluctuations in the system and the correlation functions. In the second part of... 

Here Zint is the intramolecular partition function accounting for rotations and vibrations. However, in equilibrium, the chemical potential in the gas phase is equal to that in the adsorbate, fi, so that we can write the desorption rate in (I) as... 

This is our principal result for the rate of desorption from an adsorbate that remains in quasi-equihbrium throughout desorption. Noteworthy is the clear separation into a dynamic factor, the sticking coefficient S 6, T), and a thermodynamic factor involving single-particle partition functions and the chemical potential of the adsorbate. The sticking coefficient is a measure of the efficiency of energy transfer in adsorption. Since energy supply from the... 

In order to obtain the partition function for systems of this type (where the thermal energy and potential barrier are of the same magnitude), it is necessary to have the quantum mechanical energy levels associated with the barrier. Pitzer5 has used a potential of the form... 

Dissociation equilibriums in both electrolyte and polymer gels and the ionic concentration partition (Donnand potential) between solutions and polymer gels allow189 the relaxation-oxidation current to be obtained as a function of the perchlorate concentration ... 

Quite similar equations can be formulated for AG and AH by use of the partition function f of the activated complex. It follows from equations (6) and (7) that AEp can only be evaluated if the partition functions and AEz are available from spectroscopic data or heat capacity measurements. However, if AG = AH, the entropy change AS equals zero, and if AEz also equal to zero, either AG or AH can then be identified with the potential energy change. If... 

Other thermodynamic quantities such as chemical potential and entropy also follow directly from the partition function, as we demonstrate later on. However, to illustrate what a partition function means, we will first discuss two relatively simple but instructive examples. 

Note that the zero of energy is now the bottom of the potential, and the ground state -the lowest occupied level - lies Vihv higher. As partition functions are usually given with respect to the lowest occupied state, we shift the zero of energy upward by Vihv to obtain... 

By using Eq. (35) we find the chemical potential directly from the partition function ... 

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. 

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