Systems partition functions

It is also important to keep in mind the independent (state) variables that were specified in deriving q. That is, the partition function was derived with the number of molecules N, the volume of the system V, and the temperature T specified as the independent variables. Thus, when taking the derivative with respect to temperature, as will be needed later, it is good to keep in mind that q = q(N, V, T). The partition function for the entire system of identical molecules, with independent variables N, V, and T, is denoted by a capital Q. If the molecules are indistinguishable, as would normally be the case when calculating thermochemical properties for a given species, then the system partition function is related to the molecular partition function by... 

Equation 8.78 can also be written in terms of the system partition function Q as... 

Let qs be the molecular partition function for an adsorbed species. Consider the adsorption of N molecules on some portion of the surface containing a total of M possible adsorption sites. The system partition function Qs of the collection of N adsorbed species is... 

From Eq. 8.116, which related the Helmoholtz free energy to the system partition function Qs,... 

At a given temperature, which is larger the molecular partition function, q or the system partition function, Q1... 

For the surface, we calculate the Helmholtz free energy from Eq. (45) of Chapter 5 A = —RT In Q. We assume that surface molecules are distinguishable (by their position) and noninteracting, so that the system partition function is a product of N molecular partition functions. However, because we are not interested in which of the N out of a total of M surface sites are occupied, we must include a degeneracy factor of M /N M — A) . The energy of a molecule on the surface is taken as zero. 

From a statistical thermodynamic standpoint, the description of the folding/unfolding equilibrium in proteins requires the specification of the system partition function, Q defined as the sum of the statistical weights of all the possible states of the molecule (see Freire and Biltonen, 1978a) ... 

Name of Ensemble Independent Variables Types of Contact with Next System Partition Function Fundamental Thermodynamic Equation... 

The adsorbate is now visualized as a two-dimensional gas that can move freely tangential to the surface, where the molecules are "caught" In the force field of the surface. They have potential energy (caused by this attraction), vibration energy (normal to the surface) and translational energy. In (1.3.5.15) the following equation was derived for the sub-system partition function of a monatomic adsorbate ... 

According to our approach, known in the literature as the LFHB (lattice fluid hydrogen bonding) approach, the system partition function can be factored into a physical and a chemical or hydrogen bonding term ... 

For a system containing N distinguishable molecules such as an ideal crystal, the system partition function for all N molecules is given by... 

You cannot now factor the system partition function into particle partition functions as we did before. Elere s the problem. If one particle occupied energy level 27 and other particle occupied energy level 56, you could not distinguish that from the reverse. Because of this indistinguishability, you would have overcounted by a factor of 2 . ... 

A molecule in solution can be seen as a system complying with constant F, V, and N. Let Z9) denote its partition function. Having the partition function a probabilistic interpretation, the partition function of a composite system equals the product of the components partition functions, unless they are identical and indistinguishable. Consider for instance a system composed of N identical, indistinguishable molecules in solution. In this case, the system partition function is ... 

Consider the derivative of the molecular partition function, q, while recognizing that the same form of result holds for an N-particle system partition function, Q. 

Simplifying yields for the entire A-molecule system partition function... 

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