Total partition function

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 total partition function may be approximated to the product of the partition function for each contribution to the heat capacity, that from the translational energy for atomic species, and translation plus rotation plus vibration for the diatomic and more complex species. Defining the partition function, PF, tlrrough the equation... 

This equation defines the quasienergy partition-function functional. Its use results in the total partition function written in terms of the coordinate Q alone... 

For a polyatomic molecule the total vibrational energy may be written as a sum of energies for each vibration, and the partition function as a product of partition functions. 

The partition function Q here describes the whole system consisting of N interacting particles, and the energy states Ei are consequently for all the particles (in Section 12.2 we considered N non-interacting molecules, where the total partition function could be written in terms of the partition function for one molecule, Q — /N[). More correctly... 

We run into a complication when we attempt to write an expression similar to equation (10.56) for a combination of N molecules that are not distinguishable.0 When this happens, combinations such as (ea. i + eb. ) and (ea2 + et,. i) are the same and should not be counted twice. Thus, the total number of terms in the partition function should be decreased to eliminate such duplications. To determine how to correct for this duplication, consider three gas molecules a, b, and c with energy levels we will represent as ei, e2, and e3. A total of 3 = 6 different combinations of these energy levels can be written as follows... 

Statistical weight factors are present in rroi and re eci. The multiplication involving these partition functions still works, since the total statistical weight factor is the product of the statistical weight factors for the individual units. Thus g, =g,.,rans g, g,.Mb g,. elect... 

But molecular gases also have rotation and vibration. We only make the correction for indistinguishability once. Thus, we do not divide by IV l to write the relationship between Zro[, the rotational partition function of N molecules, and rrol, the rotational partition function for an individual molecule, if we have already assigned the /N term to the translation. The same is true for the relationship between Zv,h and In general, we write for the total partition function Z for N units... 

The above two examples illustrate that the value of the partition function is an indicator for how many of the energy levels are occupied at a particular temperature. At T = 0, where the system is in the ground state, the partition function has the value q = 1. In the limit of infinite temperature, entropy demands that all states are equally occupied and the partition function becomes equal to the total number of energy levels. 

For a system of distinguishable particles the total partition function of the system is the product of all the individual partition functions, i.e. 

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 ... 

The probability distribution is normalized by ZM( p, t), which is a time-dependent partition function whose logarithm gives the nonequilibrium total entropy, which may be used as a generating function. 

The structure of a simple mixture is dominated by the repulsive forces between the molecules [15]. Any model of a liquid mixture and, a fortiori of a polymer solution, should therefore take proper account of the configurational entropy of the mixture [16-18]. In the standard lattice model of a polymer solution, it is assumed that polymers live on a regular lattice of n sites with coordination number q. If there are n2 polymer chains, each occupying r consecutive sites, then the remaining m single sites are occupied by the solvent. The total volume of the incompressible solution is n = m + m2. In the case r = 1, the combinatorial contribution of two kinds of molecules to the partition function is... 

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