Partition functions of a system

Example Partition Function of a System with an Infinite Number of Levels... 

Quantum Corrections. The obvious way to introduce quantum corrections in eq. 10 would be to interpret Za and Z as quantum partition functions however, this neglects tunneling (Z+, being the partition function of a system constrained to the top of the activation bar-... 

For the PMF (m>0, = q3m), becomes simply Z, a configurational partition function of a system of N particles. In a special case, if m = 2, then q3m for the pairwise potential V may be denoted as R - the distance between two chosen particles. Using this for calculating the intergral of Eq. (3-12) denoted in parentheses, leads to the following form ... 

These relations will be established specifically in Chapter 3. j3 = kT, where k is the Boltzmann constant, and A is the thermal de Broglie wavelength. Q Ua = 1) = ISa is the canonical ensemble partition function of a system... 

In Section IV.A, we have shown that the quantum partition function in D dimensions looks like a classical partition function of a system in (D+ 1) dimensions, with the extra dimension being the time. With this mapping and allowing the space and time variables to have discrete values, we turn the quantum problem into an effective classical lattice problem. 

Consider the grand partition function of a system of spherical particles exposed to an external potential i/r ... 

What is the importance of the partition function of a system What simplifications occur if the molecules of the system interact only weakly ... 

Once the partition function of a system is known, the thermodynamic functions are readily found. The total partition function of the system can be written in this case as... 

The partition function of a system with energy levels e(J) and degeneracies g(J) is... 

Each term A(T, P, Nu Nl) in the sum on the right-hand side of (2.4.8) is the partition function of a system in which the conversion reaction L H has been frozen in. In other words, the system is viewed as a mixture of two components L and... 

Thus, the partition function of a system with one additional water molecule at Rq is simply the sum over all the states of this... 

Knowing the partition function of a system containing N molecules occupying an area of A, the chemical potential of a molecule is ... 

Statistical mechanics. Statistical mechanics provides the tools required to deal with the physics of materials at high temperature. There are many excellent introductory texts for the interested reader (Hill 1956 Landau and Lifshitz 1980 McQuarrie 1976 Reif 1965). We focus on equilibrium thermodynamic properties because these play such an important role in our understanding of the Earth s interior. For the purposes of this discussion, we require a single result of statistical mechanics the partition function of a system of N atoms... 

In Section HI, Equation (III-53) we have derived an expression for the partition function of a system of interacting rigid hnear rotators in the approximation of small quantum corrections. 

This equation is equivalent to the statement that the partition function of a system with one additional s particle at a fixed position is the sum of the partition function of the same system with one A particle at a fixed position and the partition function of the same system with one B particle at a fixed position. 

The canonical partition function of a system composed of Ng molecules in a volume Vg at temperature T is given by ... 

Another approach for obtaining an equation of state is based on the partition function of a system derived firom statistical mechanics. One of these models is the Perturbed Hard-Chain Theory (PHCT) proposed by Beret and Prausnitz [6]. It was subsequently extended and modified by Cotterman et al. [7] and Morris et al. [8] as the Perturbed Soft-Chain Theory (PSCT). 

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 part of the chemical potential due to solvent-solute van der Waals interaction is customarily obtained by a coupling process, in which the interaction is turned on gradually by means of a coupling parameter I. The technique consists in the evaluation of the change in the partition function of a system of N solvent molecules after adding a solute molecule at a fixed position, and leads to the following expression for Gvdw ... 

The distribution of cluster sizes can be derived microscopically from statistical mechanics. The derivation is based on Refs. [10, 101, 9]. The partition function of a system containing N particles in a volume V at temperature T is given by... 

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