Basic Statistical Mechanics

The Boltzmann distribution is fundamental to statistical mechanics. The Boltzmann distribution is derived by maximising the entropy of the system (in accordance with the second law of thermodynamics) subject to the constraints on the system. Let us consider a system containing N particles (atoms or molecules) such that the energy levels of the particles are i, 2 — If there are particles in the energy level ri2 particles in 2 and so on, then there are W ways in which this distribution can be achieved  

The most favourable distribution is the one with the highest weight, and this corresponds to the configuration with just one particle in each energy level (W = Ni). However, there are two important constraints on the system. First, the total energy is fixed  

The second constraint arises from the fact that the total number of particles is fixed  

The Boltzmann distribution gives the number of particles n, in each energy level e, as  

The denominator in this expression is the molecular partition function  

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We first consider tlnee examples as a prelude to the general discussion of basic statistical mechanics. These are (i) non-mteracting spin-i particles in a magnetic field, (ii) non-interacting point particles in a box,... 

Statistical Mechanical Background. We will present a brief review of the basic statistical mechanical concepts needed in this experiment, because standard textbooks in physical chemistry vary widely in their approach. 

We begin by recalling a few basic statistical mechanical notions [37]. These are as follows. 

The mathematical and physical theory of equilibrium cooperative phenomena in crystals has been reviewed by Newell and Montroll, and Domb, and the basic statistical mechanics is reviewed in Hill s monograph. Rowlinson has given a very thorough discussion of the classical thermodynamics of the coexistence curve and the critical region, and has also appraised much of the better data on equUibrium properties (of liquids and hquid mixtures). Rice > has several times reviewed the field of critical phenomena. 

The basic statistical mechanical expressions for the chemical potential are... 

Parametrization of the thermodynamic properties of pure electrolytes has been obtained [18] with use of density-dependent average diameter and dielectric parameter. Both are ways of including effects originating from the solvent, which do not exist in the primitive model. Obviously, they are not equivalent and they can be extracted from basic statistical mechanics arguments it has been shown [19] that, for a given repulsive potential, the equivalent hard core diameters are functions of the density and temperature Adelman has formally shown [20] (Friedman extended his work subsequently [21]) that deviations from pairwise additivity in the potential of average force between ions result in a dielectric parameter that is ion concentration dependent. Lastly, there is experimental evidence [22] for being a function of concentration. There are two important thermodynamic quantities that are commonly used to assess departures from ideality of solutions the osmotic coefficient and activity coefficients. The first coefficient refers to the thermodynamic properties of the solvent while the second one refers to the solute, provided that the reference state is the infinitely dilute solution. These quantities are classic and the reader is referred to other books for their definition [1, 4],... 

In attempting to devise a comprehensive theory for micelle formation, there are two possible approaches. One may, if so inclined, begin with basic statistical mechanics, taking into account complex interactions between surfactant molecules and water, as well as solute-solute and solvent-solvent interactions. However, since the fundamental principles of the hydrophobic interactions between small molecules in water are still not clearly defined, there seems to be httle hope that such an approach will produce a satisfactory result. However, even if a theoretically satisfying model did result, the mathematical complexities would possibly obscure any clear insight based on chemical realities. 

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