Helmholtz free energy from partition function

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. 

Figure A2.2.1. Heat capacity of a two-state system as a function of the dimensionless temperature, lc T/([iH). From the partition fimction, one also finds the Helmholtz free energy as...

From the normalization condition the relationship between Helmholtz free energy and the canonical partition function... 

From the partition function (3.5), we can now find the Helmholtz free energy, entropy, and Gibbs free energy of our gas. Using the equation... 

The free energy of the two surface system is calculated from the partition function expressed in terms of Flory-Huggins solution theory. The Helmholtz free energy per site, relative to pure solvent, and at constant [Pg.175]

To further speed up this approach, one can replace the expensive explicit-solvent simulations with implicit ones. Statistical mechanical theory gives the Helmholtz free energy A, apart from the scaling constant of the classical partition function that cancels out in binding energy calculations, as... 

The free energy of the system also includes entropic contributions arising from the internal fluctuations, which are expected to be different for the separate species and for the liganded complex. These can be estimated from normal-mode analyses by standard techniques,136,164 or by quasi-harmonic calculations that introduce approximate corrections for anharmonic effects 140,141 such approaches have been described in Chapt. IV.F. From the vibrational frequencies, the harmonic contribution to the thermodynamic properties can be calculated by using the multimode harmonic oscillator partition function and its derivatives. The expressions for the Helmholtz free energy, A, the energy, E, the heat capacity at constant volume, C , and the entropy are (without the zero-point correction)164... 

A worthwhile question to ask is why there should be a phase transition from a fluid to a solid in the hard-sphere model. We can formulate an initial perspective by considering the configurational Helmholtz free energy of the hard-sphere system, which is related to the configurational partition function via... 

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 chermcal potential and the entropy itself. The difference between the mechanical and 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 Helmholtz free energy, A. These are related to the partition function by ... 

A simple way (actually the only way) to determine this low temperature amorphous phase is to use a theory that correctly predicts the behavior of the liquid and extend it to low temperatures. One obviously should use the most realistic existing equilibrium theory to obtain the low temperature phase. The predictions are the two equations of state, S-V-T and P-V-T which are each derived from the Helmholtz free energy F which is in turn obtained from the partition function (F=-kTLnQ). In obtaining the S-V-T equation of state it is discovered that the configurational entropy Sc defined as the total entropy minus the vibrational entropy, approaches zero at a finite temperature (4), This vanishing of Sc is taken as the thermodynamic criterion of glass formation (5,6). 

The significance of the partition function Q is that thermodynamic functions, such as the internal energy U and Helmholtz free energy A (A = U- TS) can be calculated from it. 

The Helmholtz free energy follows from the partition function ... 

Various attempts have been made to include interactions between adsorbed species. As pointed out by Conway et al. [1984], the correct way to handle interactions is to include the appropriate pairwise or long-range interaction term into the partition function, which allows calculation of the Helmholtz free energy and the chemical potential. These quantities are a function of due to (a) the configurational term, as included in the Langmuir case and (b) the interaction or deviation from ideality. 

From the thermodynamic law (1.12), the Helmholtz free energy at a constant temperature is given by the work dR done for stretching the end vector from 0 to R. By the relation (1.11), the partition function is given by... 

The Helmholtz free energy can be derived from the partition function. Because of Equations (1.13) and (1.25),... 

In the case of two fluids 1 and 2, three different nearest-neighbor interaaions n, 22, and 12 are present. Since incompressibility is assumed from the outset, the Helmholtz free energy F is the relevant thermodynamic function. According to statistical mechanics it follows from the partition function Qby... 

Moreover, as seen in Chapter 1, each set of variables has a corresponding characteristic function, which is the thermodynamic potential. We saw that this is the Helmholtz free energy for variables linked to a eanonioal ensemble. Also note that this function is linked to the statistical description from the relation [5.42] and to the logarithm of the partition function of the corresponding ensemble. 

Secondly, in the theory of irreversible processes, variation principles may be expected to help establish a general statistical method for a system which is not far from equilibrium, just as the extremal property of entropy is quite important for establishing the statistical mechanics of matter in equilibrium. The distribution functions are determined so as to make thermod5mamic probability, the logarithm of which is the entropy, be a maximum under the imposed constraints. However, such methods for determining the statistical distribution of the s retem are confined to the case of a system in thermodynamic equilibrium. To deal with a system out of equilibrium, we must use a different device for each case, in contrast to the method of statistical thermodynamics, which is based on the general relation between the Helmholtz free energy and the partition function of the system. 

The Helmholtz free energy for independent particles is obtained from the single particle partition function Z via... 

The derivation of this relationship from the canonical ensemble partition function is straightforward. It is given here to illustrate the type of partition function manipulations commonly used in developing simulation expressions for thermodynamic quantities. The excess chemical potential is defined as the Helmholtz free energy difference between two (N + l)-particle systems, one... 

This discussion can be extended to NVT systems. Consider the process of free expansion for a system of N particles from volume V/2 to volume V. In this case assume that the entire system is in thermal contact with a heat bath of constant temperature. The change in the Helmholtz free energies between the two equilibrium states can be computed with the help of the partition function... 

The most successful statistical theory of liquids is that derived by Simha and Somcynsky. The model considers liquids to be mixtures of voids dispersed in solid matter, i.e., a lattice of unoccupied and occupied sites. The occupied volume fraction, y (or its counterpart the free volume fraction f = 1 - y), is the principal variable y = P, T). From die configurational partition function the configurational contribution to the Helmholtz molar free energy of liquid i was expressed as [3] ... 

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