Statistical thermodynamics Helmholtz energy

To establish the molecular thermodynamic model for uniform systems based on concepts from statistical mechanics, an effective method by combining statistical mechanics and molecular simulation has been recommended (Hu and Liu, 2006). Here, the role of molecular simulation is not limited to be a standard to test the reliability of models. More directly, a few simulation results are used to determine the analytical form and the corresponding coefficients of the models. It retains the rigor of statistical mechanics, while mathematical difficulties are avoided by using simulation results. The method is characterized by two steps (1) based on a statistical-mechanical derivation, an analytical expression is obtained first. The expression may contain unknown functions or coefficients because of mathematical difficulty or sometimes because of the introduced simplifications. (2) The form of the unknown functions or unknown coefficients is then determined by simulation results. For the adsorption of polymers at interfaces, simulation was used to test the validity of the weighting function of the WDA in DFT. For the meso-structure of a diblock copolymer melt confined in curved surfaces, we found from MC simulation that some more complex structures exist. From the information provided by simulation, these complex structures were approximated as a combination of simple structures. Then, the Helmholtz energy of these complex structures can be calculated by summing those of the different simple structures. 

For situations of overlapping chains, where lateral fluctuations in the segment concentration become rather small, mean-field descriptions become appropriate. The most successful of this type of theoiy is the lattice model of Scheutjens and Fleer (SF-theoiy). In chapter II.5 some aspects of this model were discussed. This theory predicts how the adsorbed amount and the concentration profile 0(z) depend on the interaction parameters and x and on the chain length N. From the statistical-thermodynamic treatment the Helmholtz energy and, hence, the surface pressure ti can also be obtcdned. When n is expressed as a function of the profile 0(z), the result may be written as ... 

Many thermodynamic and statistical mechanical theories of fluids lead to predictions of the Helmholtz energy A with T and V as the independent variables that is, the result of the theory is an expression of the form A = A T, V). The following figure is a plot of A for one molecular species as a function of specific volume at constant temperature. The curve on the left has been calculated assuming the species is present as a liquid, and the curve on the right assuming the species is a gas. 

If one employs the partition function Z for an ideal gas, as given by equations (28-59) and (28-87), then the Helmholtz free energy and chemical potential are calculated by combining classical and statistical thermodynamics ... 

Classical thennodynamics deals with the interconversion of energy in all its forms including mechanical, thermal and electrical. Helmholtz [1], Gibbs [2,3] and others defined state functions such as enthalpy, heat content and entropy to handle these relationships. State functions describe closed energy states/systems in which the energy conversions occur in equilibrium, reversible paths so that energy is conserved. These notions are more fully described below. State functions were described in Appendix 2A however, statistical thermodynamics derived state functions from statistical arguments based on molecular parameters rather than from basic definitions as summarized below. 

Statistical thermodynamics relates the partition function to the measurable properties of the system as in Eq. [3] where, for example, the Helmholtz free energy is related to the partition function by... 

As found in any textbook on statistical thermodynamics one can write, for instance, for the Helmholtz free energy ... 

The same result may be obtained, for a variation between any other pair of quantum states. The canonical distribution (11 24), which we have derived on strictly statistical grounds with the help of our basic postulate, therefore results in the function A being at a minimumf when a body of fixed volume is at equilibrium in a thermostat. The thermodynamic function which has this property is the Helmholtz free energy A. It seems probable, therefore, that A as previously defined, is a statistical analogue of the free energy of thermodynamics. Further grounds for this will be discussed in the next section. 

Regardless of whether the property formulation for a particular fluid is explicit in pressure, Helmholtz energy, Gibbs energy, or another property, the user must be given an assessment of the uncertainty of the predicted properties so that the equation can be considered practical. The quality of a thermodynamic property formulation is best determined by its ability to model the physical behaviour to represent the measured properties of the fluid. Statistics and deviation plots are used to show how thermodynamic properties calculated from equations of state compare to experimental data. 

In order to get expressions for Gibbs energy G and the Helmholtz energy A, we will need an expression for the entropy, S. The statistical thermodynamic approach for S is somewhat different. Rather than derive a statistical thermodynamic expression for S (which can be done but will not be given here ), we present Ludwig Boltzmanns 1877 seminal contribution relating entropy S and the distribution of particles in an ensemble il ... 

Integration of Eq. (278) yields the Helmholtz free energy of the ideal gas with the question of what the integration constant is. This, however, is known from quantum statistical thermodynamics, i.e. 

The central relationship of statistical thermodynamic links a thermodynamic quantity (the Helmholtz fi ee energy F) to a stastistical property of the system (the partition function Zjy of the system with N particles) through the Boltzmann constant kg and the temperature T of the heat reservoir to which the system is in thermal contact [4] ... 

We can express the Helmholtz energy using statistical thermodynamics, remembering that the molecules of solid are considered to be discernible molecules. If we have a population of Nj molecules of the species / which each have a molecular partition function z , then the Helmholtz energy of that species / is written as ... 

In this case, equation 14 provides a rational framework to extend existing expressions of the Helmholtz free energy from the equilibrium curve to the entire space of the non-equilibrium states of the system. Any mathematical model for the equilibrium thermodynamic properties of polymeric mixtures could be used to derive an expression for the non-equilibrium Helmholtz free energy according to the procedure described above. On the basis of statistical thermodynamic arguments. 

On the basis of statistical thermodynamic arguments and the mean field approximation, the lattice fluid theory provides the following expression for the equilibrium Helmholtz free energy per polymer mass in the limit of infinite coordination number (12) ... 

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. 

As in the classical expression (25) the quantity ip can be inferred directly as representing the statistical analogue of the Helmholtz free energy. The average behaviour of the canonical ensemble thus obeys the laws of thermodynamics. 

The determination of solvation free energy of solutes in solvents is a problem of primary importance, since all thermodynamic quantities can be derived from the free energies. For a system of N particles located at r/, r2, . ., rN, the statistical mechanical expression for the Helmholtz free energy, A, reads as [2]... 

In this section we study closed systems (closed to mass transport but not energy transfer) held at constant temperature. In statistical mechanics these systems are referred to as NVT systems (because the thermodynamic variables N, V, and T are held fixed). We shall see that the Helmholtz free energy represents the driving force for NVT systems. Just as an isolated system (an NVE system) evolves to increase its entropy, an NVT system evolves to decrease its Helmholtz free energy. 

Statistical mechanics of assemblies of axially symmetric molecules.— To illustrate the way in which the contributions of the directional forces to the thermodynamic functions can be estimated by statistical mechanics, we shall limit ourselves to pure substances in this section. The extension to mixtures will be discussed in the last section. The Helmholtz free energy of an assembly of N identical molecules occupying a volume V at temperature T is given by... 

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