Thermodynamic limit molecular systems

Free energy calculations rely on a well-known thermodynamic perturbation theory [6, 21, 22], which is recalled in Chap. 2. We consider a molecular system, described by the potential energy function U(rN), which depends on the coordinates of the N atoms rN = (n, r2,..., r/v). The system could be a biomolecule in solution, for example. We limit ourselves to a classical mechanical description, for simplicity. Practical calculations always consider differences between two or more similar systems, such as a protein complexed with two different ligands. Therefore, we consider a change in the system, such that the potential energy function becomes ... 

We do not consider here the case in which a nuclear magnetic subsystem or any other subsystem that has a limited number of quantum states may be considered as a thermodynamic system separate from the other parts of the total molecular system. 

In Figure 4.2, crystallization of sucrose can only occur at conditions of temperature and composition that fall between the solubility and glass transition temperature lines. On either side of this crystallization envelope, there is either no thermodynamic driving force (dilute system) for crystallization or nucleation is constrained by kinetic effects due to limited molecular mobility. Thus, processing conditions must be controlled so that the system falls within the crystallization envelope to ensure crystallization. Furthermore, the point within the crystallization envelope at which crystallization occurs can significandy affect the nature of the crystalline phase in the food, and thus affect the material properties. 

Simulation calculations on finite systems will entail some error associated with the submacroscopic size considered (Lebowitz and Percus, 1961a b 1963). For example, periodic boundary conditions will influence molecular correlations to some extent (Pratt and Haan, 1981a b). Support of a claim of accuracy would typically involve some practical investigation of the thermodynamic limit. A claim of preference for calculations in one ensemble over another is typically made first on the basis of convenience rather than on the basis of accuracy defined in some absolute way. Thus, advantages of practical accuracy for ensemble-specialized alternatives to Eq. (3.18) are not proven typically, and they are not necessary fundamentally. 

The field of bioenergetics needs a rigorous quantitative description of processes to test proposed molecular mechanisms for these processes. Although one might think that kinetics alone would be sufficient to cover this need, in practice it is impossible to include all properties of the participating enzymes in a complex system in a complete kinetic description. Furthermore, kinetics alone will not reveal the thermodynamic limitations of the possible pathways. Therefore, a fruitful symbiosis of kinetics and thermodynamics has been developed over the past years. 

Various continuum limits of the lattice description are taken (i.e., the number of Cl molecules Nm + , the number of C2 molecules Nm2 - °°, and the number of lattice sites M + ), such that the density and the other thermodynamic and molecular ordering variables can vary continuously for the system of molecules. (In these limits, the lattice statistics can treat molecules in which the number of segments per molecule is not an integer.)... 

The aim of the descriptions and analyses presented in the foregoing sections was to illuminate some aspects of chemical lasers as molecular systems far from equilibrium. Particular emphasis was drawn on the limits of weak and strong rotational coupling since they represent extremely different kinetic schemes and consequently different kinetic behaviors. Thermodynamic considerations were employed to complement the detailed kinetic description. The thermodynamic approach can yield additional physical insights, but (at least so far) not new quantitative data. These are... 

Traditionally, the thermodynamics of fluids used in engineering is essentially macroscopic. Fluids are treated as homogeneous molecular structure and fluctuations are ignored. Size and surface effects disappear in the thermodynamic limit in which the volume V and the number of particles N tend to infinity while the molecular density of the substance, p = NjV, remains finite. Macroscopic thermodynamics often eliminates the size of the system by reducing the extensive thermodynamic properties by the number of particles, mass, or volume. The actual scale is restored only in the stage of engineering design. 

Fluctuations are spontaneous and random deviations of thermodynamic properties from their average equilibrium values. These deviations are caused by thermal molecular motion. Macroscopic thermodynamics ignores fluctuations because they do not affect thermodynamic properties in the thermodynamic limit and they are usually insignificant in finite macroscopic systems. However, the situation changes when the system becomes very small or when it is near the limit of thermodynamic stability. In these two cases, fluctuations may become very large and may play a significant role in determining thermodynamic properties. 

As was noted above, the model parameter K is the true constant whose value is determined by the molecular properties of the system. Since constant K is the characteristic of a single PQ molecule, its value does not depend on the volume of a whole system. On the other hand, the experimentally determined conventional equilibrium constant in the general case, is not a real constant. This constant, being calculated from the measured experimental concentrations of particles, will depend on the volume confining these particles. In the thermodynamic limit only, when we can neglect the fluctuations, the apparent equilibrium constant would be equal to the real one, app = = const. 

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