THERMODYNAMIC PROPERTIES OF SIMPLE SYSTEMS

These statements are "laws of experience . That is, no one has been able to find exceptions to them (although many have tried). If one assumes that these two laws are valid, then four fundamental equations, referred to as the Four Fundamental Equations of Gibbs, can be obtained. From these four, more than 50,000,000 equations relating the thermodynamic properties of the system can be derived using relatively simple mathematics. The derivations are rigorous. Thus, if the two laws are true,... 

As in the case of a one-component system, ideal-gas (IG) mixtures also enjoy having a simple and solvable molecular theory, in the sense that one can calculate all the thermodynamic properties of the system from molecular properties of single molecules. We also have a truly molecular theory of mixtures of slightly nonideal gases, in which case one needs in addition to molecular properties of single molecules, also interactions between two or more molecules. 

In practice, it is not very convenient to define equilibrium with reference to total entropy. In most problems in chemistry and biology we are interested in the properties of the system itself and it is inconvenient to have to consider the properties of the surroundings. For example, if a process is made to occur in a beaker it is usually a relatively simple matter to determine the thermodynamic properties of the system itself, but rather troublesome to determine the properties of the surroundings. For. this reason, it is important to derive conditions for equilibrium which relate to the system itself and which ignore the properties of the surroundings. This is conveniently done by combining the first and second laws of thermodynamics. If a system is at equilibrium and we provide an infinitesimal amount of heat dq, the process of heat transfer must be reversible. From the second law, for a reversible process... 

Thermodynamic properties of a system can also be obtained from the atomistic considerations. Molecular dynamics or Monte Carlo methods have been successfully used to smdy polymers. The success stems from the fact that many properties can be projected from dynamics of relatively simple, oligomeric models. Unfortunately, miscibility strongly depends on the molecular weight and so far it cannot be examined by these methods. 

Here A - Ajg is the excess Helmholtz free energy with respect to an ideal gas at the same temperature, volume, and number density of each species. Thus, because of the minus sign, the factor kT, and the factor V in the first equality, si can be regarded as a negative dimensionless excess free energy density for the system. Since both A and Aig are extensive thermodynamic properties of the system, A/V and A JV are functions only of the intensive independent variables. Thus si has been expressed as a function of only the temperature and the number density of each species. (Moreover, we have chosen to use j8 = l/Ztr, rather than T, as the independent temperature variable.) It is this quantity si which has a simple representation in terms of graphs, which will be given below. If si can be calculated (exactly or approximately), this leads to (exact or approximate) results for A and hence for all the thermodynamic properties. 

Our aim in this section is to consider some unusual thermodynamic properties of chemical systems that could reveal themselves by decreasing the system dimensions. For this purpose we will analyze a simple model simulating the chemical reaction PQ P -h Q within a small volume conflned by a closed vesicle. We will see that small dimensions of vesicles under certain circumstances can impose limitations on the conventional thermodynamic description of chemical reactions based on the mass action law. The model considered may help to simulate chemical processes in the interior of small energy-transducing organelles. 

The determination of dense fluid properties from ab initio quantum mechanical calculations still appears to be some time from practical completion. Molecular dynamics and Monte Carlo calculations on rigid body motions with simple interacting forces have qualitatively produced all of the essential features of fluid systems and quantitative agreement for the thermodynamic properties of simple pure fluids and their mixtures. These calculations form the basis upon which perturbation methods can be used to obtain properties for polyatomic and polar fluid systems. All this work has provided insight for the development of the principle of corresponding state methods that describe the properties of larger molecules. 

Modern synthetic methods allow preparation of highly monodisperse spherical particles that at least approach closely the behavior of hard-spheres, in that interactions other than volume exclusion have only small influences on the thermodynamic properties of the system. These particles provide simple model systems for comparison with theories of colloidal dynamics. Because the hard-sphere potential energy is 0 or 00, the thermodynamic and static structural properties of a hard-sphere system are determined by the volume fraction of the spheres but are not affected by the temperature. Solutions of hard spheres are not simple hard-sphere systems. At very small separations, the molecular granularity of the solvent modifies the direct and hydrodynamic interactions between suspended particles. 

The exact computation of the partition function allows us to obtain all thermodynamic properties of the system this is possible for a small number of very simple systems, such as the ideal gas. For other systems, we can reach some... 

In general we may, however, say that the situation in the field of statistical mechanics of multicomponent systems is more favourable than for simple components. For the latter, there are only a few cases where there exists a satisfactory theory which is able to predict the thermodynamic properties of the system from first principles (imperfect gas, two dimensional Ising model...). New basic ideas are necessary for progress. In the theory of multicomponent systems it is, however, possible to gain at least a partial success by aiming for a less ambitious goal. If we postulate that the solution to the problem of single-component systems is known, we may try to express the properties of multicomponent systenos as far as possible in terms of those of pure components. 

The present book has been written with the enthousiastic cooperation of my group of co-workers. Particularly important contributions are due to Dr. A. Bellemans and Dr. V. Mathot. Dr. Bellemans is associated with the recent developments of the theory of multi-component systems in the field of isotopes and pol3nner solutions as well as with the formulation of the average potential model. Moreover, he contributed actively to Ch. IX-XI on the average potential theory and to Ch. XVI-XVTI on Polymers. Dr. Mathot, who contributed to the section on Polymer Solutions in Ch. Ill, was associated with the extension of the cell model to multicomponent sterns. His main contribution in this field is his experimental work on the thermodynamical properties of simple mixtures. For these reasons I have invited Dr. Bellemans and Dr. Mathot to be co-authors. 

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