Fundamental Relations for Closed Simple Systems

Thermodynamics includes a number of useful relations that allow one variable to be replaced by another to which it is equal but which can be more easily evaluated. In this section we obtain several such relations based on expressions for the differentials of state functions. For a closed simple system and for reversible processes the first law of thermodynamics is 

Combination of these equations using the fact that T cannot be negative gives an important relation for dV for reversible processes in a simple closed system  

This equation is restricted to reversible changes of state because dqtev was used in its derivation and because P = P(transmitted) was assumed. 

Irreversible thermodynamics or nonequilibrium thermodynamics is an extended version of thermodynamics that deals with rates of entropy production and with rates of processes and their driving forces. In irreversible thermodynamics, Eq. (4.2-3) is assumed to be valid for nonequilibrium changes if the deviation from equilibrium is not too large. This assumption is an additional hypothesis and does not follow from thermodynamics. See the Additional Reading section for further information on irreversible thermodynamics. 

The equilibrium macroscopic state of a one-phase simple system is specified by c + 2 independent variables, where c is the number of independent substances (components) in the system. If the system is closed, the amounts of the substances are fixed and only two variables can be varied independently. We take 17 to be a function of S and V for a simple closed system. An infinitesimal change in U that corresponds to a reversible process is given by the fundamental relation of differential calculus  

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Section 4.2 Fundamental Relations for Closed Simple Systems... 

Several fundamental relations were obtained for closed simple systems. The first relations were for the differentials of the different energy-related state variables for closed simple systems. For example,... 

The modern discipline of Materials Science and Engineering can be described as a search for experimental and theoretical relations between a material s processing, its resulting microstructure, and the properties arising from that microstructure. These relations are often complicated, and it is usually difficult to obtain closed-form solutions for them. For that reason, it is often attractive to supplement experimental work in this area with numerical simulations. During the past several years, we have developed a general finite element computer model which is able to capture the essential aspects of a variety of nonisothermal and reactive polymer processing operations. This "flow code" has been Implemented on a number of computer systems of various sizes, and a PC-compatible version is available on request. This paper is intended to outline the fundamentals which underlie this code, and to present some simple but illustrative examples of its use. 

The Maxwell relations (5.49a-d) are easy to rederive from the fundamental differential forms (5.46a-d). However, these relations are used so frequently that it is useful to employ a simple mnemonic device to recall their exact forms as needed. Sidebar 5.7 describes the thermodynamic magic square, which provides such a mnemonic for Maxwell relations and other fundamental relationships of simple (closed, single-component) systems. 

The Taylor dispersion problem is closely related to that discussed in the previous section, but also differs from it in some important fundamental respects. In the preceding problem, we assumed that the fluid was initially at a constant temperature upstream of z = 0 and that there was a constant heat flux into (or out of) the tube for all z > 0. In that case, the system has a steady-state temperature distribution at large times, and it was that steady-state problem that we solved. In the present case, there is no steady state. If the velocity were uniform across the tube instead of having the parabolic form (3 220), the temperature pulse that is initially at z = 0 would simply propagate downstream with the uniform velocity of the fluid, gradually spreading in the axial direction because of the action of heat conduction (i.e., the diffusion of heat). After a time /, the pulse would have moved downstream by a distance Uf, and the temperature pulse would have spread out over a distance of 0(s/(K tt)). Even in this simple case, there is clearly no steady state. The temperature distribution continues to evolve for all time.21... 

This idea may be clarified by a comparison with other branches of physics. Every department of deductive science must necessarily be founded on certain postulates which are regarded as fundamental. Frequently these fundamental postulates are so closely related to experiment that their acceptance follows directly upon the acceptance- of the experiments upon which they are based, as, for example, the inverse-square law of electrical attraction. In other cases the primary postulates are not so directly obvious from experiment, but owe their acceptance to the fact that conclusions drawn from them, often by long chains of reasoning, agree with experiment in all of the tests which have been made. The second law of thermodynamics is representative of this type of postulate. It is not customary to attempt to derive the second law for general systems from anything more fundamental, nor is it obvious that it follows directly from some simple experiment nevertheless, it is accepted as correct because deductions made from it agree with experiment. It is an assumption, justified only by the success achieved by its consequences. 

Primitive models have been very useful to resolve many of the fundamental questions related to ionic systems. The MSA in particular leads to relatively simple analytical expressions for the Helmholtz energy and pair distribution functions however, compared to experiment, a PM is limited in its ability to model electrolyte solutions at experimentally relevant conditions. Consider, for example, that an aqueous solution of NaCl of concentration 6 mol dm (a high concentration, close to the precipitation boundary for this solution) corresponds to a mole fraction of salt of just 0.1 i.e. such a solution is mostly water. Thus, we see that to estimate the density of such solutions accurately the solvent must be treated explicitly, and the same applies for many other thermodynamic properties, particularly those that are not excess properties. The success of the Triolo et approach can be attributed to the incor-... 

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