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Methods of Thermodynamics
By Howard Reiss
Dover Publications, Inc.Copyright © 1965 Blaisdell Publishing Company
All rights reserved.
Some General Concepts
1. Objectives of Thermodynamics
Throughout this book, it will be assumed that the reader has had rudimentary thermodynamic training such as one might acquire in an elementary course in physical chemistry. In later chapters, we shall take time to define fundamental parameters such as temperature, surface tension, et cetera, whose aggregate forms the discipline of thermodynamics; but for the moment, we shall deal with the subject discursively rather than rigorously, assuming that the reader has some familiarity with these concepts. Later we shall deal with these same subjects more thoroughly.
Thermodynamics deals with the properties of matter in bulk. Although we know that matter is composed of elementary particles and may have a detailed organized structure, we cannot call upon this information for the purpose of thermodynamic reasoning. This does not mean that it is not possible nor desirable to use extra-thermodynamic information in conjunction with thermodynamic methods, but only that one should be careful to isolate thermodynamic from nonthermodynamic reasoning, especially because those relationships which have been established by means of thermodynamics will be invariant to all changes in our concepts of atomic and molecular phenomena. We shall refer to systems of matter in bulk as macroscopic, whereas atoms and molecules constitute microscopic systems.
At this point in time, one is usually willing to place a very high level of confidence in conclusions arrived at by thermodynamic means. Actually, there are not many things in this world which come to us bound with immutable fact; and therefore, we are led to suspect that there are not many things which can be known by thermodynamics alone. This is true. To make thermodynamics useful, it will be necessary to feed a great deal of information into its mechanisms. Our profit will be other facts which might have been obtained through the agency of additional measurement. This is one of the major services of thermodynamics—the avoidance of redundant measurements among macroscopic variables. For example, suppose that the coefficient of thermal expansion, the isothermal compressibility, the heat capacity at constant volume, and the temperature and volume of a substance have been measured. It will then prove unnecessary to measure its heat capacity at constant pressure, for thermodynamics reveals that the heat capacity is entirely determined by the former quantities. In another example, suppose that the latent heat of vaporization of a liquid and the heat capacities of both the liquid and its vapor have been measured. If the vapor pressure is known at one temperature, it becomes possible to specify its value at other temperatures through the application of thermodynamic methods.
This is the primary service afforded by thermodynamics, namely a means of transforming certain useful macroscopic data concerning a system into other useful macroscopic data on the same system. There are other applications. None of them, however, is quite so general as the one just mentioned. For example, thermodynamic methods have been utilized to investigate the stability of systems against change; or they have been used by engineers to discuss efficiencies, that is, the efficiency with which heat can be converted into mechanical work.
Originally, thermodynamics was intended to provide relationships between the parameters which describe systems at equilibrium. More recently, attempts have been made to apply thermodynamic methods to the macroscopic parameters of nonequilibrium systems. This endeavor has been rewarded with some, although limited, success. It has been suggested that what was originally thermodynamics should now be called thermostatics, and that the term thermodynamics should be reserved for the methods used to treat nonequilibrium systems. The general methods employed to interrelate macroscopic data in the manner described are conveniently referred to asphenomenological. Thus, thermostatics and thermodynamics are phenomenologies. In this book, the original meaning of the word thermodynamics will be retained, and the phenomenology of nonequilibrium processes may be denoted by the term irreversible thermodynamics.
2. The Thermodynamic System
We have already employed words like "experimental data," "system," "equilibrium," and "temperature" without adequate definitions. This is acceptable for purely discursive purposes such as these brief introductory passages, but will prove entirely inadequate in what follows. Although we shall employ the discursive style from time to time throughout the remainder of this chapter, there is something to be gained by defining a few things carefully. It is logical to begin with the definition of the thermodynamic system.
A thermodynamic system is an arbitrary geometrical portion of the universe with fixed or movable boundaries which may contain matter or energy or both.
Thus, a thermodynamic system is merely that part of the universe which we elect (arbitrarily) to focus attention upon. All the rest of the universe is then considered to be the environment.
We have mentioned that thermodynamics in its classical sense deals with systems which are in equilibrium. The definition of equilibrium requires careful discussion. Generally, there will be certain macroscopic attributes of the system (thermodynamic properties) which can be measured and assigned numerical values. Such properties might be pressure, temperature, volume, density, heat capacity, et cetera, most of which also require careful definition. Having understood the nature of thermodynamic properties we are in a position to define equilibrium:
A thermodynamic system is in equilibrium when none of its thermodynamic properties are changing with time at a measurable rate.
This definition leaves a lot to be desired, but it is difficult to furnish a better one. It provides the experimenter with a certain latitude in his assessment of what constitutes equilibrium. Its utility rests upon the idea that a continuity of phenomena exists such that if the system is not truly at equilibrium, and changes are occurring at an infinitesimal rate, then the relationships between macroscopic parameters will differ only infinitesimally from the relationships at true equilibrium.
The definition, unless further qualified, leaves the door open to still further confusion. For example, steady state phenomena are known in which matter or energy are flowing through the system at a definite rate while the local thermodynamic properties remain unchanged. The steady conduction of heat through a solid in which the local distributions of temperature and density remain invariant is an example of such a process. Such steady states must be distinguished from true equilibrium even though they seem consistent with the definition presented above. Usually the experimenter will have no difficulty in recognizing them, since obvious changes will take place in the surroundings of the thermodynamic system, even though the appearance of the system itself may remain invariant.
Another source of confusion which may arise in connection with the above definition concerns the reproducibility of the equilibrium state. Once it has been ascertained that the properties of the system are not changing with time and that a steady state is not involved, it may still be impractical to treat the system by the methods of thermodynamics. Before thermodynamics can be applied effectively, the experimenter must be able to reproduce the state at will. To accomplish this, he must know what variables determine the state; and even more important, he must be able to control these variables. As an example, a solid may be prepared in many states which meet the requirement of invariance to time (at least during the time of a measurement). But each of these states may be complicated by the fact that the solid contains an array of internal strains. To each distinct set of strains will correspond (under our definition) another state of equilibrium. The strains themselves are variables of the system, and unless we can control them and reproduce them, it will not be practical to employ the methods of thermodynamics.
4. Thermodynamic State, Variables of State
We are now in a position to define a thermodynamic state, a term which, unless otherwise specified, will be restricted to the description of systems at equilibrium. If the thermodynamic properties of a system at equilibrium are observed, it will be noted that whenever a certain number of them assumes certain numerical values all the remaining properties take on perfectly definite values. For example, in the case of a substance like water, whenever a given mass has a certain temperature and pressure, the volume of that mass will be determined in the sense that whenever the values of temperature and pressure are reproduced so will be the value of its volume. Alternatively, whenever the temperature and volume assume certain values, the pressure will be perfectly well defined. In order for all the properties to be reproduced, it is only necessary that the same number of them be reproduced. This number is called the degrees of freedom, or the number of independent variables of the system. A thermodynamic state is then defined in terms of the independent variables, each set of values for the independent variables corresponding to a state. For a macroscopic system, the number of independent thermodynamic variables is usually small. By contrast, if it were necessary to describe the microscopic state of some thermodynamic system, say a mole of water, it would be necessary to specify the values of the coordinates of position and momentum for all the elementary particles constituting that mole. Thus, the number of independent microscopic variables might be as high as 1024or 1025.
Why is so profound a simplification achieved in proceeding from the microscopic to the macroscopic mode of description? The answer (at least a crude one) seems to lie in the existence of reproducible average behavior. The rate of microscopic motion is so high that during the time required for the performance of a single macroscopic thermodynamic measurement the system has passed through a near infinity of microscopic states. At best, therefore, the thermodynamic measurement can sense the time average behavior of the system as it passes through its multitude of microscopic states. The average behavior, however, is reproducible and is determined by the specification of only a few conditions. These are the thermodynamic variables of state.
Specification of the number of independent variables, like the specification of equilibrium, must be accomplished by the experimenter. Thus, a given system having fixed composition in bulk may have temperature and volume as independent variables. Suppose now that it is pulverized into a very fine powder, or, if it is a liquid, dispersed into fine droplets. The reproducibility of the state of the system may then depend upon the experimenter's ability to reproduce its surface area as well as its temperature and volume. Under this condition there are three rather than two independent variables. At some state of subdivision the experimenter would have to decide to take account of the surface. This will be necessary whenever he discovers that reproducing the temperature and volume of the system does not guarantee the reproduction of a given state. When the experimenter decides that a property must be treated as an independent variable, he must in effect assume that this is the case, and his assumption should be a direct outgrowth of experiment.
This example may be used to point up the fact that the experimenter will have to use his judgment in deciding when a variable or property is macroscopic. Thus, if a process of subdividing is continued, the relevant surface area can become the total molecular surface (insofar as it is possible to give meaning to this concept). The total molecular surface will not do as a thermodynamic variable. Only macroscopic properties will suffice, and the experimenter must decide when a property ceases to be macroscopic.
5. Macroscopic State Space
Thermodynamics, taking no account of atomic or molecular structure, does not lend itself to discussion in terms of models and may therefore appear to be overly vague. Since models can provide valuable assistance to analytical thinking, it is worthwhile to introduce what is probably the next best thing in the abstraction of thermodynamics, namely the macroscopic state space. This is an abstract hyperspace whose coordinate axes are the several independent thermodynamic variables of the system in question. Thus, the state space of a simple one-component substance of fixed mass might be a plane along whose two cartesian axes are plotted the temperature and volume of the system. Any thermodynamic state can then be represented by a point on this plane, the coordinates of which are the temperature and volume appropriate to the state.
This example involves a two-dimensional state space. If, in addition, the mass of the substance is permitted to vary, then mass becomes a third independent variable; and it is necessary to augment the state space with a third cartesian axis. Then we have a three-dimensional state space. If there are more than three independent variables, more axes must be added so that a hyperspace of more than three dimensions results.
State space provides a framework within which, in lieu of any models, one can perform thermodynamic reasoning in a more concrete manner. Notice that all the points in state space, since they represent thermodynamic states, are equilibrium points. Any two points in this space specify a thermodynamic change of state. Notice that the change of state does not specify the path. There are many different paths which correspond to the same change of state; that is, the same end points may be connected by a plurality of paths. In fact, it may not be possible to plot some paths in state space. These are nonquasistatic (see Section 7) paths which do not consist of sequences of equilibrium states. We will employ the term process to describe a path by which two points in state space are connected. Thus, to a given change of state there may correspond several processes, whereas to a given process there corresponds only one change of state.
The reader will find it a great aid to his understanding if, whenever possible, he performs his thermodynamic reasoning in terms of state space. In our discussions throughout the book, we shall return to it again and again until it becomes a realistic domain within which we are made to feel at home.
6. Mechanical Work
The expeditious application of thermodynamics to various systems requires us to be able to express, quantitatively, the mechanical work exchanged between a system and its environment. In fact, it will emerge as the text unfolds that the method of treatment of a given system depends, more than anything else, upon the kind of work which the system may perform.
If the system is a simple substance, liquid water for example, then it may perform work on its environment, or the environment may perform work on it, when its volume changes. Suppose that the medium surrounding the system is a fluid which exerts only hydrostatic stresses at the pressure p. Then if the volume of the system is V and an infinitesimal change in volume is represented by dV, the work performed by the system on its environment may be represented by
Dw = pdV. (1.1)
This relation is easily derived if the liquid is considered to be enclosed by a piston in a cylinder of cross-sectional area A, and if its volume is increased by dV through the outward movement of the piston through the distance dl. The force on the piston is simply pA, the pressure times the area, and the work done (force times distance) is therefore
Dw = pAdl = pdV, (1.2)
since dV = Adl. Equation (1.1) may be generalized to a system of any shape.
Most of the time no difficulty is encountered in arriving at a quantitative expression such as (1.1). For example, in the case of an elastic system the incremental work performed on the elastic body may be represented as the force required to induce the extension multiplied by the incremental extension; or in the case of a system in which surface is important, the work performed in extending the surface may be expressed as the surface tension multiplied by the incremental extension in surface. These forms are all quite similar to Equation (1.1).
Excerpted from Methods of Thermodynamics by Howard Reiss. Copyright © 1965 Blaisdell Publishing Company. Excerpted by permission of Dover Publications, Inc..
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