Chapter 1: Introduction

Transcription

1 Chapter 1: Introduction Chapter 1: Introduction Why Study Thermodynamics? Outline of a Physical Theory Continuum and Atomistic Models References...5 Chapter 2: Scope and Definitions Scope Definitions The system Physical properties Thermodynamic state Equilibrium state Constitutive coordinates and constitutive equations Geometric coordinates and conjugate forces Classes, types, and particular materials Changes of state: transitions Walls and partitions Probes and reservoirs Simple systems Standard systems...15 Chapter 3: The Zeroth Law Experimental Basis The Zeroth Law The Empirical Temperature Measurability: Thermometers The Thermal Equation of State The Temperature of a Non-Simple System...19 This course is concerned with the science of thermodynamics, and, in particular, with the implications of thermodynamics for the behavior of engineering materials. 1.1 WHY STUDY THERMODYNAMICS? It is almost universally accepted that a thorough grounding in thermodynamics is an essential part of the education of a materials scientist. It is less widely understood why this should be so, particularly by those who are required to take the course. The communication problem is compounded by the fact that most graduate students in materials science have Page 1

2 had some undergraduate training in thermodynamics, and have been introduced to the subject as the "Science of Heat". While heat is certainly an important thing in many of the processes that influence materials, there is no obvious reason why it should be regarded as a central feature of the whole field of materials science. The basic reason that thermodynamics is central to materials science is that it is much less the "Science of Heat" than the "Science of Time". The most important of the laws of thermodynamics is the Second Law, and its main contribution to science is to put a direction on the variable "time" that distinguishes the past from the future. Under the constraints imposed by the Second Law, the possible states of any physical system are divisible into three sets: those that necessarily belong to the past, those that are alternate possibilities for the present, and those that lie in the future. The Second Law governs what a system may become as it evolves into the future, defines the driving forces that impel it to go there, and limits the paths it may take. As the "arrow of time" (in Max Born's phrase) the Second Law ranks among the cardinal principles of theoretical physics. [This phrasing of the Second Law may come as something of a shock to students who were introduced to it as a limit on the practical efficiency of heat engines. That the Carnot limit should be a consequence of the fact that the past is distinguishable from the future may seem genuinely weird. As we shall see, the time defined by the Second Law is evolutionary time, the "time" of the system itself. And Carnot efficiency is achieved when a heat engine is operated so that it stays in its present. While the thought may not immediately occur to the typical graduate student, those of us who have been around for a while are not surprised that the only engines that continue to work at peak efficiency are those that stay forever young.] Given that the Second Law is a cardinal principle of theoretical physics, why do materials scientists spend so much more time studying it than physicists do? The answer lies in the nature of the problems they have to solve. Physicists tend to work on problems that are reasonably well-posed and mathematically tractable. The evolution of the systems they study is ordinarily found by solving the pertinent equations of motion, and it is possible to do that, at least in an approximate way. If the solutions are correct, they are automatically consistent with the Second Law, and the physicist need not concern himself with the constraints the Second Law lays down. Materials scientists, on the other hand, are commnly faced with problems that are ill-posed and cannot be solved analytically. The properties of an engineering material are determined by its microstructure, which is, in turn, determined by the processing the material has received. Excepting trivial cases, the microstructure cannot even be described in a language that is mathematically precise. (Ask a physicist to describe his system and he will, ordinarily, write down an equation. Ask a materials scientist, and he will show you a picture, or, worse, provide a verbal description of what a picture might look like if he had one.) The microstructure is not even unique. Natural variationss in composition and processing have the consequence that nominally identical engineering materials have microstructures that are easily distinguished from one another. Page 2

3 Since most of the problems that the materials scientist worries about fall into the class that cannot be solved mathematically, he has less need to know the methods of computational physics, and more need to grasp the theories that build intuition and point the way toward solutions that will have to be found experimentally. Thermodynamics is the most powerful of these intuitive theories. This is the principal reason its study is so important to those who hope to succeed in materials science. The applications of thermodynamics range broadly through the field of materials science. Given the limited time available for this course and the variety of interests of the students who take it, our emphasis here will be on theoretical foundations and methods rather than on specific applications. The intent is to make the student competent to read and assimilate the literature that is relevant to his particular engineering interests, however those interests may change in the future. In planning any course one faces the problem of deciding where to begin, and, in particular, what background one ought to assume. The problem is particularly acute in the case of an advanced course in thermodynamics. Thermodynamics is like materials science itself in that it sits between the fundamental physical sciences and their practical applications. Thermodynamics incorporates the fundamental sciences of mechanics, chemistry and electromagnetism to describe the states and evolution of physical systems. For that reason the student of thermodynamics should already have a solid grounding in classical and quantum mechanics, chemistry, crystallography, electromagnetic theory, and the mathematical apparatus that goes with them. Very few do. But to fail to show the place of any of the fundamental sciences in thermodynamics is to present an incomplete, and possibly deceptive picture of the power of thermodynamics in materials science. The plan of the course is to cover a significant part of the theory of thermodynamics while keeping the underlying physics relatively simple. The necessary physical principles and mathematical techniques will be introduced as they are needed, but with the expectation that the student has seen them before at some undergraduate level. This approach should provide a theoretical framework that is adequate to incorporate whatever level of sophistication in the fundamental sciences the student has, or may subsequently acquire. 1.2 OUTLINE OF A PHYSICAL THEORY Since the course is theoretical, it makes sense to begin with a brief outline of what a physical theory is (and, equally important, what it is not). A physical theory attempts to understand and predict physical phenomena by modeling them. It replaces the "real world" with an ideal one whose behavior is sufficiently clear and simple that it can be described and treated mathematically. If the theory is well chosen, in the sense that its model world mimics that part of the behavior of the "real world" that the theoretician cares about and the equations that govern its behavior are tractable, then the theory is useful. Otherwise it is not. At the present stage of human development well-constructed scientific theories are not "right" or "wrong". They are useful or they are not useful. The theories we shall review in this course are known to be useful. Page 3

4 A wide variety of theoretical models are useful in materials science. However, they all have a common underlying structure. This structure contains four separate elements that must be carefully distinguished if the theory is to be understood. 1. Definitions. A theory begins with a set of definitions (or vocabulary) that delineate its subject matter, set out the model, and define the terms that are used. 2. Postulates or Assumptions. The second element of a theory is a set of postulates (laws, assumptions, axioms) that state the rules that govern the behavior of the theoretical model. The whole physical content of a theory is contained in its postulates. The behavior of the theoretical model is completely determined by its postulates. The postulates are statements that stand alone. In the abstract it does not matter whether they are true or false, reasonable or absurd. The internet is full of popular games whose features assume a set of physical laws very different from our own. But if the theory we construct is to be useful for predicting the behavior of the world around us, the postulates we select must be at least generally consistent with what we know about that world. 3. Mathematics. In this part of the theory the mathematical consequences of the postulates are developed and used to explain and predict the detailed behavior of the model. Since the mathematical development is usually the most obvious part of the theory, and often the most difficult to master, students often focus on the math as if it were the theory. This is not true. The definitions and postulates are the theory. The mathematics never add content to the theory. To the contrary, the mathematics almost always subtract content., since it is usually necessary to introduce restrictions and mathematical approximations in order to solve the equations that reveal the consequences of the theory. The results of the mathematics only apply to that part of the theoretical model that incorporates whatever additional restrictions have been introduced, and are only accurate to the extent that the mathematical approximations are valid. 4. Intuitive Extrapolations. In the final step of a theory one uses its predictions to interpret and explain the behavior of systems that fall outside the strict range of its model, including physical systems in the real world. A familiar example of a physical theory is classical or Newtonian mechanics. The theory can be formulated in several equivalent ways. A typical formulation begins with the definition of a material body that is endowed with mass and momentum. The postulates then include the conservation of mass and of linear and angular momentum, the law of gravitational attraction, and the conservation of mechanical energy (kinetic plus potential). The mathematical development of these laws predicts the behavior of specific mechanical systems, such as the trajectory of a planet in orbit around the sun and the motion of a system of elastic particles that move within a rigid box. Some of these problems can be solved exactly, such as the two examples cited above. Others, such as the trajectories of three bodies that interact gravitationally, cannot be solved exactly with available Page 4

5 mathematical techniques, but can, in principle, be solved to any desired degree of approximation by perturbation methods or other numerical techniques. All of these solutions describe the behavior of model systems that obey the assumptions of Newtonian mechanics. However, it is well known that no real systems strictly obey these assumptions. Real bodies interact through electromagnetic as well as gravitational forces, are quantized and uncertain in their mechanical states, and behave relativistically when they move with respect to one another. Nonetheless, Newtonian mechanics predicts the mechanical behavior of most physical systems of interest with very satisfactory accuracy. 1.3 C ONTINUUM AND ATOMISTIC M ODELS The models that will be studied in this course can be divided into two general categories: the macroscopic, or continuum models and the microscopic or atomistic models. A continuum model of a material ignores its underlying molecular structure. The matter that makes up the material is treated as if it were continuously distributed through space. These models are useful to analyze behavior that is relevant on a scale large compared to the effective range of atomic interaction. They offer the advantages of generality and relative mathematical simplicity. The microscopic models that are most useful in materials science treat the material as an aggregate of atoms or molecules, or, in more detail, of ion cores and electrons. The atomistic models are powerful in that they can describe material behavior on a very local scale, but achieve this power at a significant cost in mathematical complexity. Of course, quantum physics provides still more detailed models that explicitly include the internal structure of the atoms themselves. This increased precision is sometimes needed in the practice of materials science, but is often unnecessary. In the development of thermodynamics that is given below we begin with the macroscopic description, which is often called Classical Thermodynamics. Classical thermodynamics is adequate for the vast majority of the problems that are important in materials science. It rests on its own experimental base, and is sufficient to establish the familiar laws of thermodynamics. We shall then turn to the atomistic model and develop the theory of Statistical Thermodynamics. The combination of classical and statistical thermodynamics is used to develop the equilibrium thermodynamics of the four most useful classes of materials: multicomponent fluids, rigid solids, elastic media and electromagnetic media. We then develop the theory of non-equilibrium thermodynamics for systems that are close to equilibrium, which includes the non-equilibrium phenomena that are of the greatest current importance in materials science, and apply it to selected problems in diffusion, electromagnetic phenomena, and the kinetics of phase transformations. 1.4 R EFERENCES The literature of thermodynamics is very large, and bits and pieces of a great deal of it have been used in developing these notes. These are intended to be lecture notes, and I Page 5

6 have not yet had the time to incorporate detailed references or an extensive bibliography. However, I have relied on several texts to the extent that they make very useful supplementary references. The formulation of classical thermodynamics and the development of the laws of thermodynamics follows H.A. Buchdahl, Concepts of Classical Thermodynamics. A.H. Wilson, Thermodynamics and Statistical Mechanics, is also a useful reference. The development of the fundamental equation, the thermodynamics of fluids, and parts of the thermodynamics of elastic solids largely follows J.W. Gibbs, The Equilibrium of Heterogeneous Substances, which is also worth reading because it is the work that invented the subject. The development of statistical thermodynamics tends to follow Landau and Lifshitz, Statistical Physics (Vol. 1), although there are many good alternate sources. Parts of the material included in the thermodynamics of fluids and solids is also taken from Landau and Lifshitz. The thermodynamics of crystalline solids and elastic media is drawn in part from A.G. Khachaturyan, Theory of Structural Transformations in Solids, and from the work of J.D. Eshelby, much of which is summarized in Progress in Solid Mechanics, 2, 89 (1961). The thermodynamics of electromagnetic media is taken in part from Landau and Lifshitz, Electrodynamics of Continuous Media. The most important reference for non-equilibrium thermodynamics is DeGroot and Mazur, Non-Equilibrium Thermodynamics. Other useful references are cited in the text. Page 6

7 Chapter 2: Scope and Definitions 2.1 SCOPE It is useful to think of classical thermodynamics as the complement and completion of the other classical physical sciences: mechanics, electromagnetism, and chemistry. Adopting this point of view, we shall take the other physical sciences as given, recognize that, together, they provide an incomplete description of material behavior, and develop thermodynamics to complete the classical theory. In accordance with the outline of a physical theory that was given in the Introduction, we begin with a set of definitions that specify the scope of the theory and the meaning of the terms that will be used. These include the concepts of systems, physical properties, equilibrium states, constitutive variables, and transitions between states. The definitions allow us to construct the model to which the theory applies. Next we develop the Laws of Thermodynamics, of which there are four. The socalled Zeroth Law defines the empirical temperature and the thermal equation of state. It is redundant, but useful. The First Law defines the internal energy and the caloric equation of state. The Second Law defines the entropy, the absolute temperature, and the fundamental equation. The Third Law defines the absolute entropy. Each of the laws is based on a central postulate. The central postulates can be chosen in many different ways. The set of postulates selected here is chosen from the perspective that classical thermodynamics should stand on its own, without depending on any underlying theoretical model. It is therefore important that each of the laws of thermodynamics rest on a base of experimental data, and that their central postulates be phrased so that they can be disproved experimentally. Similarly, each of the quantities defined by the laws of thermodynamics (temperature, energy, entropy) must be defined so that it is experimentally measurable for any system that falls within the purview of the theory. The formal development of thermodynamics is completed by using the laws of thermodynamics to formulate the fundamental equation and establish the conditions of equilibrium and stability. The fundamental equation of a substance is a single relation that contains all of its equilibrium properties. The existence of the fundamental equation, which was first established by Gibbs and is the basis for his Equilibrium of Heterogeneous Substances, is the central element of thermodynamics that allows a compact theory of material behavior. The fundamental equation governs the equilibrium states of a material. The conditions of equilibrium and stability govern the equilibrium between different phases within a system and the transitions they may undergo. Page 7

8 The applications of thermodynamics to materials science are the mathematical consequences of the fundamental equation in light of the conditions of equilibrium and stability. To explore these applications in a general way it is useful to consider classes of materials, which are defined by particular sets of variables that appear in the fundamental equation. The classes of materials we shall specifically consider are multicomponent fluids, ideal crystalline solids, thermoelastic solids, and electromagnetic media. 2.2 DEFINITIONS The system Thermodynamics applies to systems, which are simply collections of physical objects. The system must be defined so as to separate it from all else (the surroundings or environment). Its specification includes verbal constraints that describe what it is (e.g., a block of iron, a volume of gas), and operational constraints that define the nature of its boundaries and any internal divisions (e.g., quantities of oxygen and nitrogen separated by an impermeable membrane and enclosed within a thermos bottle). The verbal constraints are immutable parts of the definition of the system. The operational constraints can be changed or manipulated to induce transitions in the state of the system. The system may be open in the sense that the physical objects contained within it change with time so long as the set of objects is defined at any given time (e.g., the air in this room) Physical properties Physical properties are the observable aspects of the system. The properties of interest include the physical content of the system, such as its size, shape, weight, chemical composition, magnetic moment, etc.; the intensity with which it interacts with other systems, that is, its pressure, stress, chemical potential, electric field, etc.; and the material properties that govern its response to changes in the forces imposed on it, such as its compressibility, elastic moduli, coefficient of thermal expansion, specific heat, magnetic susceptibility, etc. For mathematical specificity we will designate the k th physical property of interest by the symbol Q k. The set of all interesting physical properties is {Q}. The only physical properties that can be treated theoretically are those that are objectively measurable in the sense that independent observers can determine their values with the same result. The term "physical property" contains this limitation. For example, color can be a physical property but, in the absence of a very restrictive definition, beauty is not. The definition of a physical property must therefore include an experimental recipe for finding its value. Such a recipe is given or assumed to be known for all of the physical properties that will be considered here. The adjective "interesting" was introduced because one usually only cares about some subset of all of the aspects of a system that can be measured, and defines the system so that only that subset of properties ever needs to be considered. For example, sufficiently dilute gases can be approximated as ideal gases. The behavior of an ideal gas is affected by Page 8

9 the quantity of material present (the number of moles or atoms) but is not affected by the identity of the chemical species. Similarly, the behavior of an ideal gas depends on its volume, but is insensitive to its shape, which is simply the shape of the container that encloses it. If the system is defined to be an ideal gas then its molar content and volume must be specified, but its chemical identity, color, shape, etc. are irrelevant Thermodynamic state The thermodynamic state of the system (which we shall designate by the symbol, Í) is specified by a particular set of values of the interesting physical properties {Q}. Two states Í and Í' are different if they differ in any physical property, that is, in the value of any member of the set {Q} Equilibrium state An equilibrium state is a state that does not change perceptibly with time. The definition contains some subtlety since every real physical system fluctuates on a microscopic scale and evolves on the macroscopic scale, and we would like to construct a theory that predicts the behavior of real systems. The equilibrium states we shall consider represent the conditions of real systems whose microscopic fluctuations can be ignored and whose macroscopic evolution is sufficiently slow that any change in physical properties during the time of observation is small enough to be neglected. The behavior of the system is then approximately the same as the behavior of a model system in which the macroscopic properties are time averages and the reactions that cause the real system to change are prevented by constraints. The theory is written for the ideal, constrained system rather than the real, evolving one. This definition of equilibrium implicitly includes a sense of time. For example, let a block of iron be exposed to the weather. The iron gradually rusts away. If the period of time that we care about is months or years this rusting cannot be ignored, and the iron is not, initially, in an equilibrium state. If, on the other hand, the time period of interest is minutes, hours, or days then the slight rust that develops can usually be ignored and the behavior of the block of iron is essentially the same as it would be if the iron were sealed in an impermeable membrane that prevented corrosion. While the block of iron is not, strictly, in equilibrium, a model system that contains the block of iron within a membrane that is impermeable to oxygen is in equilibrium, and is essentially identical in its behavior over the time of interest. We may treat the behavior of the representative system and use it to predict the short-term behavior of the block of iron. However, since the model system has well-defined values of each of its properties, the real system it represents must remain unchanged on a time scale that is long compared to the relaxation times of the microscopic fluctuations that occur within it. Because of microscopic fluctuations, if the properties of the real system are measured on too fine a scale in space and time they will not be constant, but will rather vary noisily about their mean values. The mean values, or time averages of the microscopic measurements, define the properties of the model system. The real system must exist for a long enough time that Page 9

10 its time average properties have meaning. This sense of time is implicit in the definition of a physical property, which is assumed to carry with it a prescription for consistently measuring its value. Macroscopic properties can only be measured on a time scale that is long compared to the relaxation times of microscopic fluctuations Constitutive coordinates and constitutive equations To specify the state of a system it is generally only necessary to know the values of a subset of its properties, which we shall denote by the general symbol {x}. The values of all the other properties can be found from these. A subset of properties that is sufficient to determine the state is called a complete set of constitutive coordinates or constitutive variables. The other properties are determined by the constitutive coordinates through two kinds of relations: tautological relations and constitutive equations. The tautological relations express the fact that some properties can be defined in terms of others. For example, the volume of a cube is determined by its length; the density of a monatomic gas is the product of its molecular density and its molecular weight. The constitutive equations are physical equations that are obtained from fundamental theory or experiment and relate tautologically independent properties of the system to one another. Two familiar constitutive equations are the ideal gas law: P = P (T,V,N) = NRT V 2.1 that gives the pressure of an ideal gas in terms of its mole number (N), temperature (T), and volume (V), and Hooke's Law for an isotropic linear elastic solid ß = ß ( ) = E 2.2 which relates the tensile stress, ß, to the tensile strain,, through Young's modulus, E, for a sample tested at constant temperature. The constitutive coordinates must be chosen so that they uniquely define the state by uniquely determining the value of all other properties of interest. If there is more than one distinguishable state with the same values of the constitutive coordinates, the set of coordinates is insufficient and more must be added. The precise coordinates that are needed to describe any particular system is, hence, a matter for experiment, and differ from one system to another. In developing the theory of thermodynamics we assume that it is possible to define a finite, complete set of constitutive coordinates Geometric coordinates and conjugate forces There are many possible choices for the constitutive variables of a given system. However, one natural scheme for choosing at least part of the set of constitutive coordinates follows from classical mechanics. In classical mechanics the properties of the system include a set of geometric coordinates that relate to the size or shape of the system (the vol- Page 10

11 ume of a fluid, the strain in an elastic solid) and a set of forces conjugate to the geometric coordinates (the pressure in a fluid, the elastic stresses in an elastic solid), so that the work done by the forces {p} during an infinitesimal change in the geometric coordinates {q} is W = k p k q k 2.3 where p k is the force conjugate to the geometric coordinate q k. As we shall see, a similar result holds for chemical changes, in which case the geometric coordinates are the mole numbers {N} or masses {M} of each of the chemical species present and the conjugate forces are the chemical potentials {µ}. Because the work done in a change of state is simply related to the change in the geometric coordinates it is usually preferable to use the geometric coordinates as constitutive coordinates to the extent possible Classes, types, and particular materials The properties of a given system are governed by a specific set of constitutive equations. But the set of constitutive coordinates that appear in these equations is common to a broad class of materials. For example, all fluids are mechanically isotropic, deform to fit the shape of their containers, and will not support a shear stress at equilibrium. These properties have the consequence that the equilibrium state of a fluid depends on its size and shape only through its volume, V. An elastic solid, on the other hand, depends on the shape as well as the volume, and requires the nine elastic strains, ij (only six of which are independent), in its list of constitutive coordinates. As we shall see, it is possible to say a great deal about the thermodynamic behavior of a system simply from a knowledge of its constitutive coordinates. It is hence useful to define classes of materials on the basis of the constitutive coordinates that characterize their states. The most familiar material class is the multicomponent fluid, whose states are specified by the variables T, V, {N}, or by any equivalent set, where {N} is the set of atom or mole numbers of the chemical species present. The ideal crystal is described by the variables T, V, Nº, {N} (or any equivalent set), where Nº is the number of lattice sites. The linear elastic solid and the electromagnetic material are other important classes whose constitutive coordinates will be identified in the following. The class of a material provides important, but limited information. For example, the constitutive equation for the internal energy of a one- component fluid can be written E = E (T, V, N) 2.4 (an equation of this form is also called a caloric equation of state). This equation does not reveal the functional dependence of E on the constitutive variables and does not permit a computation of the energy of any particular state. Page 11

12 A material type denotes a group of materials that not only have the same constitutive variables, but whose constitutive equations have the same functional form. For example, an ideal gas is a type of multicomponent fluid. Whatever the chemical composition of an ideal gas, its pressure is determined by the constitutive equation 2.1, where N is the total molar content, and its internal energy is given by the constitutive equation E = VCT 2.5 where C is a constant. The members of a particular type of linear elastic solid obey the constitutive equation 2.2. These are linear elastic solids with isotropic symmetry. The type of a material still does not specify it completely since the coefficients that appear in its constitutive equations are left unknown and the numerical values of its properties are, therefore, undetermined. A particular material is designated by setting the numerical values of the constants that appear in each of its governing constitutive equations Changes of state: transitions A transition is a change of state. The notation we shall usually use to denote a transition is Í Í', indicating that a system in state Í, which is specified by the constitutive variables {x} and has properties {Q}, undergoes a transition to the state Í' in which the constitutive variables have the values {x'} and the properties are {Q'}. It is useful to define several idealized kinds of transitions. a. Infinitesimal transitions. An infinitesimal transition is one in which the initial and final states differ infinitesimally in the values of all constitutive coordinates. b. Continuous and discontinuous transitions. The path of a transition is specified by the sequence of values the constitutive variables assume as the transition proceeds. However, the constitutive variables do not necessarily have definite values during the transition. For example, the volume of a gas is undefined during its free expansion into a vacuum. If the path of a transition is such that the constitutive variables always have definite values then the transition is said to be continuous. Otherwise the transition is discontinuous. A continuous transition can be described mathematically by the set of equations x k = x (t) 2.6 k that give the value of the k th constitutive coordinate as a function of time (or any equivalent continuous variable that measures the progress of the transition). It follows that a continuous transition can always be broken up into a sequence of infinitesimal transitions. c. Quasi-static transitions. Even when a transition is continuous it may introduce forces or changes, such as friction, turbulence, electrical currents or chemical hetero- Page 12

13 geneities, that cause work or heat flows that cannot be accounted for from a knowledge of the equilibrium states. When such effects can be neglected we call the transition quasistatic. A quasi-static transition is necessarily continuous. During a quasi-static transition the system is always instantaneously in an equilibrium state in the sense that its equilibrium constitutive equations are obeyed. In particular, in a quasi-static transition the applied mechanical forces are given by the constitutive equations that govern equilibrium and the work done in an infinitesimal segment of the transition is given by equation 2.3. d. Reversible Transitions. A reversible transition is a transition that can be reversed so that both the system and its environment return to their original states Walls and partitions A transition from one equilibrium state to another is brought about by an interaction between the system and its environment or between one distinct subpart of the system and another. To describe and control transitions we use the concept of walls and partitions. The wall is the outer boundary of the system. It governs the interaction between the system and its environment by permitting certain types of interaction and preventing others. A partition is a wall that is internal to the system, dividing one part of it from another. The common types of walls are used to restrict mechanical, chemical, or thermal interactions (since a thermal interaction is defined by the laws of thermodynamics, the term is used here simply to designate a change in the state of the system that is not due to a mechanical, chemical or electromagnetic interaction). The common types of walls include the following: a. An isolating wall is an enclosure that prevents any interaction between the system and its environment. b. An adiabatic wall is an enclosure that permits mechanical interactions between the system and its environment, but prohibits non-mechanical interactions. c. A deformable wall is an enclosure that permits mechanical interactions. An nondeformable or rigid wall prohibits mechanical interactions. d. A diathermal wall permits thermal interactions. e. An impermeable wall prohibits the exchange of chemical species. A permeable wall permits chemical interactions. A semipermeable wall is permeable to some chemical species but not to others. f. An open wall is an enclosure that permits both thermal and chemical interactions. An imaginary boundary is an example of an open wall. Transitions are controlled by manipulating walls and partitions. For example, a sample is heated by bringing it into contact with a hotter system across a diathermal wall. Page 13

14 A sample is deformed by bringing it into contact with a loading mechanism across a deformable wall. If the deformation is to be done under isothermal conditions the wall should be diathermal as well. If the deformation is done so fast that the heat generated is retained in the specimen, the wall is best assumed adiabatic. If chemical interactions can be ignored, the walls are impermeable Probes and reservoirs The states of well-behaved systems should be both measurable and controllable in the macroscopic sense. To accomplish this we introduce two idealized types of systems: probes and reservoirs. A probe is a hypothetical system that is so small compared to the system of interest that it can be brought into equilibrium with the system of interest without measurably perturbing its state. A pressure probe, for example, is brought into contact with the system across a deformable membrane. Mechanical equilibrium requires that the pressure in the probe equal that in the system. If the probe is sufficiently small compared to the system the equilibrium can be achieved without measurably changing the pressure in the system. If the probe includes a dial that reads its pressure, the pressure within the system is known from the pressure of the probe. A reservoir is a hypothetical system that is so large compared to the system of interest that it can be brought into equilibrium with the system of interest without significantly changing its own state. A reservoir is used to fix the properties of a system. For example, a thermal reservoir is brought into contact with a system across a diathermal wall to fix its temperature. Thermal equilibrium requires that the temperature of the system change to equal that of the reservoir, which is so large compared to the system that its own temperature is unchanged Simple systems It is easy to show experimentally that the states of a system cannot be specified by the geometric coordinates {q} alone. The geometric coordinates can be fixed by enclosing the system within a rigid, impermeable wall. A strong, metallic wall will usually suffice. But if the system is then exposed to a flame or immersed in ice water its internal state changes. Such experiments show that something is missing, and that at least one nongeometric coordinate must be included in the set of constitutive coordinates. A simple system is a system whose constitutive coordinates can be chosen to be the geometric coordinates plus only one other, usually taken to be the force conjugate to one of the geometric coordinates: {x} = {{q},p k } 2.7 Most physically simple systems are thermodynamically simple to a good approximation, provided that they satisfy the following constraints: Page 14

15 1: They contain no isolated parts. A composite system that is made up of two simple systems separated by an isolating wall is not a simple system. 2: Surface effects can be neglected. As we shall see, the internal state of a system that contains non-planar interfaces depends on the precise shapes of the interfaces. 3: Long-range interactions between different parts of the system can be neglected. If different parts of the system have non-negligible interactions with one another then the state of the system may depend on the precise internal arrangement of the parts. For example, an inhomogeneous solid that is elastically stressed is not a simple system, though it can ordinarily be treated as a composite of simple systems by breaking it up into homogeneous parts. The non-geometric coordinate of a simple system cannot always be chosen arbitrarily. A classic example is liquid water, which has a maximum in its density near 0ºC when it is subject to atmospheric pressure. It follows that water is not a simple system if the pressure is used as the non-geometric coordinate, since the variables {p,v,n} do not determine a unique state when the pressure is near 1 atmosphere and the volume is near its minimum value. Water is a simple system if a different choice is made for the non-geometric coordinate, for example, the chemical potential, µ, which is the force conjugate to the mole number, N, and is measured to a good approximation by the vapor pressure in equilibrium with the liquid Standard systems A standard system is a simple system that satisfies the additional constraint that its infinitesimal transitions are quasi-static. Most simple systems are standard systems to a good approximation, provided that the transitions occur slowly. Viscous effects intrude in virtually any change of state, but vanish in the limit of zero rate of change. Static friction is not usually a problem in thermodynamic transitions. Page 15

16 Chapter 3: The Zeroth Law The Zeroth Law of thermodynamics is based on the experimental fact of thermal equilibrium, and defines the empirical temperature. It is called the "Zeroth Law" because it is unnecessary and redundant; the concepts of temperature and thermal equilibrium are consequences of the First Law. But temperature and thermal equilibrium are so central to thermodynamics that it is useful to separate them out. 3.1 EXPERIMENTAL B ASIS The concept of a thermal interaction is a familiar one. If two systems are prepared separately in adiabatic enclosures and are then brought into contact with one another, for example, across a thin, rigid metallic wall, then the states of the systems may change, but eventually reach an equilibrium that can be maintained indefinitely. Since the geometric coordinates of both systems are fixed their states change without mechanical or chemical work and must, therefore, respond to some other kind of interaction. We call this interaction a thermal interaction which leads to thermal equilibrium.. Thermal equilibrium does not establish a unique relation between the states of the interacting systems. Many distinct states of a given system can be in thermal equilibrium with the same state of a reference system. For example, consider a dilute monatomic gas. The variables {p,v,n} are good constitutive coordinates. It is easily shown that all states of the gas that have the same value of the grouping t = pv N 3.1 are in thermal equilibrium with the same state of a reference system. However, thermal equilibrium between simple systems can be made unique by fixing the values of all but one of the constitutive coordinates (e.g., V and N in equation 3.1). Let K 1 and K 2 be simple systems that are governed, respectively, by the constitutive coordinates {{q 1 },p 1 }, {{q 2 },p 2 }, where the members of the set {q i } are the geometric coordinates of system K i and p i is the force conjugate to one of them. Suppose that the geometric coordinates of both systems are fixed, and that they are then brought into diathermal contact. Thermal interaction changes the forces, p 1 and p 2. It is found experimentally that the equilibrium values of p 1 and p 2 are uniquely associated; for given values of the geometric coordinates {q 1 } of K 1 and {q 2 } of K 2 there is a particular value of p 1 that is in equilibrium with each value of p 2. Moreover, an infinitesimal change in any of the geometric coordinates of either system creates a new equilibrium pairing in which the value of p 1 associated with each value of p 2 differs infinitesimally from the previous one. Page 16

17 These experimental observations are precisely the conditions needed to establish the existence of a continuous function connecting the constitutive coordinates of the two systems when they are in thermal equilibrium, which has the form Ï 12 [{q 1 },p 1,{q 2 },p 2 ] = where the function Ï 12 is such that p 1 and p 2 are uniquely related for given values of the sets {q 1 } and {q 2 }. A second important observation about thermal equilibrium is that it is transitive. If a state Í 1 of a system K 1 is in thermal equilibrium with a state Í 2 of a system K 2, and if Í 2 of K 2 is also in thermal equilibrium with the state Í 3 of system K 3, then Í 1 of K 1 and Í 3 of K 3 do not change if they are brought into diathermal contact with one another. Mathematically, Í 1 (K 1 ) = Í 2 (K 2 ) Í 2 (K 2 ) = Í 3 (K 3 ) => Í 1 (K 1 ) = Í 3 (K 3 ) THE ZEROTH LAW The Zeroth Law codifies these experimental observations in three statements: 1: Systems in contact can interact thermally, that is, can influence one another in the absence of any mechanical or chemical interaction. The walls that permit thermal interactions are called diathermal walls. 2: Systems in diathermal contact evolve into compatible states that are said to be in thermal equilibrium, and satisfy a relation of the form : Thermal equilibrium is transitive. 3.3 THE EMPIRICAL TEMPERATURE The transitive character of thermal equilibrium has the mathematical consequence that the two relations and imply the relation Ï 12 [{x 1 },{x 2 }] = Ï 23 [{x 2 },{x 3 }] = Page 17

18 Ï 13 [{x 1 },{x 3 }] = where the members of the set {x i } are the constitutive coordinates of K i. If this is true it must be possible to use equations 3.4 and 3.5 to eliminate all n of the variables in the set {x 2 }. But, algebraically, one can only eliminate one independent variable by solving two simultaneous equations. It follows that it must be possible to gather the set of variables {x 2 } in equation 3.4 and 3.5 into a single variable, or function, and that this function must be the same in both equations. Mathematically, Ï 12 [{x 1 },{x 2 }] = t 1 [{x 1 }] - t 2 [{x 2 }] 3.7 Ï 23 [{x 2 },{x 3 }] = t 2 [{x 2 }] - t 3 [{x 3 }] 3.8 The function t 2 can then be eliminated by addition to yield equation 3.6. Equations 3.4 and 3.5 can be rewritten and imply equation 3.6 in the form t 1 [{x 1 }] = t 2 [{x 2 }] 3.9 t 3 [{x 3 }] = t 2 [{x 2 }] 3.10 t 3 [{x 3 }] = t 1 [{x 1 }] 3.11 If we define the empirical temperature of the i th system by the relation t i = t i [{x i }] 3.12 we are led to the result that a simple system has the property temperature, given by the constitutive equation 3.12, that governs thermal equilibrium in the sense that the two simple systems K 1 and K 2 are in thermal equilibrium if and only if t 1 = t M EASURABILITY: THERMOMETERS It would be enormously difficult to work through the formal recipe given above to find constitutive equations for the empirical temperatures of all the systems of interest, and it would be unworkable to have to use such equations to discuss thermal equilibrium. We avoid the problem by using the concept of a thermometer and the transitivity of thermal equilibrium. A thermometer is a simple system that can be made small enough to use as a probe and is (preferably) so designed that it has only one free coordinate. Common examples are Page 18

19 a small quantity of a dilute gas under fixed pressure and a small quantity of liquid mercury confined in an evacuated cylinder. In both cases the only free variable is the volume, which can be measured as height in the case of mercury confined in a column. If a thermometer has only one free variable, x, then the value of that variable gives a unique measure of the temperature of any other system that is in thermal equilibrium with it. If the thermometer is small enough to be used as a probe then it can be brought into thermal equilibrium with another system without disturbing its state, and can hence be used to measure temperature. Since the choice of the empirical temperature is arbitrary, the temperature of the system can be set equal to the value, x, of the free parameter of a standard thermometer, which is essentially what was done in defining the Kelvin, Celsius and Fahrenheit temperature scales. We shall see later that there is a preferred, or absolute temperature scale. The absolute temperature is defined by the Second Law of thermodynamics. The absolute temperature is measured, for example, by the quantity pv/n of an ideal gas, which differs by a constant factor (which, for random historical reasons, has dimensions) from the Kelvin scale. 3.5 THE THERMAL EQUATION OF STATE Let a simple system be referred to the constitutive coordinates {{q},p}, where p is a thermodynamic force. The empirical temperature obeys the constitutive equation t = t ({q},p) 3.14 If we assume that t varies monotonically with p (we shall show later that it does) equation 3.14 can be inverted to give p = p ({q},t) 3.15 Since the right hand side of equation 3.15 can be substituted for p wherever p appears in any other constitutive equation, the temperature, t, can replace p as a constitutive coordinate. It follows that the set {{q},t} is a complete set of constitutive coordinates for a simple system.. An equation of the form 3.15 that relates a thermodynamic force to the temperature and geometric coordinates is called a thermal equation of state. 3.6 THE TEMPERATURE OF A NON-SIMPLE SYSTEM The reasoning used above is based on the assumption that the system is thermodynamically simple, and does not strictly apply when it is not. But it is still possible to define the temperature and use it as a constitutive coordinate for a very general class of non-simple systems: systems that are composites of simple systems. Let the system, K, be the union of subsystems, K i, each of which is simple. Then the temperature of each subsystem is Page 19

20 defined and is a good constitutive coordinate for that subsystem. In the most general case the constitutive coordinates for the whole system include one temperature for each subsystem. However, if the partitions that divide the subsystems are diathermal then the temperatures of the subsystems are equal. The whole system then has a uniform temperature and requires only a single thermal coordinate. We shall consider many non-simple systems in the course of this study, but shall assume that all of them can be regarded as composites of simple systems. In fact, we shall almost always make the stronger assumption that all the subsystems are standard systems. Page 20