Chapter 5: The Second Law

Transcription

1 Chapter 5: The Second Law CHAPTER 5: THE SECOND LAW EXPERIMENTAL BASIS THE SECOND LAW DEFINITION OF THE ENTROPY THE ENTROPY AS A CONSTITUTIVE COORDINATE THE NATURE OF ENTROPY ENTROPY AND HEAT METRICAL ENTROPY AND ABSOLUTE TEMPERATURE Definition of the metrical entropy Measurability ENTROPY AS A FUNCTION OF COMPOSITION Chemical equilibrium and the measurement of entropy The energy function and the chemical potential Heat and wor The energy function and the overall wor Comment on the units of temperature and entropy THE ENTROPY MAXIMUM PRINCIPLE Maximum entropy Spontaneous processes; non-equilibrium thermodynamics The Carnot inequality THE CONDITIONS OF EQUILIBRIUM AND STABILITY THE FUNDAMENTAL EQUATION THE ARROW OF TIME EXPERIMENTAL BASIS Many different experiments and analyses contributed to the development of the Second Law. Most concerned the nature of heat, its "storage" within a system and its conversion into mechanical wor. This research was typically done with the highest of human motives: if heat were somehow stored in a body then it should be possible to extract it in the form of useful energy that could be sold for cash money. Others who contributed to the theory were motivated by philosophical considerations that have a rather gloomy cast to them, and center on such morbid issues as human impotence and the inevitability of death. It is simply not possible to do everything. Worse, while a human being can do very little, he is ineradicably stuc with what he has done. While the laws of mechanics are reversible the laws of nature are not. One's hopes may vanish, but his deeds are eternal, however foolish they may be. The scientific principle that relates the economic aspirations of the Pacific Gas and Electric Company to the spiritual lamentations of Holy Hubert is the Second Law of ther- page 31

2 modynamics. The Second Law is offered to explain and interpret the observations that certain physical changes, lie converting the Sierra to electric power, cannot be done; others, lie growing old, can be done easily, but not undone; still others, lie moving conduction electrons through a superconductor, can be done bacwards and forwards as often as one lies. We shall formulate the Second Law from a single, universal experimental observation. Consider a simple system that is in an adiabatic enclosure that fixes its geometric coordinates. It is still possible to do mechanical wor on the system, for example, by inserting a paddle wheel and turning it. But if the system is left in an equilibrium state it will not turn the paddle wheel bac again. We can add an arbitrary amount of energy to an adiabatic system whose geometric coordinates are fixed (that is, in an isometric transformation), but we cannot extract energy from it. This experimental observation leads to the definition of a state property, called the entropy. The entropy is an additive property that is a good constitutive variable. It is, in fact, the geometric coordinate for heat, or thermal wor, and, simultaneously, the thermodynamic measure of time. The entropy governs the reversibility of transformations and the direction of spontaneous changes in physical systems. Its properties establish the conditions for equilibrium and stability and permit the formulation of the fundamental equation, a single constitutive equation that contains all of the relevant thermodynamic information about a system. 5.2 THE SECOND LAW The Second Law has been written in several equivalent ways. Some formulations, such as that of Caratheodory, produce the Second Law independently, without reference to the Zeroth or First. But it is simplest to use the First Law as a foundation for the Second. The Second Law of Thermodynamics is then contained in the following postulate: Given two isometric states, Í and Í', of an adiabatically isolated simple system, K, Í < Í' if and only if E(Í') > E(Í). In simple words, one can always add energy to a simple system in an adiabatic enclosure but cannot extract energy without changing the values of its geometric coordinates. 5.3 DEFINITION OF THE ENTROPY The Second Law postulates that a state Í' is adiabatically accessible from an isometric state Í if its energy is higher, but is inaccessible if its energy is lower. For standard systems it is easy to extend the concept of accessibility to other states. Let K be a standard system in an adiabatic enclosure, and Í and Í' be isometric states with E(Í') > E(Í). Since quasi-static adiabatic transitions are reversible any state, Í 0, that can be reached from Í' through a quasi-static adiabatic transition is accessible from it, and is hence also accessible page 32

3 from the state Í. But Í is inaccessible from any of the Í 0 ; if Í could be accessed from Í 0 then it could also be reached from Í'. For the same reason, any state Í 1 that can be reached through a quasi-static adiabatic transition from Í has the property Í 1 < Í 0, but Í 0 > Í 1. These relations are illustrated in Fig. 5.1 Í' Í 0 E Í Í 1 q Fig. 5.1: Two-dimensional representation of the state space of a simple system. It follows that adiabatic accessibility defines a hierarchy of states. Those states that can be reached through quasi-static adiabatic transitions from a given reference state occupy the same level in the hierarchy. In the sense of accessibility they precede all states that have higher energies for given values of the geometric coordinates, and come after all isometric states that have lower energies. The relation of adiabatic accessibility can be represented geometrically as follows. Let the system be controlled by the n constitutive coordinates {{q},e}, and choose a particular set of values of the {q}, labeled {q*}, as a reference set (for example, the dotted vertical line in Fig. 5.1). The variables {q} can be altered continuously to achieve quasistatic transitions. Their infinitesimal changes determine the associated change in E according to the differential relation E = W = p q 5.1 It follows that the states, {Í*}, that are accessible by quasi-static adiabatic transitions from a given reference state, {E*,{q*}}, lie on an n-1 dimensional hypersurface that cuts through the line fixed by {q*} at E*. This hypersurface divides all the states of the system into two sets: those that lie on or above the hypersurface are adiabatically accessible from any state in {Í*}, those that lie below the hypersurface cannot be reached by an adiabatic transition from any state in {Í*}. The states {Í*} therefore have a common property; we say they have the same entropy. The entropy, S, is a property of a state that defines its accessibility by adiabatic transitions. We choose the values of S so that, for any two states Í and Í, page 33

4 Í < Í' if and only if S(Í) S(Í') Í < Í' and Í' < Í if and only if S(Í) = S(Í') 5.4 THE ENTROPY AS A CONSTITUTIVE COORDINATE S 4 E S 3 S 2 q q* S 1 Fig. 5.2: Constant entropy curves in the state space {E,q} The entropy of a standard system can be chosen to be a single-valued and continuous function of state that varies monotonically with the energy, and can hence replace the energy as a constitutive variable. The geometric definition is illustrated in Fig. 5.2, and is justified by the following reasoning. The entropy is defined for every state since every state lies on one of the hypersurfaces of constant entropy that can be found by freely varying the geometric coordinates to generate all possible quasi-static adiabatic connections. Barring pathological cases, which can be handled by a separate treatment, the geometric coordinates can always be adjusted to a reference set {q*}, so that all the hypersurfaces of constant entropy cut through the isometric line at {q*}. The entropy is single-valued since the hypersurfaces of constant entropy cannot intersect. If they were to cross the states on the line of intersection would be accessible from neighboring states that are isometric with one another but have different values of the energy. The situation is illustrated in Fig The Second Law prohibits an adiabatic transition from state Í 2 to the isometric state Í 1, which has lower energy. But the state Í' is accessible from Í 2 since it is on the same entropy surface. Í 1 is accessible from Í' for the same reason. Hence the transition Í 2 Í' Í 1 is achievable, in violation of the Second Law. It follows that hypersurfaces of constant entropy cannot intersect. Since every state lies on some isentropic surface and since the surfaces cannot cross, a single value of the entropy can be assigned to every state. One can, for example, define the empirical entropy of the state Í by the simple relation page 34

5 S* = E* 5.2 where E* is the energy at which the constant-entropy hypersurface that contains Í cuts through the isometric line {q*}. Since the value of E* that is associated with an arbitrary state is measurable, by adjusting its geometric coordinates to the set {q*} in a quasi-static adiabatic transition, the empirical entropy is a well-defined physical property. Since the energy is continuous along the isometric line {q*}, the entropy is a continuous, monotonically increasing function of the energy, S* = S ({q},e) 5.3 S E > Given equation 5.4, equation 5.3 can be inverted into the form E = E ({q},s*) 5.5 which shows that the empirical entropy can replace the internal energy as a constitutive variable. Í 2 E Í 1 Í' q* q Fig. 5.3: A hypothetical system with intersecting entropy surfaces 5.5 THE NATURE OF ENTROPY The results of the previous section show that the empirical entropy can be defined and measured for an adiabatic system and that it can replace the internal energy as a constitutive coordinate. We must now show how it can be defined and measured for a system that is not adiabatically isolated. It is also important to clarify the physical meaning of the material property called entropy. As we shall see, the entropy is intimately associated with the two central concepts that are introduced into physical science by thermodynamics: heat and evolutionary time. page 35

6 The concept of heat was brought into the theory by the First Law. It is a quantity that is conceptually parallel to mechanical wor as a form of energy. In a quasi-static process the mechanical wor is done by applied forces which act through "distance", the change in the values of the conjugate geometric coordinates: W = p q 5.6 If we thin of the heat, Q, as a ind of "thermal wor" that is conceptually parallel to the mechanical wor done by the applied forces, as the First Law suggests, then it ought to be possible to define a thermal force, p T, and geometric coordinate, q T, so that the heat transmitted in a quasi-static process is given by the form: Q = p T q T 5.7 We shall show below that these quantities can be defined. The thermal force is the absolute temperature, T, and the geometric coordinate for thermal wor is the metrical entropy, S. The notion of time is introduced into physics by classical mechanics, and appears in the equations of motion of classical mechanical systems. But the "time" of classical mechanics has only some of the properties of the physical time with which we are familiar. In classical mechanics time is reversible. Precisely the same mechanical laws are obeyed whether time changes in a positive or negative sense, and there is no reason to prefer one direction over the other. Physical time, on the other hand, is an arrow that always points from past to future. Given sufficient information about the state of the world when a particular event occurred, one can (within the constraints imposed by relativity) not only determine whether the event is in the past or the future, but can unambiguously fix the time at which it happened in relation to other events. The first, and still only place where the real time enters physical science in an unambiguous form is the Second Law of thermodynamics. The idea is implicit in the postulate we used to set the Second Law. Since one can add energy to an adiabatically isolated simple system but cannot get it bac again, it is always possible to identify which state of the system came first. As we shall see below, it is possible to pursue this line of reasoning to a point where the concepts of entropy and time essentially merge into one. To develop these ideas we begin with an exploration of the connection between entropy and heat, and then discuss its relation to physical time. 5.6 ENTROPY AND HEAT Let a system be enclosed in an impermeable wall that fixes its composition. When the system undergoes a quasi-static adiabatic transition the change in its entropy ( S*) and page 36

7 the heat evolved ( Q) are both zero. It can be shown that this correspondence implies a mathematical relation between the heat and the entropy change of the form Q = S* 5.8 where the coefficient is a state function that depends on the temperature and the entropy only: The proof is as follows. = (S*,t) 5.9 Let the system undergo an infinitesimal, quasi-static transition. The system is controlled by the r coordinates {x} = {{q},t}, where the {q} are the geometric coordinates that govern its mechanical response. The differential quantity of heat evolved during the transformation is given by the linear differential form Q = E - W = X x 5.10 where X = E q - p ( =1,..., r-1) 5.11 X = E t ( = r) 5.12 Since the empirical entropy is a property of the state its value is fixed by the same coordinates S* = S ({x}) 5.13 and its change during the transition is given by the linear differential form S* = S x x 5.14 where S x = x [S ({x})] 5.15 page 37

8 When the transition is adiabatic, Q = 0, which implies S* = 0 for a reversible transition. To find the mathematical consequence of this, let the transition involve only the first two of the constitutive coordinates. Then the two linear differential equations and X 1 x 1 + X 2 x 2 = S x x 1 + S 1 x x 2 = hold simultaneously. If these equations are solved for the ratio x 1 / x 2 the result is X 1 S x -1 = X2 S 1 x -1 = where is a constant for the given state. But the same result holds whatever pair of constitutive coordinates is chosen. Thus X = S x 5.19 where has the same value for every. Since the same coefficients govern infinitesimal quasi-static transitions whether or not they are adiabatic, it follows from equations 5.10 and 5.14 that for all quasi-static transitions. Q = S* 5.20 Equation 5.20 superficially suggests that is a constant number. But cannot be constant. The reason is that S* is independent of the path of the transition (S* is a function of the state) while Q is not. If were a constant then it would be possible to integrate both sides of equation 5.19 with the result which is clearly wrong. Q = S* 5.21 If is not a constant then it must be a function of the state, = [{x}] 5.22 page 38

9 since equation 5.19 shows that it multiplies a function of the state, ( S / x ), to yield a second function of the state, X. In fact, can depend on the state only through its entropy and its temperature, as can be shown by the following argument. Let K be a compound system that consists of two standard systems, K 1 and K 2, that are in thermal equilibrium across an impermeable, diathermal partition. Let the states of K 1 be controlled by the n + 2 constitutive coordinates {x 1,..., x n, S * 1,t}, where the empirical entropy has been used in place of one of the other constitutive coordinates, and let the states of K 2 be controlled by the m + 2 coordinates {y 1,..., y m, S * 2, t}. Then K is a standard system controlled by the set {x 1,..., x n, y 1,..., y m, S * 1, S* 2, t}. Now let K undergo a quasi-static transition. Since and Q = S*, or Q = Q 1 + Q S* = 1 S * S * S* = 1 2 S * 1 + S * where we have used the fact, evident from 5.20, that is not zero. Since S* is a state function, S* = S ({x},{y},s * 1,S* 2 written in terms of its constitutive variables:,t) and its variation can be S* = i S x i x i + j S y j y j + S S * 1 S * 1 + S S * 2 S * 2 + S t t 5.26 Equations 5.25 and 5.26 both hold for arbitrary choices of the differential changes of the constitutive variables. This can only be true if S* does not depend on the variables {x}, {y} or t, since otherwise arbitrary changes in the variables {x}, {y} and t at constant values of S * 1 and S* 2 would yield non-zero values for the right-hand side of equation 5.26 while equation 5.25 has the consequence that the left-hand side is zero. It follows that which has the consequence that the quotient S* = S (S * 1, S* 2 ) 5.27 i = S S * i = f(s * 1, S* 2 ) (i = 1,2) 5.28 page 39

10 is also a function of the entropies of the subsystems only. If the quotients 1 / and 2 / depend only on the entropies of the subsystems, then the variables {x}, {y}, and t either do not appear or are eliminated by division. Since 1 is independent of {y} and 2 is independent of {x}, cannot depend on {x} or {y}. It follows that 1 does not depend on {x} and 2 does not depend on {y}. The coefficients 1, 2 and can depend on their common coordinate, t, but they must all depend on t through a single multiplicative function that divides out. Hence the most general form for the is = T(t)ç(S * 1,S* 2 ) = T(t)ç 1 (S * 1 ) = T(t)ç 2 (S * 2 ) METRICAL ENTROPY AND ABSOLUTE TEMPERATURE Definition of the metrical entropy Since any two standard systems can be combined into a composite system in the manner described in the previous section, the function T(t) that appears in equations must be a universal function of the empirical temperature. The function is arbitrary, of course, to within a multiplicative constant, but, once this constant is chosen, the function T(t) is the same for every system. Having found this function we can use it as a measure of the temperature. We hence define the absolute temperature T = T(t) 5.32 After dividing out the absolute temperature equation 5.24 becomes ç(s * 1,S* 2 ) S*(S* 1,S* 2 ) = ç 1(S * 1 ) S* 1 + ç 2(S * 2 ) S* Consider the first term on the right hand side. If we define the entropy S 1 by the integral S 1 = [ç 1 (S * 1 )]ds* then ç 1 (S * 1 ) S* 1 = S If S 2 is defined in the same way equation 5.33 reduces to ç(s * 1,S* 2 ) S*(S* 1,S* 2 ) = (S 1 + S 2 ) 5.36 page 40

11 If we now define the entropy, S, of the composite system by the integral of the right-hand side of equation 5.36 then S = S 1 + S The quantity S is called the metrical entropy (or simply the entropy). By its derivation it is a good measure of the entropy. Moreover, it is additive in the sense that the entropy of a composite system is the sum of the entropies of its parts (to mae this statement general we mae use of the assumption that any system may be regarded as a composite of standard systems). Finally, it has the property that, for any quasi-static transition Q = T S 5.38 where T is the absolute temperature. Comparing equations 5.38 and 5.7, we see that, at least for systems of fixed composition, the metrical entropy, S, can be regarded as the geometric coordinate for the heat, or thermal wor done in a quasi-static transition, and the absolute temperature, T, is its conjugate force. The metrical entropy and absolute temperature contain an arbitrary multiplicative constant that can be used to set the sign and the units of the temperature. To set the sign recall that the empirical entropy was defined by the Second Law so that it increases or remains the same in an adiabatic process. The addition of energy increases the entropy. Since the metrical entropy is a good measure of the entropy it should have the same property. It follows that the multiplying constant in the absolute temperature must be chosen so that T is positive Measurability The absolute temperature and metrical entropy can be calculated from the empirical temperature and entropy by applying the analysis that was developed above. However, it is only necessary to perform that analysis for a sufficient number of systems to define the absolute temperature over the range of experimental interest. Once the absolute temperature is nown, the metrical entropy is directly measurable. Good thermometers are relatively easy to find. For example, the quantity (pv/n) of an ideal gas is a good measure of the absolute temperature (I leave it as an exercise to show this). Since all gases approach ideality in the dilute limit, the ideal gas thermometer is satisfactory for most temperatures of interest. It fails at very low temperature since no gas is ideal in this limit. Other substances can be used in this temperature range (though the practical problem of finding a good thermometric substance does become more difficult as T 0). Once a suitable thermometer has been identified any convenient thermometric substance can be calibrated and used. page 41

12 Given the absolute temperature, the metrical entropy of a system of fixed composition can be readily measured to within an additive constant, which, as we shall see, is set by the Third Law of Thermodynamics. The simplest method is to establish the entropy along an isometric line by fixing the geometric coordinates at some reference values, {q*}, and changing the state through quasi-static thermal interactions with a suitable reservoir. The entropy of the system in the state specified by the variables ({q*},t) is given by the integral S({q*},T) = S 0 ({q*},t 0 ) + T 0 T dq T 5.39 where S 0 is the value of the entropy in the reference state at {{q*},t 0 }. The entropy function S = S ({q},t) 5.40 can then be found for arbitrary {q} by transforming {q} to the reference set {q*} through a quasi-static adiabatic transition. The transition {q} {q*} will generally induce a change in the temperature to some new value, T*. The entropy of the state {{q},t} is equal to that of the state {{q*},t*}, which is set by equation ENTROPY AS A FUNCTION OF COMPOSITION The change of entropy with composition involves some subtlety because there is no simple means for changing the composition of a system without changing the entropy at the same time. Since all permeable walls are also diathermal, chemical species cannot be added or subtracted from a system without the possibility of a simultaneous transfer of heat. In fact, the meaning of heat is ambiguous for transitions that involve chemical changes; two very different definitions can be made, each of which has its peculiar advantages and disadvantages Chemical equilibrium and the measurement of entropy The prescription developed in the previous section is adequate to measure the entropy of any system of fixed composition. To develop the entropy as a function of composition we might imagine independently measuring the entropy functions of systems of all possible compositions. However, this would not wor since each function contains an arbitrary constant, the entropy of the reference state, and we would not now how to relate these constants to one another. We therefore need a different procedure that directly relates the entropies of systems that have different compositions. To measure the entropy as a function of composition we use the fact that systems of different composition can be brought into chemical equilibrium with one another across semi-permeable or permeable walls. Perhaps the simplest example is the equilibrium between a condensed phase and the vapor above it when the condensed phase does not fill the page 42

13 system. The compositions of the homogeneous systems, or phases, that remain in equilibrium in thermochemical contact are not arbitrary, but are constrained by conditions of chemical equilibrium that we shall develop below. The equilibrium compositions depend on the other coordinates, and will generally change if the state of either system is perturbed, for example, by doing wor on it or adding heat. Consider a composite system containing two subsystems of different composition that are separated by a diathermal partition that is permeable to at least some of the chemical components present. The system might, for example, contain a condensed phase and its vapor, in which case the partition could be taen to be the surface of the condensed phase. Since the temperature of the two subsystems is the same the composite system is a simple system. It has an entropy, S, which is the sum of the entropies of the subsystems, S 1 and S 2. Assume that the entropy function of one of the subsystems (S 1 ) is nown. We wish to measure the other (S 2 ). To do this let the composite system undergo a quasi-static adiabatic transition that involves some chemical exchange between the subsystems. In this transition S = 0 = S 1 + S Since S 1 is nown from the entropy function of K 1 and the initial and final states of the transition, S 2 is determined, and fixes the change in entropy of system K 2 during a change of state that includes a variation of the composition. Experiments of this sort relate the entropies of systems of different composition. The procedure is particularly simple when it is possible to mae K 2 disappear entirely in a quasi-static adiabatic transition. In that case the entropy of its original state is measured directly by the change in the entropy of K 1. To give this procedure a physical context, recognize that, as we shall show below, it is straightforward to calculate the entropy of a mixture of ideal gases in terms of the entropies of its components in pure form. Since every vapor approaches ideality in the dilute limit, the entropies of all real gases can be found by adiabatic compression from a sufficiently dilute state. Any condensed phase that does not fill the container in which it is placed is in equilibrium with a vapor and can be made to evaporate by heating or simply expanding the size of the container, which fixes its entropy in terms of the entropy of the resulting vapor and the heat (if any) which had to be added to the system to induce evaporation. It is hence possible to measure the entropy of any condensed phase in terms of the entropies of its pure components. It follows that the entropy is measurable as a function of composition for any system The energy function and the chemical potential Let the states of a system of variable composition be specified by the constitutive coordinates {E,{q},{N}}, where {q} is the set of mechanical geometric coordinates and {N} is the set of chemical contents (atoms, moles or masses of the chemical constituents). Since the entropy is a physical property it is given by the equation S = S(E,{q},{N}) 5.42 page 43

14 The change in entropy during an isometric change of state can be calculated directly from equation 5.42, and is S = S E E 5.43 Equation 5.43 holds for an arbitrary isometric transition, which may involve both heat and mechanical wor. In a quasi-static transition, however, from which it follows that E = Q = T S 5.44 S E = 1 T 5.45 Since the absolute temperature is positive, equation 5.45 shows that equation 5.42 can be inverted to the form E = E(S,{q},{N}) 5.46 in which the entropy appears as a constitutive variable. Equation 5.46 is called the Energy Function, and, as we shall see below, is one form of the Fundamental Equation. Its partial derivatives have important properties. Using equation 5.46 the variation of the energy in an arbitrary change of state can be written E = E S S + E q q + E N N 5.47 It follows from equation 5.45 (or, equivalently, from the change in energy in a quasi-static isometric transition) that E S = T 5.48 that is, the partial derivative of the energy function with respect to the entropy is the absolute temperature. To find the partial derivatives of the energy function with respect to the mechanical geometric coordinates let the system undergo a quasi-static, adiabatic transition. The change in the energy is the quasi-static mechanical wor: E = p q 5.49 Comparing equation 5.49 to the form taen by 5.47 when S = N = 0, page 44

15 E q = p 5.50 which shows that the partial derivative of the energy function with respect to the th deformation coordinate is the conjugate force, p. If we now define the chemical potential, µ, of the th chemical constituent by the relation E N = µ 5.51 then the variation of the energy in an arbitrary change of state can be written in the simpler form Heat and wor E = T S + p q + µ N 5.52 Equation 5.52 has a transparent meaning for quasi-static transitions at constant composition. The first term on the right-hand side is the heat delivered and the second is the mechanical wor done. When the transformation involves chemical changes as well the assignment of the third term on the right is ambiguous; the chemical energy change can be regarded either as part of the wor done or as part of the heat added in the usual form of the First Law. Treating the chemical energy term as part of the heat, as is done in a number of texts on non-equilibrium thermodynamics and fluid flow, has the advantage that the total heat is easily measured for an arbitrary transition, since the energy change is nown from the terminal states and the mechanical wor can be measured independently from the laws of mechanics. However, it has the important disadvantage that the usual association between the heat and the entropy is lost. For a quasi-static transition, if the chemical contribution is regarded as heat the change in entropy is Q - µ N S = T 5.53 in place of the usual relation. A similar problem occurs in the definition of the Carnot inequality, which we shall define below, which is basic to the non-equilibrium theory. The view that will be taen here follows the tendency in the chemical literature to treat the chemical contribution to the energy change as a chemical wor that is conceptually parallel to the mechanical wor. With this definition the heat and wor in a quasi-static process are page 45

16 Q = T S 5.54 W = p q + µ N 5.55 and the simple relation between entropy and heat is always preserved. When the transition is not quasi-static the chemical wor can still ordinarily be defined from the chemical potentials of the reservoir with which the species are exchanged. With the definition of chemical wor equation 5.52 has a simple and very useful interpretation, since each term in the expression has a form that parallels that taen by the mechanical wor done in a quasi-static transition. We can consistently regard the entropy, mechanical coordinates and chemical contents as the geometric coordinates that govern, respectively, the thermal, mechanical and chemical interactions. Their coefficients are the conjugate forces: temperature for the thermal interaction, the mechanical forces, p, for the mechanical interaction, and the chemical potentials, µ, for the chemical interaction. The change in energy is then the sum of three types of wor, thermal, mechanical and chemical, which are mathematically parallel. It is no longer necessary to mae a distinction between heat and wor The energy function and the overall wor The states of a simple system can always be characterized by the complete set of geometric coordinates {u}, where The energy function is then {u} = {S, {q}, {N}} 5.56 E = E({u}) 5.57 and the partial derivatives of the energy function with respect to its coordinates, {u}, give the forces ({f})that act on the system: the thermal force, T, the mechanical forces, {p}, and the chemical forces, {µ}. The change in energy in an arbitrary infinitesimal change of state is E = f u = T S + p q + µ N 5.58 Equation 5.58 gives the incremental wor done (thermal plus mechanical plus chemical) in an arbitrary change of state. The difference between a quasi-static transition page 46

17 and one that is not quasi-static is that in the former case the thermal, mechanical and chemical wor can be individually equated to the three terms on the right-hand side of In an arbitrary transition they cannot be; for example, the mechanical wor may appear as an increase in the entropy rather than a change in the values of the mechanical coordinates Comment on the units of temperature and entropy In the international system of units the temperature is assigned a unit of its own: degrees Celsius (or Kelvin). This has the consequence that the entropy has units of energy per degree. But from the perspective of thermodynamics there is absolutely no point in having a separate unit for the temperature. Temperature never appears except as one term in a product that has the dimensions of energy. Moreover, the use of a separate unit, the degree, forces one to create units for the entropy and formulate conversion factors relating it to the energy, and compels countless students to spend innumerable hours on such useless exercises as memorizing the precise value of Boltzmann's Constant. We can avoid the problem by adopting the simple expedient of measuring the temperature in units of energy. The metrical entropy is then dimensionless and Boltzmann's constant is superfluous. (The Russian physicists Landau and Lifshitz do this throughout their classic texts.) Of course, if we were really on a roll we would find a way to define the geometric coordinates, {q }, in dimensionless form as well. Then the forces, p, would also have the units of energy. 5.9 THE ENTROPY MAXIMUM PRINCIPLE In the following four sections we explore the consequences of the Second Law for the entropy of an isolated system. This investigation leads to the entropy maximum principle, the conditions of equilibrium and stability, the fundamental equation, and the thermodynamic definition of time Maximum entropy The Second Law of Thermodynamics states that the entropy can only increase in an adiabatic change of state. However, implicit in the formulation is the stronger statement that the entropy must change if it can do so. This statement leads to the Entropy Maximum Principle: An isolated system in an equilibrium state has the greatest entropy that is consistent with the constraints imposed on it. The entropy maximum principle follows from the Second Law and the definition of the constitutive coordinates. If a system is isolated then its internal energy is fixed along with the values of its geometric coordinates. But it is possible to imagine many different states of the system that are consistent with given values of the coordinates. For example, a gas in a container might be found to occupy only one-half of the container, leaving the rest page 47

18 a vacuum. The entropy of the inhomogeneous state can be defined since it is possible to construct such a system by joining a system that contains the gas in one-half the container volume to a similar volume of vacuum across an impermeable wall. The Second Law obviously rules out the possibility of the gas spontaneously dividing itself into a gas and a vacuum from an initially homogeneous state, since it can easily be shown that the entropy of the inhomogeneous state is lower and the entropy cannot decrease. But this is not sufficient to specify the state of the isolated system. It is possible to set up the system in the inhomogeneous state and then remove the impermeable partition. The gas can then expand to fill the container. But it can also avoid decreasing its entropy by remaining divided or by evolving into any other inhomogeneous state that has higher entropy. The problem of defining the ultimate state of the gas is resolved in a somewhat circuital way by referring to the definition of the constitutive coordinates. We assumed at the beginning that it is possible to choose a finite set of measurable properties of the system whose values are sufficient to specify the state in the sense that they determine the values of all other properties. If the system is simple these coordinates can be chosen to be the energy and the geometric coordinates. It follows that a specification of the energy and the geometric coordinates uniquely determine the state. If they do not do so, the system is not a simple system. It then can, by hypothesis, be regarded as a composite of simple systems separated by partitions. Consistent with this definition any system, such as the gas in the example above, that can exist in a number of different states that have the same values of the constitutive coordinates must select one. That one must be the state of maximum entropy since the system can evolve into the maximum entropy state, but cannot evolve out of it. If the system fails to reach the state of maximum entropy consistent with the values of its constitutive coordinates then one of two statements is true: either the list of constitutive coordinates is insufficient or the system is a composite system with internal constraints. The Maximum Entropy Principle must be true if the constitutive coordinates are to specify the state Spontaneous processes; non-equilibrium thermodynamics It follows from the Maximum Entropy Principle that the entropy of a system cannot decrease in any spontaneous process, that is, any process that can occur within the system without the intervention of an outside agency. Since spontaneous processes are internal to the system, they would proceed even if the system were isolated from its environment. Hence, for any spontaneous process, ÎS However, in order to apply the Maximum Entropy Principle to spontaneous processes both the initial and final states must be equilibrium states for which the entropy is defined. This requirement is automatically satisfied when the initial state is maintained by constraints, such as internal partitions, that are removed so that the system can relax. In other cases it is satisfied by constructing a representative model in which the initial state is page 48

19 replaced by one that is constrained to be in an equilibrium state that is as nearly lie the equilibrium state as possible. The latter construction is used to apply equilibrium thermodynamics to systems that are slowly evolving toward equilibrium, and is the basis of the Non-Equilibrium Thermodynamics we shall develop at a later point in the course. We model the instantaneous state of a system that is spontaneously evolving toward equilibrium by imposing constraints that fix it in an equilibrium state that is as near as possible to its instantaneous condition. The thermodynamic properties of the constrained state are well defined. We then model evolution by gradually modifying or relaxing the constraints so that the system evolves in a series of small steps The Carnot inequality In a general change of state the system is not isolated, and may receive heat or wor from its environment. The generalization of the Maximum Entropy Principle to this case is called the Carnot Inequality. It can be established in the following way. In the most general case an infinitesimal transition includes the transfer of a quantity of heat, the performance of mechanical and chemical wor, and internal changes within the system. The transition can be represented by an equivalent process that includes a quasistatic transition followed by an adiabatic one. Using the energy function, equation 5.52, the entropy change in the quasi-static step can be written S 1 = 1 T ( E - p q - µ N ) = 1 T ( E - W) = Q T 5.60 where {p} and {µ} are the mechanical and chemical forces in the system, T is its temperature, and W is the wor done. Since the second step in the transition is adiabatic, the associated entropy change is non-negative: Summing the two, S S Q T 5.62 The inequality 5.62 is the Carnot Inequality, and holds for arbitrary transitions. page 49

20 Using the first form of equation 5.60 the total entropy change can also be written S = Q+ W' T 5.63 where W' is the irreversible wor, W' = W - p q - µ N 5.64 It follows from the Carnot inequality that The irreversible wor is never negative. W' Note that the form 5.62 of the Carnot inequality uses a definition of the wor that includes the chemical contribution. If the chemical wor is assigned to the heat, as is often done, the Carnot inequality taes the somewhat more complicated form Q - µ N S T 5.66 While it is easier to measure the heat when it is defined to include the chemical wor, the latter must still be measured independently before the Carnot inequality can be applied THE CONDITIONS OF EQUILIBRIUM AND STABILITY The Principle of Maximum Entropy asserts that an isolated system is at equilibrium if there is no possible way to reconstitute it so that its entropy increases. Let the system be characterized by the coordinates {E,{q},{N}}. The admissible states of the system are different ways of distributing the energy, entropy and chemical content while eeping their total values constant. The entropy of these hypothetical rearrangements can be found, for example, by regarding the system as a composite of subsystems whose contents are individually fixed by means of suitable partitions. With this understanding, a system is in equilibrium if every possible change of state causes the entropy to decrease or remain the same, that is, if the Global Condition of Equilibrium ÎS {E,{q},{N}} is satisfied for every possible rearrangement of the system that has the same values of the coordinates {E,{q},{N}}. page 50

21 The inequality 5.67 is a sufficient condition for equilibrium, but it is not necessary. When the initial state of the system does not provide an absolute maximum of the entropy it may happen that states of higher entropy can only be reached by a major reconstruction of it, for example, a change in crystal structure or a decomposition into separate phases. The transition to the high-entropy state is favored, but the inetics of the transition may be such that it does not occur in reasonable time. In this case we say that the system is preserved in metastable equilibrium. It is useful to have a criterion for equilibrium that includes the possibility of metastable states. To phrase a necessary condition for equilibrium we use the physical fact that real systems are in constant fluctuation on the microscopic scale. Their local properties oscillate in a noisy way about the mean values that are specified by their macroscopic states. These fluctuations are macroscopically small, but have the consequence that a system is constantly sampling all admissible states that differ infinitesimally from the current one. If any of these raises the entropy the system necessarily transforms; but if the system is stable with respect to all infinitesimal changes then it may be preserved. It follows that a system can be in equilibrium only if it is in equilibrium with respect to every infinitesimal change of state, that is, if it satisfies the Local Condition of Equilibrium S {E,{q},{N}} The inequality 5.68 is a necessary condition for equilibrium but is clearly not sufficient since a state of the system may satisfy 5.68 without satisfying the stronger condition imposed by equation The condition 5.68 is not even sufficient to guarantee that the state is stable with respect to small perturbations. For example, suppose that the parameter, x, is a continuous parameter that measures the difference between two hypothetical states that have the same overall values of the constitutive coordinates. If S is written as a function of x then 5.68 requires that S = ds dx x If x can have either sign this yields the usual mathematical condition for an extremum: ds dx = But an extremum may be a minimum or a saddle point rather than a maximum. In either of these cases there is at least one direction of change of x for which S increases monotonically, and the system necessarily transforms. These cases are examples of unstable equilibrium. To differentiate metastable from unstable equilibrium the Local Condition of Equilibrium is supplemented by the Local Condition of Stability. If the system is to be in stable equilibrium with respect to local fluctuations it is necessary that page 51

22 2 S {E,{q},{N}} where the variation is taen to second or higher order in whatever parameters characterize the local change of state. For example, if x is a continuous variable that characterizes a possible transition path then, to second order, ÎS = ds dx x d 2 S dx 2 x A local equilibrium state satisfies the condition of stability if In the special case d 2 S dx 2 < d 2 S dx 2 = the system is said to be in neutral equilibrium, and its stability is governed by derivatives of higher order. Continuing the Taylor expansion started in equation 5.72 it follows that if the transition is governed by the single continuous variable, x, and if d n S/dx n is the first non-vanishing derivative of S with respect to x, then the equilibrium is stable only if n is even and if d 2 S dx 2 < When more than one variable is required to specify the change of state the inequality 5.71 still governs stability, but the detailed mathematical development of it is somewhat more complicated. We shall defer a detailed discussion to a later section THE FUNDAMENTAL EQUATION The physical properties of a material are given by choosing a complete set of constitutive coordinates and phrasing an appropriate set of constitutive equations. The equilibrium behavior of the system is determined by the constitutive equations, and the equilibrium states are fixed by he conditions of equilibrium and stability. It is, in fact, possible to gather all the information that pertains to equilibrium behavior into a single constitutive equation, called the Fundamental Equation, from which the other constitutive equations can be derived. Consider an isolated system. Let its constitutive coordinates be E, {q} and {N}. A given set of values of these coordinates determines an equilibrium value of the entropy, S, page 52

23 which is the maximum value that is consistent with the constraints on the system. The association between the values of the constitutive coordinates and the maximal value of the entropy defines a function, the entropy function S = S(E,{q},{N}) 5.76 This function is one form of the fundamental equation of the system, and is the appropriate form when the system is isolated. It has three important features 1. Its coordinates are the natural coordinates for an isolated system; the energy, geometric coordinates and chemical content are precisely the variables that can be controlled in an isolated system since their values can be set before the isolation is imposed. 2. Its values correspond to the equilibrium states defined by the Entropy Maximum Principle. 3. It contains all of the information that pertains to the equilibrium behavior of an isolated system. The energy and the geometric coordinates are the independent constitutive coordinates whose values are fixed for an isolated system. The applied forces applied are determined by simple differentiation. From the first form of equation 5.60, S E S q S N = 1 T = - p T = - µ T that is, the partial derivatives of the entropy function determine the constitutive equations for the forces. The material properties that govern the changes in the forces with the constitutive coordinates are governed by the second derivatives of the entropy function, as we shall discuss further in a later section. The fundamental equation is the natural starting point for the applications of thermodynamics. We shall discuss it in more detail in a later section, and show that it can be cast in various forms that define the familiar thermodynamic potentials, such as the energy, the Gibbs and Helmholtz free energies and the wor function. Each of the various thermodynamic potentials is equivalent to the fundamental equation when it is written as a function of a particular set of coordinates, and is the appropriate form of the fundamental equation when those coordinates are the independent, controllable coordinates of the system. page 53

24 The fundamental equation of a material defines it. Classes of materials are specified by the variables that appear in it, types of materials are characterized by its functional forms, and particular materials are identified by its exact forms. As we shall see, the theory of statistical thermodynamics is formulated to calculate the fundamental equation from the behavior of the elementary particles that mae up the system THE ARROW OF TIME In introducing the Second Law we mentioned that it is the principle of physics that sets a direction to the evolution of a physical system; what Max Born (The Natural Philosophy of Cause and Chance) has called the "arrow of time". This notion follows from the principle of maximum entropy, and can be easily understood in the case of an adiabatic system. Let an adiabatic system be probed at two times and found to be in states Í and Í' where S(Í) < S(Í'). Assuming that the system remained isolated it is necessarily true that state Í preceded state Í'; therefore the time of Í preceded that of Í'. The Second Law establishes a temporal hierarchy in the states of an adiabatic system, and, hence, in the states of any system since any system can be made into an adiabatic system by joining its surroundings to it. This temporal sequence of states can be used to define a cloc that always runs forward and sets the times of physical processes. It is possible to go forward in time but not bac again. In the case of the adiabatic system this is obvious, since Í < Í' but Í' > Í. But time cannot be reversed even if adiabatic isolation is broen. To reestablish Í from Í' one must let the system interact with a second system that accepts at least the entropy difference S(Í')-S(Í). The composite system carries a record of the time of Í', and records an even later time if there is any net entropy increase in the process that restores state Í. A change of state that involves a net increase in the entropy is irretrievable; its consequences can be transferred to other systems, but they can never be reversed. The entropy hence defines the thermodynamic time, and is the only parameter in physical science that carries with it a sense of time that has a definite direction. In fact, it is not unreasonable to equate entropy with the physical time. While it is true that we usually measure time with devices such as mechanical clocs or planetary motions that can (at least in theory) be run bacward as well as forward, to establish a direction of time, that is, to state unequivocally that one position of the cloc hands came after the other, the cloc must be calibrated against some evolutionary process that involves a progression of states of increasing entropy. The degree of evolution, or increase in the entropy, measures the physical time. Familiar choices for the evolving system are the aging of humans, the growth of plants and the decay of radioactive species. These conclusions are straightforward consequences of the Second Law, and, as such, have stimulated speculations on cosmological subjects such as the ultimate state of the universe and the end of time. Such speculations are fun, but one must be cautious about page 54