8 Lecture 8: Thermodynamics: Principles

Transcription

1 8. LECTURE 8: THERMODYNMICS: PRINCIPLES 69 8 Lecture 8: Thermodynamics: Principles Summary Phenomenological approach is a respectable way of understanding the world, especially when we cannot expect microscopic mechanical detailed description is impossible (such as heat exchange, very long time behavior, etc.) Equilibrium states may be described by the thermodynamic coordinates consisting of the internal energy and the work coordinates. The space spanned by thermodynamic coordinates is called the thermodynamic space. Every equilibrium state is a point in this space. Even if the change between two equilibrium states is irreversible, by devising a quasistatic path between them, changes in thermodynamic quantities may be computed thermodynamically. The first law is essentially the conservation of energy, but to describe it in terms of small number of macroscopic variables, changes (or processes) must be sufficiently slow. The second law implies that the thermodynamic space is foliated into adiabats = constant-entropy surfaces. Under adiabatic conditions S = 0, if change is reversible, and S > 0, if irreversible. Key words 62 Phenomenological approach, zeroth law, thermodynamic coordinates, work coordinates, thermodynamic space, quasiequilibrium process, reversible process, state function, thermal contact, thermal equilibrium, temperature, conjugate pair, Clausius law, Kelvin s law, Planck s law, adiabatic process, adiabat, stability criterion, evolution criterion What you should be able to do Explain why the thermodynamic coordinates are privileged coordinates. Show that all three expressions of the second law mentioned here are equivalent. Demonstrate that we can introduce a state function (= entropy) that is constant on the adiabat and is increasing as a function of E under constant work coordinates. them. 62 There are many today, but they are important, so at least once you should try to memorize

2 70 s we have discussed in the last lecture, important characterization of macroscopic observables (e.g., extensivity of internal energy) and irreversibility may be understood from mechanics, so perhaps you may think mechanics can explain everything thermal as well. Unfortunately, however, we cannot confine ourselves to the discussion of closed (or isolated) systems. For example, we must discuss heat transfer, which is hard to describe in terms of mechanics. 63 We have learned that, although rownian dynamics is due to underlying particle mechanics, we could set up a reasonable model without directly referring to the underlying mechanics. Then, it should be a wise approach to try to describe a system without referring to the underlying smaller scale events. This is called the phenomenological approach. The phenomenological theory for macroscopic systems we have is thermodynamics. The reader must clearly recognize that thermodynamics survived the quantum revolution without any revision in contrast to the supposedly more fundamental statistical mechanics. We know that classical mechanics is not compatible with thermodynamics. So far it seems that quantum mechanics has not produced any contradiction with thermodynamics. Even if someday quantum mechanics will be replaced by something else, thermodynamics will remain intact. In physics, the term phenomenology is not necessarily respected; it is almost a pejorative. However, notice that when underlying microscopic descriptions are impossible or only approximate, phenomenology may be the only realistic rational approach allowed to the human being. Phenomenology is not an approximation or crude description of something more accurate, just as thermodynamics is not an approximation to something else. To explain thermodynamics, first we must characterize a macroscopic system in equilibrium. If a macroscopic system (a system with extremely many particles) is isolated and is left undisturbed for a long time, the system would reach a macroscopic state (that is characterized by macroscopic observables) which would not change any further. This final state is called a thermal equilibrium state. Macro-observables observed at any time in equilibrium take unchanging values. 63 lso we saw that the effect of the external world of the system cannot be completely eliminated. The long time limit (t ) and the external noise zero limit (i.e., the pure mechanical limit) are not commutative, so purely mechanical description may well fail to describe the system reaching equilibrium after a long time.

3 8. LECTURE 8: THERMODYNMICS: PRINCIPLES 71 macroscopic system in equilibrium is partitioning-rejoining invariant: if a macroscopic system in equilibrium state is divided into two halves, the halves are themselves in equilibrium and if they are joined again, the result is indistinguishable from the original system as long as the thermodynamic observables are concerned (Fig. 8.1). 1 Equilibrium as a whole in isolation 2 3 Each piece is in equilibrium in isolation even after separation Combining and recovers a macrostate indistinguishable from 1 Figure 8.1: Partitioning-rejoining invariance of equilibrium states s already discussed intuitively, macroscopically important observables are extensive or intensive. ll thermodynamic observables are either extensive or intensive. This is the fourth law of thermodynamics. n equilibrium state of a macroscopic system may be described in terms of a small number of variables called thermodynamic coordinates. They are extensive quantities (= additive quantities) and consist of internal energy E and other variables (called work coordinates) that we can control mechanically. The system volume V is often among them. For a magnetic system, magnetization M is included. These work coordinates can describe macroscopic work done to or done by the system. The thermodynamic coordinates uniquely specifies the equilibrium state. Notice that they make a quite special set of variables. T or P is not among them; clearly recognize how privileged the set is ( Fig. 8.2). The space spanned by the thermodynamic coordinates is called the thermodynamic space. For a given macroscopic system, its each equilibrium state uniquely corresponds to a point in the thermodynamic space of its own. 64 Thermodynamics wishes to study changes of equilibrium states by various processes. Not all (actually most) processes allowed to the system cannot be described 64 Some readers might question that there are much more macroscopic observables we can observe for a given object, shapes, orientation, etc. Precisely speaking, thermodynamic states are equivalence classes of macroscopically distinguishable states according to the values of the thermodynamic coordinates.

4 72 C Figure 8.2: -C are the same amount of water at 0 C. However, their internal energies are distinct; has less than C. In elementary thermodynamics, often the temperature T appears as a key variable instead of internal energy E, but these examples clearly tell us that T cannot distinguish equilibrium states that are clearly distinct. nalogously, in the liquid-gas phase transition under constant pressure P and T, E and V change. These examples clearly indicate that thermodynamic coordinates are the fundamental privileged set of variables to describe thermodynamic equilibrium state; generally speaking intensive variables such as T and P fail to describe states uniquely. in the thermodynamic space, because every point actually realizable in the thermodynamic space describes an equilibrium state of the system. Only processes that are extremely (infinitesimally) close (experimentally indistinguishably close) to equilibrium states at every moment may be expressed as a continuous curve in the thermodynamic space. Such processes are called quasistatic processes. quasistatic process is synonymous to a process that has a curve representation in the thermodynamic space (Fig. 8.3). thermodynamic space noneq process quasistatic equilibrium states Figure 8.3: and are equilibrium states. quasistatic process connecting and is in the thermodynamic space. From to a process need not be quasistatic. Then, such a process cannot be described in the thermodynamic space (red). Whether it is reversible (retraceable) or not is not directly related to quasistatic nature (being slow or not) of the process, 65 although not quasistatic processes are very likely to be irreversible. Certainly, the processes not described as curves in the thermodynamic space are 65 Think of a hot coffee in a thermos. Whether a given quasistatic process is reversible or not depends on the context. In the case of thermos, the process is undoubtedly irreversible. However, you could cool your coffee by removing heat reversibly by producing work

5 8. LECTURE 8: THERMODYNMICS: PRINCIPLES 73 nonequilibrium processes. For example, if a process is sufficiently rapid, it cannot have any corresponding path in the thermodynamic space, because states along the process are not infinitesimally close to equilibrium states. If the value of a macroscopic quantity of an equilibrium state is uniquely specified by the corresponding point in the thermodynamic space, the macroscopic quantity is called a state function.. That is, any observable that is a function defined on the thermodynamic space is a state function. Its value is indifferent to how the state is realized. For example, the equilibrium volume of a system is a state function; temperature is another example. When the initial and final equilibrium states are given, the change of a state function between these two states does not depend on the actual process but only on the initial and final equilibrium states. Even if the actual process connecting these two states is not a quasistatic process (i.e., does not lie in the thermodynamic space), we can thermodynamically compute the variation of any state function during the process with the aid of an appropriate (appropriately devised) quasistatic process connecting the same end points, because we can devise a quasistatic reversible process between any realizable points in the thermodynamic space. This makes thermodynamics extremely useful in practice. There is a special way of making contact between two systems called thermal contact. Thermal contact is a contact through a special wall which does not allow the systems to exchange work, matter, or any systematic macroscopic interaction (such as electromagnetic interactions), but still energy is exchanged as heat through it. If two systems and are in thermal contact and are in equilibrium as a compound system, we say and are in thermal equilibrium. If the systems and are in thermal equilibrium, and if and C are in thermal equilibrium, then so are the systems and C. That is, the thermal equilibrium relation is an equivalence relation. Thermal equilibrium is temperature equilibrium: The second assertion implies the existence of a scalar quantity called temperature (or more precisely, an empirical temperature): there is a quantity called temperature which takes identical values for two systems in thermal equilibrium. demonstration follows, but if you think the conclusion is natural, you can skip the following fine-lettered discussion. The thermal equilibrium condition between and may be expressed as a relation between the thermodynamic coordinates of these systems: The equivalence relation of thermal equilibrium implies F C (E, X, E, X ) = 0. (8.1) F C (E, X, E, X ) = 0, F (E C, X C, E, X ) = 0 F (E, X, E C, X C ) = 0. (8.2)

6 74 From F C = 0 and F = 0, we can solve as E = φ (X, E, X ) = φ C (X, E C, X C ). (8.3) The second equality is an identity with respect to X, so we may choose a convenient particular values for the work variables X to obtain a functional relation nalogously, we can have (see Fig. 8.4) a (E, X ) = a C (E C, X C ). (8.4) b (E, X ) = b C (E C, X C ), (8.5) c (E, X ) = c (E, X ). (8.6) = c c a = ac b = C b C Figure 8.4: Thermal equilibrium relation (8.4), (8.5) and (8.6). We must demonstrate that we may choose b = c, a = c, and c = c. From (8.5) and (8.6) we have b C (E C, X C )/c (E, X ) = b (E, X )/c (E, X ). (8.7) The right-hand side of this equation depends only on the coordinates, but the left is not, so this ratio must be a constant, which may be set to unity, since we may multiply any number to (8.6). Therefore, b C (E C, X C ) = c (E, X ), (8.8) b (E, X ) = c (E, X ). (8.9) Repeating an analogous argument for (8.5) and (8.4) we can obtain b C (E C, X C ) = a C (E C, X C ). Thus, all the values are identical and we may introduce an empirical temperature t X for system X as t (E, X ) = b (E, X ) = c (E, X ), (8.10) etc., satisfying t (E, X ) = t (E, X ) = t C (E C, X C ). (8.11) The first law of thermodynamics is essentially the conservation of mechanical energy (= internal energy) of the system. We now know heat as well as work is a way

7 8. LECTURE 8: THERMODYNMICS: PRINCIPLES 75 to transfer energy. Mayer, Joule, Helmholtz and others established that the conservation of mechanical energy implies there is a state function E called internal energy satisfying E = W + Q. (8.12) Notice that although E is a state function, neither W nor Q is a state function; they depend explicitly on the path connecting the initial and the final equilibrium states (the path may not be in the thermodynamic space 66 ). Let us make the sign convention explicit. The sign is seen from the system s point of view: everything imported to the system is positive, and exported negative. For example, if you do work to the system, W > 0. If you get some useful work from the system W < 0. When not only heat but matter can be exchanged between the system and its environment (in this case the system is called an open system), (8.12) does not hold anymore. To rescue the equality, we introduce the term called mass action Z E = W + Q + Z. (8.13) Z is not a state function, either. Now, we can summarize the first law of thermodynamics as: The internal energy E defined by (8.13) is a state function. For an infinitesimally small change, we write (8.13) as follows: de = d W + d Q + d Z, (8.14) where d is used to emphasize that these changes are not the changes of state functions (not the path-independent changes). 67 Let us study some concrete expressions of the work. When the change is quasistatic, W, Q and Z in (8.13) are determined by the equilibrium states of the system along the quasistatic path. For example, let us consider the work required to change the system volume from V to V + dv (V is clearly a state function, so d is not used here). The necessary work supplied to the system reads (See Fig. 1.2) 66 Notice that W is purely mechanically defined, irrespective of the nature of the process, and it is macroscopically measurable, but Q is usually not directly measurable in nonequilibrium processes. It is computed, when the process is finished, to satisfy (8.12). 67 Mathematically, it is not a total differential or not a closed form.

8 76 dl P F Figure 8.5: Work done by volume change. d W = F dl = P dv, (8.15) where P is the pressure and the force F is given by the following formula, if the process is sufficiently slow F = P, (8.16) where is the cross section of the piston. Here, we use the sign convention such that the energy gained by the system becomes positive. Hence, in the present example, d W should be positive when we compress the system (i.e., when dv < 0). If the process is fast, there would not be sufficient time for the system to equilibrate. For example, when we compress the system, the force necessary and the force given by (8.16) can be different; the pressure P may not be well defined. Consequently, (8.15) does not hold (the work actually done is larger than given by (8.15)). Thus, although the first law is essentially the conservation of every, to write it in terms of a small number of variables, the change must be slow (quasiequilibrium). The electromagnetic work can be written as d W = H dm, (8.17) d W = E dp, (8.18) where H is the magnetic field, M the magnetization, E the electric field, and P the polarization. The mass action is empirically written as d Z = i µ i dn i, (8.19) where N i is the number of i-th particles (or the molar quantity of the i-th chemical species), and µ i is its chemical potential. Notice that the work and the mass action can always be write as the product of intensive variable d(extensive variable). These pair of state variables is called a conjugate pair. Their product has a dimension of energy.

9 8. LECTURE 8: THERMODYNMICS: PRINCIPLES 77 We know empirically that not all conceivable processes are realizable in Nature. Carnot and Clausius established that thermal energy is special. Joule demonstrated that work can be converted into heat, but Carnot showed that the reverse is impossible. Only when there are hotter and colder heat baths can we produce work. There is a fundamental asymmetry between heat and work. Clausius understood this as follows. Temperature is the price of unit energy. You cannot simply promote the price of energy (you cannot transfer energy from a colder to a hotter bath). Work corresponds to heat at T =. The second law of thermodynamics summarizes this as follows. Two famous expressions are: Clausius law: Heat cannot be transferred from a colder to a hotter body spontaneously. Kelvin s law: process cannot occur whose only effect is the complete conversion of heat into work. (No existence of perpetum mobile of the second kind; there is no engine which can produce work without a radiator.) Notice that Clausius law contains Kelvin s law, if we understand work is the heat energy from T = heat bath. Here, we use the second law in the following form: Planck s law: In the adiabatic process if all the work coordinates return to their original value, E 0. Here, adiabatic process must be explained. In short, it is a process without any exchange of heat with the surroundings (a process realized in a Dewar jar). 68 The first law implies adiabatically de = i x i dx i, (8.20) where (x i, X i ) are conjugate pairs for work coordinates (non-thermal variables). The variables E and X i span thermodynamic space. 68 There is a special wall called an adiabatic wall such that for a system surrounded by this wall the necessary work to bring the system from a given initial equilibrium state to a specified final equilibrium state is independent of the actual process but is dependent only on these two end states of the process. process that can be realized in a system surrounded by adiabatic walls is an adiabatic process. Furthermore, even if the process is realized without surrounded by adiabatic walls but the same process can be realized surrounded by adiabatic walls, it is an adiabatic process. This turned about to be identical to the process without heat exchange with its environment. Thus, even if a system is attached to a heat bath, a process in the system can be adiabatic. Notice that adiabatic process need not be describable in terms of pure mechanics.

10 78 E X 1 Figure 8.6: path in the thermodynamic space corresponds to a quasistatic process (notice, however, generally that Planck s law does not require quasistatic processes). vertical move implies a purely thermal process. diabatically, there is no way to move from a state to another state that is vertically below it according to Planck s law. Here, X 1, X 2 represent work coordinates. X 2 Planck s law, Kelvin s law and Clausius law are equivalent: If Planck s law is violated, then adiabatic work can reduce the system energy. That is, work can be produced by a single heat bath. Therefore, Kelvin s law is violated. If Kelvin s law is violated, then we can get work from a cold bath and do work on a hotter bath to increase its energy. Thus, Clausius law is violated. If Clausius law is violated, then we can convert a uniform system into colder and hotter portions and produce work to make the portions to be the same temperature again by a cyclic change of the work coordinates. Thus, Planck s law is violated. Thus, we have demonstrated the equivalence of all the laws mentioned here (since we have demonstrated the chain of contrapositions). The second law implies that there is a state variable that is constant along the adiabatic reversible process, which Clausius called entropy, and ritish people resisted to recognize for a while. Even Maxwell misunderstood it. English speaking scientists were rescued by Gibbs who correctly understood thermodynamics. The constant entropy surfaces foliate the thermodynamic space. These surfaces are called adiabats or isentropic hypersurfaces. Roughly speaking, if state has larger entropy than state, then we can never go from to adiabatically. I hope at least once in your life you try to reproduce the following explanation of the existence of entropy to your lay friends. Choose an arbitrary point P in the thermodynamic space and a quasistatic adiabatic and reversible path 69 connecting P and L, a line parallel to the energy axis (a 69 Why is such an awkward description of the path? Reversibility does not logically guarantee

11 8. LECTURE 8: THERMODYNMICS: PRINCIPLES 79 E L Q P adiabatically inaccessible from P X X' Figure 8.7: If can be reached by an adiabatic reversible process from P, then adiabatically we can go from Q to via P, violating Planck s law. Thus, the gray-shaded portion is inaccessible. If can be reached by an adiabatic reversible process from P, then adiabatically we can go from to Q via P, violating Planck s law. Thus, Q is unique. We can adiabatically go from to P and from P to, but they are irreversible processes under adiabatic condition. constant work coordinate line; Fig. 8.7). Suppose the path lands on L at point Q. Can we reversibly and adiabatically go from P to or distinct from Q on the line L? Planck tells us is inaccessible; if possible, we can adiabatically go to from Q via P or to Q from via P, contradicting Planck. Now, moving the stick L throughout the space keeping it parallel to the energy axis, we can construct a hypersurface consisting of points adiabatically, quasistatically and reversibly accessible from point P. This is an adiabat containing P (the totality of the state that can be quasistatically and reversibly reachable from P without any heat exchange with the environment). diabats foliate the thermodynamic space. That is, no two different adiabatic surfaces cross each other. See Fig. 8.8 to understand that these sheets = adiabats cannot cross; crossing means Planck is violated. This implies that we can define a state function S, whose level sets are given by these sheets (S = constant defines an adiabat). The adiabats do not have any overhangs. reason is the same as that for no crossing. s you can see from Fig. 8.9, the Therefore, adiabats can be parameterized by a number that increases continuously and monotonically with energy E. This number S is a state function. How can we quasiequilibrium; quasiequilibrium does not mean reversible. This si the reason. However, intuitively, we may say reversible path, because not quasistatic reversible process is not very realistic.

12 80 P' P Q Figure 8.8: If two adiabats cross or touch, then we can make a cycle that can be traced in any direction, because P Q, P Q can be traced either directions (reversibility of quasistatic processes), so can be P P with the aid of an appropriate heat bath. Planck s law is violated. P' P Figure 8.9: Just as Fig. 8.8, an overhang violates Planck s law. Q change this value? We can change entropy (reversibly) keeping all work coordinates constant (by going up or down along the line L); that is, we can change S by supplying or removing heat Q. Since energy is extensive, so is Q. 70. If d Q > 0, ds > 0 should be required. We may choose these two differentials to be proportional: ds d Q. This automatically implies that we assume S to be extensive, and the proportionality constant must be intensive. Suppose two systems are in contact through a wall that allows only the exchange of heat (that is, in thermal contact), and they are in thermal equilibrium. Exchange of heat d Q between the system is a reversible process (say, system I gains d Q I = d Q and II d Q II = d Q), so this process occurs within a single adiabat of the compound system (i.e., the two systems considered together as a single system). If we write 70 That is, if we double the system, we must double the heat to reach the same thermodynamic state characterized by the same intensive parameters and densities (= extensive variables per volume).

13 8. LECTURE 8: THERMODYNMICS: PRINCIPLES 81 d Q X = θ X ds X (X = I or II), ( 1 0 = ds I + ds II = d Q 1 ). (8.21) θ I θ II This implies θ I = θ II. That is, when two systems have the same temperature, the proportionality constants are also the same. Hence, we may interpret the proportionality factor as a temperature (cf. the zeroth law). 71 The introduced temperature can be chosen as a universal temperature T called the absolute temperature. Hence, in the quasistatic process we can write 72 d Q = T ds. (8.22) Now we can write down the infinitesimal version of the first law of thermodynamics for the quasistatic process as follows: de = T ds P dv + µdn + H dm +. (8.23) This is called the Gibbs relation. because it was first written down by Gibbs. Notice that each term consists of a product of a conjugate pair: an intensive factor and d[the corresponding (i.e., conjugate) extensive quantity]. Suppose a (perhaps compound) system is in equilibrium. If we modify its external field (say, the electric field), its volume, or if we remove a wall between the subsystems, the equilibrium state of the system may or may not change. Since all spontaneous processes are irreversible, we may say that for a system to change (under an adiabatic or isolated condition) from an initial state to a final state, the entropy change (which is completely determined by these end points in equilibrium) must be positive. If not, there cannot be any spontaneous change. Thus, for an isolated system S < 0 the state is thermodynamically stable, (8.24) S > 0 the state spontaneously evolves. (8.25) Therefore, the second law gives us a variational principle (entropy maximization principle) to find a stable equilibrium state for an isolated system If you do not like this explanation, go to a much more complicated standard argument in the posted lecture notes. 72 Precisely speaking, we must show that this T is identical to the T appearing in the ideal gas law. 73 We will return to the system stability question in Chapter 4.