Understanding Entropy Using the Fundamental Stability Conditions Karen S. Sanchez
Department of Natural Sciences, Florida Community College at Jacksonville, Jacksonville, FL 32246 Robert A. vergenzi
Department of Natural Sciences, University of North Florida, Jacksonville, FL32224-2645 In this paper we introduce the second law of thermodynamics in a way that provides chemical intuition about the thermal meaning of entropy and is rigorously based on mathematical logic. A perennial student question is What IS Entropy (WISE)?!
We derive an equation, hereafter called the WISE equation, that refocuses the student's attention to the more useful question, Which IrreversibilitiedSourees of Entropy (WISE) apply in this problem?
Answering the second WISE question also answers the f r s t WISE question. The WISE equation,
expresses a small change in the entropy S of the universe in terms of differences, T - Tawand P - P , in temperature and pressure between system and surroundings, and also in terms of small changes in S, and in the system volume V. It is valid for constant-composition systems doing only PV work. A simple analysis of this equation using fundamental empirical conditions for thermal and mechanical stabilitv directly makes clear both the f a d and the causes of the increase in entropy of the universe. The approach is efficient because the stability conditions are ordinary experiences of chemistry students. More remote examples, such as heat engines, can be avoided. When the equation is extended to describe phase changes, it provides insight about chemical potential as it relates to irreversibility and the second law. Two different approaches to defming entropy are commonly encountered in chemistry curricula ( I ) .Most textbook calculations rely on the thermal definition, given for reversible heat transfer Eq,,, by (2)
'Author to whom correspondence should be addressed.
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The second approach is statistical. It states that S = kgln M
where k~ is the Boltzmann constant; and M is the number of microstates, the indistinguishable ways in which the energy of the system can be stored (3). This is the source of the common but misleadin~sttltement that entrow is a measure of the disorder of the svskrn (1I. In practice, the theoreticallypredicted behavior of s"tatistica1en~~OD matches Y the measured behavior of thermal entroov (4). . . the statistical definition of entropy is based on a theoretical wnstmdion, using microstates that are not directly observable, whereas thermodynamics is a fundamentally empirical subject. Because the thermal mean in^ of entropy is the more direct experimentally obsewable decnition, an intuitive understand in^- of it is crucial to understanding thermodynamics (5). Begining chemistry textbooks provide more intuitive coverage about the statistical entropy than the thermal concept because the simplicity and vividness of word images for the statistical interpretation better fit their level. Consequently, most students who take beginning chemistry courses understand entropy exclusively in terms of disorder. Lowe ( I ) provides several examples of how oversimplification of statistical entropy has perpetuated errors in the teaching literature. In more-advanced courses, rigorous development of the thermal concept classically starts with discussions of heat engines and refrigerators, and proceeds through a tortuous chain of reasoning involving complicated reversible cycles, as illustrated inFigure l ( 6 8).It is easy to lose the forest for all the trees. A simple, rigorous way to teach about the second law involves convincing students of two fundamental principles, and unfolding their consequences (2).Principle 1 is that entropy is a state function, and principle 2 is that spontaneous processes increase the entropy of the universe. Principle 1is addressed well elsewhere (2,9,10).In this paper we only develop principle 2, the universal increase of entropy The WISE equation provides a rigorous mathematical connection between cogent images, based on the thermal definition, and the principle ofincrease of entropy. The resulting logical simplification is clearly seen by comparing Figures 1 and 2.
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Increase of Figure 1. A traditional logic scheme for developing the second law. ('The Kelvin-Planck statement is that all heat engines operate at less than 100% efficiencv:thev reiect waste heat. The Clausius statement is that heat flow frokcold to 'hot requires the input of work.)
Figure 2. Logic for developing the second law using the WISE equation.
In the next section we present the fundamental stability conditions as a solid empirical basis from which to teach the second law. Then we derive the WISE equation, and show its usefulness. We discuss phase changes aRer that, as an example of other sources of irreversibility.
condition for thermal stability. It is a mathematical statement of our normal experience that heat spontaneously flows from hot to cold. Another fundamental stability condition involves mechanical work resulting from pressure gradients across the system boundary. Our everyday experience tells us that things get smaller when squeezed. Thus, in a closed system under isothermal conditions, if P,,, < P, then the system expands until mechanical equilibrium is reached and Pa,, = P. Under these conditions dP < 0 and dVz 0, and the isothermal compressibility KT defined by
~herrnaland Mechanical Stability Conditions
Entropy and the second law of thermodynamics can be easily understood using the fundamental stability conditions. These are intuitive and empirical, but classically are treated as consequences of the second law (11, 12). They are at the root of a recent focus on the "driving forces" of irreversibility (13-16).One of these stability conditions involves temperature gradients across a system boundary. We will consistently designate properties of the surroundings and universe with appropriate subscripts and leave system variables unsubscripted. For a closed system, if T, > T and no work is done, then heat q flows from the surroundings to the system. Experience teaches that heating a system makes it hotter, with the T increase continuing until thermal equilibrium is reached and T = T,,. With the standard sign convention, when E q > 0, it follows that dT > 0. Then the definition of constant volume heat capacity, C, 6q .C,dT (3) implies that C">O
(4)
Analogous reasoning applies for the case when T., < T, so eq 4 is perfectly general. This is called the fundamental
is positive. The negative of KT is the relative change in volume dV experienced when the pressure of the system changes by dP at constant temperature., Similar reasoning for increases in pressure leads us to the general condition for mechanical stability. This is a mathematical expression of our normal experience that expansion occurs in the direction of lowest pressure. Classical treatments of thermodynamics derive the stability conditions from the second law. This commonly is not done until graduate-level courses in thermodynamics, where thev ~rovideconfirmation for more traditional second law statements. We suggest that the stability conditions are more intuitively obvious to the neophyte chemisVolume 71
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try student than either the Clausius or Kelvin-Planck statements of the second law. (See the caption for Fig. 1.) They therefore offer a more efficient and intuitive, empirical basis for the second law. There are other stability conditions of more limited scope that can be used in developing the principle of increase in entropy. We introduce one of these below when discussing phase changes. Derivation of the WlSE Equation
The logical flow and context of what follows is shown in Figure 2. Our initial system consists of a fxed quantity of gas contained by a frictionless piston and a thermally conductive cylinder. This derivation applies equally to any closed system capable of controlled volume changes and of heat transfer with the surroundings. The system is allowed to irreversibly expand (or contract) as well as to absorb heat from (or expel heat to) the surroundings. We assume in this section that onlv PV work is done. For internal energy U, the fundamental equation ofthermodynamics, dU=T&-PdV (7) follows fmm the first law, dU = Eq - P., dV, and fmm the fact that S is a state function W), as illustrated in Figure 2. Substituting dU from the first law into eq 7, and solving for dS,we get
Heat Transferand Lost Work
Equation 8 is a u s e l l separation of the entropy change of the system into a heat transfer term and a "lost work" term (18).It is clear that the heat transfer term can be positive or negative, depending on the direction of heat flow. However, as a consequence of the mechanical stability condition, the lost work term is always nonnegative. In expansion we know that P > P, and dV > 0, whereas in contraction clearly P < P, and dV < 0. As implied by the name, the numerator of the lost work term is the energy expended as waste heat that could have been work if the process had been performed reversibly, that is, if P = P,,. For this case, the lost work term is zero. The fact that the lost work term is always nonnegative implies the Clausius inequality.
We believe that eq 8 is a more insightful expression of the principle expressed by eq 9 because it describes S and its sources more intuitively. To determine the entropy change of the universe, defined by ds-=dS+ds, (10) we must consider the entropy change of the surroundings. From eq 2,
because heat gained by the surroundings is lost from the system, and vice versa. By definition the surroundings are large enough that T,,, will not be affected by energy transfers with the system. Thus, any heat transfer is reversible with respect to the surroundings. Solve eq 11 for Eq, and substitute this into eq 8. Then substitute eq 8 into eq 10, so that 564
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Grouping the coefficients of dS, and finding their common denominator results in the WISE equation for constant composition processes having only W work. The First WlSE Equation
This equation is obtained without reeourse to the physical content of the second law of thermodynamics, as illustrated in Figure 2. The WISE equation expresses a change in entropy of the universe a s a sum of thermal and mechanical irreversibility components. To see this, fwst consider the case in which a mechanical stop in the cylinder prevents volume changes so that the second term ofthe WISE equation is zero. According to the thermal stability condition, if T., > T, heat will be transferred from surroundings to the system, and from eq 11, CIS.,,< 0. Because (T - T,,) < 0, the first term of the WISE equation is positive. Likewise, if T, < T ,then both (T- T,,) and dS, are positive. We have thus shown that dS-, > 0 in these cases. If T, T, then any heat transfer is revenible, and dS-,. = 0. Let us now consider the more complicated situation in which the volume can change while heat is transferred. The argument in the previous paragraph for the first term of the WISE equation still applies. The discussion after eq 8 showing that (P-P,J dV/T is nonnegative also still applies. The WISE equation thus contains the principle of increase of entropy,
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Temperature and Pressure Gradients
However, the WISE equation is actually a stronger statement than eq 13 because it clarifies the driving forces for irreversibility If there is a temperature gradient between system and surroundings, T, + T, then an irreversible heat transfer contributes to increasing S-" due to the thermal stability condition. If there is a pressure gradient, P., # P, during a volume change, then irrevenible work due to the mechanical stacontributes to increasing Sum, bility condition. The physical content of the second law, in the form of the stability conditions, applied to an equation based only on the first law, immediately leads to the princiole of increase of entroov. 'Comparison of the WSE equation with eq 8 also helps to answer the auestion "What is entroov?" bv emohasizine the role of the s-undings. The heat &fei te& of the ~ S E equation is always nonnegative, whereas the analogous tenn in eq 8 can be positive, negative, or zero. This term contributes to an increase inS,, only ifit is driven by a temperature gradient a m the system boundruy, in which case it is irreversible. The entropy of a thermodynamic universe, that is, any isolated system, is thus a cumulative measure of theirreversibilities occurring within it. Other Sources of Irreversibility
There are many other potential sources of irreversibility, including phase changes, chemical reactions, concentration gradients, friction, and electrical resistance. The pattern of the previous section can be helpful for understanding these phenomena. In each case, an additional term in the WISE equation results, and an applicable stability condition helps us to see that the new contribution is
nonnegative. Unfortunately, in many of these cases, the stability conditions are problem-specific, rather than general, and are not common everyday experiences. That the WISE equation cannot be written in a completely general way, as in eq 13, is a fault that prevents it from replacing eq 13. The WISE equationnevertheless complementseq 13 by providing explanation and insight. Phase Changes As an illustration, we now discuss phase changes. The effect of changes in composition on dSmc,.can be evaluated using chemical potential, defined for substance and phase i as
I where ni is the number of moles of i. For a single component, p i s the molar internal energy for constant S and V. A general chemical stability condition is that the chemical potential of a system tends to a minimum. Classically, this is a consequence of the second law (19,20). The specific form of this condition depends on the nature of the materials in the initial and final states, and may vary with each specific process. For phase changes there is a simple, observable stability condition: The heats of sublimation, fusion, and vaporization are positive. Considering Snowflakes Consider a system in which snowflakes form or sublime in moist air in a thermally conductive container of constant volume with T = T,. The equilibrium is
For systems with composition as a variable, eq 7 must be modified to
where the sum is over all phases and substances in the system. Because the system is closed, the derivation proceeds using dU = 6q -P,, dV, as in the last section. For the case of dn, = 0 for allj the fwst two terms on the right of eq 16 lead to the two terms already found in the WISE equation. Because dn, = +Inp, eq 16 leads in general to an extended WISE equation.
I n this example, the first two terms of the extended WISE equation are zero because T - T,, = 0 and dV= 0.It is a weakness of eq 17 that the third term cannot easily be evaluated quantitatively by students. However, the direction of the process is determined by the value of T relative to the sublimation temperature Tsub.Therefore, qualitative evaluation of the third term of eq 17 requires an understanding of the temperature dependence of chemical potential. Ascbematic of the relative rates of change of p with T for the solid and gaseous phases are shown in F i r e 3. Using only the first law and the fact that entropy is a state func-
Temperature
Figure 3. Schematic of the temperature dependence of the chemical potential ilfor sublimation. tion, we can show that the rate of change of p, with T is the negative of the molar entropy S, (20).
Our chemical experience for constant-pressure processes is that heat must be added for sublimation to o m , that is, q = Ai&d = H, - H. > 0.Analogous obsewations apply to constant-volume processes, where q = AUd = Ug- Us> 0. Therefore from eq 2, S, > S., and from eq 18, the curve for the gas in Figure 3 drops more steeply than that for the solid. From experience, we know that a t T < Taubsnowflakes form spontaneously, so under these conditions dn. > 0.Figure 3 shows that p. < p, in this temperature region, so the third term of eq 17 is positive. When T > Ts,,b,snowflakes spontaneously sublime and dn. < 0. Under these conditions > k,and the third term of eq 17 is again positive. Only when T = Teubdoes the phase change occur reversibly, and the third term of eq 17 does not contribute to
G-". With eq 17, we have extended the WISE equation to account for phase changes. The general stability condition, minimization of chemical potential, takes a very specific form in this case. This specific stability condition,that sublimations (as well as fusion and evaporation) are always endothe&, is already a common experience of chemistry students. Analysis of eq 17 shows this to be a direct manifestation of the second law. For other possible sources of irreversibility,appropriate additional terms can be derived for the WISE equation, and stability conditions make clear that the additional contributions cause dSuni., to increase. Conclusions The thermal meaning of entropy has been misunderstood and underused in chemical education. The reasoning traditionally used to develop it is long, circuitous, and based on experiences largely unfamiliar to chemistry students. We have developed the WISE equation based only on first law considerations. Using mathematical statements of such common experiences as "objects get hotter when heated" and "things get smaller when squeezed", we show that the WISE eauation directlv leads to the ~rinciple of the universal increase of entropy Thc resultingsimplification in the lopical develo~mentof thc suhirct is clear h m comparison o c ~ i ~ u r1e and s 2. Volume 71 Number 7 July 1994
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In addition, the WISE equation provides physical insight into the meaning of entropy by highlighting the roles of irreversibility and property gradients between system and surroundings. The various causes of irreversibility, like the terms of the WISE eouation and its extensions. are seen to operate independently. The WISE equation can be extended to include sources of irreversibility other than temperature and pressure gradients. When phase changes are discussed in this way, the principle of icrease of entropy minimization of chemicd pot e n t i a l ) is seen to be manifested in yet another stability c o n d i t i o n , which i s already familiar to most students after h l g h school or beginning college courses. For chemistry students. the stat~ilitvconditmns are thus a much better empirical basis than traditional statements about heat eneines and refrigerators for teachine the second law of thermodynamics ina mathematically Xgorous way.
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Acknowledgment
The authors wish to thank J. C. for many insightful and insoiring conversations, and Jawueline Simms and Todd ~ e b o efor r thoughtful critiques oithe manuscript.
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Literature Cited 1. Lowe, J. P J. C h . E d u c 1953,65,40M06. 2. Atkins, P W Physical Chemistry, 4th ed.; Freeman: New York, 1990; Section 3.2. 3. bid., S~etion5.1. 4. Though see Huyakena, P. L.; Siegel, G. G. Bull. Sce Chirn. Belg 1988,97,809-814. 5. Ibid.. s n d references therein, discuss a thirdmeaning. based on information theory Chem. Edw. 1970,47,353356. 6. Nseh, L.'.tI 7. Raman, Y V.J Chem. E d u e 1970.47,331. 8. Noggle, J. H. Phyaiml Chemistry, 2nd ed.: Scott, Foresman: Glenview, IL, 1989; Chapter 3. 9. ~em&palli, G. K J Chom. E d u c 1886.63.846, 10. Faamntroversial view on thls subject,see Guy, A. G.ApplipdPhysies Communim~ t b n s 1990.9.305. Addisan-Wesley: Reading, MA, 1983: 11. Castellan, G. W Physicel Chemistry, 3rd d.; p 172. 12. K i k w m d , J. G.: Oppenheim, I. Chamiml Thomodynomics: Mdjraw-Hill: New York.. 1961:..o 64. 13. Bent,H.A.J ChemEduc. 1870.47,317441 14. Redlich.0. J. Chom. Educ. 1975.52.374,
17. Noggle, op. tit., p 130. 18. Van Wylen, G. J.;Sonntag, R. E. Fundomntols ed.; Wlley: New Ymk, 1986; p 205. 19. Kirkwod, op. cit.. p 58. 20. Atkinins. op. eit, Section 6.3.
of
Cbssrml Thomwdynomia, 3rd
