Visualizing the Entropy Change of a Thermal Reservoir

Visualizing the Entropy Change of a Thermal ReservoirElon Langbeheim,* Samuel A. Safran, and Edit Yerushalmi

Department of Science Teaching and Department of Materials and Interfaces, Weizmann Institute of Science, Rehovot, Israel 76100

*S Supporting Information

ABSTRACT: When a system exchanges energy with a constant-temperature environment, the entropy of the surroundingschanges. A lattice model of a fluid thermal reservoir can provide a visualization of the microscopic changes that occur in thesurroundings upon energy transfer from the system. This model can be used to clarify the consistency of phenomena such ascrystallization or similar phase transitions with the second law of thermodynamics; in those phenomena, students intuitivelygrasp that the system entropy decreases, but may not have a clear picture of how it is compensated by an increase in the reservoirentropy. The model may be used in the classroom to visually demonstrate how processes in which the entropy of the systemdecreases can occur spontaneously; specifically, it shows how the reservoir temperature affects the magnitude of the entropychange that occurs upon energy transfer from the system.

KEYWORDS: First-Year Undergraduate/General, Upper Division Undergraduate, Physical Chemistry, Thermodynamics,Statistical Mechanics, Liquids

Processes in which unstable systems spontaneously evolveto a lower-entropy, equilibrium state are often accom-

panied by the dissipation of heat to the surroundingenvironment (e.g., the crystallization of a solution of sodiumacetate trihydrate used in “heating pads”). The explanation ofsuch processes is based on the laws of thermodynamics thatrelate the changes in the system entropy and internal energy tothose of the reservoir. The second law of thermodynamicsmandates that a system in thermal contact with a reservoir(with both isolated from the rest of the universe) togethercomprise a composite unit whose total energy is constant andwhose total entropy must increase for spontaneous processes.The main property of the reservoir is that it is much larger thanthe system. Thus, any energy exchange between the system andthe reservoir is negligibly small compared to the reservoir’s totalinternal energy, and does not change the final temperature ofthe reservoir perceptibly. In isothermal processes in which thesystem evolves from a state such as a fluid in which the particlepositions are not localized to a crystal where the particlepositions are localized at lattice sites, the system entropy usuallydecreases. The second law of thermodynamics then mandatesan entropy increase in the reservoir that must more thancompensate for the decrease in the system entropy. Thereservoir entropy is an extensive property and is thusproportional to the size of the reservoir, whereas the reservoirtemperature is an intensive property of the reservoir and thusremains unchanged in the thermodynamic limit in which thesystem size compared to the reservoir size tends to zero.Studies of undergraduate students’ understanding of various

topics in thermodynamics have documented a commonconceptual difficulty: students tend to apply the second lawof thermodynamics to the system and ignore the surroundingreservoir.1,2 As a result, some students might (mistakenly)conclude that in the case of reduction of system entropy, thesecond law of thermodynamics does not hold. Even if thosestudents would acknowledge the presence of the reservoir theymight still overlook the change in its entropy because of

another misconception that posits that no heat transfer takesplace when the initial and final reservoir temperatures areequal.3

These difficulties also occur for students who take courses inwhich the system entropy is derived from a statisticalperspective. In such courses that include both introductorylevel4−11 as well as upper division courses12,13 the statisticalviewpoint provides students with intuitive, microscopicallybased insight into the meaning of entropy. However, the studydescribed by Sozbilir and Bennett4 reported that introducingentropy using the statistical perspective does not detachstudents from their limited view of entropy in terms of spatialdisorder. The association of entropy with disorder is consideredinadequate because it does not capture the meaning of entropyas the dispersal of energy14,15 and may lead to erroneousanalysis of problems.16 Some authors17,18 also opposeintroducing configurational (positional) entropy altogether inintroductory courses, as it does not convey the idea that everychange in entropy entails a spatial redistribution of energy.Nevertheless, these authors agree that when analyzing systemswith interparticle interactions, configurational entropy cannotbe avoided.19 Such systems are encountered by the students ineveryday phenomena as well as in materials of chemicalinterest. The inclusion of configurational entropy also plays animportant role in shaping conceptual understanding. Researchshowed that the grasp of configurational entropy is intuitive forstudents20 and should not be therefore overlooked but ratherharnessed, elaborated, and refined. That is, the presentation ofentropy can build upon intuitive, configurational visualizationsand then be refined by reframing the visual representations interms of probabilities and energy dispersion. Finally, themicroscopic changes and the thermodynamic definition ofentropy should be related by discussing the interplay betweenchanges in energy distributions in the system and thesurroundings.

Published: February 4, 2014

Article

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© 2014 American Chemical Society andDivision of Chemical Education, Inc. 380 dx.doi.org/10.1021/ed400180w | J. Chem. Educ. 2014, 91, 380−385

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Indeed, a recent study21 conducted in an introductory-levelcourse that introduced entropy from a statistical perspective22

indicates that students easily conclude that the system entropydecreases when a system undergoes a transition from a single,homogeneous phase to phase separation. However, most ofthem did not reconcile this observation with the overallincrease in entropy mandated by the second law ofthermodynamics. Rather, students perceived such processes asdriven by a (false) “energy minimization” law that “supersedes”the second law of thermodynamics. Similar conceptions ofspontaneity driven by enthalpy decrease were expressed byundergraduate chemistry majors taking a physical chemistryclass.23

Visualizations and spatial representations were shown tosupport the development of mental models; they also helpstudents to retain information.24 Thus, the relatively goodmental models students have of the system may be related tosaliency of snapshots of possible configurations of the systemthat provide a microscopic explanation of the change in itsentropy. We suggest that the excessive focus on the entropy ofthe system in student reasoning can be remedied by adding tothe statistical picture a detailed microscopic model of thereservoir. This suggestion is in line with Boltzmann’s originaltreatment that treated an isolated, composite entity comprisedof a system of interest plus a reservoir. This approach has beenused in a recent textbook25 that develops the statisticaltreatment of a composite system following Boltzmann, ratherthan the common method of considering a single, isolatedsystem. In the following we first discuss the limitations of themodels of reservoirs that are based only on kinetic energy forillustrating changes in the entropy of a reservoir, and suggest adifferent microscopic model based on a lattice representation ofa fluid. We then discuss how the lattice model can be used tovisualize the change in entropy of the reservoir that occurswhen the system undergoes a phase transition.

■ A THERMAL RESERVOIR BASED ON THE IDEALGAS MODEL

Fluids such as liquid nitrogen or water commonly serve asthermal reservoirs in temperature-controlled experiments. In afluid reservoir, the kinetic energy is associated with the randommotion of the particles and the potential energy is associatedwith the interactions between them. Therefore, when energy istransferred to or from the reservoir, it can change either thedistribution of kinetic energies among the particles and/or thespatial configurations of the particles by altering the potentialenergy associated with the interactions. Contrary to the latticemodel that focuses on particle configurations, in most modelsof a thermal reservoir such as the two-state, noninteracting spinreservoir, the Boltzmann reservoir (a reservoir with an energyspectrum in which the spacing between energy levels increasesexponentially), and the ideal gas reservoir,26 the spatialconfigurations of the particles do not contribute to the entropycalculation. Before introducing our model for a fluid reservoir,we discuss the ideal gas model for a reservoir in order tohighlight the pedagogical advantages and problems of using amodel that has only kinetic energies. Specifically, we examinethe extent to which this model can be useful for students tovisualize and quantify the entropy increase in the reservoir thatresults from energy transfer from a system with which it is incontact.The main strength of the ideal gas model lies in its simplicity

and the variety of thermodynamic properties that it can

illustrate. The ideal gas model can be used to quantify the factthat the increase in the reservoir entropy more thancompensates for the decrease in the system entropy in aspontaneous process. The ideal gas reservoir comprises Nrparticles, each of which is characterized by its kinetic energywith no interactions or potential energy. The total kineticenergy, Er (equivalent in this case to the internal energy), isgiven by

=E N k T32r r B (1)

Consider a system (which is much smaller than the reservoir,but which also comprises an ideal gas with Ns particles) with aninitial temperature that is higher than that of the reservoir sothat Tfinal − Tinitial = ΔT < 0. When the two are brought intothermal contact, the system transfers some of its energy to thereservoir in order to equilibrate the temperature differencebetween them. The ideal gas is kept in a vessel with a constantvolume, so that the change in energy due to work is zero. Thetemperature equilibration results in a decrease of the totalenergy in the system by an amount ΔEs = 3/2NskBΔT. Thetransfer of energy to the reservoir has a negligible effect on itstemperature because the number of particles (and hence theenergy) in the reservoir is much larger than that of the systemso that in the thermodynamic limit, Ns/Nr → 0. The change inthe entropy of an ideal gas system for fixed particle number andvolume is given by ΔSs = (3/2)NkB ln (Tfinal/Tinitial) (see ref 8,pp 145−147). In our case the change in entropy of the systemis ΔSs = (3/2)NskB ln[T/(T + |ΔT|)].How is this process, in which the system entropy decreases,

consistent with the second law of thermodynamics? Duringtemperature equilibration, the system transfers heat of theamount Qr = (−ΔEs) to the reservoir (note that Qr is the heatadded to the reservoir). As a result, the increase in entropy ofthe reservoir is given by ΔSr = Qr/T. Inserting the value of thechange in the system energy one gets ΔSr = (−ΔEs)/T = (−3/2)NskB lnΔT/T. The total entropy change, ΔSt = ΔSr + ΔSs is

Δ = − + |Δ | − Δ ≥S N k T T T T(3/2) [ ln(1 / ) / ] 0t s B (2)

So that when ΔT = 0, ΔSt = 0. In the case considered here,where the system entropy decreases, the initial systemtemperature is larger than its final temperature, so that (ΔT/T) < 0, the total entropy change in entropy is positive because[−(ΔT/T)] > ln(1 + |ΔT|/T) for any value of |ΔT|/T > 0.Thus, the ideal gas model applied to both the system andreservoir gives an explicit proof based on the kinetic energiesthat the entropy change of the reservoir more thancompensates for the entropy decrease of the system as itequilibrates from its initially higher temperature to that of thereservoir.However, using this model for visualizing the increase in

entropy of the reservoir is difficult, because all the argumentsfocus on the kinetic energy changes, which are not readilyrelated to the entropy in a visual manner, and not on theparticle configurations, which are much more visual. Thus, themicroscopic configurations of the particles that comprise theideal gas reservoir show no evident change when energy istransferred from the system to the reservoir, because theaverage kinetic energy does not change as the temperature isunchanged in the thermodynamic limit. Due to the limitationsof visual representation of the changes in the ideal gas reservoir,we believe it should be supplemented by a model that enables avisual representation of the particle configurational changes in

Journal of Chemical Education Article

dx.doi.org/10.1021/ed400180w | J. Chem. Educ. 2014, 91, 380−385381

the reservoir. We therefore next introduce a model of a fluidreservoir of interacting particles that allows us to visualize theparticle configurations responsible for the increase in thereservoir entropy. The model also reflects the commonexperimental situation in which fluids (as opposed to idealgases) are used as thermal baths.

■ ENTROPY INCREASE OF RESERVOIR UPON HEATTRANSFER: INTERACTING PARTICLE MODEL

Lattice models were used to predict properties of simple fluidssuch as liquid argon27 and more sophisticated ones such ascarbon tetrachloride.28 A key feature of these models is that theparticles do not completely fill the lattice sites, so that there aresome vacant cells in the lattice. The dense fluid reservoir ismostly filled with particles but has a small number of vacantsites. The lattice model allows us to visualize how a smallamount of energy, transferred from a system to the reservoir,results in an increase in the configurational entropy of thereservoir. The model enables a vivid display of the change inparticle configurations in the reservoir to quantify the entropychange. First, the properties are described qualitatively; then weprovide a quantitative analysis and finally, we make a fewsuggestions for using the model in the classroom.Qualitative Description of the Reservoir Model

In our model, the particles are restricted to the sites of a lattice.The lattice reservoir is kept at constant volume and is notcompletely filled with particles, as a result it contains a fixedvolume fraction of vacancies. In a fairly dense reservoir, thetotal volume fraction of vacancies is small (far from the liquid−gas critical point). Thus, the number of particles and vacanciesare both fixed and the total number of vacancies cannot changeupon energy transfer from the system. Therefore, one mustconsider a degree of freedom that can reflect the configurationalchanges that occur in the reservoir upon energy transfer, suchas the correlations among the vacancies.For simplicity, the model includes only the two most

important configurations: (i) “monomer” vacancies, all ofwhose nearest neighbor sites are occupied by particles and (ii)“dimer” vacancies in which two adjacent sites in the lattice areboth unoccupied, while all remaining nearest neighbor sites areoccupied. These are illustrated in Figure 1. Of course, there can

be trimers and higher-order clusters of vacancies, but if theoverall particle density is relatively high, these are expected tobe negligible (this is because in the lattice model at highdensities, the volume fraction of vacancies cv is small, so that theprobability of finding a trimer vacancy is of the order of cv

3 andcan be neglected). In this model, the important degree of

freedom is the relative number of monomer and dimervacancies, which can vary depending on the energy transferto the reservoir.The stability of a relatively dense, lattice-fluid phase

presupposes attractive interactions among the particles. Thesecan be isotropic van der Waals interactions in the case of fluidssuch as argon, xenon, and so forth, or more complexinteractions between dipoles in fluids such as in carbontetrachloride.28 The two types of vacancies, monomer anddimers, have different numbers of uncoupled “dangling” bondsand, thus, different contributions to the potential energy. On asquare lattice as shown in Figure 1, a dimer vacancy mandates 6“dangling” bonds of particles with no near neighbors, while twoseparated monomer vacancies mandate 8 such danglingbonds.29 Assuming that each bond contributes a negativeenergy of (−J) to the reservoir where J > 0, each open,“dangling” bond has a positive energy cost. This means that adimer vacancy is a state with lower potential energy (with apositive cost of 6J) and is thus energetically preferred over twoseparated monomer vacancies (with a positive cost of 8J). Theequilibrium number of dimers that will actually form dependson the competition between the dimer energy and thetemperature times the entropy.Energy transferred to a reservoir may change the kinetic

energy associated with the random motion of its particles andthe potential energy associated with the interactions betweenthem. However, a lattice model of the reservoir does notaccount for the contribution of kinetic energies to its internalenergy (which was the focus of the ideal gas reservoir in theprevious section); in the lattice model of a fluid, the focus is onthe change in potential energy associated with the interparticleinteractions. An increase in the reservoir’s energy will decreasethe number of dimer vacancies, thus increasing the number ofmonomer vacancies (as the total number of vacancies, which isthe sum of the number of monomer vacancies and dimervacancies, is fixed). It is easy to show that monomer vacancieshave larger entropy per lattice site than dimer vacancies as thereare more possible configurations allowed when one places Nmmonomers on Nr lattice sites compared with the situation inwhich one places Nd = Nm/2 dimers in the same volume (thisassumption is valid when Nd, Nm ≪ Nr). Thus, decreasing thenumber of dimer vacancies and increasing the number ofmonomer vacancies leads to an increase in the reservoirentropy.30 The magnitude of the entropy change of thereservoir upon energy transfer is determined by its temper-ature: at low temperatures the change in entropy of thereservoir is higher than at high temperatures, as illustrated inFigure 2. The illustration of the configurational changes in thereservoir via the changes in the relative numbers of monomerand dimer vacancies provides a visual representation of theentropy change and relates it to the temperature. At highertemperatures the initial dimer volume fraction is smallercompared with lower temperatures. Therefore, the transfer ofenergy from the system that converts a dimer vacancy to twomonomer vacancies (with higher energy) does not change thenumber of configurations to the larger extent that it occurs inthe low temperature case, when dimers are abundant. Thisallows students to understand the effect of temperature on themagnitude of the entropy change and eventually reinforcestheir understanding of the second law of thermodynamics thataddresses the contributions of entropy changes in both thesystem and the reservoir.

Figure 1. In the cubic lattice model (left) and the square model(right), each cell can contain at most one particle; single vacancies are“monomer” vacancies and two adjacent vacancies are “dimers”. Thedimers have two possible configurations on a square lattice and threeon a cubic lattice.



In the next section, the entropy and internal energy of thereservoir as a function of the number of monomer and dimervacancies are explicitly calculated. These quantities are thenused for deriving the temperature dependence of theequilibrium dimer and monomer volume fractions.

Quantitative Analysis of Reservoir Properties

The first step in the quantitative analysis of the model is tocalculate the entropy of the reservoir for a given number ofdimers and monomers. The reservoir model comprises Nrlattice sites most of which are filled with particles. Nm sitescontain monomer vacancies, and 2Nd contain dimer vacancies(the factor of 2 represents the two lattice sites occupied by adimer). Finding the exact number of spatial configurations is anunresolved mathematical challenge, because in principle onemust account for the restrictions that the already existingmonomers and dimers impose on the placement of thesubsequent dimers.31 However, because this discussion isrestricted to the relatively high density case in which thenumber of dimers and monomers is very small, the case ofoverlapping dimers is disregarded. The number of spatialconfigurations of the reservoir is estimated as follows: Threeentities are placed on a lattice: Nm monomer vacancies, Nddimer vacancies, and Nr − Nm − 2Nd fluid molecules so that thetotal number of entities placed on the lattice is Nm + Nd + (Nr− Nm − 2Nd) = Nr − Nd. Note that the number of lattice sitesNr is fixed and the number of fluid molecules, Nr − Nm − 2Nd,is also fixed, which guarantees that total of number vacanciesNm + 2Nd is fixed too, and only the numbers of monomers anddimers, Nm, Nd, are variable.Because all the elements of each entity type (monomers,

dimers, fluid molecules) are indistinguishable, the total numberof permutations (Nr − Nd)! can be divided by the number ofpermutations of each entity among itself. In addition, eachdimer has z/2 distinct orientations, where z is the coordinationnumber of the lattice. The orientational configurations of adimer in a square lattice, in which z = 4 is shown in Figure 1.Thus the number of permutations should be multiplied by thenumber of orientations of each dimer, which is (z/2)Nd. Theoverall number of configurations, Ω, is

Ω =− !

! ! − − !

⎛⎝⎜

⎞⎠⎟

N NN N N N N

z( )( 2 )

( /2)Nr d

m d r m d

d

(3)

The entropy per lattice site, sr, is calculated using theBoltzmann equation sr = kB/Nr ln Ω. Using Stirling’sapproximation ln N! = N ln N − N (see ref 8, p133) in thethermodynamic limit of large Nr, Nm, 2Nd (even though thevolume fraction of vacancies is small, their amount in thethermodynamic limit is large enough to justify the Stirlingapproximation), the entropy can be written in terms of therelative fraction of dimers and monomers respectively cd = (Nd/Nr) and cm = (Nm/Nr) as

= − − − − −

− − + − − +

s k c c c c c c

c c c c c z

[ ln ln (1 2 )

ln(1 2 ) (1 ) ln(1 ) ln /2]r B m m d d m d

m d d d d(4)

The monomer and dimer vacancy fractions are related by thefact that the total number of vacancies, cv is fixed so that: cm +2cd = cv where the factor of 2 accounts for the fact that a dimervacancy occupies two sites of the lattice. Using this relationbetween cm and cd in the previous equation for the entropyallows us to represent sr as a function of only the monomervacancy fraction (for a fixed overall number of vacancies).Analyzing or plotting this function (the entropy) as portrayedin Figure 2 makes apparent that the entropy of the reservoir isnot a monotonically increasing function of the monomervolume fraction; sr has a maximum at a certain volume fractionof monomer vacancies (which depends on the overall numberof vacancies). The meaning of this maximum point and theunphysical region illustrated in Figure 2 is discussed below andin the Supporting Information (see the Supporting Informa-tion, file 1, section III).At fixed temperature and volume, the change in internal

energy depends only on the potential energy associated withthe interparticle interactions. Each particle forms cohesivebonds with energy (−J) where J > 0, with its z nearestneighbors, so that if there were no vacancies, each particlewould have a potential energy of −zJ/2. (The factor of 1/2arises from the fact that a bond is shared between twoneighboring particles). Using the mean-field approximation(see ref 22) the internal energy of the reservoir can becalculated as the number of particles multiplied by their averageinteraction energy.Assuming a fairly dense packing of the reservoir particles, all

the nearest neighbors of a monomer vacancy are particles witha total of z shared “dangling bonds” so that a monomer vacancy

Figure 2. (Left) An illustration of the change of the configurations of dimer and monomer vacancies in the reservoir. (The pictures represent onerepresentative configuration out of many equivalent ones.) (Right) The entropy of the reservoir (Nr = 135) calculated as a function of the number ofmonomer vacancies. The configurations are shown at both low temperature (Nm = 2 initially) and high temperature (Nm = 6 initially) thatcorrespond to the two different slopes marked on the entropy/energy graph.



increases the potential energy by zJ. Two monomer vacanciesincrease the potential energy by 2zJ; however, for a dimervacancy, the interaction energy of the reservoir is increased by asmaller amount, (2z − 2)J, as there are only 2z − 2 “danglingbonds” surrounding a dimer vacancy. Therefore, the totalinternal energy of both monomer and dimer vacancies is zJNm+ (z − 1)J2Nd and the internal energy, Ur of the reservoir isgiven by

= − − − −U JN z zc z c[ /2 ( 1)2 ]r r m d (5)

Because J > 0, a reservoir with only dimer vacancies (cm = 0,2cd = cv) is energetically favorable compared with a system ofpurely monomer vacancies (cd = 0, cm = cv). If heat istransferred to the reservoir, it increases the reservoir internalenergy, which serves to reduce the number of dimer vacanciesand to therefore increase the number of monomer vacancies ascan be seen from eq 5.Because the internal energy of the system is proportional to

the monomer volume fraction (∂U/∂cm = 1/NrJ), there is anunphysical regime (indicated by a dashed line in Figure 2 inwhich the derivative of entropy with respect to energy isnegative (∂S/∂UV;N) < 0. The equilibrium state of the reservoiralways lies in the physical region where (∂S/∂UV;N) > 0. Theunphysical region originates from the simplifying assumptionsmentioned above that are appropriate only for low volumefractions of vacancies.32 In any case, the state of the reservoir isnot determined by the maximum of the reservoir entropy butrather, the minimum of the Helmholtz free energy. The freeenergy calculated for low values of cv always predicts values ofthe monomer and dimer vacancy volume fraction that lie in thephysical regime (see the Supporting Information, file 1, sectionIII).Assuming that the overall density and therefore the vacancy

volume fraction are constant, the monomer vacancy volumefraction can be written as a function of dimer vacancies cm = cv− 2cd and simplify eq 5 to get: Ur = −JNr [z/2 − zcv + 2cd]. In aconstant volume reservoir in which the total number ofmonomer and dimer vacancies is a constant, the Helmholtz freeenergy is a function of only one degree of freedom, which wechoose to be the fraction of dimer vacancies cd. This degree offreedom is determined within a mean-field approximation byminimizing the free energy. Minimization of the free energy ofthe reservoir is equivalent to maximizing the total entropy ofthe reservoir in contact with a thermal bath that maintains fixedtemperature. In our case in the thermodynamic limit, one partof the reservoir can act as the “bath” for another part. TheHelmholtz free energy Fr = Ur −TSr, which combines eqs 4 and5 is then minimized to yielding the following equation of state:

−≈

cc c

zJ k T

( 2 ) 2exp(2 / )d

v d2 B

(6)

From eq 6 it is clear that the ratio between the dimer andmonomer volume fractions decreases as temperature increases.Solving eq 6 yields the equilibrium volume fraction of dimervacancies (see the Supporting Information, file 1, section I).Examining this expression in the limit of very weak interactionsJ → 0 (when the system free energy is dominated by theentropy) predicts that the dimer vacancy fraction isapproximately equal to cv

2; this is what one expects from therandom placement of single vacancies where the probabilitythat two vacancies will be adjacent is, for a dilute system, justthe square of their volume fractions.

Other properties of reservoir such as its heat capacity arecalculated in the Supporting Information. It is also shown thatthe entropy change in the reservoir model is equivalent to thatpredicted by the thermodynamic relation: δSr = δE/T. Finally, acalculation of the properties of the sample system and itsreservoir is used to demonstrate that the entropy increase of thereservoir at low temperatures more than compensates for thedecrease in the system entropy (see the SupportingInformation, file 1, section IV).

■ USING THE MODEL IN THE CLASSROOM

The reservoir properties may be used in a physical chemistrycourse (preferably in modern courses that present entropy froma statistical perspective as mentioned in refs 4−13) to explainwhy processes in which the system equilibrates from an initiallyunstable high entropy state, to a lower entropy state areconsistent with the second law of thermodynamics. We suggestthat the instructor initiate a discussion by first presenting amotivating problem of a system such as such as the region of acandle near the wick that is initially in a fluid state and thencrystallizes. Next, one could elicit from the students theapparent contradiction between the decrease of the system’sentropy and the second law of thermodynamics that mandatesentropy increase in spontaneous processes. To resolve this, thereservoir model of Figure 2 allows students to visualize theeffect of energy transfer on the change in the configurationalentropy of the reservoir comparing the situation for both lowand high temperatures. Finally, the changes in the reservoir canbe linked back to the process in the system so that the studentsare prompted to re-examine the apparent contradiction basedon what they have learned about the reservoir. The entiresequence can be integrated into a worksheet which requiresstep-by-step analysis of the thermodynamic changes in thesystem and the reservoir (see the Supporting Information, file2).In a graduate level course, the equations describing the

equilibrium dimer volume fraction in the reservoir may bederived, and the reservoir entropy and energy may be used todemonstrate how the reservoir entropy is increased as heat(energy) is transferred from the system.To conclude, the model presented above shows how an

increase in the reservoir internal energy due to heat transferfrom a system increases the number of monomer vacanciesrelative to the number of dimer vacancies in the reservoir. Thisserves to increase the reservoir entropy so that the second lawis obeyed even though the system entropy has decreased in thisprocess. Doing this allows us to show students in a quantitativeand pictorial manner how the reservoir entropy reacts to thechanges that occur in the system and specifically how theincrease in the entropy of the reservoir can, at lowtemperatures, compensate for the entropy decrease in thesystem.

■ ASSOCIATED CONTENT

*S Supporting Information

The supporting information for this paper includes the fullderivation of the fluid reservoir model and a sample studentworksheet utilizing the model that may be used in anintroductory level classroom. This material is available via theInternet at http://pubs.acs.org.



http://pubs.acs.org

■ AUTHOR INFORMATIONCorresponding Author

*E-mail: [email protected]

The authors declare no competing financial interest.

■ ACKNOWLEDGMENTSWe thank Benoit Palmieri for his comments on the manuscript.S.A.S. is grateful to the Israel Science Foundation. Weappreciate the support of the Department of Science Teachingof the Weizmann Institute of Science, and of the DavidsonInstitute of Science Education.

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(19) In equilibrium descriptions of interacting, many particle systemsthe potential energy is crucial in determining the system mesoscalestructure and thermodynamics. In contrast to ideal gases, the energylevels of these systems cannot be calculated exactly and the physicaldegrees of freedom that enter the statistical calculations are indeed theparticle configurations.(20) Shultz, T. R.; Coddigton, M. Development of the concept ofenergy conservation and entropy. J. Exp. Psychol. 1981, 31, 131−153.(21) Langbeheim, E.; Livne, S.; Safran, S.; Yerushalmi, E. Evolutionin students’ understanding of thermal physics with increasingcomplexity. Phys. Rev. Spec. Top.Phys. Educ. Res. 2013, 9, 020117.(22) Langbeheim, E.; Livne, S.; Safran, S.; Yerushalmi, E.Introductory physics going soft. Am. J. Phys. 2012, 80, 51−60.(23) Seventy-five percent of the interviewees in the study presentedin ref 1 said that endothermic reactions cannot be spontaneous andmore than a third of them claimed that the system proceeds towardequilibrium because the internal energy of the system is lower inequilibrium.(24) Schwartz, D. ; Heiser, J. Spatial Representations and Imagery inLearning. In The Cambridge Handbook of the Learning Sciences; Sawyer,R. K., Ed.; Cambridge University Press: New York, 2006; pp 283−298.(25) Swendsen, R. H. An Introduction to statistical Mechanics andThermodynamics: Oxford University Press: New York (2012)(26) Prentis, J. J; Andrus, A. E; Stasevich, T. J. Crossover from theexact factor to theBoltzmann factor. Am. J. Phys. 1999, 67, 508−515.(27) Widom, B. Intermolecular forces and the nature of the liquidstate. Science 1967, 157, 375−382.(28) Shinmi, M.; Huckaby, D. A. A lattice gas model for carbontetrachloride. J. Chem. Phys. 1986, 84, 951−954.(29) This is because the reservoir is characterized for simplicity byshort-range (nearest neighbor), attractive interactions between theparticles, consistent with the stability of the condensed (fluid-like)phase. Thus, unbonded sites incur a positive energy cost.(30) We focus only on the internal energy and entropy of thereservoir whose temperature remains constant. This is because in thethermodynamic limit in which Ns/Nr → 0, the temperature of thereservoir remains unchanged by the heat transfer as discussed above.(31) Fowler, R. H.; Rushbrooke, G. S. The statistical theory of perfectsolutions. Trans. Faraday. Soc. 1937, 33, 1272−1275.(32) Within our approximation, this is attributed to the fact that theloss of rotational entropy of a dimer is greater than the increase inentropy of two monomers.



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Visualizing the Entropy Change of a Thermal Reservoir