Article pubs.acs.org/jchemeduc
Visualizing the Entropy Change of a Thermal Reservoir Elon 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 surroundings changes. A lattice model of a fluid thermal reservoir can provide a visualization of the microscopic changes that occur in the surroundings upon energy transfer from the system. This model can be used to clarify the consistency of phenomena such as crystallization or similar phase transitions with the second law of thermodynamics; in those phenomena, students intuitively grasp that the system entropy decreases, but may not have a clear picture of how it is compensated by an increase in the reservoir entropy. The model may be used in the classroom to visually demonstrate how processes in which the entropy of the system decreases can occur spontaneously; specifically, it shows how the reservoir temperature affects the magnitude of the entropy change that occurs upon energy transfer from the system. KEYWORDS: First-Year Undergraduate/General, Upper Division Undergraduate, Physical Chemistry, Thermodynamics, Statistical Mechanics, Liquids
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another misconception that posits that no heat transfer takes place when the initial and final reservoir temperatures are equal.3 These difficulties also occur for students who take courses in which the system entropy is derived from a statistical perspective. In such courses that include both introductory level4−11 as well as upper division courses12,13 the statistical viewpoint provides students with intuitive, microscopically based insight into the meaning of entropy. However, the study described by Sozbilir and Bennett4 reported that introducing entropy using the statistical perspective does not detach students from their limited view of entropy in terms of spatial disorder. The association of entropy with disorder is considered inadequate because it does not capture the meaning of entropy as the dispersal of energy14,15 and may lead to erroneous analysis of problems.16 Some authors17,18 also oppose introducing configurational (positional) entropy altogether in introductory courses, as it does not convey the idea that every change in entropy entails a spatial redistribution of energy. Nevertheless, these authors agree that when analyzing systems with interparticle interactions, configurational entropy cannot be avoided.19 Such systems are encountered by the students in everyday phenomena as well as in materials of chemical interest. The inclusion of configurational entropy also plays an important role in shaping conceptual understanding. Research showed that the grasp of configurational entropy is intuitive for students20 and should not be therefore overlooked but rather harnessed, elaborated, and refined. That is, the presentation of entropy can build upon intuitive, configurational visualizations and then be refined by reframing the visual representations in terms of probabilities and energy dispersion. Finally, the microscopic changes and the thermodynamic definition of entropy should be related by discussing the interplay between changes in energy distributions in the system and the surroundings.
rocesses in which unstable systems spontaneously evolve to a lower-entropy, equilibrium state are often accompanied by the dissipation of heat to the surrounding environment (e.g., the crystallization of a solution of sodium acetate trihydrate used in “heating pads”). The explanation of such processes is based on the laws of thermodynamics that relate the changes in the system entropy and internal energy to those of the reservoir. The second law of thermodynamics mandates that a system in thermal contact with a reservoir (with both isolated from the rest of the universe) together comprise a composite unit whose total energy is constant and whose total entropy must increase for spontaneous processes. The main property of the reservoir is that it is much larger than the system. Thus, any energy exchange between the system and the reservoir is negligibly small compared to the reservoir’s total internal energy, and does not change the final temperature of the reservoir perceptibly. In isothermal processes in which the system evolves from a state such as a fluid in which the particle positions are not localized to a crystal where the particle positions are localized at lattice sites, the system entropy usually decreases. The second law of thermodynamics then mandates an entropy increase in the reservoir that must more than compensate for the decrease in the system entropy. The reservoir entropy is an extensive property and is thus proportional to the size of the reservoir, whereas the reservoir temperature is an intensive property of the reservoir and thus remains unchanged in the thermodynamic limit in which the system size compared to the reservoir size tends to zero. Studies of undergraduate students’ understanding of various topics in thermodynamics have documented a common conceptual difficulty: students tend to apply the second law of thermodynamics to the system and ignore the surrounding reservoir.1,2 As a result, some students might (mistakenly) conclude that in the case of reduction of system entropy, the second law of thermodynamics does not hold. Even if those students would acknowledge the presence of the reservoir they might still overlook the change in its entropy because of © 2014 American Chemical Society and Division of Chemical Education, Inc.
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Indeed, a recent study21 conducted in an introductory-level course that introduced entropy from a statistical perspective22 indicates that students easily conclude that the system entropy decreases when a system undergoes a transition from a single, homogeneous phase to phase separation. However, most of them did not reconcile this observation with the overall increase in entropy mandated by the second law of thermodynamics. Rather, students perceived such processes as driven by a (false) “energy minimization” law that “supersedes” the second law of thermodynamics. Similar conceptions of spontaneity driven by enthalpy decrease were expressed by undergraduate chemistry majors taking a physical chemistry class.23 Visualizations and spatial representations were shown to support the development of mental models; they also help students to retain information.24 Thus, the relatively good mental models students have of the system may be related to saliency of snapshots of possible configurations of the system that provide a microscopic explanation of the change in its entropy. We suggest that the excessive focus on the entropy of the system in student reasoning can be remedied by adding to the statistical picture a detailed microscopic model of the reservoir. This suggestion is in line with Boltzmann’s original treatment that treated an isolated, composite entity comprised of a system of interest plus a reservoir. This approach has been used in a recent textbook25 that develops the statistical treatment of a composite system following Boltzmann, rather than the common method of considering a single, isolated system. In the following we first discuss the limitations of the models of reservoirs that are based only on kinetic energy for illustrating changes in the entropy of a reservoir, and suggest a different microscopic model based on a lattice representation of a fluid. We then discuss how the lattice model can be used to visualize the change in entropy of the reservoir that occurs when the system undergoes a phase transition.
illustrate. The ideal gas model can be used to quantify the fact that the increase in the reservoir entropy more than compensates for the decrease in the system entropy in a spontaneous process. The ideal gas reservoir comprises Nr particles, each of which is characterized by its kinetic energy with no interactions or potential energy. The total kinetic energy, Er (equivalent in this case to the internal energy), is given by Er =
3 NrkBT 2
(1)
Consider a system (which is much smaller than the reservoir, but which also comprises an ideal gas with Ns particles) with an initial temperature that is higher than that of the reservoir so that Tfinal − Tinitial = ΔT < 0. When the two are brought into thermal contact, the system transfers some of its energy to the reservoir in order to equilibrate the temperature difference between them. The ideal gas is kept in a vessel with a constant volume, so that the change in energy due to work is zero. The temperature equilibration results in a decrease of the total energy in the system by an amount ΔEs = 3/2NskBΔT. The transfer of energy to the reservoir has a negligible effect on its temperature because the number of particles (and hence the energy) in the reservoir is much larger than that of the system so that in the thermodynamic limit, Ns/Nr → 0. The change in the entropy of an ideal gas system for fixed particle number and volume 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 system is 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? During temperature equilibration, the system transfers heat of the amount Qr = (−ΔEs) to the reservoir (note that Qr is the heat added to the reservoir). As a result, the increase in entropy of the reservoir is given by ΔSr = Qr/T. Inserting the value of the change in the system energy one gets ΔSr = (−ΔEs)/T = (−3/ 2)NskB lnΔT/T. The total entropy change, ΔSt = ΔSr + ΔSs is
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A THERMAL RESERVOIR BASED ON THE IDEAL GAS MODEL Fluids such as liquid nitrogen or water commonly serve as thermal reservoirs in temperature-controlled experiments. In a fluid reservoir, the kinetic energy is associated with the random motion of the particles and the potential energy is associated with the interactions between them. Therefore, when energy is transferred to or from the reservoir, it can change either the distribution of kinetic energies among the particles and/or the spatial configurations of the particles by altering the potential energy associated with the interactions. Contrary to the lattice model that focuses on particle configurations, in most models of a thermal reservoir such as the two-state, noninteracting spin reservoir, the Boltzmann reservoir (a reservoir with an energy spectrum in which the spacing between energy levels increases exponentially), and the ideal gas reservoir,26 the spatial configurations of the particles do not contribute to the entropy calculation. Before introducing our model for a fluid reservoir, we discuss the ideal gas model for a reservoir in order to highlight the pedagogical advantages and problems of using a model that has only kinetic energies. Specifically, we examine the extent to which this model can be useful for students to visualize and quantify the entropy increase in the reservoir that results from energy transfer from a system with which it is in contact. The main strength of the ideal gas model lies in its simplicity and the variety of thermodynamic properties that it can
ΔSt = (3/2)NskB[−ln(1 + |ΔT | /T ) − ΔT /T ] ≥ 0
(2)
So that when ΔT = 0, ΔSt = 0. In the case considered here, where the system entropy decreases, the initial system temperature 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 and reservoir gives an explicit proof based on the kinetic energies that the entropy change of the reservoir more than compensates for the entropy decrease of the system as it equilibrates from its initially higher temperature to that of the reservoir. However, using this model for visualizing the increase in entropy of the reservoir is difficult, because all the arguments focus on the kinetic energy changes, which are not readily related to the entropy in a visual manner, and not on the particle configurations, which are much more visual. Thus, the microscopic configurations of the particles that comprise the ideal gas reservoir show no evident change when energy is transferred from the system to the reservoir, because the average kinetic energy does not change as the temperature is unchanged in the thermodynamic limit. Due to the limitations of visual representation of the changes in the ideal gas reservoir, we believe it should be supplemented by a model that enables a visual representation of the particle configurational changes in 381
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the reservoir. We therefore next introduce a model of a fluid reservoir of interacting particles that allows us to visualize the particle configurations responsible for the increase in the reservoir entropy. The model also reflects the common experimental situation in which fluids (as opposed to ideal gases) are used as thermal baths.
freedom is the relative number of monomer and dimer vacancies, which can vary depending on the energy transfer to the reservoir. The stability of a relatively dense, lattice-fluid phase presupposes attractive interactions among the particles. These can be isotropic van der Waals interactions in the case of fluids such as argon, xenon, and so forth, or more complex interactions between dipoles in fluids such as in carbon tetrachloride.28 The two types of vacancies, monomer and dimers, have different numbers of uncoupled “dangling” bonds and, thus, different contributions to the potential energy. On a square lattice as shown in Figure 1, a dimer vacancy mandates 6 “dangling” bonds of particles with no near neighbors, while two separated monomer vacancies mandate 8 such dangling bonds.29 Assuming that each bond contributes a negative energy of (−J) to the reservoir where J > 0, each open, “dangling” bond has a positive energy cost. This means that a dimer vacancy is a state with lower potential energy (with a positive cost of 6J) and is thus energetically preferred over two separated monomer vacancies (with a positive cost of 8J). The equilibrium number of dimers that will actually form depends on the competition between the dimer energy and the temperature times the entropy. Energy transferred to a reservoir may change the kinetic energy associated with the random motion of its particles and the potential energy associated with the interactions between them. However, a lattice model of the reservoir does not account for the contribution of kinetic energies to its internal energy (which was the focus of the ideal gas reservoir in the previous section); in the lattice model of a fluid, the focus is on the change in potential energy associated with the interparticle interactions. An increase in the reservoir’s energy will decrease the number of dimer vacancies, thus increasing the number of monomer vacancies (as the total number of vacancies, which is the sum of the number of monomer vacancies and dimer vacancies, is fixed). It is easy to show that monomer vacancies have larger entropy per lattice site than dimer vacancies as there are more possible configurations allowed when one places Nm monomers on Nr lattice sites compared with the situation in which one places Nd = Nm/2 dimers in the same volume (this assumption is valid when Nd, Nm ≪ Nr). Thus, decreasing the number of dimer vacancies and increasing the number of monomer vacancies leads to an increase in the reservoir entropy.30 The magnitude of the entropy change of the reservoir upon energy transfer is determined by its temperature: at low temperatures the change in entropy of the reservoir is higher than at high temperatures, as illustrated in Figure 2. The illustration of the configurational changes in the reservoir via the changes in the relative numbers of monomer and dimer vacancies provides a visual representation of the entropy change and relates it to the temperature. At higher temperatures the initial dimer volume fraction is smaller compared with lower temperatures. Therefore, the transfer of energy from the system that converts a dimer vacancy to two monomer vacancies (with higher energy) does not change the number of configurations to the larger extent that it occurs in the low temperature case, when dimers are abundant. This allows students to understand the effect of temperature on the magnitude of the entropy change and eventually reinforces their understanding of the second law of thermodynamics that addresses the contributions of entropy changes in both the system and the reservoir.
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ENTROPY INCREASE OF RESERVOIR UPON HEAT TRANSFER: INTERACTING PARTICLE MODEL Lattice models were used to predict properties of simple fluids such as liquid argon27 and more sophisticated ones such as carbon tetrachloride.28 A key feature of these models is that the particles do not completely fill the lattice sites, so that there are some vacant cells in the lattice. The dense fluid reservoir is mostly filled with particles but has a small number of vacant sites. The lattice model allows us to visualize how a small amount of energy, transferred from a system to the reservoir, results in an increase in the configurational entropy of the reservoir. The model enables a vivid display of the change in particle configurations in the reservoir to quantify the entropy change. First, the properties are described qualitatively; then we provide a quantitative analysis and finally, we make a few suggestions 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 not completely filled with particles, as a result it contains a fixed volume fraction of vacancies. In a fairly dense reservoir, the total volume fraction of vacancies is small (far from the liquid− gas critical point). Thus, the number of particles and vacancies are both fixed and the total number of vacancies cannot change upon energy transfer from the system. Therefore, one must consider a degree of freedom that can reflect the configurational changes that occur in the reservoir upon energy transfer, such as the correlations among the vacancies. For simplicity, the model includes only the two most important configurations: (i) “monomer” vacancies, all of whose nearest neighbor sites are occupied by particles and (ii) “dimer” vacancies in which two adjacent sites in the lattice are both unoccupied, while all remaining nearest neighbor sites are occupied. These are illustrated in Figure 1. Of course, there can
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”. The dimers have two possible configurations on a square lattice and three on a cubic lattice.
be trimers and higher-order clusters of vacancies, but if the overall particle density is relatively high, these are expected to be negligible (this is because in the lattice model at high densities, the volume fraction of vacancies cv is small, so that the probability of finding a trimer vacancy is of the order of cv3 and can be neglected). In this model, the important degree of 382
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Figure 2. (Left) An illustration of the change of the configurations of dimer and monomer vacancies in the reservoir. (The pictures represent one representative configuration out of many equivalent ones.) (Right) The entropy of the reservoir (Nr = 135) calculated as a function of the number of monomer vacancies. The configurations are shown at both low temperature (Nm = 2 initially) and high temperature (Nm = 6 initially) that correspond to the two different slopes marked on the entropy/energy graph.
The entropy per lattice site, sr, is calculated using the Boltzmann equation sr = kB/Nr ln Ω. Using Stirling’s approximation ln N! = N ln N − N (see ref 8, p133) in the thermodynamic limit of large Nr, Nm, 2Nd (even though the volume fraction of vacancies is small, their amount in the thermodynamic limit is large enough to justify the Stirling approximation), the entropy can be written in terms of the relative fraction of dimers and monomers respectively cd = (Nd/ Nr) and cm = (Nm/Nr) as
In the next section, the entropy and internal energy of the reservoir as a function of the number of monomer and dimer vacancies are explicitly calculated. These quantities are then used for deriving the temperature dependence of the equilibrium dimer and monomer volume fractions. Quantitative Analysis of Reservoir Properties
The first step in the quantitative analysis of the model is to calculate the entropy of the reservoir for a given number of dimers and monomers. The reservoir model comprises Nr lattice sites most of which are filled with particles. Nm sites contain monomer vacancies, and 2Nd contain dimer vacancies (the factor of 2 represents the two lattice sites occupied by a dimer). Finding the exact number of spatial configurations is an unresolved mathematical challenge, because in principle one must account for the restrictions that the already existing monomers and dimers impose on the placement of the subsequent dimers.31 However, because this discussion is restricted to the relatively high density case in which the number of dimers and monomers is very small, the case of overlapping dimers is disregarded. The number of spatial configurations of the reservoir is estimated as follows: Three entities are placed on a lattice: Nm monomer vacancies, Nd dimer vacancies, and Nr − Nm − 2Nd fluid molecules so that the total number of entities placed on the lattice is Nm + Nd + (Nr − Nm − 2Nd) = Nr − Nd. Note that the number of lattice sites Nr is fixed and the number of fluid molecules, Nr − Nm − 2Nd, is also fixed, which guarantees that total of number vacancies Nm + 2Nd is fixed too, and only the numbers of monomers and dimers, Nm, Nd, are variable. Because all the elements of each entity type (monomers, dimers, fluid molecules) are indistinguishable, the total number of permutations (Nr − Nd)! can be divided by the number of permutations of each entity among itself. In addition, each dimer has z/2 distinct orientations, where z is the coordination number of the lattice. The orientational configurations of a dimer in a square lattice, in which z = 4 is shown in Figure 1. Thus the number of permutations should be multiplied by the number of orientations of each dimer, which is (z/2)Nd. The overall number of configurations, Ω, is ⎛ ⎞ (Nr − Nd)! Ω=⎜ ⎟(z /2)Nd ⎝ Nm! Nd ! (Nr − Nm − 2Nd)! ⎠
sr = kB[− cm ln cm − cd ln cd − (1 − cm − 2cd) ln(1 − cm − 2cd) + (1 − cd) ln(1 − cd) + cd ln z /2] (4)
The monomer and dimer vacancy fractions are related by the fact 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 dimer vacancy occupies two sites of the lattice. Using this relation between cm and cd in the previous equation for the entropy allows us to represent sr as a function of only the monomer vacancy fraction (for a fixed overall number of vacancies). Analyzing or plotting this function (the entropy) as portrayed in Figure 2 makes apparent that the entropy of the reservoir is not a monotonically increasing function of the monomer volume fraction; sr has a maximum at a certain volume fraction of monomer vacancies (which depends on the overall number of vacancies). The meaning of this maximum point and the unphysical region illustrated in Figure 2 is discussed below and in the Supporting Information (see the Supporting Information, file 1, section III). At fixed temperature and volume, the change in internal energy depends only on the potential energy associated with the interparticle interactions. Each particle forms cohesive bonds with energy (−J) where J > 0, with its z nearest neighbors, so that if there were no vacancies, each particle would have a potential energy of −zJ/2. (The factor of 1/2 arises from the fact that a bond is shared between two neighboring particles). Using the mean-field approximation (see ref 22) the internal energy of the reservoir can be calculated as the number of particles multiplied by their average interaction energy. Assuming a fairly dense packing of the reservoir particles, all the nearest neighbors of a monomer vacancy are particles with a total of z shared “dangling bonds” so that a monomer vacancy
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increases the potential energy by zJ. Two monomer vacancies increase the potential energy by 2zJ; however, for a dimer vacancy, the interaction energy of the reservoir is increased by a smaller amount, (2z − 2)J, as there are only 2z − 2 “dangling bonds” surrounding a dimer vacancy. Therefore, the total internal energy of both monomer and dimer vacancies is zJNm + (z − 1)J2Nd and the internal energy, Ur of the reservoir is given by Ur = −JNr[z /2 − zcm − (z − 1)2cd]
Other properties of reservoir such as its heat capacity are calculated in the Supporting Information. It is also shown that the entropy change in the reservoir model is equivalent to that predicted by the thermodynamic relation: δSr = δE/T. Finally, a calculation of the properties of the sample system and its reservoir is used to demonstrate that the entropy increase of the reservoir at low temperatures more than compensates for the decrease in the system entropy (see the Supporting Information, file 1, section IV).
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(5)
USING THE MODEL IN THE CLASSROOM The reservoir properties may be used in a physical chemistry course (preferably in modern courses that present entropy from a statistical perspective as mentioned in refs 4−13) to explain why processes in which the system equilibrates from an initially unstable high entropy state, to a lower entropy state are consistent with the second law of thermodynamics. We suggest that the instructor initiate a discussion by first presenting a motivating problem of a system such as such as the region of a candle near the wick that is initially in a fluid state and then crystallizes. Next, one could elicit from the students the apparent contradiction between the decrease of the system’s entropy and the second law of thermodynamics that mandates entropy increase in spontaneous processes. To resolve this, the reservoir model of Figure 2 allows students to visualize the effect of energy transfer on the change in the configurational entropy of the reservoir comparing the situation for both low and high temperatures. Finally, the changes in the reservoir can be linked back to the process in the system so that the students are prompted to re-examine the apparent contradiction based on what they have learned about the reservoir. The entire sequence can be integrated into a worksheet which requires step-by-step analysis of the thermodynamic changes in the system and the reservoir (see the Supporting Information, file 2). In a graduate level course, the equations describing the equilibrium dimer volume fraction in the reservoir may be derived, and the reservoir entropy and energy may be used to demonstrate 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 transfer from a system increases the number of monomer vacancies relative to the number of dimer vacancies in the reservoir. This serves to increase the reservoir entropy so that the second law is obeyed even though the system entropy has decreased in this process. Doing this allows us to show students in a quantitative and pictorial manner how the reservoir entropy reacts to the changes that occur in the system and specifically how the increase in the entropy of the reservoir can, at low temperatures, compensate for the entropy decrease in the system.
Because J > 0, a reservoir with only dimer vacancies (cm = 0, 2cd = cv) is energetically favorable compared with a system of purely monomer vacancies (cd = 0, cm = cv). If heat is transferred to the reservoir, it increases the reservoir internal energy, which serves to reduce the number of dimer vacancies and to therefore increase the number of monomer vacancies as can 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 an unphysical regime (indicated by a dashed line in Figure 2 in which the derivative of entropy with respect to energy is negative (∂S/∂UV;N) < 0. The equilibrium state of the reservoir always lies in the physical region where (∂S/∂UV;N) > 0. The unphysical region originates from the simplifying assumptions mentioned above that are appropriate only for low volume fractions of vacancies.32 In any case, the state of the reservoir is not determined by the maximum of the reservoir entropy but rather, the minimum of the Helmholtz free energy. The free energy calculated for low values of cv always predicts values of the monomer and dimer vacancy volume fraction that lie in the physical regime (see the Supporting Information, file 1, section III). Assuming that the overall density and therefore the vacancy volume fraction are constant, the monomer vacancy volume fraction 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 a constant volume reservoir in which the total number of monomer and dimer vacancies is a constant, the Helmholtz free energy is a function of only one degree of freedom, which we choose to be the fraction of dimer vacancies cd. This degree of freedom is determined within a mean-field approximation by minimizing the free energy. Minimization of the free energy of the reservoir is equivalent to maximizing the total entropy of the reservoir in contact with a thermal bath that maintains fixed temperature. In our case in the thermodynamic limit, one part of the reservoir can act as the “bath” for another part. The Helmholtz free energy Fr = Ur −TSr, which combines eqs 4 and 5 is then minimized to yielding the following equation of state: cd 2
(c v − 2cd)
≈
z exp(2J /kBT ) 2
(6)
From eq 6 it is clear that the ratio between the dimer and monomer volume fractions decreases as temperature increases. Solving eq 6 yields the equilibrium volume fraction of dimer vacancies (see the Supporting Information, file 1, section I). Examining this expression in the limit of very weak interactions J → 0 (when the system free energy is dominated by the entropy) predicts that the dimer vacancy fraction is approximately equal to c2v ; this is what one expects from the random placement of single vacancies where the probability that two vacancies will be adjacent is, for a dilute system, just the square of their volume fractions.
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ASSOCIATED CONTENT
S Supporting Information *
The supporting information for this paper includes the full derivation of the fluid reservoir model and a sample student worksheet utilizing the model that may be used in an introductory level classroom. This material is available via the Internet at http://pubs.acs.org. 384
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(19) In equilibrium descriptions of interacting, many particle systems the potential energy is crucial in determining the system mesoscale structure and thermodynamics. In contrast to ideal gases, the energy levels of these systems cannot be calculated exactly and the physical degrees of freedom that enter the statistical calculations are indeed the particle configurations. (20) Shultz, T. R.; Coddigton, M. Development of the concept of energy conservation and entropy. J. Exp. Psychol. 1981, 31, 131−153. (21) Langbeheim, E.; Livne, S.; Safran, S.; Yerushalmi, E. Evolution in students’ understanding of thermal physics with increasing complexity. 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 presented in ref 1 said that endothermic reactions cannot be spontaneous and more than a third of them claimed that the system proceeds toward equilibrium because the internal energy of the system is lower in equilibrium. (24) Schwartz, D. ; Heiser, J. Spatial Representations and Imagery in Learning. 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 and Thermodynamics: Oxford University Press: New York (2012) (26) Prentis, J. J; Andrus, A. E; Stasevich, T. J. Crossover from the exact factor to theBoltzmann factor. Am. J. Phys. 1999, 67, 508−515. (27) Widom, B. Intermolecular forces and the nature of the liquid state. Science 1967, 157, 375−382. (28) Shinmi, M.; Huckaby, D. A. A lattice gas model for carbon tetrachloride. J. Chem. Phys. 1986, 84, 951−954. (29) This is because the reservoir is characterized for simplicity by short-range (nearest neighbor), attractive interactions between the particles, 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 the reservoir whose temperature remains constant. This is because in the thermodynamic limit in which Ns/Nr → 0, the temperature of the reservoir remains unchanged by the heat transfer as discussed above. (31) Fowler, R. H.; Rushbrooke, G. S. The statistical theory of perfect solutions. Trans. Faraday. Soc. 1937, 33, 1272−1275. (32) Within our approximation, this is attributed to the fact that the loss of rotational entropy of a dimer is greater than the increase in entropy of two monomers.
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS We thank Benoit Palmieri for his comments on the manuscript. S.A.S. is grateful to the Israel Science Foundation. We appreciate the support of the Department of Science Teaching of the Weizmann Institute of Science, and of the Davidson Institute of Science Education.
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REFERENCES
(1) Thomas, P. L.; Schwenz, R. W. College physical chemistry students’ conceptions of equilibrium and fundamental thermodynamics. J. Res. Sci. Teach. 1998, 35, 1151−1160. (2) Carson, E. M.; Watson, J. R. Undergraduate students’ understandings of entropy and Gibbs free energy. Univ. Chem. Educ. 2002, 6, 4−12. (3) For example: in ref 1, 60% of the students stated that no heat flow occurs under isothermal conditions. A similar percentage of students in an introductory physics course stated that no heat flow occurred during a process of isothermal compression. See: Meltzer, D. E. Investigation of students’ reasoning regarding heat, work, and the first law of thermodynamics in an introductory calculus-based general physics course. Am. J. Phys. 2004, 72, 1432−1446. (4) Sozbilir, M.; Bennett, J. M. A Study of Turkish Chemistry Undergraduates’ Understandings of Entropy. J. Chem. Educ. 2007, 84, 1204−1208. (5) Baierlein, R. Entropy and the second law: A pedagogical alternative. Am. J. Phys. 1994, 62, 15−26. (6) Reif, F. Thermal physics in the introductory physics course: Why and how to teach it from a unified atomic perspective. Am. J. Phys. 1999, 67, 1051−1062. (7) Chabay, R. W.; Sherwood, B. Bringing atoms into first-year physics. Am. J. Phys. 1999, 67, 1045−1050. (8) Hanson, R. M.; Green, S. Introduction to Molecular Thermodynamics; University Science Books: Sausalito, CA, 2008. (9) Kozliak, E. I. Introduction of entropy via the boltzmann distribution in undergraduate physical chemistry: A molecular approach. J. Chem. Educ. 2004, 81, 1595−1598. (10) Novak, I. The microscopic statement of the second law of thermodynamics. J. Chem. Educ. 2003, 80, 1428−1431. (11) Bindel, T. H. Teaching entropy analysis in the first-year high school course and beyond. J. Chem. Educ. 2004, 81, 1585−1594. (12) Dill, K. A. ; Bromberg, S., Molecular Driving Forces: Statistical Thermodynamics in Chemistry and Biology; Taylor and Francis: New York, 2003. (13) Cartier, S. F. An integrated, statistical molecular approach to the physical chemistry curriculum. J. Chem. Educ. 2009, 12, 1397−1402. (14) Lambert, F. L. Shuffled cards, messy desks, and disorderly dorm roomsExamples of entropy increase? Nonsense! J. Chem. Educ. 1999, 76, 1385−1387. (15) Lambert, F. L. DisorderA cracked crutch for supporting entropy discussions. J. Chem. Educ. 2002, 79, 187−192. (16) See for example: Styer, D. F. Insight into entropy. Am. J. Phys. 2000, 68, 1090−1096. Leff, H. S. Removing the mystery of entropy and thermodynamicsPart V. Phys. Teach. 2012, 50, 274−276. Jeppsson, F.; Haglund, J.; Amin, T. G.; Stromdahl, H. Exploring the use of conceptual metaphors in solving problems on entropy. J. Learn. Sci. 2013, 22, 70−120. (17) Lambert, F. L. Configurational entropy revisited. J. Chem. Educ. 2007, 84, 1548−1550. (18) Kozliak, E. I. Overcoming misconceptions about configurational entropy in condensed phases. J. Chem. Educ. 2009, 86, 1063−1068. 385
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