In the Classroom
Mass-Elastic Band Thermodynamics: A Visual Teaching Aid at the Introductory Level William C. Galley Department of Chemistry, McGill University, Montreal, Quebec H3A 2K6, Canada;
[email protected] Helping students acquire an intuitive grasp of the basics of chemical thermodynamics that is sufficient for them to employ the concepts in later courses and in their overall scientific thinking represents a continuing challenge. In many large introductory physical chemistry courses a significant fraction of the students are majoring in the biological sciences. These students are familiar with systems at a descriptive molecular level, but they are not by and large quantitatively oriented. They see the difficulties with thermodynamics as being with the mathematics rather than the rather abstract relationships that exist within the subject. Students have difficulty with the concept of employing heat effects that are measured for very specific paths, that is, irreversible or reversible, to obtain values of changes in state functions such as ∆H and ∆S that are independent of the path. Establishment of a conceptual link between the mathematical description of a process and a molecular picture is often facilitated with the introduction of an intermediate step involving a graph or illustration. In my experience with students I have found that a simple model of thermodynamic behavior involving masses and elastic bands serves as an effective link. It allows students to make the transition from measured heat effects to changes in state functions and then to equilibrium and molecular behavior with relative ease. The model presented here is not employed or intended to replace the usual introduction of the laws of thermodynamics and the manner in which measurements are made, but serves as an adjunct to the standard treatment of the subject matter. The mass–elastic band processes can be used to illustrate a number of fundamental aspects of chemical thermodynamics such as the shift in equilibrium with temperature, the relationship between maximum work and ∆G, the nature of coupled processes, and so forth. In the present article the connection between measurements of q to obtain changes in thermodynamic parameters and the five spontaneous processes involving a mass and elastic band is considered. The contraction of a rubber band, or elastic band, has long been employed as an illustration of entropy-driven processes (1–5), and it continues to be emphasized in recent texts (6–8). A mass falling in a gravitational field has been used on occasion to represent a spontaneous process involving a simple energy change (9). The change in potential energy associated with a mass in a particular gravity field is dependent on the change in elevation, the acceleration due to gravity and the mass, with no entropic involvement in the system. A rising or falling mass has more often been employed to account for the mechanical changes that occur in the surroundings when work is done on a system (9). It is useful in these circumstances as a representation of non-PV work. The present approach differs from earlier treatments in that both the falling or rising mass1 as well as the contracting or stretching elastic band are considered to be part of the system itself and are employed to represent the ∆U and ∆S contributions to the processes that occur within these systems. www.JCE.DivCHED.org
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The Model Spontaneous Processes Model spontaneous processes that are associated entirely with a decrease in energy and an increase in entropy are represented by (a) a falling mass and (b) a contracting elastic band, respectively. The three remaining spontaneous processes that involve a combination of energy and entropy changes are depicted with (c) a falling mass stretching an elastic band, (d) a contracting elastic band lifting a mass, and (e) a falling mass assisted by a contracting elastic band. Following the introduction of the first law of thermodynamics and the relation between energy, or enthalpy changes and measured heat effects, the process of a freely-falling mass is utilized as an illustration of a process in which the decrease in potential energy under isothermal irreversible conditions appears in the form of a heat effect in the surroundings employing, for example, isothermal heat conduction calorimetry (10). In the consideration of measurements of ∆U or ∆H the contraction of an elastic band is employed to illustrate how, as with the isothermal expansion of an ideal gas, processes can occur spontaneously in the absence of a change in the energy function. Attention is drawn to the fact that there is essentially no volume change associated with the contracting elastic band and hence no distinction is made between ∆U and ∆H (4). When the concept of entropy and its measurement from qrev兾T have been dealt with the falling mass and the contracting elastic band are reintroduced as an example in the first case of a spontaneous process in which there is no entropy change in the system and in the second of one that is associated entirely with a positive ∆S. A brief discussion of the structure of rubber and how contraction is considered to result from the randomization of the linked polymer chains (3) is required at this stage. It is when the total entropy change of the system and its surroundings and the Gibbs energy have been introduced as criteria of spontaneity that the model processes become most useful. Under conditions of constant temperature and pressure the change in Gibbs energy given by
∆G = ∆H − T ∆S = qirr − qrev which emphasizes that the spontaneity of a particular process at constant T and P can be established from heat effects obtained when a process is carried out between the same initial and final states, but under two very specific and welldefined sets of conditions: (i) irreversibly when there is no work other than expansion done and (ii) reversibly, that is, when there is an opposing force infinitesimally less than the driving force. At this point it becomes useful to consider all five processes. Recalling that the data show that a falling mass represents an energy change in the system and a contracting elastic band is an entirely entropy-associated process in the system, the three remaining spontaneous processes are generated by
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In the Classroom
connecting these two together. An important aspect of these five processes is in the demonstration of how measurements of q under two limiting sets of conditions lead to an identification of the signs and relative magnitudes of the ∆H and T∆S involved, and the connection of these to equivalent molecular behavior. In class the individual process are demonstrated, but the values of the “measured heat effects” are presented to the students so in this sense the demonstration is a Gedanken experiment. The implements used to demonstrate the five processes can be carried in the lecturer’s pocket in the form of a mass and a heavy elastic band. The falling mass and contracting elastic band are demonstrated first and appropriate values of q attributed to each. To demonstrate the three remaining processes the rubber band is then tied to the mass and the spontaneous processes (c), (d), and (e) are demonstrated and values of q are assigned here as well. Alternatively, I have constructed a model containing five “towers” depicted in Figures 1 and 2 that illustrate the processes referred to above. In Figure 1 the five systems appear with their inner-workings hidden from view. The initial state depicted in Figure 1 is generated by raising the small fishing float on top of each system by hand, the float being attached with monofilament fishing line to the interior. The towers are composed of acrylic tubes 25 cm in length with an inside diameter of 4 cm and a one-hole rubber stopper at the top. The spontaneous process in each case occurs when the float is simply released and it snaps down onto the stopper in seemingly the same way with each system. The hand of the demonstrator depicted in the Figure is not part of the system but is within the surroundings. For simplicity it only appears holding back the spontaneous process in system (a), but is involved in each case. To an observer the processes that occur within the five systems appear indistinguishable, that is, the float descends. Each system is allowed to undergo a transition at constant temperature and pressure2 from the initial to the final state twice: (i) irreversibly, by simply releasing the float allowing the process to occur with no work being done and then, (ii) reversibly, again permitting the system to proceed to the final state but performing maximum work in the process. This is accomplished by allowing the float to descend only very slowly by hand or against an opposing force in the surroundings that remains infinitesimally less than the driving force. The value of q measured hypothetically is assigned for both processes. It is pointed out that qrev could be obtained after the irreversible process had been allowed to occur and then measuring q (hypothetically) when maximum work is done raising the system by hand back to the initial state. The sign of the measured qrev would then be reversed to obtain T∆S for the process in the spontaneous direction. The q obtained for the irreversible process (qirr) provides a measure of ∆H [∆H = ∆U since ∆(PV ) = 0], which in this model is only associated with a falling or rising mass. When maximum work is being done at a constant temperature, the hypothetical measurement of q (qrev), yields T∆S, which in the mass–elastic band systems is associated only with the contracting or stretching elastic, the mass of the small float being considered negligible. The demonstration begins with the falling mass and the contracting elastic band processes, and these are followed by the three remaining spontaneous processes all of which are 1148
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Figure 1. The five spontaneous processes involving a mass or an elastic band hidden from view. The arrow shows the spontaneous motion observed with each of the systems when it is released by the hand in the surroundings.
Figure 2. The five spontaneous processes involving a mass or an elastic band uncovered. The large arrow indicates the direction of the process within each system. The small arrows at the bottom of system (d) appear on either side of a pulley, which allows the direction of the spontaneous contracting elastic band and rising mass to appear identical to the other four processes when they are hidden from view.
hidden from view. Appropriate hypothetical values of q for the two paths are assigned. A sample of the values of q that could be given to the students for each of the processes in Figure 1 appears in Table 1. The sign of qirr and qrev in each case appropriately reflects that of the ∆H and T∆S involved. The values of q are restricted for simplicity to 0, ±5, and ±10. The absolute values are arbitrary with the restriction that their relative magnitudes must combine to result in negative ∆G values associated with spontaneous processes. In processes (c) and (d) the value of ∆G is smaller in magnitude in that the ∆H and T∆S oppose each other while the ∆G value of ᎑15 appears in (e) to reflect the enhanced spontaneity of the process in which both ∆H and T∆S contribute to a negative ∆G. Students are told that, in the absence of real data, qirr and qrev values of ᎑5 and +5, respectively, resulting in ∆G of ᎑10 would equally reflect the nature of process (e). Furthermore students are also told that the actual measurements of the heat effects for the systems depicted in Figure 2 would be small and difficult to measure and that the thermal reservoir involved in obtaining the data is not defined. It is emphasized to the students that it is the collection of signs of qirr, qrev, and qirr − qrev associated with each of the five fundamental processes that is unique, and that the signs and relative magnitudes of qirr and qrev clearly identify any of the processes even when hidden from view.
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In the Classroom Table 1. Sample Values of q Given to Students for Each of the Five Isothermal Mass–Elastic Band Processes Along with Analogous Molecular Events qirr (∆H) ᎑10
Model Process (a) Falling mass
qrev (T∆S) 0
qirr − qrev (∆G)a ᎑10
Analogous Molecular Events (i) CH3CN → CH3NCb (ii) CH4(g) + 2O2(g) → CO2(g) + 2H2O(g)c
(b) Contracting elastic band
0
10
᎑10
(i) Isothermal expansion of an ideal gas (ii) Diffusion from high to low concentration
(c) Falling mass stretching an elastic band
᎑10
᎑5
᎑5
(i) 2N(g) → N2(g) at 25 oCd (ii) dDNAe → nDNA for T < Tm
(d) Contracting elastic band lifting a mass
5
10
᎑5
᎑10
5
᎑15
(i) Melting of ice for T > 0 oC and P = 1 bar (ii) nDNA → dDNA for T > Tm
(e) Falling mass plus a contracting elastic band
(i) glucose + 6O2(g) → 6CO2(g) + 6H2O(g) (ii) ATP + H2O → ADP + Pi (under standard conditions)
a
The increase in the entropy of the system plus surroundings for the irreversible process can be utilized here in lieu of ∆G. bReaction suggested by a reviewer. cBased on ∆H0and T∆S0 of ᎑800 and ᎑2 kJ mol᎑1, respectively, this combustion reaction should be assigned to process (c), however, due to the dominance of the magnitude of ∆H over the T∆S contribution, it is listed as with the isomerization reaction, as an example of process (a). dThe lecturer emphasizes that for most of the processes of interest the ∆S is obtained in an indirect manner rather than directly from qrev/T as imagined in the weight/elastic processes. enDNA and dDNA refer to the double-stranded native and denatured forms of DNA, respectively.
When the ∆H and T∆S values have been obtained from qirr and qrev, respectively, the cardboard cylinders that cover the systems are removed revealing, as shown in Figure 2, the inner workings of the systems and the nature of the processes involved. The signs of ∆H, T∆S, and ∆G appear again beneath each of the unique processes in Figure 2 to emphasize that the “measured heat effects obtained under the appropriate conditions” are used to evaluate the changes in the state functions that are associated with the spontaneity of the processes involved. Students can then readily make connections between the mass–elastic band processes and analogous chemical reactions and phase transitions of the type presented in Table 1 in a facile manner as long as the association of ∆H and ∆S with molecular events such as bond breaking and making and randomization has been emphasized along the way. The mass–elastic band processes serve as a readily visible intermediate step between measured heat effects and a description of spontaneous processes at the molecular level. It helps to avoid the confusion that is generated when students attempt to look for direct connections between qirr and qrev and, for example, molecular behavior. The processes depicted in Figure 2 can alternatively be discussed in terms of a global formalism involving the entropy changes in the system and the surroundings in lieu of ∆G. Students can be encouraged to interpret the results using both formalisms and do so with no seeming confusion. With the global formalism recently stressed as a preferred approach in teaching thermodynamics (11), values of qrev兾T and ᎑qirr兾T that represent ∆Ssystem and ∆Ssurroudings, respectively, are employed. It has been satisfying to note that from examination questions based on the associations appearing in Table 1 that students are readily able to grasp the significance of the connections between the measurement of heat effects obtained under appropriate conditions and events that are often considered at the molecular level such as chemical reactions and phase transitions. Students at the normal second-year level would typically average 8 out of 10 on this type of question. Students have been questioned on anonymous course evaluation forms at the end of term on the usefulness of the mass–elastic band demonstrations to their understanding of the fundamental aspects of www.JCE.DivCHED.org
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chemical thermodynamics. Over more than a 10-year period the majority of student comments indicate that they find the demonstration useful. An average numerical score of 4.5兾5 is found, invariably the highest mark of the other 24 standard questions that appear on the course evaluation sheet. Notes 1. In formal thermodynamic treatments the influence of external fields is introduced as an additive contribution at the level of the Gibbs or Helmholtz energy of the system. In the present approach changes in the position of the mass in the gravitational field of the earth are considered to represent changes in energy in the system. To satisfy this requirement one can consider that the earth is included within these systems. 2. The various forms of calorimetry used in measuring heat effects are discussed elsewhere in the course. It is pointed out to students that the experimental determination of the small heat effects anticipated with these five different processes would preclude the use, for example, of the type of isothermal calorimetry (10) utilized with reacting systems and hence the Gedanken nature of the “data” in the demonstration.
Literature Cited 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
Gilbert, G. L. J. Chem. Educ. 1977, 54, 754–755. Byrne, J. P. J. Chem. Educ. 1994, 71, 531–533. Nash, L. K. J. Chem. Educ. 1979, 56, 363–368. Farris, R. F. Polym Eng. Sci. 1977, 17, 737–744. Hirsch, W. J. Chem. Educ. 2002, 79, 200A–200B. Mortimer, R. G. Physical Chemistry; Benjamin/Cummings Publishing Co., Inc.: Menlo Park, CA, 1993; pp 1002–1006. Atkins, P. W. Physical Chemistry, 5th ed.; Freeman & Co.: New York, 1994; pp A37–A38. Chang, R. Physical Chemistry, for the Chemical and Biological Sciences; University Science Books: Sausalito, CA, 2000; pp 186–189. Barrow, G. W. Physical Chemistry, 4th ed.; McGraw-Hill Book Co.: New York, 1979; pp 121–127. Wadsö, L.; Smith, A. L.; Shirazi, H.; Mulligan, S. R.; Hofelich, T. J. Chem. Educ. 2001, 78, 1080–1086. Craig, N. C. J. Chem. Educ. 2005, 82, 827–828.
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