In the Classroom
From Bunsen Burners to Fuel Cells Invoking Energy Transducers To Exemplify “Paths” and Unify the Energy-Related Concepts of Thermochemistry and Thermodynamics Paul W. Hladky Department of Chemistry, University of Wisconsin-Stevens Point, Stevens Point, WI 54481-3897;
[email protected] Students encounter several energy-related topics while studying general chemistry. Internal energy, work, heat, calorimetry, enthalpy, state functions, path dependence and independence, standard states, and formation reactions are introduced as part of thermochemistry. Entropy (reaction, surroundings, total), the Gibbs function, and the criteria for spontaneity of chemical reactions appear later, often much later, when students reach the chapter on thermodynamics. Electrochemistry, which may precede thermodynamics (1) but usually follows it (2–4), presents electrochemical cells, cell potentials, and practical applications such as batteries and fuel cells.1 Students revisit these energy-related topics, in much greater detail, in physical chemistry when the main results that were simply presented in general chemistry are derived from the laws of thermodynamics. Paths of a process, the path dependence or independence of thermodynamic quantities, the use of exact and inexact differentials, and path integration are important topics in physical chemistry (5). An article published a few years ago claims that students taking physical chemistry do not understand the concept of paths (6). While we agree with these authors’ criticisms and suggestions, which are aimed at physical chemistry, we also think that the general chemistry introduction to paths could be developed beyond the map or map-like analogies and Hess’s law applications that appear in the thermochemistry chapter of many textbooks (1–4). In particular, we suggest that Bunsen
ter mC
B mA
fuel cell
C
A eed
Figure 1. A nine-part isolated system. The three substances need not be at the same pressure; they only need to be maintained at their own specific constant pressure by the masses connected to their pistons. The substances are placed so that the center of mass of the chemical species does not change height during the reaction. The thermal energy reservoir (ter) guarantees that the product and the reactants are all at the same temperature and the electrical energy device (eed) can be connected to the fuel cell. Also, the ideal wires that connect the masses to the pistons are rigid enough to carry the load yet flexible enough to turn the corner and the isolated system is in a vacuum so that the movements of the pistons do not cause any compressions or expansions of an inert “spectator” gas.
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The System and Process Consider an oxidation–reduction reaction of the form
isolated system mB
burners and fuel cells, or more generally, reaction chambers attached to matter reservoirs, are well-suited for exemplifying a continuum of paths that a chemical reaction can take and unifying the energy-related concepts that students have already encountered in the thermochemistry and thermodynamics chapters. Our approach, like that of Williamson and Morikawa (6), uses a chemical system to illustrate thermodynamic ideas and quantities. What we have done differently is used the efficiency of producing electrical work to tie the thermodynamic quantities together by plotting them on one graph; a graph that has only straight lines—horizontal for path independent quantities and sloped for path dependent quantities—that are readily constructed from quantities that general chemistry students are already expected to calculate. Furthermore, our approach is not restricted to fuel cells, or even electrochemical cells, which means that any reaction in which chemical energy is used to perform nonpressure–volume work can be treated in the same manner. A final benefit to the use of a fuel cell, or any other energy transducer, is that the difference between the thermodynamics of chemical processes and the technological challenges of designing and building a working device can be differentiated from each other and better appreciated.
aA(g or l) + bB(g or l)
cC(g or l)
(1)
and let the reactants and product occupy separate containers (cylinders fitted with frictionless pistons) in the isolated composite system shown in Figure 1. The reactants are fluids so that it is easy to imagine them being fed into the fuel cell (energy transducer or reaction chamber). The single product species is also a fluid so that it is easy to imagine collecting it in a separate container. Furthermore, the masses connected to the pistons can be selected independently, which means that the pressure (i.e., state or condition) of each species can be chosen independently. We will choose the masses so that the pressure of each chemical species is 1 atm (or more correctly 1 bar); each pure species is now in its standard state. Also, assume that a thermal energy reservoir and an electrical energy device are both connected to the reaction chamber. Connecting an electrical energy device is useful because it sets the stage for introducing the efficiency of the reaction at producing electrical energy. In particular, three cases of reaction efficiency will be considered. The first case is the extreme situation in which the fuel cell is shorted out (efficiency = 0) so that all of the chemical energy released during the reaction is converted to thermal energy. In this case the fuel cell is essentially a Bunsen burner (Bb). The opposite extreme is the best (ideal or reversible) fuel cell that produces the
Journal of Chemical Education • Vol. 86 No. 5 May 2009 • www.JCE.DivCHED.org • © Division of Chemical Education
In the Classroom
maximum quantity of electrical energy possible (efficiency = 1). All practical fuel cells lie between these two extremes and have efficiencies between 0 and 1. This efficiency, once we specify how to calculate it, provides a quantitative scale that labels the different paths that the reaction can take. Before any reaction occurs, we may write
U tot, i° = U A, i° + U B, i° + U C, i° + UmA, i° + UmB , i° + UmC, i° + Uter, i° + Ueed, i° + U fc, i°
(2)
where each of the nine parts of the composite system—three matter reservoirs (A, B, C), three masses (mA, mB, mC) situated on frictionless pistons that maintain constant pressures, a thermal energy reservoir (ter), an electrical energy device (eed), and the fuel cell (fc)—has an initial internal energy. The superscript “0” on each term is a reminder that the chemical reaction is occurring under standard state conditions. This particular starting point follows the approach advocated by Barrow (7). After some amount of reaction occurs, say c moles of C is produced (i.e., one mole of reaction has occurred), we can write an analogous expression for the final state of the isolated composite system. Taking the difference between the final and initial states gives
The preceding analysis also reveals several aspects of energy: its localization, change, transformation, and exchange. Equation 2 and its final state analog highlight energy localization. A term of the form ΔU accompanies the change in the quantity of energy in each compartment. The appearance of heat and work terms signify that energy has been transformed. In other words, chemical energy has been converted into some combination of thermal, mechanical, and electrical energies. The heat and work terms also signify that energy has been exchanged; that it has moved from one location to another. Recognizing these different aspects of energy should help students make sense of the energy-related terms that appear in thermochemistry and thermodynamics. Equation 3 can also be used to introduce the enthalpy change of the chemical reaction. Rearranging the third equality gives 0 = Δ rxnU A° + pA° Δ rxnV A + Δ rxnU B° + pB° Δ rxnVB + Δ rxnU C° + pC° Δ rxnVC + Δ rxnU ter° + Δ rxnU eed°
= Δ rxn HA ° + Δ rxn H B ° + Δ rxn HC° + q ter° + w eed°
Δ rxnU tot = 0
= Δ rxn H ° − q rxn° − w rxn, elec°
= Δ rxnUA° + Δ rxn U B° + Δ rxn UC ° + mA° g Δ rxn hA + m ° g Δ rxn hB B
+ mC° g Δ rxn hC + Δ rxn U ter°
+ Δ rxnU eed° + Δ rxnU fc°
(3)
= Δ rxnUA° + Δ rxnU B° + Δ rxn UC ° + pA° Δ rxn VA + pB° Δ rxnVB + pC° Δ rxnVC + Δ rxnU ter° + Δ rxnU eed ° = Δ rxnUA° + Δ rxnU B° + Δ rxn UC ° + wpV ° + q ter° + w eed°
where this sequence of equalities results from the following considerations:
• the total energy of an isolated system is conserved,
• energy changes can occur in each part of the system,2
• the fuel cell has a constant internal energy and the mass–height terms can be written as pressure–volume terms,3
• internal energy changes of the reservoirs (masses, ter, eed) can be expressed in terms of their energy exchanges (heat and work), and
• energy exchanges can be rewritten from the point of view of the reaction.
The final equality in eq 3 can be rewritten as Δ rxnU ° = w rxn, pV ° + q rxn° + w rxn, elec°
which has utilized the definition of enthalpy (H = U + pV ) for each chemical substance, recognized that the pressure of each substance is held constant during the reaction, and combined the individual enthalpy changes to give the standard enthalpy change of the reaction. The last equality of eq 5 together with eq 4 leads to
Δ rxn H ° = q rxn° + w rxn, elec° = Δ rxnU ° − w rxn, pV °
(6)
Equation 6 can be applied to each of the efficiency cases mentioned earlier (Bunsen burner, practical, or best) and leads to the following set of equalities:
= Δ rxnU ° − w rxn, pV ° − q rxn° − w rxn, elec°
(5)
(4)
where the presence of the electrical work term makes this expression more general than most introductory chemistry textbooks present when introducing the first law of thermodynamics.
q rxn, Bb° = Δ rxn H ° = q rxn, prac° + w rxn, elec, prac° = q rxn, best° + w rxn, elec, best°
(7)
The first equality is the basis for the calorimetric measurement of reaction enthalpies and draws attention to the fact that the standard enthalpy change for one mole of reaction has a single value at the temperature chosen. For practical fuel cells and for a best possible fuel cell, the sum of the heat and work term must be equal to the enthalpy change of the reaction as given by the first equality. In other words, the enthalpy change is path independent while the heat and electrical work terms are individually path dependent. We should mention at this time that the pressure–volume work accompanying this reaction is also independent of the path of the reaction since the pressures are all constant and we are considering a constant amount of reaction. Equation 7 illustrates the limitation of the first law of thermodynamics; even though the enthalpy change of the reaction is known, the quantity of electrical work that can be obtained from the reaction under the best conditions cannot yet be predicted. To make further progress, the idea of entropy
© Division of Chemical Education • www.JCE.DivCHED.org • Vol. 86 No. 5 May 2009 • Journal of Chemical Education
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In the Classroom
must be introduced. For an isothermal process, the entropy change is given by ΔS =
q rev T
(8)
where the requirement that heat is exchanged reversibly is an exceedingly important part of the definition. Considering the three cases of efficiencies, it should be obvious that the “Bunsen burner” extreme is not a reversible mode for a fuel cell. In addition, any fuel cell being operated in a practical fashion is not reversible. Although we will not prove it, the best case fuel cell will be one that operates reversibly,4 which gives T Δ rxn S ° = q rxn, best°
(9)
Combining eq 9 with the familiar Gibbs function Δ rxnG ° = Δ rxn H ° − T Δ rxn S °
(10)
for the standard state chemical reaction allows eq 7 to be extended as
q rxn, Bb° + 0 = Δ rxn H ° = q rxn, prac° + wrxn, elec, prac° (11) = q rxn, best° + wrxn, elec, best° ° ° = T Δ rxn S + Δ rxnG
where the “0” on the left hand side is an explicit reminder that the “Bunsen burner” mode produces no electrical energy. Equa-
tion 11 captures several important thermodynamic relationships and indicates that the allowed values of the practical heat and work terms lie between their respective “Bunsen burner” and “best” values. Furthermore, the last equality highlights the fact that the change in the Gibbs function is equal to the maximum quantity of nonpressure–volume work, electrical in this case, that the reaction can provide. The total entropy change, the entropy change of the reaction, and the entropy change of the thermal energy reservoir are related to each other by
⎧ −q rxn, Bb° efficiency = 0 ⎪ T Δ rxn Ster° = q ter° = ⎨ −q rxn, prac° 0 < efficiency < 1 (13) ⎪ −q ⎩ rxn, best° efficiency = 1
Energy / [kJ/(mol rxn)]
TΔrxnSter°
∙ΔrxnG°
∙qrxn,best°
TΔrxnStot°
wrxn,elec,Bb°
qrxn,best° = qrxn,rev°
TΔrxnS°
wrxn,elec,prac° qrxn,prac°
ΔrxnG°
wrxn,elec,best° = wrxn,elec,rev°
ΔrxnU°
ΔrxnH°
qrxn,Bb° 0.4
0.6
0.8
1.0
Efficiency of Producing Non-pV (Electrical) Work Figure 2. Thermodynamic quantities for a spontaneous chemical reaction producing electrical work at various efficiencies. Horizontal lines correspond to path independent quantities and sloped lines correspond to path dependent quantities. In this particular example, the reaction is exothermic and the standard entropy change of the reaction is negative.5
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and it is path dependent. Furthermore, eqs 12 and 13 dictate that the total entropy change is also path dependent. The efficiency of the reaction, from the perspective of the quantity of electrical energy it can produce, can be defined as
wrxn, pV°
0
0.2
(12)
The entropy change of the reaction was introduced earlier in connection with eq 9 and although it can only be equated to the heat exchanged under the best or reversible conditions, this entropy change is a property of the reactants and products and it is independent of the path of the reaction. The entropy change of the thermal energy reservoir is often referred to as the entropy change of the surroundings. There are some advantages to choosing a thermal energy reservoir. First, the thermal energy reservoir can be chosen to be so large that it stays at a constant temperature for the amount of reaction considered. Second, it can only exchange thermal energy with the system, which means that for a given quantity of thermal energy, say the heat exchanged when one mole of reaction occurs, the entropy change of the reservoir is the same whether the exchange occurs reversibly or not. Therefore, the entropy change of the thermal energy reservoir is given by
∙qrxn,Bb°
0.0
T Δ rxn Stot° = T Δ rxn S ° + T Δ rxn Ster°
0 ≤ efficiency =
w rxn,elec°
wrxn, elec, best°
=
w rxn,elec° ≤ 1 (14) Δ rxnG °
Using this efficiency as the scale for the horizontal axis, all of the quantities appearing in eqs 11 and 12 can be placed on one graph as shown in Figure 2. In addition, the internal energy change and the pressure–volume work term are also included. The reader might wonder why eq 12 does not contain an entropy term for the electrical energy device. There are two answers to this question. In the special case of a reversible reaction, we imagine an ideal fuel cell connected to an ideal electrical device. The ideal electrical device can receive and store electrical energy without an entropy change or it can deliver previously stored electrical energy without an entropy change.6 In practical situations, the electrical device could be as simple as a resistor that converts the incoming electrical energy entirely to heat or the device could be more complex such as a rechargeable battery that stores some of the electrical energy as chemical energy. Both of these practical electrical devices will have entropy changes associated with their operation yet these entropy changes can and should be ignored because the total entropy change of the process of interest is determined entirely by the quantities and forms of energy that exit directly from the reaction chamber and is independent of any and all subsequent changes.
Journal of Chemical Education • Vol. 86 No. 5 May 2009 • www.JCE.DivCHED.org • © Division of Chemical Education
In the Classroom Table 1. Standard State Redox Reaction That Involves Aqueous Solutions Rxn
2KMnO4 + 16HBr →
2MnBr2
+
5Br2
+
2KBr
+
8H2O
H2O
State
aq
aq
aq
liq
aq
liq
solvent
Molality/(mol kg‒1)
1
1
1
–
1
–
–
Amount/mol
2
16
2
5
2
8
–
Mass water/kg
2
16
2
–
2
0.14
13.86
Note: The shaded solvent-balance column keeps track of all of the water.
The compartmentalization of the reaction and the role of the fuel cell can now be better appreciated. Eight of the nine parts of the system serve as reservoirs; each one devoted to a specific type or form of energy. The three masses store mechanical energy some of which is exchanged with the reaction as the reactants are pushed into the fuel cell and the product is collected. The matter reservoirs store chemical energy; some of the reactants’ chemical energy is converted to thermal energy that goes to the thermal reservoir, some is converted to electrical energy that goes to the electrical device, and the remainder is stored as chemical energy in the product. Matter and energy flow through the fuel cell; all of the mass that enters must exit and all of the energy that enters must exit. Even though the fuel cell is the site of the chemical reaction, there is no need for it to be at constant temperature or constant pressure during the process as long as it returns to its initial state after one mole of reaction has occurred. Consequently, although the inner workings of the fuel cell are of great practical interest to the company that builds the device, they are not part of the thermodynamic analysis. Finally, since all of the pressures were set to 1 atm or 1 bar (standard state conditions), if the temperature is set at 298 K, then almost all of the quantities appearing in Figure 2 can be obtained from the thermodynamic information (Δf H°, Δf G°, S°) that is contained in general chemistry textbooks. The internal energy change and the pressure–volume work require additional information (ideal gas law for gases, densities for liquids) to calculate the necessary volume changes.
own container. Rather than invoking highly selective, semipermeable membranes for each product species, it is much more reasonable to imagine the reaction of interest—pure reactants to pure products—occurring as a two-stage process; pure reactants to a product mixture in the first stage and the mixing of pure products to give the same product mixture in the second stage. The enthalpy change for the reaction is now written as
prod mix
2. Allowing solid species complicates the picture of matter flowing into and out of the transducer. Although this complication makes it more difficult to imagine a practical device, it does not really affect the analysis. In other words, the engineering challenges have become more formidable but the scientific issues are unchanged.
3. Allowing more than one product also complicates the engineering aspects of the system because it is difficult to imagine a practical device that can route each of the products into its
(15)
prod mix
An analogous expression for the entropy of reaction, TΔrxnS°, can be written in terms of two reversible heat exchanges and an analogous expression for the Gibbs energy change of the reaction can be written in terms of two reversible non-pV work terms. While the experimental work may not be done this way, this picture leads to a correct analysis and interpretation of the process.
4. Solution reactions can be included without much difficulty although it is easier to deal with them using molality, the physical chemistry concentration unit of choice, rather than molarity. Item 3 above has already dealt with product mixtures and that approach can be easily adapted to reactions involving solutions. However, standard states for solutions introduce difficulties that cannot be adequately addressed in a general chemistry course; a one molal or one molar concentration of an ionic solute is not an ideal solution. If the rules are bent a bit, then a general chemistry-level picture of solutions can be presented. Table 1 illustrates the essential features for a reaction involving four solutions and two pure liquids. A solvent-balance term is included so that the dissolved compounds, written as neutral formula units, are always at a concentration of 1 molal. Additional solvent containers, when present, really do not complicate the analysis even though they may seem peculiar.
5. All of the discussion and analysis up to this point has been based on a redox reaction. Since the inner workings of the energy transducer and the non-pV work reservoir are unimportant for the thermodynamic analysis, we can actually consider any chemical reaction as long as we imagine that a suitable transducer exists. General chemistry students can now construct a graph analogous to Figure 2 for any reaction as long as they replace the electrical work term with a generic, nonpressure–volume work term.
6. The assumption of standard-state conditions is convenient because it allows general chemistry students to construct a graph like Figure 2 for any reaction that is spontaneous under those conditions. For many situations involving nonstandard
The oxidation–reduction reaction used in the preceding analysis has two pure reactants and one pure product and all of them are fluids. We will now remove several restrictions and show that the picture of a reacting chemical system, as developed above, has broad applicability. 1. The restriction to two reactants is easily removed; just add another container for each additional reactant and add the appropriate pair of internal energy terms to eqs 1, 2, and the early equalities of eq 3. The last equality of eq 3 and eqs 4–14 will not be altered.
pure prods
= Δ rxn Hpure reacts to − Δ rxn Hpure prods to
Some Generalizations
Δ rxn H ° ≡ Δ rxn Hpure reacts to°
© Division of Chemical Education • www.JCE.DivCHED.org • Vol. 86 No. 5 May 2009 • Journal of Chemical Education
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In the Classroom state conditions such as gases at pressures other than 1 atm (or 1 bar) and concentrations other than 1 molar (or 1 molal), general chemistry students already know how to calculate the change in the Gibbs function using
Δ rxnG = Δ rxnG ° + RT ln Q
(16)
where Q is the reaction quotient and T is 298 K. If the enthalpy change for the reaction has been measured under the nonstandard conditions, then the entropy change can be calculated from
T Δ rxn S = Δ rxn H − Δ rxnG
(17)
and a graph like Figure 2 can be constructed. Superimposing the plots for the nonstandard state conditions on the purely standard state figure adds a visual component to the students’ calculations that may help them recognize that the reactants’ conditions determine the starting point of the reaction path, the products’ conditions determine the ending point of the reaction path, and the reaction chamber (Bunsen burner, fuel cell, ...) determines the path that connects the starting point to the ending point. In other words, choosing the conditions of the reactant(s) and product(s) sets the values of ΔrxnG = wrxn,non-pV,rev, ΔrxnH = qrxn,Bb, TΔrxnS = qrxn,rev, ΔrxnU, and wrxn,pV, which are then independent of the choice of reaction chamber or path.
Summary Picturing a chemical system as having each reactant and product in its own container, so that the reaction itself occurs in an energy transducer yields several benefits. Each compartment of the composite system has its own internal energy so that the energy quantities (U’s), energy changes (ΔU’s), energy transformations (chemical to thermal or electrical or ....), and energy exchanges (q and w’s) can all be tracked and distinguished from each other. This in turn helps make the first law expressions easier to understand and apply correctly. Introducing the efficiency of the reaction brings practical devices into the analysis and uses them to bridge the gap between “Bunsen burners” and ideal energy transducers. As a result, path dependent and path independent thermodynamic quantities can be plotted as a function of process efficiency, which should help general chemistry students better understand the significance of each quantity and the connections between them. Notes 1. References 1–4 mention fuel cells on pages 540, 647–648, 750–751, and 798–799, respectively. References 1, 3, and 4 also include diagrams. 2. The energy change of each mass is its change in gravitational potential energy, which is the mass, m, multiplied by the acceleration due to gravity, g, and the change in its height, Δh. 3. Divide the force by the area of the piston to get pressure, p° = m°g/A = 1 atm or 1 bar and then multiply the change in height by the area to get the volume change, ΔV = AΔh. The mass is chosen so that the pressure is 1 atm, which is usually used as the standard state in general chemistry texts. 4. The proof that a reversible process produces the maximum quantity of nonpressure–volume work and its relationship to the
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Gibbs function is an important topic in physical chemistry (see ref 5, pp 98–99). 5. All of the lines in Figure 2, apart from those for the internal energy change and the pressure–volume work, can be draw as soon as two appropriate pieces of information (say ΔrxnH° and TΔrxnS°) are known. Three examples of reactions, with ΔrxnH° and TΔrxnS° in kJ∙mol (8), that involve fluids (gases or liquids) and are spontaneous under standard conditions at 298 K are the formation of liquid H2O (‒286, ‒49), formation of gaseous HBr (‒36, +17), and the formation of gaseous BrCl (+14.6, +15.6). 6. One possible ideal electrical device is a parallel-plate capacitor in which the plates are superconductors and in which the area of the plates or the distance between the plates can be varied infinitely slowly. Note that the electrostatic energy stored in a parallel-plate capacitor (ppc) is given by Uppc = ε0 AV 2/2s where ε0 is the permittivity of free space, A is the area of each plate, V is the potential difference between the plates, which is determined by the conditions of the chemical species (half-cell potentials) and remains constant during the chemical reaction, and s is the distance between the plates (9). Reversibly increasing the area of the plates or decreasing the distance between the plates allows electrical energy (charge) into the capacitor without an entropy change occurring in the device. Keeping the plate areas and separation constant will stop the flow of charge and stop the reaction. Slowly decreasing the plate area or increasing the plate separation will push electrical energy (charge) back to the reaction chamber and force the reverse reaction to occur all without an entropy increase in the capacitor. Of course, all parts of the process must be occurring infinitely slowly.
Literature Cited 1. Spencer, J. N.; Bodner, G. M.; Rickard, L. H. Chemistry—Structure and Dynamics, 3rd ed.; Wiley and Sons: New York, 2006; Chapters 13 and 12, respectively. 2. Atkins, P. W.; Jones, L. L. Chemistry—Molecules, Matter, and Change, 3rd ed.; W. H. Freeman: New York, 1997; Chapters 16 and 17, respectively. 3. Brown, T. L.; LeMay, H. E., Jr.; Bursten, B. E. Chemistry—The Central Science, 7th ed.; Prentice Hall: Upper Saddle River, NJ, 1997; Chapters 19 and 20, respectively. 4. Ebbing, D. D.; Gammon, S. D; General Chemistry, 9th ed.; Houghton-Mifflin: Boston, 2009; Chapters 18 and 19, respectively. 5. Atkins, P. W.; de Paula, J. Atkins’ Physical Chemistry, 8th ed.; W. H. Freeman: New York, 2006; pp 57–58. 6. Williamson, B. E.; Morikawa, T. J. Chem. Educ. 2002, 79, 339–342. 7. Barrow, G. E. J. Chem. Educ. 1988, 65, 122–125. 8. Handbook of Chemistry and Physics, 80th ed.; Lide, D. R., Ed.; CRC Press: Boco Raton, FL, 1999; section 5, Standard Thermodynamic Properties of Chemical Substances. 9. Tipler, P. A. Physics for Scientists and Engineers, 3rd ed.; Worth Publishers: New York, 1991; Vol. 2, pp 708–709.
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