Article Cite This: J. Chem. Educ. 2019, 96, 1407−1411
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Enthalpy and the Second Law of Thermodynamics David Keifer* Department of Chemistry, Salisbury University, Salisbury, Maryland 21801, United States
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S Supporting Information *
ABSTRACT: The change in enthalpy of a chemical reaction conducted at constant pressure is equal to the heat of the reaction plus the nonexpansion work of the reaction, ΔH = qP + wadditional. After deriving that relationship, most general and physical chemistry textbooks set wadditional = 0 to arrive at the claim that ΔH = qP, and nearly all further discussion of enthalpy assumes that ΔH = qP. Setting wadditional = 0 is viable for spontaneous reactions, but for nonspontaneous reactions, wadditional ≠ 0 as a consequence of the second law of thermodynamics. Therefore, ΔH ≠ qP for nonspontaneous reactions. Moreover, nonexpansion work is important for many interesting and important spontaneous reactions in biology (e.g., muscular movement, nerve signal transmission) and in modern society (e.g., batteries); incorporating wadditional into ΔH allows for a more accurate discussion of the energy flow in these reactions. In this paper, I show that ΔH ≠ qP for nonspontaneous reactions, and I discuss how ΔH must be partitioned between qP and wadditional for several kinds of reactions according to the second law of thermodynamics. Finally, I suggest how the discussion of enthalpy could be corrected in general and physical chemistry textbooks. KEYWORDS: First-Year Undergraduate/General, Upper-Division Undergraduate, Physical Chemistry, Misconceptions/Discrepant Events, Textbooks/Reference Books, Thermodynamics, Calorimetry/Thermochemistry Therefore, ΔH ≠ qP for nonspontaneous reactions, making “heat of formation”, or “heat of reaction” more generally, a misnomer for any nonspontaneous reactions. Textbooks are technically correct in their claim that ΔH = qP whenever wadditional = 0, but they neglect to mention that this condition is impossible for nonspontaneous reactions. I suggest that this practice be changed in textbooks and in courses.
W
hen chemical reactions are conducted in a laboratory setting, such as in beakers and flasks, they are most often spontaneous reactions (i.e., reactions for which ΔG < 0) occurring under constant-pressure conditions. (Note that the familiar claim that the sign of ΔG indicates spontaneity is not quite correct.1,2 Even so, this claim will be used here to avoid confusion.) When these reactions occur, energy may flow into or out of the system via heat and/or expansion work (i.e., P V work), both of which change the internal energy of the system, U. The heat flow is usually a much larger quantity than the expansion work. Enthalpy, H, is defined to cancel the usually small amount of expansion work so that the change in enthalpy for these reactions is the heat under constant pressure, ΔH = qP. Therefore, the change in enthalpy conveniently quantifies the heat flow of reactions carried out in typical laboratory settings. However, enthalpy is not defined to cancel nonexpansion work, so when nonexpansion work is done, such as in electrical work, muscular activity, nerve signal transmission, or mechanical work like pulling a string, the change in enthalpy is given by ΔH = qP + wadditional, where wadditional is nonexpansion work. In common practice in general and physical chemistry textbooks, it is often stated that ΔH = qP for any reaction for which wadditional = 0. For example, ΔH is typically described as heat under constant pressure for all enthalpy of formation reactions. In fact, enthalpy of formation is often colloquially referred to as “heat of formation”.3−6 The problem is that wadditional ≠ 0 for nonspontaneous reactions as a consequence of the second law of thermodynamics, as I will show below. © 2019 American Chemical Society and Division of Chemical Education, Inc.
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TRADITIONAL TEXTBOOK APPROACH TO ENTHALPY
General and physical chemistry textbooks typically introduce enthalpy by its definition: H = U + PV, where H is enthalpy, U is internal energy, P is pressure, and V is volume. Under constant-pressure conditions, the change in enthalpy is given by eq 1. ΔH = ΔU + P ΔV
(1)
The change in internal energy is equal to the sum of work and heat. Work can be divided into expansion work, which is equal to −PΔV,7 and nonexpansion work such as electrical work or muscular activity. The following equation incorporates these terms into eq 1, where w is total work, wadditional is nonexpansion work, and qP is heat under constant pressure: Received: April 3, 2019 Revised: May 16, 2019 Published: June 4, 2019 1407
DOI: 10.1021/acs.jchemed.9b00326 J. Chem. Educ. 2019, 96, 1407−1411
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Figure 1. Line at the top of each figure represents the size of ΔH, while the boxes below represent the sizes of ΔG and TΔS. The bold vertical line shows that the values of ΔG and TΔS are fixed. The white part of each box represents the portion of the change in enthalpy that is composed of nonexpansion work, and the shaded part of each box represents the portion of the change in enthalpy that is composed of heat. The two panels represent (A) the water-splitting reaction shown in eq 4 and (B) the reverse of that reaction.
in enthalpy can be composed of heat and how much must be composed of nonexpansion work. For example, consider the water-splitting reaction in eq 4.
ΔH = w + qP + P ΔV = wadditional − P ΔV + qP + P ΔV = wadditional + qP ΔH = wadditional + qP
2H 2O(l) → 2H 2(g) + O2 (g)
(2)
At this point in most general and physical chemistry textbooks, wadditional is set to 0 so that ΔH = qP.8−17 Many general chemistry textbooks neglect to discuss wadditional entirely in this derivation to get straight to ΔH = qP, implying that expansion work is the only type of work available.3,4,18−22 Setting wadditional = 0 is consistent with the first law of thermodynamics for any chemical reaction because wadditional and qP are path functions, so their values can change as long as ΔH stays constant. However, I will show below that wadditional cannot be set to 0 for nonspontaneous reactions as a consequence of the second law of thermodynamics. This fact is widely overlooked in discussions of enthalpy by general chemistry textbooks,3,4,8−14,18−22 physical chemistry textbooks,15−17 and chemical engineering textbooks.5,6
For this reaction at room temperature under standard conditions, ΔH > 0, ΔG > 0, and ΔS > 0. Figure 1A visualizes the relationship between all of the terms in eq 3 for the forward reaction. The change in enthalpy is represented by the length of the line at the top. It is equal to the sum of ΔG and TΔS, represented by the rectangles beneath the ΔH line. The bold vertical line shows that the sizes of the ΔG and TΔS boxes are constants, regardless of the degree of irreversibility. As eq 3 shows, when the reaction occurs the change in enthalpy can be composed of a combination of heat and nonexpansion work. The white portion of each rectangle is the amount of nonexpansion work that is done on the system; the shaded portion is the amount of heat added to the system. Under reversible conditions, q = TΔS and wadditional = ΔG as described above, so the entire TΔS rectangle is shaded, and the entire ΔG rectangle is white. If the reaction is conducted under irreversible conditions so that q < TΔS, the heat portion must shrink, meaning that a larger portion of the change in enthalpy must be composed of nonexpansion work. In the extreme case of complete irreversibility (i.e., the reaction occurs so rapidly that heat has no time to flow into the system from the surroundings), all of the change in enthalpy must be composed of nonexpansion work. Thus, for this reaction (and any reaction for which ΔG > 0), nonexpansion work must be supplied for the reaction to proceed because wadditional ≥ ΔG. The reaction cannot proceed at room temperature if heat is the only energy source. (The reaction could theoretically proceed by heat alone if the temperature were increased so that T ΔS > ΔH, making ΔG < 0.) The necessity of nonexpansion work is acknowledged by scientists working on water splitting.24,25 In fact, general and physical chemistry textbooks often correctly describe ΔG as the minimum nonexpansion work that must be supplied to drive a nonspontaneous reaction, implying that heat alone is insufficient. Yet, these textbooks simultaneously neglect nonexpansion work to arrive at the claim that ΔH = qP.9−12,14−17,20−22 A quick example provides an intuitive justification for why nonexpansion work is necessary for nonspontaneous reactions to proceed: Can you recharge a battery by heating it up on a stove? Obviously not. Electricity (one form of nonexpansion work) must flow through the battery to recharge it. Figure 1B represents the reverse of the water-splitting reaction, for which ΔH < 0, ΔG < 0, and ΔS < 0. It is still true
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ΔH CANNOT EQUAL HEAT FOR NONSPONTANEOUS REACTIONS In reference to eq 2, the first law of thermodynamics would suggest that ΔH for any reaction may equal qP, or wadditional, or a combination, depending on how the reaction is carried out. However, the second law of thermodynamics places limits on what portion of ΔH can be composed of heat and what portion must be composed of nonexpansion work. To better understand the partitioning of this energy between nonexpansion work and heat, the common ΔG = ΔH − TΔS equation can first be rearranged into eq 3. Equation 2 is also incorporated. ΔH = ΔG + T ΔS = wadditional + qP
(4)
(3)
ΔH, ΔG, and ΔS are all state functions, so their values are constant regardless of how the reaction is carried out as long as the reaction occurs at constant temperature and pressure. However, wadditional and qP are path functions and can vary depending on the degree of irreversibility of the reaction. According to the second law of thermodynamics (in the form of the Clausius inequality), q ≤ TΔS. Under reversible conditions (i.e., theoretical conditions for which the reaction occurs at a rate of 023), q = TΔS, and under irreversible conditions, q < TΔS. The consequence of this according to eq 3 is that wadditional ≥ ΔG. Under reversible conditions, wadditional = ΔG. Under irreversible conditions, wadditional > ΔG. These relationships put a limit on how much of the required change 1408
DOI: 10.1021/acs.jchemed.9b00326 J. Chem. Educ. 2019, 96, 1407−1411
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discuss the relationship among ΔH, nonexpansion work, and heat for nonspontaneous reactions, as described above.
that q ≤ TΔS and wadditional ≥ ΔG, which in this case means that, for irreversible conditions, q must be more negative than TΔS, and wadditional must be less negative than ΔG. Thus, as Figure 1B shows for completely irreversible conditions, it is possible for enthalpy to change exclusively by releasing heat during the reverse reaction, even though it is impossible for the enthalpy to change exclusively by absorbing heat during the forward reaction. This is all a consequence of the second law of thermodynamics in the form q ≤ TΔS. In other words, the required change in enthalpy can occur via heat alone only for spontaneous reactions (ΔG < 0). In practice, if no apparatus is present to harness the energy released by these reactions and put it to work, then all of the ΔH is released as heat. This is the basis of using calorimetry to determine ΔH for spontaneous reactions. Similar arguments can be made for reactions for which not all of ΔH, ΔG, and TΔS have the same sign. For example, a particular reaction might have ΔH < 0, ΔG > 0, and TΔS < 0. Figure 2A represents possible values of those quantities. Recall
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MOTIVATION FOR CHANGING THE TEXTBOOK APPROACH TO ENTHALPY The most important reason to change how general and physical chemistry textbooks approach enthalpy is that the current approach (setting wadditional = 0 to arrive at the conclusion that ΔH = qP) is valid for spontaneous reactions, but it is inaccurate for all nonspontaneous reactions, as shown in the previous section. This is because wadditional ≠ 0 for nonspontaneous reactions, so ΔH ≠ qP by eq 2. One might suspect that wadditional is usually very small compared to qP so that setting wadditional = 0 is simply an approximation. This is a poor approximation for many important reactions. For example, over 80% of the total enthalpy required to split water at room temperature would have to be supplied as wadditional, as suggested by the size of the bars in Figure 1. (Several other numerical examples and their detailed calculations are given in the Supporting Information.) One might also argue that instructors and textbook authors already know that ΔH ≠ qP for nonspontaneous reactions but that they choose to focus on the simpler case of spontaneous reactions for the students’ sakes. There are two problems with that suggestion. One problem is that textbooks and instructors do not focus exclusively on spontaneous reactions. For example, many enthalpy of formation equations that students may need to use are for nonspontaneous reactions. The second problem is that I suspect many instructors do not have a complete grasp of the relationship between enthalpy and the second law of thermodynamics. These instructors cannot be blamed: I have yet to find a chemistry or chemical engineering book that addresses this connection. Even physical chemistry textbooks make claims like, “The enthalpy of reaction, ΔHR... is defined as the heat withdrawn from the surroundings as the reactants are transformed into products.”17 As this paper intends to make clear, that statement cannot be true for nonspontaneous reactions. A second reason to change how general and physical chemistry textbooks approach enthalpy is that wadditional is an important part of many spontaneous reactions, such as those involved in muscle activity and batteries. Setting wadditional = 0 so that ΔH = qP is valid for spontaneous reactions occurring on the benchtop in beakers, flasks, and calorimeters, where nonexpansion work is not done. Those reactions occur very frequently in chemistry laboratories, so it is understandable that ΔH and heat are so often conflated. However, those are only a subset of spontaneous reactions. I recommend that textbooks and instructors use ΔH = qP + wadditional as the default view of change in enthalpy and only set wadditional = 0 in the specific cases for which it is appropriate to do so. Finally, undergraduate students at both general and physical chemistry levels hold a wide variety of misconceptions about enthalpy as it is currently addressed in textbooks and courses.28−30 Updating the discussion of enthalpy in textbooks and courses in the way that this paper proposes might eliminate some of those misconceptions, but it might also lead to different misconceptions. Thus, the main purpose of this paper is to improve the accuracy of what we are teaching students, not necessarily to improve student understanding of what they are being taught.
Figure 2. Diagrams showing visually how ΔG and TΔS sum to give ΔH. Part A represents a reaction for which ΔH < 0, ΔG > 0, and ΔS < 0, in which case |qP| > |ΔH|. Part B represents a reaction for which ΔH > 0, ΔG > 0, and ΔS < 0, in which case qP and ΔH have opposite signs.
that q ≤ TΔS and wadditional ≥ ΔG, where the equalities hold under reversible conditions. Those relationships are represented by the arrows alongside the bars for ΔG and TΔS; the wadditional arrow must be at least the size of the ΔG bar for reactions with ΔG > 0, and the qP arrow must be at least the size of the TΔS bar for reactions with TΔS < 0. Figure 2A shows that when this reaction occurs, the heat released must be greater in magnitude than ΔH. Figure 2B represents a reaction for which ΔH > 0, ΔG > 0, and ΔS < 0. In this case, heat must actually be negative for the reaction to proceed, showing that ΔH and qP do not even necessarily have the same sign. (Therefore, a reaction cannot be classified as “exothermic” or “endothermic” solely on the basis of the sign of ΔH.) Similar figures can be made for any type of reaction. These figures show that ΔH does not typically equal qP. In fact, ΔH = qP only under very specific conditions, namely, when pressure and temperature are constant, ΔG < 0, and no nonexpansion work is involved. It should be noted that several articles have described the partitioning of ΔH into qP and wadditional much more thoroughly than most textbooks do.23,26,27 These articles have focused on harnessing wadditional from spontaneous reactions but do not 1409
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RECOMMENDATIONS FOR TEXTBOOKS AND THE CLASSROOM
above. Preferably, the relationship between enthalpy and the second law of thermodynamics, shown in the ΔH Cannot Equal Heat for Nonspontaneous Reactions section above, would be discussed in detail in chemical thermodynamics courses and textbooks. After that theoretical background has been established, several numerical examples could be given (or thermodynamic data could be given to students so they could do the calculations themselves) that illustrate how important wadditional can be for both spontaneous and nonspontaneous reactions. Several examples are given in the following paragraphs. Unique insights that can be drawn from each example are included. Detailed calculations and thermodynamic data are provided in the Supporting Information. ATP hydrolysis is a spontaneous reaction. This reaction is the most important source of the wadditional needed for muscle contraction.
Enthalpy in General Chemistry
Enthalpy is an important topic in both general and physical chemistry, so the partitioning of ΔH into qP and wadditional is relevant in general and physical chemistry courses. However, instructors and textbook authors might think that the full discussion of the relationship between enthalpy and the second law of thermodynamics shown above is too complicated for the general chemistry audience. Even so, the most important conclusions of the above discussion can still be addressed in undergraduate general chemistry. In general chemistry, enthalpy is introduced in the thermochemistry unit. This is often many chapters before Gibbs free energy and entropy, so spontaneity would not have been discussed yet. To avoid having to rearrange the order of topics in general chemistry, only a small change needs to be made to how enthalpy is discussed in thermochemistry. Rather than teaching students that ΔH is equal to the heat involved in a constant-pressure reaction, instructors and authors can say that ΔH is the sum of heat and nonexpansion work of the reaction (eq 2). In this way, ΔH can be compared to the change in internal energy equation ΔU = q + w, with which many students would already be familiar. So internal energy can change by heat and/or any kind of work, whereas enthalpy can change by heat at constant pressure and/or nonexpansion work. We can tell students that the utility of using ΔH rather than ΔU is that, in many cases, heat at constant pressure and nonexpansion work have more important consequences than expansion work. For example, when a battery powers a smartphone, the nonexpansion work is used to run the apps on the phone, and the heat at constant pressure that is generated is why the phone feels warm. Any expansion work is inconsequential. For the rest of thermochemistry, students could simply conceptualize ΔH as the net amount of heat at constant pressure and nonexpansion work (rather than just the amount of heat at constant pressure) released or absorbed by the reaction. In the thermodynamics chapter, when ΔG, ΔS, and spontaneity are introduced, instructors and authors could say that nonspontaneous reactions require nonexpansion work to proceed, so ΔH ≠ qP. As mentioned above, the example of trying to recharge a battery by heating it up on a stove could be used to try to convince students that nonspontaneous reactions require nonexpansion work. Then, instructors and authors could say that spontaneous reactions are able to proceed without any nonexpansion work, so it is possible for ΔH = qP. Any mention of ΔH as heat should be eliminated unless the instructor or author is explicitly referencing a spontaneous reaction conducted under constant pressure in which no nonexpansion work is done. In other words, the default view of ΔH should be the sum of heat and nonexpansion work, rather than just heat. In general chemistry textbooks and courses, there is probably no need to rigorously justify these conclusions using the detailed argument given above. That justification can be incorporated into textbooks or into advanced or honors general chemistry courses if desired.
ATP(aq) + H 2O(l) → ADP(aq) + P(aq) i
For this reaction, ΔH° = −24.10 kJ/mol, and ΔG° = −38.17 kJ/mol. Typically about 40% of ΔG is used as work to move the muscle,31 leading to wadditional = −15.27 kJ/mol. Therefore, about 63% of the loss of enthalpy is used to do useful work. Water splitting is a nonspontaneous reaction. Driving this reaction is an area of research relevant to green energy because it produces hydrogen fuel, which does not give off significant amounts of greenhouse gases. 2H 2O(l) → 2H 2(g) + O2 (g)
For this reaction, ΔH° = +571.6 kJ/mol, and ΔG° = +474.2 kJ/mol. Therefore, ΔG°/ΔH° × 100% = 82.96%, meaning that over 80% of the total enthalpy required to split water would have to be supplied as wadditional (because wadditional ≥ ΔG). Recharging an alkaline battery is nonspontaneous. For a zinc battery, the reaction is ZnO(s) + Mn2O3(s) → Zn(s) + 2MnO2 (s)
For this reaction, ΔH° = +272 kJ/mol, and ΔG° = +274.9 kJ/mol. Therefore, ΔG°/ΔH° × 100% = 101%. This example shows that in some cases the wadditional required for nonspontaneous reactions is actually larger than the ΔH needed. This is because ΔS° is negative for the reaction (see the Supporting Information), necessitating that heat leaves the reaction according to q ≤ TΔS, so the wadditional must make up for that loss. Finally, several enthalpy of formation reactions are nonspontaneous, such as that of dinitrogen tetroxide, which is often used as a component for rocket fuel. N2(g) + 2O2 (g) → N2O4 (l)
For this reaction, ΔH° = −19.5 kJ/mol, and ΔG° = +97.5 kJ/mol. This is the only numerical example provided here for which ΔH and ΔG have opposite signs. Recall that wadditional ≥ ΔG and ΔH = wadditional + qP. Therefore, for this reaction to occur, at least 97.5 kJ of wadditional would have to be supplied, but for the enthalpy to decrease as needed, at least 117.0 kJ of heat would have to be released to the surroundings. I chose those four examples for several reasons. First, they all show that wadditional can be a sizable portion of ΔH, so it really should not be ignored. Second, students hopefully find the reactions relevant and interesting, making thermodynamics less dry for them. Third, they represent a variety of combinations
Enthalpy in Physical Chemistry
At a minimum, physical chemistry textbooks and courses could be corrected in the way that I describe for general chemistry 1410
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wexpansion = −PΔV, leading to the cancellation of PΔV and −PΔV in the ΔH equation. (8) Zumdahl, S. S.; Zumdahl, S. A. Chemistry; Houghton Mifflin: Boston, MA, 2003. (9) Brady, J. E.; Senese, F. Chemistry: Matter and Its Changes; Wiley: Hoboken, NJ, 2004. (10) Zumdahl, S. S. Chemical Principles; Houghton Mifflin: Boston, MA, 2005. (11) Kotz, J. C.; Treichel, P. M.; Townsend, J. R. Chemistry & Chemical Reactivity; Brooks/Cole: Belmont, CA, 2012. (12) Ebbing, D. D.; Gammon, S. D. General Chemistry; Brooks/ Cole: Belmont, CA, 2013. (13) Flowers, P.; Theopold, K.; Langley, R.; Robinson, W. R. Chemistry; Rice University (OpenStax), 2015. https://cnx.org/ contents/
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of signs of ΔH, ΔG, and ΔS, so students could get practice explaining or thinking about the role of qP and wadditional in many kinds of reactions.
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SUMMARY The main suggestion in this paper is that the default view of ΔH as presented in general and physical chemistry textbooks and courses should be ΔH = wadditional + qP rather than ΔH = qP. Textbooks and instructors typically adopt the latter view after claiming that wadditional is often 0, but they rarely point out that wadditional ≠ 0 for nonspontaneous reactions or for a variety of important spontaneous reactions. A justification of the fact that wadditional ≠ 0 for nonspontaneous reactions has been presented, as well as a discussion of how that connects to enthalpy. The paper also suggests how textbooks and courses could be adapted to address this and includes several numerical examples that illustrate the main concepts.
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ASSOCIATED CONTENT
* Supporting Information S
The Supporting Information is available on the ACS Publications website at DOI: 10.1021/acs.jchemed.9b00326. Mathematical details for muscle activity, water splitting, alkaline batteries, and dinitrogen tetraoxide enthalpy of formation (PDF, DOCX)
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
David Keifer: 0000-0003-0770-0213 Notes
The author declares no competing financial interest.
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ACKNOWLEDGMENTS I thank my colleague Dr. Anthony Rojas for providing insightful comments on how to strengthen this paper. I also thank the members of Salisbury University’s chemistry department for letting me raid their offices and borrow a selection of textbooks.
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REFERENCES
(1) Quilez, J. First-Year University Chemistry Textbooks’ Misrepresentation of Gibbs Energy. J. Chem. Educ. 2012, 89, 87−93. (2) Raff, L. M. Spontaneity and Equilibrium: Why “ΔG < 0 Denotes a Spontaneous Process” and “ΔG = 0 Means the System Is at Equilibrium” Are Incorrect. J. Chem. Educ. 2014, 91, 386−395. (3) Tro, N. J. Chemistry: A Molecular Approach; Pearson: Boston, MA, 2017. (4) Hill, J. W.; Petrucci, R. H.; McCreary, T. W.; Perry, S. S. General Chemistry; Pearson: Upper Saddle River, NJ, 2005. (5) Felder, R. M. Elementary Principles of Chemical Processes; Wiley: New York, NY, 2000. (6) Smith, J. M.; Van Ness, H. C.; Abbott, M. M. Introduction to Chemical Engineering Thermodynamics; McGraw-Hill: New York, NY, 2001. (7) One detail that many general and physical chemistry books neglect in this derivation is that “constant pressure” means not only that the system and surrounding pressures are constant, but also that they are equal. This is because, for constant-pressure surroundings, wexpansion = −Psurr ΔV, where Psurr is the pressure of the surroundings. It is only if the system pressure, P, is equal to Psurr that we can say 1411
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