First-Year University Chemistry Textbooks' Misrepresentation of Gibbs

ARTICLE pubs.acs.org/jchemeduc

First-Year University Chemistry Textbooks’ Misrepresentation of Gibbs Energy Juan Quílez* Departamento de Física y Química, IES Benicalap, Nicasio Benlloch, 46015 Valencia, Spain ABSTRACT: This study analyzes the misrepresentation of Gibbs energy by college chemistry textbooks. The article reports the way first-year university chemistry textbooks handle the concepts of spontaneity and equilibrium. Problems with terminology are found; confusion arises in the meaning given to ΔG, ΔrG, ΔG°, and ΔrG°, which results in many textbooks not differentiating between ΔG and ΔrG. Also, there is confusion over when standard conditions apply and when they do not. A problem with the proper use of units is also found. Finally, it is suggested that most of these difficult concepts could be removed from the first-year university chemistry syllabus because (i) an accurate presentation of Gibbs energy would be far beyond an introductory chemistry level and (ii) current attempts to introduce those difficult concepts in first-year university chemistry courses are usually full of misleading formulations. KEYWORDS: First-Year Undergraduate/General, Curriculum, Physical Chemistry, Misconceptions/Discrepant Events, Textbooks/Reference Books, Equilibrium, Thermodynamics

ΔG°, ΔrG, and ΔrG°. The different meanings of these quantities will be discussed with the help of one figure. We will focus mainly on discussing the meaning of spontaneity; that is, this concept refers both to determining whether a reaction is product or reactant favored and to predicting the direction in which a reacting system shifts in response to a disturbance. This foundation addresses some current misrepresentations. Finally, keeping this analysis in mind, we will study how general chemistry textbooks deal with all these related concepts and report some of the possible sources of misleading thermodynamic treatments. Although we are going to deal with some advanced thermodynamic concepts, the main purpose of this study is not to provide a full background in these concepts. Nonetheless, our review of prior work in the aforementioned areas may be useful for those who need an extended and more detailed mathematical or conceptual approach.

R

esearch on learning difficulties associated with thermodynamics is well documented. These studies have characterized student conceptions of energy,1 phase changes,2,3 equilibrium,47 and the second law of thermodynamics.810 Comprehensive reviews covering students’ conceptual difficulties about several thermodynamic ideas such as chemical equilibrium,7 chemical energetics, chemical thermodynamics,1 and entropy11 establish that students have significant learning difficulties with thermodynamics. Research studies12,13 suggest that one of the sources of the students’ learning difficulties in physical chemistry lies in how textbooks and teachers deal with key chemistry concepts. For example, several authors4,1420 have made an inventory of university students’ misconceptions due to a poor understanding of the spontaneity concept. Some of those misunderstandings may have their origin in the way this concept is taught. Project 2061’s Benchmarks for Science Literacy21 and the National Standards for Science Education22 called for the inclusion in textbooks of terms meaningfully defined and of appropriate representation of key ideas.23 These documents recognized the importance of textbooks and their evaluation. Thus, evaluation of science textbooks has become an important area of research and the inconsistencies in presentation of the subjects among textbooks are a major concern for both teachers and learners. In a recent study,24 it was found that chemistry textbooks often do not explicitly distinguish between thermodynamic (K) and practical equilibrium constants (Kc and Kp). For example, in many cases, Kc (or Kp), instead of K, were used to calculate ΔrG° and ΔrG. Thus, it was asserted that students could not be introduced appropriately to Gibbs energy. Hence, this work is aimed at analyzing if Gibbs energy is misrepresented by college chemistry textbooks. This article deals with the concepts of spontaneity and equilibrium. First, a thermodynamic discussion will allow us to differentiate some important thermodynamic quantities: ΔG, Copyright r 2011 American Chemical Society and Division of Chemical Education, Inc.

’ METHODOLOGY The analysis of textbooks has involved a qualitative approach for achieving the aim described above. For this purpose, 30 firstyear university chemistry textbooks2554 have been analyzed. These texts are well-known first-year chemistry textbooks that have gone through several editions, thereby showing their acceptance by chemistry teachers. Although the textbooks were originally written in English and their authors are mainly from United States and Great Britain, most of them are found on the shelves of the libraries of many chemistry colleges in countries where English is not the first language. They are (or were) usually recommended to first-year university chemistry students, and most of them have been translated into several languages. Also, Published: October 25, 2011 87

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Table 1. Summary of Both Spontaneous and Equilibrium Conditions in Usual Chemical Reaction Systems Constant variables

T, V

T, p

Spontaneity condition

dA < 0

dG < 0

Equilibrium condition

dA = 0

dG = 0

Table 2. Values of ΔrG and Their Meaning [aA(g) + bB(g) a rR(g) + sS(g)] ΔrG

dξ

0

forward

>0

0: products f reactants). Furthermore, the equilibrium condition corresponds to (∂G/∂ξ)T,p = ΔrG = 0 (minimum value of G). This way, the minimum value of Gibbs energy (Geq) and vanishing ΔrG are approached from either direction. ΔG is a finite difference; two cases are illustrated: ΔG1 = Geq Greactants and ΔG2 = Geq Gproducts. But, ΔrG is not a finite difference: it is the rate of change of Gibbs energy with respect to the advancement of reaction.

’ SPONTANEITY, EQUILIBRIUM, AND THE MEANING OF ΔG, ΔRG, ΔG°, AND ΔRG° To ground our textbook study, first the different meanings of ΔG, ΔrG, ΔG°, and ΔrG° need to be explained. Doing so involves the presentation of two thermodynamic energies: Gibbs (G) and Helmholtz (A). The fundamentals of this introduction are usually developed with more detail in physical chemistry textbooks,5559 as well as in advanced thermodynamics textbooks.6063 Ultimately, those current approaches are rooted on the modern thermodynamic definition of affinity due to de Donder.64 That is, the fundamentals of Gibbs energy6567 will serve as a basis for discussion. Eventually, this previous analysis will help when we later review how first-year university chemistry textbooks deal with both equilibrium and spontaneous reactions. The change in the Gibbs energy (G) at constant pressure and temperature for an ideal gas chemical reaction (i.e., spontaneous process) is presented. This procedure may also be applied to the thermodynamic energy A (i.e., chemical reaction at constant T and V). The development and integration of these mathematical treatments lead to the determination of the general condition for spontaneity. Keeping in mind this general condition, we will be able to account for the different meanings of ΔG, ΔrG, ΔG°, and ΔrG°.68 All this previous discussion will serve as a proper reference when we analyze how first-year textbooks define and use the aforementioned Gibbs quantities. The change in the Gibbs energy (dG) is given by dG ¼ SdT þ V dP þ Δr Gdξ

constant T and V, ∂G ∂A ¼ Δr G ¼ ∂ξ T, p ∂ξ T, V

The meaning of this equation must be emphasized: ΔrG is a derivative and not an ordinary difference despite the use of “Δ”, as signaled by the subscript r feature. If there is a proper control of the variables involved, the conditions that the second law establishes for both spontaneous processes and chemical equilibrium (Table 1) can be obtained. Furthermore, as ðdGÞp, T ¼ ðdAÞT, V ¼ Δr Gdξ

ð4Þ

a general equation that embodies the two conditions for spontaneity outlined in Table 1 can be obtained: Δr Gdξ < 0

ð5Þ

That is, for a spontaneous reaction from left to right [aA(g) + bB(g) f rR(g) + sS(g)], dGp,T < 0 [or dAT,V < 0], and because dξ > 0, then ΔrG < 0. For the reaction to reverse spontaneously [aA(g) + bB(g) r rR(g) + sS(g)], dGp,T < 0 [or dAT,V < 0], and because dξ < 0, then ΔrG > 0. Thus, the sign of ΔrG predicts the direction of the spontaneous chemical reaction. Similarly, the general equilibrium condition can finally be written as follows

ð1Þ

Similarly, the change in the Helmholtz energy (dA) is as follows dA ¼ SdT PdV þ Δr Gdξ

ð3Þ

ð2Þ

Δr Gdξ ¼ 0

where ΔrG = ∑iνiμi is the so-called free energy of reaction, and represents the rate of change of G with respect to the advancement of reaction (ξ), at constant T and p, and also the rate of change of A with respect to the advancement of reaction, at

ð6Þ

That is, if ΔrG = 0, equilibrium has been attained. The general conditions of forward and backward reaction, as well as that of equilibrium, are summarized in Table 2. It must be stressed that 88

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measured. In Figure 1, and as has been stated above, ΔG is a finite difference. Also, one should realize that ΔG is an extensive quantity; it is expressed in energy units only, kJ. Conversely, in Figure 1, the slope of the curve at a particular value of ξ is ΔrG. That is, ΔrG is not a finite difference: it is an instantaneous rate of change of G with respect to the degree of advancement of reaction. It is an intensive quantity and is normally reported in units of kJ/mol, where ξ has units of mol. At equilibrium (ξeq), the rate of free energy change is zero, ΔrG = 0, but this is not the case of ΔG. It should be stressed in Figure 1 that, when the reaction has reached equilibrium, the change in the Gibbs energy has been

those conditions are not restricted to reactions at constant p and T, for they apply also to systems at constant V and T. Keeping in mind the main purpose, the discussion will exclusively be concentrated on Gibbs energy (G). The variation of G as a function of ξ in an ideal gas chemical process is shown in Figure 1. In this particular case, it is assumed that G°products G°reactants > 0. When starting with unmixed reactants in their standard states at the same temperature T and there is no mixing of reactants with products, the Gibbs energy changes linearly as the reaction progresses, and eventually the reactants are completely converted to unmixed products in their standard states at the same temperature T. But when reactants are mixed, their pressures drop below their standard state values and thus there is a drop in their Gibbs energy values (Greactants < G°reactants). As the reaction proceeds, the change in the Gibbs energy is not linear. The discussion is going to focus mainly on the meaning of the curve obtained in this last case. These valley graphs can help to clarify the different meanings of ΔG, ΔrG, ΔG°, and ΔrG°.5,63,69,70 Although modern physical chemistry textbooks usually draw and discuss the variation of G as a function of the extent of reaction, the corresponding figures are much less detailed, for their authors tend only to concentrate on the meaning of ΔrG. Hence, each of the different quantities represented in Figure 1 will be discussed. When a reaction takes place at constant p and T, the Gibbs energy decreases, and the reaction continues until G has reached a minimum value (Geq). Moreover, keeping in mind that ∂G Δr G ¼ ¼ νi μ i ð7Þ ∂ξ T, p i

ΔG ¼ Geq Greactants 6¼ 0

Finally, let us recall the meaning of ΔrG°. It is the rate of change of standard free energy, viz. ∂G° ¼ νi μ°i ð11Þ Δr G° ¼ ∂ξ T

∑

Therefore, it is the slope of the standard line and is constant, Δr G° ¼

and recalling that the chemical potentials (μi) depend on the composition, we can conclude that as the chemical reaction proceeds the values of the chemical potentials change and, therefore, so does the slope of G, viz. ΔrG. In this way, the different meanings of ΔrG outlined in Table 2 can be visualized,

∂G ∂ξ

ð12Þ

¼ Δr G < 0; that is; forward reaction allowed

Δr G ¼ Δr G° þ RT ln Q

ð13Þ

Δr G° ¼ RT ln K

ð14Þ

p, T

∂G ξ > ξeq : ¼ Δr G > 0; that is; backward reaction allowed ∂ξ p, T

ξ ¼ ξeq :

ðG°products G°reactants ÞðkJÞ ΔG°ðkJÞ ¼ ΔξðmolÞ ð1 0ÞðmolÞ

ΔrG° is an intensive quantity and is expressed in kJ/mol. Neither can the negative value of ΔrG° be used as the general condition for spontaneity, nor is it true that a positive value of ΔrG° means that a chemical reaction will not proceed. As has been discussed previously, it is the sign of ΔrG that should always be considered for that purpose. Students can be presented selected examples71,72 in which despite ΔrG° > 0, the forward reaction is spontaneous (ΔrG < 0). A proper calculation of ΔrG makes use of the following equations55,63 (in which the intensive function nature of both ΔrG and ΔrG° should not go unnoticed)

∑

ξ < ξeq :

ð10Þ

where, Q is the reaction quotient, which has the form of the equilibrium constant, K, but is not equal to the equilibrium constant [notice that when ΔrG = 0 (equilibrium), then Q = K], !r !s pðRÞ pðSÞ p° p° Q ¼ ð15Þ !a !b pðAÞ pðBÞ p° p°

∂G ¼ Δr G ¼ 0; that is; equilibrium ∂ξ p, T

Notice that ΔG 6¼ ΔG°

ð8Þ

for ΔG° ¼ G°products G°reactants

We must remark that only when Q = 1 does ΔrG = ΔrG°. Of course, this is not the case for most chemical reactions. Thus, we must stress that in most cases the sign of ΔrG° does not serve as a criterion for the spontaneity of a chemical process. The relationship of ΔrG° with K may be used to state how far the reaction has gone before equilibrium has been attained, for K may be obtained from ΔrG°

ð9Þ

and ΔG is a finite difference in the Gibbs energy between two states. In Figure 1, it is, for example, the finite difference in the Gibbs energy between the final state (i.e., the equilibrium mixture) and the initial one (i.e., the mixture of reactants). Thus, ΔG differs from ΔG°, which is also a finite difference, but now ΔG° is the difference in the Gibbs energies of the products and reactants when each is in its standard state. Also, it should be noticed that ΔrG 6¼ ΔG. Not only is there a conceptual distinction between these two quantities, but there is also a distinction in the units with which each quantity is

K ¼ eΔr G°=ðRTÞ

ð16Þ

If ΔrG° < 0, then K > 1 (i.e., the process is product favored). Conversely, if ΔrG° > 0, then K < 1 (i.e., the process is reactant favored). This determination does not depend on the isothermal 89



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Table 3. A Simple Condition for Both Spontaneity and Equilibrium

Table 4. Summary of General Chemistry Textbooks’ Misrepresentation of Gibbs Energy • The equilibrium condition is often defined as ΔG = 0 (also, ΔG° = 0),

ΔrG

Q/K

instead of ΔrG = 0.

Spontaneous Reaction

0

> 1 (Q > K)

backward

=0

= 1 (Q = K)

equilibrium

• The condition for spontaneity is always defined as ΔG < 0 (instead of ΔrG < 0), and it is usually exemplified calculating ΔG°. • In connection with the previous point, in many cases the discussion of spontaneous reactions is implicitly restricted to standard conditions. Thus, it is often assumed that ΔG° < 0 corresponds to a general criterion for spontaneity.

conditions (p = constant or V = constant) under which K is calculated,73,74 as suggested by Antonik.75 ΔrG differs from ΔrG°, for at a given temperature the value of ΔrG° is fixed, but the value of ΔrG is determined by two terms: ΔrG° and a concentration-dependent term. These two terms can be joined into only one term, which depends on the quantity Q/ K. That is, making use of eqs 13 and 14, the following equation can be obtained Δr G ¼ RT ln

Q K

• It is usually assumed that if ΔG° > 0 the forward reaction is forbidden. • In many cases, the value of ΔrG° is reported in kJ in calculations involving the equation ΔGo = RT ln K, for authors usually do not pay attention to the correct units of R (in this case, kJ K1 mol1). • QK inequalities are normally employed to decide the direction of a disturbed equilibrium system. This discussion is usually based on the following equation ΔG = RT ln(Q/K), instead of ΔrG = RT ln(Q/K). Consequently, ΔG is reported in kJ units. Once again, some authors do not

ð17Þ

pay attention to the correct units of R.

This expression allows us to calculate the value of ΔrG and, therefore, to discuss the direction of the spontaneous reaction (Table 2). That is, the value of the quotient Q/K may be the basis of an easier condition for spontaneity (Table 3). The limitations of Le Châtelier’s principle, as well as the misconceptions students and teachers hold when trying to apply it, have received great attention in the literature.7,12,67,7684 QK inequalities (Table 3) are also easy-to-apply conditions in the prediction of the evolution of disturbed equilibria. They have the advantage of having no limitations, which is a powerful alternative to Le Châtelier’s qualitative rules.12,67,8184

to not using correct units of R (that is, usually the incorrect units are kJ/K). All the possible sources of this error have been discussed previously at greater detail.72 Four textbooks36,39,46,47 report ΔG° in kJ/mol. In addition, QK inequalities are normally employed to decide the direction of a system disturbed from equilibrium. This discussion is usually based on the following equation ΔG ¼ RT ln

Q K

ð27Þ

(instead of ΔrG = RT ln(Q/K)). In those cases, ΔG is usually reported in kJ units,25,29,32,35,41,42,51,87 for authors have not paid attention to the correct units of R. But, in other cases, authors36,37,45,48 make use of ΔG having kJ/mol units. Still, Umland and Bellama45 explain

’ SPONTANEITY AND EQUILIBRIUM: TEXTBOOKS’ MISREPRESENTATIONS Authors of first-year university chemistry textbooks scarcely pay attention to the basis of the above discussion. In most of those textbooks, there is confusion in terminology, for the terms used in the treatment of Gibbs energy are usually misrepresented (Table 4), which may lead to imprecise or even incorrect conclusions. Some of these misleading statements are reported. The equilibrium condition is often defined as ΔG = 0, instead of ΔrG = 0.25,28,30,33,38,41,4446,48,51,52,54 Freemantle36 states that this condition is ΔG° = 0. Also, the condition for spontaneity is always defined as ΔG < 0 (instead of ΔrG < 0), which is usually exemplified by calculating ΔG°. Thus, the discussion of spontaneous reactions is normally restricted to standard conditions, although this situation is not always stated explicitly. That is, these presentations do not stress that restriction, which often leads to the assumption that ΔG° < 0 corresponds to a general condition for spontaneity: two textbooks31,50 include a section entitled “ΔG° as a criterion for spontaneity”; conversely, it is assumed that if ΔG° > 0, the forward reaction is forbidden. Moreover, the values of ΔG° are usually reported in kJ units for calculations involving the equation ΔG° = RT ln K.2528,30,32,33,38,41,49,85,86 Moore et al.43 and Whitten et al.40 report ΔG° in kJ in one exercise, but in the next one, it is expressed in kJ/mol. Gilbert et al.54 calculate ΔG° in kJ, but these units change to kJ/mol when introducing the calculated value in the above equation. Reporting ΔrG° in units of kJ is mainly due

We have been writing J or kJ for the units of ΔG° and ΔG. ... However, ΔG° and ΔG are extensive properties and really do include units of mol1 because, in thermodynamics, equations are always interpreted in terms of moles. In calculations that involve both ΔG° or ΔG and R, the unit J/mol (or kJ/mol) must be used for ΔG° and ΔG0 . (authors’ emphasis) On the other hand, authors do not enlarge this topic to cases in which Le Châtelier’s principle is limited. On the contrary, it is used to demonstrate its supposed validity. For example, a QK discussion can help when considering the limited character of that principle when predicting the evolution of a disturbed chemical equilibrium system when adding a reactant at constant p and T.12,61,67,71,72,76,81,84

’ CONCLUDING REMARKS AND SUGGESTIONS FOR TEACHING The misleading assumptions reported in this study arise from the quantitative and mathematical emphasis given to the thermodynamic concepts involved, but without explaining them in a proper way. A sound qualitative discussion would help in the clarification and differentiation of ΔG, ΔrG, ΔG°, and ΔrG°. This way, many authors have proposed a revision of 90



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QK inequalities (i.e., QcKc or QpKp) could be introduced at this level as a first basic criteria for spontaneity in isothermal conditions.100 That is, one does not need thermodynamics to distinguish between Q and K for a reaction. Indeed, it is probably a mistake to wait until the discussion of thermodynamics to make this point. One merely needs to distinguish between the ratio of partial pressures or concentrations at a degree of reaction progress (Q) and the value when the system is at equilibrium (K). Eventually, this mathematical discussion could be justified in an advanced course dealing with the second law of thermodynamics.81 Still, defenders of the current first-year thermodymics emphasis would argue, among other things, that engineers need thermodynamics earlier than suggested in this article or that biologists could never get to Gibbsian thermodynamics; also, many teachers would support the view that directionality and spontaneity rationalized by the meaning of QK inequalities are enriched if Gibbs energy is introduced. Therefore, they may think that Figure 1 could be used to provide beginning students with conceptual understanding that could be expanded on in later courses. Moreover, they would also add that Gibbs energy is useful in teaching electrochemical phenomena. Unfortunately, the scarcity of time at the introductory level would not allow the Gibbs energy to be taught well and thus might not favor meaningful learning. If Gibbs thermodynamics do have to remain in the general chemistry syllabus, it must be stressed to authors that any attempt to introduce them to first-year students must simplify the conceptual approach given in this study, but without continuing current textbook errors. The teaching of thermodynamics should pay careful attention to the precise meaning of the terminology involved as well as to the demanding difficulty of the concepts to be learned. Thus, the debate about what, how, and when thermodynamics should be taught is still open, challenging future research on this topic.1

the symbolism,5,8891 but their suggestions have not been heeded. Still, modern physical chemistry textbooks usually comply with the IUPAC recommendations.68 The main conclusion of this study is that in first-year chemistry textbooks ΔG, ΔrG, ΔG°, and ΔrG° are not properly defined and are only used in algorithmic calculations. Hence, this teaching approach would promote robotic learning rather than conceptual understanding and even might be the origin of students’ misunderstandings. Still, this article cannot go further stating in which specific way this widespread misleading textbook approach can lead to student misconceptions, for our attempt has been to contribute to both the distinction and proper use of each of the quantities involved in the discussion of spontaneous reactions. With respect to ΔrG and ΔG, the confusion in textbooks arises due to the overuse of the symbol “Δ” in teaching thermodynamics in introductory university chemistry courses. That is, it is assumed that ΔG plays the role of ΔrG. Also, there is a great deal of confusion when reporting their units. Moreover, some authors state that ΔG° < 0 is a general condition for spontaneity. Significant differences were not found over time in the way college chemistry textbooks deal with ΔG, ΔrG, ΔG°, and ΔrG°. The thermodynamic foundation outlined in this article goes beyond the scope required for an introductory course.5564 However, current college chemistry textbooks include a great amount of information on difficult thermodynamic concepts. The stress on the Gibbs energy has not been the case historically. A glimpse at chemistry textbooks published 50 years ago9295 reveals that authors did not include Gibbs energy. The “chemical principles” shift that occurred in the mid-1960s96,97 and the increasing number of pages of each new textbook edition98,99 may explain the current emphasis given to thermodynamics in first-year university textbooks. At this point, it is interesting to note that some authors have written general chemistry textbooks32,42 as well as physical chemistry textbooks.55,57 It is rather surprising that the misleading assumptions reported in this study (e.g., confusing ΔG with ΔrG) are only present in introductory textbooks. Perhaps, those authors have tried to simplify the thermodynamic topics they introduce to students at the first-year level, but their attempts might have gone too far. Therefore, maybe, it would be better to remove most of those difficult topics from the first-year university syllabus, leaving them for an advanced treatment, instead of continuing to teach them using oversimplified and misleading thermodynamic statements. The accurate thermodynamic approach given by some first-year chemistry authors53 to the concepts analyzed in this study would seem to be far beyond the level required in an introductory college chemistry course. A balanced general chemistry course should not be focused mainly on thermodynamics because there would not be enough time to develop properly all the other essential topics to be covered. Thus, general chemistry students would not be capable of understanding those difficult thermodynamics topics, in spite of being accurately treated in their textbook, because teachers could not devote the proper time to develop such a large quantity of information. In a recent study,24 it was suggested that first-year university students should be introduced to practical equilibrium constants only, leaving the discussion of the thermodynamic equilibrium constant (and its relationship to ΔrG° and ΔrG) to an advanced level. This recommendation is also suggested in this study because a sound introduction to the Gibbs energy may be too difficult for first-year students to understand.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

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First-Year University Chemistry Textbooks' Misrepresentation of Gibbs

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