Article pubs.acs.org/jchemeduc
Reaction Extrema: Extent of Reaction in General Chemistry Jonathon E. Vandezande, Douglas A. Vander Griend, and Roger L. DeKock* Department of Chemistry and Biochemistry, Calvin College, Grand Rapids, Michigan 49546, United States S Supporting Information *
ABSTRACT: Nearly 100 years ago de Donder introduced the term “extent of reaction”, ξ. We build on that work by defining the concept of reagent extrema for an arbitrary chemical reaction, aA + bB ⇄ yY + zZ. The central equation is ξî = −ni,0/νi. The symbol ξî represents the reagent extremum for the chemical entity i; ni,0 represents the initial molar amount of entity i, and νi is its stoichiometric number, which is positive for products and negative for reactants. A reagent extremum exists for each reactant and each product; those of reactants are zero or positive, and the least positive of these is the reaction extremum to the right, ξmax; those of products are zero or negative, and the least negative is the reaction extremum to the left, ξmin. These two boundary values, called reaction extrema, indicate the maximum extent to which the reaction can progress in the forward or reverse direction, respectively. The ξmax and ξmin values are an important pedagogical tool for a quantitative understanding of chemical reaction stoichiometry. A graphical presentation in which the amounts of reagents are depicted versus extent of reaction is useful for helping general chemistry students understand intuitively the extent to which chemical reactions can progress. The Supporting Information includes a student handout, a student exercise, and a Web-enhanced object, Reaction Progress, which can be used to help students understand the concept of extent of reaction and reaction extrema. KEYWORDS: First-Year Undergraduate/General, Upper-Division Undergraduate, Physical Chemistry, Internet/Web-Based Learning, Problem Solving/Decision Making, Stoichiometry he concept of extent of reaction, ξ, was introduced by de Donder1 in 1920 and can be translated from the French as “how many complete reactions have taken place”. (The Greek letter “xi” can be pronounced as “ksee”, or “could see” while leaving out the sound of “ld”.) Even though the concept has nearly a one hundred year history in the discipline, extent of reaction is often underdeveloped or bypassed entirely in introductory chemistry in favor of simple recipe models. Whereas students may readily grasp that it takes two wheels and one set of handlebars for each bicycle that is produced, these types of analogies for chemical reactions fall short in comprehensively depicting chemical reactions. Chemical reactions have at least four features that make stoichiometric calculations more involved than might first meet the eye: (1) reactions initially may have both reactants and products; (2) reactions may proceed backward from the initial state; (3) reactions may not go to completion; and (4) reactions initially may not have stoichiometric amounts of reactants or products. As a result, there may be a limiting reagent to the right and a limiting reagent to left. Because of these features, it is important to examine the concept of “extent of reaction”, which underpins all four features above, in order to provide a general framework for thoroughly understanding chemical stoichiometry. We take de Donder’s construct one step further and introduce the concept of reaction extrema, or “how many complete reactions can take place given the amounts of species initially present”.
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© XXXX American Chemical Society and Division of Chemical Education, Inc.
Many articles in this Journal have addressed the concept of extent of reaction.2−8 Dumon et al.,3 Peckham,6 and Croce7 formulated equations for reactions that go from zero to the maximum extent of reaction. Other authors, for example, Treptow,4 used a definition such that the reaction extent ranges from “zero” at the beginning of the reaction to “one” at the end. Sostarecz and Sostarecz8 employ the term “number of reaction events” in place of “extent of reaction”. This term was earlier employed by Garst.2 Either term is equally valid. The IUPAC definition of “extent of reaction” employs the term “number of chemical transformations” divided by “the Avogadro constant”. Most pertinent to this presentation is the work of Sostarecz and Sostarecz,8 who employ a similar graphical approach but do not consider the possibility of back reaction. The establishment of the concept of reaction extrema allows for the thorough treatment of the potential movement of chemical reactions in even the most general cases. Furthermore, the connection between extent of reaction and percent completion of a reaction can then be definitively handled. In addition, in the Supporting Information we include a student handout, a student exercise, and a Web-enhanced object, Reaction Progress, which can be used to help students understand the concept of extent of reaction, percent completion, and reaction extrema. In the Appendix in the Supporting Information we present the formal implementation that leads to reaction extrema.
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dx.doi.org/10.1021/ed400069d | J. Chem. Educ. XXXX, XXX, XXX−XXX
Journal of Chemical Education
Article
We find it perplexing that extent of reaction is not more commonly employed in general chemistry. For example, under the topic of “General Chemistry” there are twenty-five items listed in a comprehensive IUPAC publication.9 All of these items are introduced in most general chemistry courses and textbooks except for two, “extent of reaction” and “degree of reaction”. Our work is designed to make the term “extent of reaction” more accessible to the beginning student. The concept of extent of reaction should be introduced in the curriculum at the same place that limiting reactant is presently introduced. The current method of introducing limiting reactant (e.g., try the reaction two different ways with dimensional analysis, etc.) can be replaced with an introduction to extent of reaction. The curriculum will not be lengthened by this substitution, and the payoff will be increased student comprehension of the underpinnings of chemical stoichiometry. The ensuing example serves to introduce the concepts of extent of reaction and reaction extrema. These chemical constructs will allow students to handle all four chemical scenarios that were described above.
reactant; likewise, H2 is the limiting reagent to the left, or what can be accurately called the limiting product. Pedagogically, all of the above can be done without the introduction of any Greek symbols.
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GRAPHICAL INTRODUCTION TO EXTENT OF REACTION Consider again the reaction of methane with hydrogen sulfide for an initial state allowing movement in either direction, as shown in eq 1. Figure 1 shows the amount of each of the four
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EXAMPLE REACTION Consider the reaction of methane with hydrogen sulfide; it has an initial state that allows for reaction movement in either direction: CH4(g) + 2H 2S(g) ⇄ CS2(g) + 4H 2(g) 1.0 mol
1.5 mol
0.5 mol
Figure 1. Graphical representation of the progress of the chemical reaction CH4(g) + 2H2S(g) ⇄ CS2(g) + 4H2(g) from arbitrary starting amounts of reactants and products (diamonds) as a function of extent of reaction, ξ. The reaction extrema (squares), ξmax and ξmin, define the effective range for ξ, which is marked with a double arrow from −0.25 to 0.75. The straight lines depicting the change in moles of each reagent are described in the Appendix in the Supporting Information. For a reactant A with coefficient a: nA = nA,0 − aξ. For a product Y with coefficient y: nY = nY,0 + yξ.
(1)
1.0 mol
Consider the following questions: what is the maximum extent to which the chemical reaction can proceed to the right, and likewise to the left, given the amounts shown below the reaction? (The pair of full arrows, forward and reverse, are prescribed by IUPAC.9 This notation does not imply chemical equilibrium, which is done with half arrows. It only implies that the reaction can shift to the left or to the right.) The maximum extent to which the reaction can progress to the left or to the right can be determined easily. Simply put, divide the initial amount of each substance, ni,0, by the stoichiometric number9,10 of that substance, vi, and change the sign: ξ̂i = −ni,0/νi. This yields a reagent extremum for each reactant and each product, which we will label ξî (pronunciation “ksee-hat”). (See the Appendix in the Supporting Information for details.) We illustrate the application of the equation for reaction extrema in Table 1. As shown in the table,
reagents at any extent of reaction. The y intercepts (at ξ = 0, initial conditions), marked by diamonds, indicate the initial moles, ni,0, of each reactant and product. The x intercepts, marked by squares, are the four reagent extrema values (ξî ) that were calculated above and that are the key features for quantitatively understanding the maximum extent to which a chemical reaction could progress, either forward (ξmax) or backward (ξmin) from the initial amounts. Pedagogically, we believe this graph is essential to helping students grasp the concept of reaction limits. In the accompanying Supporting Information, we present a student exercise that leads the student to build such a diagram for themselves, starting with the diamonds and then determining the location of the squares. This allows students to work through the concepts of reaction extrema, without concerning themselves with the mathematics that is presented in the Appendix in the Supporting Information. Whereas the horizontal axis in Figure 1 extends indefinitely, it is a meaningful representation of extent of reaction only insofar as no reactant or product is less than zero. In simple chemistry examples there are no products in the initial conditions, and therefore the reagent extremum of each product is zero. Furthermore, if stoichiometric amounts are present, all reactant extrema will be identical. For more general reactions such as the example above (eq 1), ξmin is the least negative reagent extremum for the reverse reaction (Figure 1, −0.25, not −0.50) and ξmax is the least positive reagent extremum for the forward reaction (Figure 1, 0.75, not 1.00). This range is marked in Figure 1 with a double arrow from
Table 1. Construction of the Reaction Extrema for Reaction 1 Quantity
CH4(g)
2H2S(g)
CS2(g)
4H2(g)
ni,0, initial amount/mol νi, stoich. no./mol/(mol rxn) ξî = −ni,0/νi/(mol rxn)
1.0 −1 1.0
1.5 −2 0.75 ξmax
0.5 1 −0.5
1.0 4 −0.25 ξmin
there are four values of reagent extrema ξ̂i, one for each reactant and each product: 1.0, 0.75, −0.5, and −0.25. The smaller of the positive values tells us that the maximum extent to which the reaction can proceed to the right is 0.75 mol rxn; we will label this ξmax, the reaction extremum to the right. The least negative value tells us that the maximum extent to which the reaction can proceed to the left is −0.25 mol rxn; we will label this ξmin, the reaction extremum to the left. The chemical H2S is identified as the limiting reagent to the right, or limiting B
dx.doi.org/10.1021/ed400069d | J. Chem. Educ. XXXX, XXX, XXX−XXX
Journal of Chemical Education
Article
−0.25 to 0.75. Whereas each reagent in a reaction has an extremum associated with it, the reaction has only one extremum to the right, ξmax, and only one extremum to the left ξmin. The existence of a limiting reactant is now clearly delineated as any case in which the extrema of multiple reactants are not identical. Considering the reaction for Figure 1, it is useful to refer to both H2S (reactant) and H2 (product) as limiting reagents, the former for the forward reaction and the latter for the reverse reaction.
equilibrium calculations. If the concept of extent of reaction is introduced in first semester general chemistry, it will make the task of introducing chemical equilibrium much simpler in the second semester course.
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Appendix outlining the mathematics behind this qualitative introduction; a student handout; a student exercise; a Webenhanced object, Reaction Progress, that can be used to help students understand the concept of extent of reaction and reaction extrema. This material is available via the Internet at http://pubs.acs.org.
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PERCENT COMPLETION Whereas extent of reaction is an extensive quantity, the intensive term of percent completion of a reaction, ε, can be defined with respect to extent of reaction, ξ. ε=
ASSOCIATED CONTENT
S Supporting Information *
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AUTHOR INFORMATION
Corresponding Author
ξ − ξmin × 100% ξmax − ξmin
*E-mail:
[email protected]. Notes
This term is dimensionless and is defined such that at ξmin its value will be 0%, and at ξmax its value will be 100%.11,12 Figure 1 also depicts ε graphically. Notice that if ξmin is negative, then the percent completion for the initial state will not be zero. For example, the initial state represented in Figures 1 is at 25% completion of the reaction.
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The authors thank Herbert Fynewever, Calvin College, for critical reading of the manuscript, and for the suggestion of the term reaction extrema. The authors thank Calvin College for a Thedford P. Dirkse Summer Research Fellowship to J.E.V.
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CONCLUSION Teachers routinely graph one variable versus another to illustrate connections between variables. For example, when discussing the behavior of ideal gases, a graph of P versus V at constant temperature illustrates Boyle’s law. However, when students are introduced to stoichiometry concepts, they are often not familiarized with the crucial second variable, extent of reaction, which is needed to obtain a graphical overview of chemical stoichiometry. The introduction of a formal name and a symbol for both extent of reaction, ξ, and for reaction extrema, ξmax and ξmin, is important to make these terms accessible and useful to students. Figure 1 is designed especially for students so that they can readily visualize the extent to which a chemical reaction can proceed in the most general cases. Included in the Supporting Information are a student handout, a student exercise, and a Web-enhanced object, Reaction Progress, which can be used to help students understand the concept of extent of reaction and reaction extrema. The student handout and student exercise have been employed in a classroom setting. Reaction Progress has been used in a laboratory setting in conjunction with a chemical synthesis in which there is a limiting reactant. Research data has not yet been collected to indicate how useful the software is in helping students to understand stoichiometry concepts. Readers who wish more detail about extent of reaction and stoichiometry concepts in general can examine refs 13 and 14. Extent of reaction can be thought of as a one-dimensional vector. One method to introduce this concept to students might involve placing two strips of tape on the classroom floor. One could be labeled “minimum” position and another “maximum” position. One’s starting position need not be at the minimum position. Walking in one direction could be labeled “negative”, whereas walking in the other could be labeled “positive”, and so forth. In closing, we wish to point out that extent of reaction as presented here is nothing more than the famous “x” that students encounter in simple chemical
REFERENCES
(1) de Donder, T.; van den Dungen, F. H.; van Lerberghe, G. J. M. Leçons de thermodynamique et de chimie physique; Gauthier-Villars et cie.: Paris, 1920. (2) Garst, J. F. The extent of reaction as a unifying basis for stoichiometry in elementary chemistry. J. Chem. Educ. 1974, 51, 194− 196. (3) Dumon, A.; Lichanot, A.; Poquet, E. Describing Chemical Transformations: From the Extent of Reaction to the Reaction Advancement Ratio. J. Chem. Educ. 1993, 70, 29−30. (4) Treptow, R. S. Free Energy versus Extent of Reaction. J. Chem. Educ. 1996, 73, 51−54. (5) Canagaratna, S. G. The Use of Extent of Reaction in Introductory Courses. J. Chem. Educ. 2000, 77, 52−54. (6) Peckham, G. D. The Extent of Reaction − Some Nuts and Bolts. J. Chem. Educ. 2001, 78, 508−509. (7) Croce, A. E. The Application of the Concept of Extent of Reaction. J. Chem. Educ. 2002, 79, 506−509. (8) Sostarecz, M. C.; Sostarecz, A. G. A Conceptual Approach to Limiting Reagent Problems. J. Chem. Educ. 2012, 89, 1148−1151. (9) IUPAC Quantities, Units and Symbols in Physical Chemistry, 3rd ed.; Cohen, E. R., Cvitas, T., Frey, J. G., Holstrom, B., Eds.; Royal Society of Chemistry: 2007. (10) In IUPAC parlance, the stoichiometric number is equivalent to the stoichiometric coefficient for a product, but it is the negative of the stoichiometric coefficient for a reactant. In conservation of mass terms, products minus reactants equals zero. Hence the minus sign on the coefficient of reactants. (11) This definition follows from that of H. B. Callen that he presents in both the first and second editions of his textbook on thermodynamics. He called it “degree of reaction”, and the scale was zero to one. (12) Callen, H. B. Thermodynamics; J. Wiley & Sons, Inc.: New York, 1960. (13) Alberty, R. A. Chemical Equations are Actually Matrix Equations. J. Chem. Educ. 1991, 68, 984. (14) Smith, W. R.; Missen, R. W. Chemical Reaction Equilibrium Analysis; John Wiley & Sons: 1982.
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dx.doi.org/10.1021/ed400069d | J. Chem. Educ. XXXX, XXX, XXX−XXX
