In the Classroom
The Use of Extent of Reaction in Introductory Courses
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Sebastian G. Canagaratna Department of Chemistry, Ohio Northern University, Ada, OH 45810;
[email protected] Stoichiometric calculations can be carried out in several ways. The most popular method in first-year chemistry texts is the “factor-label” method. There are several variations of methods using equations (1, 2). At the higher level, especially for theoretical derivations in thermodynamics, the quantity called extent of reaction proves very useful. This concept can be introduced early and used systematically for the treatment of enthalpy of reaction and other reaction quantities as well as chemical equilibrium (3) or introduced only in the context of chemical equilibrium (4 ). It appears to be generally recognized among the authors of physical chemistry texts that the disadvantages of having to deal with one more concept are more than outweighed by the benefits it affords in terms of dimensional correctness and compactness. There are many articles in this Journal that make free use of the extent of reaction (5, 6 ). Over the past few years I have experimented with the introduction of the concept of the extent of reaction in my introductory course directed to students majoring in biology, chemistry, and pharmacy. This article describes this experience. Balanced Equation: “Unit of Reaction” and Reference Sample As a consequence of the law of constant proportion and the fact that amount is extensive, any change in a reacting system, however large, can be described by a balanced equation. As a consequence, the ratio of changes in amounts of any two species in a chemical reaction is the same for any two samples. Thus for any two species i and j taking part in a reaction change in amount of i change in amount of j
= sample 1
change in amount of i change in amount of j
(1) sample 2
More precisely, the proportion of the changes in amounts of the various reactants and products is the same for all reacting samples. Thus experiments show that when nitrogen combines with hydrogen to yield ammonia, the changes in amounts of nitrogen, hydrogen, and ammonia are in the proportion 1:3:2. It is this that enables us to represent the reaction by means of the balanced chemical equation 1N2(g) + 3H2(g) = 2NH3(g) The term coefficient is used to describe the numbers used in the balanced equation. It is unitless. Thus the coefficient of N2 is 1 and those of H2 and NH3 are 3 and 2, respectively. The balanced equation represents not the actual changes taking place in a given sample, but the proportion in which these changes occur. The balanced chemical equation now becomes the “reference sample”, which summarizes the information concerning the changes in the amounts of the reactants and products. Changes in other extensive properties may be deduced from this.
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The change (on the mole scale) represented by the balanced equation may be taken as a reference with which all observed changes are compared. That is, the balanced equation may be considered to represent a “unit change” or a “unit of reaction”. Taking the ammonia reaction above as an example, the “unit change” corresponds to a change of 1 mol of N2, 3 mol of H2, and 2 mol of NH3. Thus the coefficient of any species represents the change in that species when the balanced equation takes place once. Conversely, the total change in a species equals the coefficient multiplied by the number of times the balanced equation has occurred. Using n to denote amount or “number of moles” n(i)reacted = coefficient of i × number of units of reaction (2) It follows from eq 2 that the number of units of reaction may be calculated by dividing the change in the amount of any species by its coefficient. The fact that the ratio of the change in amount of i divided by the coefficient of i is independent of i is known as the stoichiometric law. This ratio is numerically equal to the “number of units of reaction” that has taken place and is termed the extent of reaction, ∆ξ, or more properly, the change in the extent of reaction ξ:
n j reacted n i reacted = = coefficient of i coefficient of j (3)
n k reacted = … = ∆ξ coefficient of k From its definition, ∆ξ has units of “mol”. The extent of reaction ξ may be regarded as a macroscopic reaction coordinate, which varies as reaction proceeds. The term extent of reaction seems preferable to the term “number of units of reaction” which has been used for it, but the physical meaning may not be evident to the beginning student. Consequently, it is useful to repeatedly reinforce the physical significance behind the term extent of reaction every time it is used in class. The information contained in eq 3 is also contained in eq 1 if sample 2 in eq 1 is taken as the balanced equation. Conversely, the information contained in eq 3 may be rearranged to give
n i n j
reacted reacted
= coefficient of i coefficient of j
(4)
This is of course the basis of the ratio method or the factorlabel method, where the emphasis is in the connection between any two reactants. The method of extent of reaction uses the same information but with a different emphasis, the reaction as a whole: first the “number of units of reaction” or extent of reaction is calculated, and then the amount reacted of any species is related to this. The relevant equations for this
Journal of Chemical Education • Vol. 77 No. 1 January 2000 • JChemEd.chem.wisc.edu
In the Classroom
method are
∆ξ =
n i reacted , any i coefficient of i
n( j)reacted = coefficient of j × ∆ξ
(5) (6)
When the calculation involves only one species, the method of extent of reaction has little advantage over the other methods. However, when several calculations have to be made, this method is more compact than the other methods. As will be shown later, determining the limiting reagent is particularly simple. Also, it will be shown that it has other applications that tie in with the approach in later courses. Using the extent of reaction is rather more efficient than other methods, especially if several reagents are involved.
Example 1 The reaction of ammonium perchlorate with aluminum is expressed by a balanced equation of the form: 6A + 10B → 5C + 3D + 6E + 9F At the beginning the reaction vessel contained 0.050 mol of A and 0.095 mol of B. The reaction was stopped after some time, and it was found that 0.065 mol of B remained. Calculate the amounts of all reactants and products at the end.
All stoichiometric calculations start with the knowledge of the reacted amount of at least one species. Since the initial and final amounts of B are known, Amount of B reacted = Initial amount of B – Final amount of B = 0.095 mol – 0.065 mol = 0.030 mol
The extent of reaction can now be calculated using eq 5 with i = B.
n B reacted ∆ξ = = 0.030 mol = 0.0030 mol coefficient of B 10 Equation 6 is now used to calculate the changes in amounts of all other species. For example,
nremaining = 0. This allows us to do stoichiometric calculations with the initial amount of the limiting reagent, because it is equal to the amount reacted. For any reaction that goes to completion, the first step is to determine the limiting reagent. The concept of extent of reaction allows this to be done conveniently even when the number of reactants is greater than 2. Consider the ammonia reaction again, with the reaction vessel containing 4.5 mol of nitrogen and 9.0 mol of H2. If N2 reacted completely, the number of units of reaction using eq 3 would be 4.5 mol/1 = 4.5 mol. This is the maximum extent of reaction based on N2 and is properly denoted by ∆ξmax(N2). Similarly, ∆ξmax(H2) = 9.0 mol/3 = 3.0 mol. Thus there is enough N2 to make the balanced reaction go 4.5 times, whereas the amount of H2 present can make it go only 3.00 times. H2 is therefore the limiting reagent, and the balanced equation takes place 3.00 times. ∆ξmax for species i is given by n i initial ∆ξmax i = (7) coefficient of i The species that gives the smallest value is the limiting reagent and the actual extent of reaction is the extent of reaction corresponding to the limiting reagent.
Example 2 Consider the reaction 2 A + B + 3 C → products which takes place in a reaction vessel in which 0.650 mol of A, 0.300 mol of B and 0.700 mol of C are initially mixed together. Determine the amounts remaining at the end of the reaction.
The limiting reagent has to be determined first. Using eq 7
Similarly,
∆ξmax B = 0.300 mol = 0.300 mol 1
n(A)reacted = coefficient of A × ∆ξ = 6 × 0.0030 mol = 0.018 mol
∆ξmax C = 0.700 mol = 0.233 mol 3
n(C)produced = coefficient of C × ∆ξ = 5 × 0.0030 mol = 0.0150 mol
For the amount remaining, n(A)remaining = n(A)initial – n(A)reacted = 0.050 mol – 0.018 mol = 0.032 mol Since C is a product, the equation for the amount remaining is n(C)remaining = n(C)initial + n(C)produced = 0.000 mol + 0.0150 mol = 0.0150 mol
n A initial = 0.650 mol = 0.325 mol coefficient of A 2
∆ξmax A =
Since ∆ξmax(C) is the smallest, C is the limiting reagent and ∆ξ for the reaction is the value for C, that is, 0.233. The calculation can now be completed as before or by means of a “stoichiometric table”, thus: ∆ξ(i)m a x = n(i)in itia l/cft(i)
The calculations for the other species are similar.
0.650 mol/2 = 0.325 mol 2A
+
Initial amount, n(i)in itia l 0.650 mol
Limiting Reagent Calculation If even one of the reactants is not present, the reaction clearly cannot take place. The reaction will proceed till at least one reactant is used up: this reactant is referred to as the limiting reagent. For a limiting reagent, nreacted = ninitial and
Change in n(i) Final amount, n(i)
0.300 mol/1 0.300 mol B
+
0.300 mol
0.700 mol/3 0.233 mol 3C
→
products
0.700 mol
᎑2 × 0.233 mol ᎑1 × 0.233 mol ᎑0.700 mol = ᎑0.466 mol
᎑0.233 mol
0.184 mol
0.067 mol
0.00 mol
EXPLANATION. The ∆ξmax are calculated at the top of the table, the formula (eq 7) being written down as a reminder.
JChemEd.chem.wisc.edu • Vol. 77 No. 1 January 2000 • Journal of Chemical Education
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In the Classroom
C is the one that gives the least value and is therefore the limiting agent. Therefore the amount of C remaining at the end of the reaction is 0, and change in n(C) = ᎑ original amount of C. ∆ξ is the one corresponding to the limiting reagent C and is used in conjunction with eq 6 to complete the 2nd row. The final row can be completed, since it is the sum of the first two rows. In the traditional method of determining the limiting reagent, two species, for example A and B, would be selected to determine which was limiting with respect to the other. If B was limiting, B would then be compared with C. This is clearly not as efficient as the method used here. The method that determines the amount of some product formed from each of the reagents comes close to the present method conceptually. Other Approaches It is not essential to use equations in stoichiometric calculations using the extent of reaction. For a change corresponding to the balanced equation, ∆ξ = 1 mol, this may be referred to as 1 mol of reaction. For the ammonia reaction, for example, this would give conversion factors of 1 mol reaction ≡ 2 mol NH3, 1 mol reaction ≡ 3 mol H2, and so forth. Thus it is possible to have factor-label analogs of eqs 5 and 6. Some may prefer other terminology, but once some meaningful terminology has been created, implementation is easy. Summary and Conclusion The approach to stoichiometry and limiting reagent calculations discussed here uses the physical significance of the extent of reaction as being a measure of the “number of times the change equivalent to the balanced equation has taken place” and the coefficient of a reagent as being the change in amount every time this change takes place. The symbols used in the method make, in contrast to the traditional methods, a much needed distinction between changes in amounts and
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initial amounts. The method can be used at various levels of sophistication, depending on the level of preparation of the students. It can be used both with equations and with the factor-label method. The calculations using this method are generally more compact than those in traditional methods. For simple stoichiometric calculations the method of extent of reaction is just an alternative to the traditional method, but for limiting-reagent calculations it is more efficient. It is a good preparation for higher work where the concept of extent of reaction is beginning to be used extensively. The concepts associated with the method find application in the proper definition of the rate of reaction in kinetics and are helpful in clarifying problems of dimensional correctness that arise in connection with molar reaction quantities. The method can be used in enthalpy calculations. This method has been used in my classes for the past five years and student response to it has been favorable. When the method is properly and carefully introduced, in particular when the physical meaning of the extent of reaction is repeatedly reinforced, students will find it no more difficult than the traditional method. W
Supplemental Material
An expended version of this article, including sections on stricter notation and specific treatment in other applications, is available in this issue of JCE Online. Literature Cited 1. DeToma, R. P. J. Chem. Educ. 1994, 71, 568. 2. Canagaratna, S. G. J. Chem. Educ. 1992, 69, 957. 3. Alberty, R. A.; Silbey, R. J. Physical Chemistry, 2nd ed.; Wiley: New York, 1997; p 56. 4. Atkins, P. Physical Chemistry, 6th ed.; Freeman: New York, 1997; p 216. 5. Gerhartl, F. J. J. Chem. Educ. 1994, 71, 539. 6. Treptow, R. S. J. Chem. Educ. 1996, 73, 51.
Journal of Chemical Education • Vol. 77 No. 1 January 2000 • JChemEd.chem.wisc.edu
