In the Classroom
The Use of Limits in an Advanced Placement Chemistry Course Paul S. Matsumoto,* Jonathan Ring, and Jia Li (Lily) Zhu Galileo Academy of Science and Technology, San Francisco, CA 94109; *
[email protected] The content of this article is based on the novel use of limits in a high school advanced placement (AP) chemistry course, which is equivalent to a first-year college chemistry course (1); therefore, the content of this article would be applicable to both courses. As the concept of limits is taught in both precalculus (2, 3) and calculus (4, 5) courses, most students in a first-year college chemistry course would have the necessary mathematical background, while there would be a larger variation in the number of students with such a background in an AP chemistry course. In addition to the use of equations to solve chemistry problems, an equation is an abstract summary of the relationships between the variables in a function (6). Many students have difficulty recognizing such relationships by merely “examining” the function. A useful tool to recognize such relationships would be to examine the graphical representation of the function. Another tool would be to evaluate the limits of the function under various conditions. While AP chemistry textbooks (7) use graphs to illustrate chemical concepts, the use of limits is absent. Only one past article in this Journal was related to the use of limits in the chemistry classroom (8). In this article, we will demonstrate the use of limits in topics that are typically covered in an AP chemistry course. For a simple continuous function, a limit may be viewed as the value of a function at a specific value in its domain (3, 5). The notation of the limit of a function, f (x),
the function has an asymptote or in extrapolating a function to an unattainable experimental condition, such as an infinite temperature. The domain and range of functions would be restricted to the first quadrant since chemical quantities of interest in this article have only positive values. Case 1: Horizontal Asymptote of a Function An example of a function that has an asymptote is the relationship between fraction ionized of a weak acid and its equilibrium constant, Ka (9):
K a → ∞
x → 3
x → 3
(x
− 3)
2
means that as the value of x approaches 3, the value of the function approaches infinity. While the behavior of a function may be seen graphically, an evaluation of its limit provides the same information often in a more time-efficient manner. In addition, students may not be able to afford a graphing calculator. Another situation where the evaluation of a limit may be of value is in a function with multiple independent variables, which may be difficult or even impossible to visualize using a graph. An evaluation of the limit of a function at an arbitrary value is not valuable, for example,
lim (5 x ) = 10
x → 2
since it provides no insight into the behavior of the function. However, the value of the limit of a function at its extremes, where its value is not defined but does approach some value, may be useful. Situations of interest would be where www.JCE.DivCHED.org
•
K a2 + 4 K a [HA ]i
−2 lim
= ∞
−Ka −
(ii) then multiply the numerator and denominator by 1Ka ,
K a → ∞
1
(1)
−2 K a
lim
lim f ( x ) = f (c )
lim f ( x ) = lim
2 [HA ]i
where [HA]i is the initial concentration of the weak acid, HA. The evaluation of the limit (3, 5) of fraction ionized as Ka approaches infinity is as follows: (i) multiply the numerator and denominator of eq 1 by [᎑Ka − (Ka2 + 4Ka[HA]i)1/2],
x → c
is read as “the limit of f (x) as x approaches c is f (c)”. However, a function may not exist for a specific value of x, for example, f (x) = 1(x − 3)2 would not be defined at x = 3, but
K a2 + 4 K a [HA ]i
−K a + fraction ionized =
−1 −
1 +
4 [ HA ]i
= 1
Ka
Therefore,
lim
fraction
K a → ∞ ionized
lim
K a → ∞
= −K a +
K a2 + 4 K a [HA ]i 2 [HA ]i
= 1
which is consistent with its graph seen in Figure 1A that has a horizontal asymptote, where fraction ionized = 1. A similar argument may be used to evaluate the relationship between the fraction ionized of a weak base and its equilibrium constant, Kb (not shown). Either inspection of the graph or evaluation of the limit of eq 1 implies that as the value of Ka increases, the fraction ionized increases and approaches 1. An alternative verbal rationale of this behavior involves two steps. First, as the equilibrium constant for the reaction between water and a weak acid is the ratio of the equilibrium concentration of the products (i.e., [H3O+]e and [A−]e ) divided by the reactants (i.e., [HA]e ), an increase in the equilibrium constant implies that there is a relative increase in [H3O+]e and [A−]e with a corresponding decrease in [HA]e. Second, as the fraction ionized
Vol. 84 No. 10 October 2007
•
Journal of Chemical Education
1655
In the Classroom A
B
Figure 1. Examples of functions that contain an asymptote: (A) The relationship between the fraction ionized of a weak acid and its equilibrium constant Ka. In this simulation, the value of [HA]i = 0.1 M. (B) The relationship between pH and the volume of base in a strong acid–strong base titration. In this simulation, [HCl] = 50 mM, VHCl = 50 mL, and [NaOH] = 100 mM.
of a weak acid is the ratio of [H3O+]e (or [A−]e ) to [HA]i, an increase in the value of the equilibrium constant results in an increase in the fraction ionized. This example illustrates the benefit of using a concise mathematical argument rather than a complex verbal argument to illustrate a point. A second example of a function containing an asymptote describes how pH is affected by adding a strong base to a strong acid, which may be simulated using the following function,
[acid ] Vacid
− [base ] Vbase + Vbase for Vbase < Vequiv
Vacid
which is the pH of the base alone and is consistent with the graph in Figure 1B. A similar analysis would be used in the simulation of adding a strong acid to a strong base (not shown). Case 2: Extrapolation of a Function In a laboratory exercise involving Boyle’s law, students used an absolute pressure probe, a 20.0 mL syringe, and tubing to connect the pressure probe to the syringe. The system may be described by the ideal gas law (7, 10), P Vsystem = nR T
point
[H + ]
10 −7 M
=
for Vbase = Vequiv
or
point
10 −14
(
P Vsyringe + Vtubing
for Vbase > Vequiv
[OH− ]
point
Vsyringe = nR T
[OH− ]
=
Vacid
− [ acid ] Vacid + Vbase
The evaluation of the limit of this function as the volume of the base (Vbase ) approaches infinity would be
lim pH =
Vbase → ∞
lim
Vbase → ∞
− log
10 −14 ( Vacid + Vbase ) [base] Vbase − [acid ] Vacid
The evaluation of the limit may be done by using L’Hôpital’s rule (5, 8) or multiplying the numerator and denominator of the function by 1Vbase
= nR T
Upon rearrangement,
where
[base] Vbase
)
1 P
− Vtubing
(2)
Thus, in a graph of Vsyringe on the y axis and 1P on the x axis (Figure 2A), there would be a straight line, where the slope and y intercept lead to an estimate of the number of moles of gas in the system and the volume of the tubing, respectively. Since the value of the y intercept cannot be obtained experimentally, that is, the pressure in the syringe cannot be infinite, we need to extrapolate the line to find its y intercept. The evaluation of the limit of eq 2 is lim Vsyringe = lim Vsyringe
1 → 0 + P
P → ∞
= liim nR T P → ∞
1 − Vtubing P
= −Vtubing
10 −14 1 + − lim log
Vbase → ∞
1656
[base ]
−
Vacid Vbase
Vacid [acid ] Vbase
Journal of Chemical Education
10 −14 = − log [base]
•
which corresponds to the (extrapolated value of the) y intercept (i.e., volume of the tubing). A second example describes the relationship between fraction ionized and the concentration of a weak acid (see eq 1 and Figure 2B). Since the value of the “y intercept” can
Vol. 84 No. 10 October 2007
•
www.JCE.DivCHED.org
In the Classroom A
B
Figure 2. Examples of functions, where the limit may be used to extrapolate to the y axis: (A) Experimental data showing the relationship between the volume of a syringe and reciprocal of the pressure in the system. (B) The relationship between fraction ionized and concentration of a weak acid. In this simulation, Ka = 1.8 × 10᎑5.
not be experimentally determined, that is, the fraction ionized would not be defined when [HA]i = 0, we need to extrapolate the curve to the y axis, which is the limit of the fraction of the weak acid ionized as the concentration of the weak acid approaches zero (9), fraction lim = [HA ] → 0 + ionized lim [ HA ] → 0 +
−K a +
K a2
+ 4 K a [HA ]i
= 1
which is solved using L’Hôpital’s rule (5, 8) and is consistent with the graph in Figure 2B. A third example would be the transformed Arrhenius equation (7)
T → ∞
T → ∞
which implies that if the concentration of the products is sufficiently low, then ∆G would be negative, hence the reaction is spontaneous. The context of this argument was that in a sequential (or coupled) reaction, where a thermodynamically unfavorable reaction (A → B) may be “driven” by reducing the concentration of its products by a thermodynamically favorable reaction (B → C):
Ea 1 + ln A R T
where a graph of ln k on the y axis and 1T on the x axis (graph not shown) would have a y intercept = ln A. The value of the y intercept occurs as T approaches ∞, which is experimentally impossible, so we need to evaluate the limit, lim ln k = lim −
[products] lim ∆G = lim ∆G ° + R T ln + + [reactants] [P] → 0 [P] → 0 = −∞
2 [HA ]i
ln k = −
A biochemical application of this function (11) is that a reaction with a positive ∆G ⬚ may be spontaneous if the concentration of the products (P ) is sufficiently small. That is,
Ea 1 + ln A R T
B
C
The second example involves the Arrhenius equation (7), −
k = Ae
Ea RT
where an evaluation of its limit as a function of T Ea RT
−
lim k = lim A e
Case 3: Implications of a Function In addition to determining the value of its asymptote or extrapolating to the y axis, one may evaluate the limit of a function to uncover the implications of the function. The first example involves the relationship between Gibbs’ free energy (7), a criterion of a reaction’s spontaneity, and the reaction quotient, Q: ∆G = ∆G ° + R T lnQ = ∆G ° + RT ln
•
B
= ln A
which corresponds to the (extrapolated value of the) y intercept.
www.JCE.DivCHED.org
A
[products] [reactants]
T → 0
T → 0
= 0
implies that as the temperature decreases, the rate constant, k, decreases. Also, the evaluation of its limit as a function of Ea −
lim k =
Ea → ∞
lim
E a → ∞
Ae
Ea RT
= 0
implies that as the activation energy, Ea, increases, the rate constant decreases. Changes in the rate constant lead to corresponding changes in the rate of the reaction. The third example demonstrates that the equation describing a nonideal gas simplifies to the ideal gas equation at low number of gas molecules in the system. The Van der
Vol. 84 No. 10 October 2007
•
Journal of Chemical Education
1657
In the Classroom
Waals equation
P +
n2 a
(V
V2
− nb) = n R T
(3)
describes the behavior of nonideal gases (7), where the values of a and b account for the attraction (i.e., intermolecular forces) between gas molecules and the finite volume of a gas molecule, respectively. Examining the left-side of eq 3, as n approaches zero (i.e., as there are fewer gas molecules in the system), lim P +
n → 0
n2 a V2
(V
− nb)
= lim P V − n bP + n → 0
n2a n 3 ab − V V2
= PV
or multiple intelligence (12) and therefore strengthen students’ comprehension of the implications of an equation and increase the number of students understanding a concept. The use of limits in a chemistry course reinforces students’ knowledge of limits learned in mathematics and may serve as a basis for collaboration among chemistry and mathematics teachers. Such collaboration among teachers is encouraged by the National Science Education Standards (13), which leads to integration of topics that may result in students’ deeper understanding of chemistry and create an opportunity for professional development among teachers. Besides serving as a tool to summarize the relationship between variables, functions are the basis of a simulation, a mathematical model that describes the behavior of a system. Examples in this article serve to introduce the concept of a simulation and mathematical models, a very powerful and useful tool in science. Acknowledgments
that is, P +
n2 a V2
(V
− nb) ≈ P V
(4)
as n approaches zero. Substitution of eq 4 into eq 3, yields for values of n close to 0, P V ≈ n R T
The experimental data in Figure 2A were provided by students in an honors chemistry course (Amos Fung, Ho Yeung Lo, and Edmond Yee), while the Pasco pressure probes were a loan from the San Francisco Unified School District Office of Teaching and Learning purchased through a NSF USP (National Science Foundation; Urban Systemic Program) grant. We would also like to acknowledge Leighton Izu (University of Kentucky) for reviewing the manuscript.
which is the ideal gas equation (7, 10). Literature Cited
Discussion The contents of this article provide examples involving chemistry, where an evaluation of the limit of a function supports the information in its graph, but such information on the behavior of the function may be obtained faster and cheaper by evaluating its limit than by generating its graph. Using limits and mathematics in a chemistry course may also benefit English as a Second Language (ESL) students. Chemical concepts may be expressed using a mathematical equation or a verbal description. Immigrant students either do not speak English or have a weak English background, but they are familiar with math concepts and equations, the comprehension of which is independent of the support from language. As a result, such students may find it easier to learn chemistry presented from a mathematical viewpoint than using verbal statements. A similar argument would be valid for native English learners who are stronger in mathematics than English. Consistent with this idea, based on anecdotal evidence, more ESL students understand LeChâtelier’s principle when it is taught using mathematics (e.g., rate law, reaction quotient, and equilibrium constant) than simply using a series of verbal arguments to illustrate the concept. An analysis of the function using limits utilizes students’ logical or mathematical intelligence and ability to think abstractly. A graphical analysis of a function uses students’ visual intelligence. The use of a verbal argument benefits students with verbal or linguistic intelligence. The purpose of using such multiple modes of presentation is to provide alternative pathways that suit differing aptitudes, learning styles,
1658
Journal of Chemical Education
•
1. AP Chemistry Home Page. http://www.collegeboard.com/student/testing/ap/sub_chem.html (accessed Jul 2007). 2. NCTM (National Council of Teachers of Mathematics). Curriculum and Evaluation Standards for School Mathematics; NCTM: Reston, VA, 1989. 3. Sullivan, M. Precalculus; Prentice Hall: Upper Saddle River, NJ, 2001. 4. College Board. AP Calculus Course Description; College Board: New York, 2005. 5. Stewart, J. Calculus, 5th ed.; Brooks/Cole: Belmont, CA, 2003. 6. Kean, E.; Middlecamp, C. How to Survive and Even Excel in General Chemistry; McGraw Hill: New York, 1994. 7. (a) Brown, T. L.; LeMay, H. E.; Bursten, B. E. Chemistry. The Central Science, 7th ed.; Prentice Hall: Upper Saddle River, NJ, 1997. (b) Chang, R. Chemistry, 8th ed.; McGraw Hill: New York, 2005. (c) Zumdahl, S. S.; Zumdahl, S. A. Chemistry, 6th ed.; Houghton Mifflin: Boston, 2003. 8. Missen, R. W. J. Chem. Educ. 1977, 54, 488–490. 9. Matsumoto, P. S. J. Chem. Educ. 2005, 82, 1150. 10. Zumdahl, S. S.; Zumdahl, S. L.; DeCoste, D. J. World of Chemistry; McDougal Littell: Evanston, IL, 2002. 11. Lenhinger, A. L. Biochemistry, 2nd ed.; Worth Publ.: New York, 1975. 12. (a) Multiple Intelligences and Learning Styles. http:// www.coe.uga.edu/epltt/mi-ls.htm (accessed Jul 2007). (b) Nakhleh, M. B.; Mitchell, R. C. J. Chem. Educ. 1993, 70, 190–192. (c) Holme, T. J. Chem. Educ. 2001, 78, 1578–1580. 13. National Science Education Standards. http://www.nap.edu/ readingroom/books/nses/4.html (accessed Jul 2007).
Vol. 84 No. 10 October 2007
•
www.JCE.DivCHED.org
