ARTICLE pubs.acs.org/jchemeduc
An Analogy Using Pennies and Dimes To Explain Chemical Kinetics Concepts Jose E. Cortes-Figueroa,*,† Wanda I. Perez,† Jose R. Lopez,† and Deborah A. Moore-Russo‡ †
Organometallic Chemistry Research Laboratory, Department of Chemistry, University of Puerto Rico, P.O. Box 9019, Mayag€uez, Puerto Rico 00681 ‡ Department of Learning and Instruction, University at Buffalo, State University of New York, Buffalo, New York 14260-1000, United States
bS Supporting Information ABSTRACT: An analogy is presented that uses coins and graphical analysis to teach kinetics concepts and resolve pseudo-first-order rate constants related to transition-metal complexes ligandsolvent exchange reactions. KEYWORDS: Graduate Education/Research, Upper-Division Undergraduate, Inorganic Chemistry, Analogies/Transfer, Coordination Compounds, Kinetics, Rate Law
ctivation parameters (ΔHq, ΔSq, and Ea) used to assess and probe reaction mechanisms are obtained using the appropriate version of the general equation, eq 1, which is based in theories of reaction rates.1 Depending on the model employed, values such as 0, 1/2, 1, 1/2, 3/2, and 2 can be assigned to m in eq 1. For example, for the Arrhenius model (eq 2) and the transition-state model (eq 3) to conform with eq 1, the general equation’s parameters must be as follows: m = 0, a = A, E0 = Ea, (ΔSq/R) , and E0 = ΔHq, respectively, where and m = 1, a = (kB/h)e A is the frequency factor, kB is the Boltzmann constant, and h is the Plank constant.
A
0
k ¼ aT m eE =ðRTÞ
ð1Þ
k ¼ AeEa =ðRTÞ
ð2Þ
kB T ΔSq =R ΔHq =RT e e h
ð3Þ
k¼
ignored, the estimated activation parameters obtained may be meaningless because the solvent order may not be zero and the rate constant values correspond to pseudo-nth-order rate constants.3 Solventsolvent and solventligand exchanges are fundamental reactions that should be taken into account to understand mechanistic and reactivity aspects related to transition-metalmediated catalysis.49 Educational activities where mechanisms are proposed or existing mechanisms are tested should take into consideration that solvent molecules may act as ligands.49 For example, the rate law for the mechanism of the hypothetical ligand exchange described in Scheme 1 is
ð4Þ
Enthalpy of activation (ΔHq) and entropy of activation (ΔSq) values can be estimated by plotting ln(ktrue/T) versus (1/T), whereas values obtained by plotting ln(kobsd/T) versus (1/T) (when n 6¼ 0) lack physical meaning. For example, when the order of a solvent in a specific elementary step is unknown or Copyright r 2011 American Chemical Society and Division of Chemical Education, Inc.
ð5Þ
kobsd ¼ k1, solvent1 ½solvent1 þ k1, solvent2 ½solvent2
ð6Þ
where m is the coordination number and LmM is the transitionmetal complex. Experiments in binary solvent mixtures (solvent1 and solvent2) can be designed to assess the solvent role in the mechanism described in Scheme 1 that predicts that kobsd values will be dependent on [solvent1] and [solvent2] as described by eq 6. In the following activity, we address the concern presented above. We use pennies and dimes to teach kinetics and mechanistic concepts related to solventsolvent and solventligand exchange reactions on transition-metal complexes.
Meaningful activation-parameter values require knowledge of “true rate constant” values.2 Here a “true rate constant” is kobsd in eq 4 where the order with respect to species L, n, is zero or a rate constant without unresolved concentration dependences (ktrue): kobsd ¼ ktrue ½Ln
d½Lm M ¼ kobsd ½Lm M dt
Published: April 15, 2011 932
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Scheme 1. Proposed Mechanism for C60 Displacement by Solvent Molecules and Subsequent Solvent Displacement by a Ligand from (η2-C60)M(CO)5a
a
S1 = solvent1, S2 = solvent2, and L = ligand.
Table 1. Mass of Various Coin Collections Number of
Number of
Total Mass
Penniesa (NP)
Dimesa (ND)
(MC)/g
10 9
0 1
25.022b 24.792
8
2
24.550
7
3
24.388
6
4
24.004
5
5
23.835
4
6
23.617
3
7
23.377
2 1
8 9
23.150 22.839
10
22.655b
0 a
Figure 1. (A) Plot of the mass of the coin collection (MC) versus number of dimes (ND) (blue circles) and plot of the mass of the coin collection (MC) versus number of pennies (NP) (black squares). The function that best describes the relation between MC and NP is MC = 0.239(4) NP þ 22.65(2). Because the slope and intercept of MC versus NP plot are (PAM DAM) and 10 DAM, respectively, then the estimated average mass of pennies (PAM) and dimes (DAM) are 2.504(1) g and 2.265(2) g, respectively. Likewise, the function that best describes the relation between MC and ND is MC = 0.239(4) ND þ 25.03(2). Because the slope and intercept of MC versus ND plot are (DAM PAM) and 10 PAM, respectively, then the estimated average mass of pennies (PAM) and dimes (DAM) are 2.503(2) g and 2.264(4) g, respectively. (B) Plot of kobsd versus [cyclohexane] (red circles) and plot of kobsd versus [benzene] (black squares) for solventligand exchange on MLm.a The function that best describes the relation between kobsd and [cyclohexane] is kobsd = 0.00029(1) [cyclohexane] þ 0.00516(9). Because the slope and intercept of kobsd versus [cyclohexane] are (k1,cyclohexane 1.22(6) k1,benzene) and [benzene]0k1,benzene, respectively, then the estimated k1,cyclohexane and k1,benzene values are 0.00033(4) s1 and 0.00051(2) s1, respectively. Similarly, the function that best describes the relation between kobsd and [benzene] is kobsd = 0.000242(9) [benzene] þ 0.00271(5). Because the slope and intercept of kobsd versus [benzene] are (k1,benzene 0.816(7) k1,cyclohexane) and [cyclohexane]0 k1,cyclohexane, respectively, then the estimated k1,cyclohexane and k1,benzene values are 0.000328(6) s1 and 0.00051(1) s1, respectively.b.
b
Total number of coins is 10. These two data points, not shown to students in the activity, are included here to show the pennies average mass (2.5022 g) and the dimes average mass (2.2655 g).
’ DESCRIPTION OF THE ACTIVITY In a typical activity, eq 6 and kobsd values as function of benzene and cyclohexane concentrations (see the Supporting Information) are presented to the students.a The students are instructed to anonymously work the following exercise: estimate k1,benzene and k1,cyclohexane using eq 6 and the information provided. The students seldom proposed graph construction and analysis to answer the posed exercise. For example, in one of the activities, only 3 out of 16 students proposed kobsd versus [benzene] (or kobsd versus [cyclohexane]) plots as a solution to the posted exercise. Graphic interpretations were partially correct in the answers of the three students who proposed graph constructions. Two of the students incorrectly answered that the plots’ slopes were equal to k1,benzene or k1,cyclohexane, depending on the actual plot (no intercept interpretation was provided). One of the students correctly answered that plots’ intercepts were equal to k1,solvent[solvent]0 (k1,benzene[benzene]0 or k1,cyclohexane[cyclohexane]0, depending on
the plot) but incorrectly answered that plots’ slopes were equal to k1,benzene or k1,cyclohexane. The students were then instructed to sketch kobsd versus [benzene] and kobsd versus [cyclohexane] plots and describe and interpret the slopes and intercepts in terms of the exercise posted. The same three students did the correct sketch and the same partially correct interpretation as above. In the next part of the activity, the students were presented with various collections of coins (pennies and dimes), each collection containing 10 coins. The varying number of pennies (NP) and number of dimes (ND) in each collection were counted (NP þ ND = 10) and each collection containing 10 coins was weighed. The information contained in Table 1 was presented to the students in the activity. The students then were asked to estimate the 933
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Figure 2. Plot of [benzene] versus [cyclohexane] (red circles). The function that describes the relation between [benzene] and [cyclohexane] is [benzene] = 1.22(1) [cyclohexane] þ 10.13(7); the ratio (d[benzene]/d[cyclohexane]) = 1.22(1). Plot of [cyclohexane] versus [benzene] (black squares). The function that describes the relation between [cyclohexane] and [benzene] is [cyclohexane] = 0.816(7) [benzene] þ 8.27(3). The ratio (d[cyclohexane]/d[benzene]) = 0.816(7).
pennies’ and dimes’ average mass using the information provided in Table 1. This specific exercise sparked lively discussions but few students constructed a graph or used graphical analysis to answer the posed question. For example, 5 out of 16 students proposed to construct plots of total mass versus number of pennies and total mass versus number of dimes. The students either disregarded interpretation or provided an incorrect interpretation of the slope, but consistently provided the correct interpretation of the intercept. After about 10 min of discussion, the instructor guided the students to justify the following relationship MC ¼ ðNP PAMÞ þ ðND DAMÞ
Figure 3. (A) Plot of (MC/ND) versus (NP/ND) (blue squares). The equation that describes the relation between (MC/ND) and (NP/ND) is (MC/ND) = 2.50(1) (NP/ND) þ 2.265(4). Plot of (MC/NP) versus (ND/NP) (red triangles). The equation that describes the relation between (MC/NP) and (ND/NP) is (MC/NP) = 2.260(2) (NP/ND) þ 2.508(5). (B) Plot of (kobsd/[cyclohexane]) versus ([benzene]/[cyclohexane]. The equation that describes the relation between (kobsd/[cyclohexane]) and ([benzene]/[cyclohexane] is (kobsd/[cyclohexane]) = 0.000517(4) ([benzene]/[cyclohexane]) þ 0.00032(2).
ð7Þ
where MC is the total mass of the collection containing the 10 coins, PAM is the pennies’ average mass, and DAM is the dimes’ average mass. Students were asked again to estimate the pennies’ and dimes’ average mass employing the information in eq 7. Having used this activity six times, the authors’ experience is that students invariably suggest MC versus NP and MC versus ND graphs. At this point, the instructor guided the students to do the following: (i) sketch the graphs, (ii) describe the graphs, and (iii) interpret the graphs in terms of the original question (estimate the average mass of the pennies and dimes). In this exercise, the students’ incorrect interpretation persisted that the slope of MC versus NP plot is equal to the pennies’ average mass. To confront them with this incorrect interpretation, the instructor asked the students to interpret the decreasing plot of MC versus ND. Because mass must be a positive value, the students readily realized that the slope of MC versus ND plot can not be equal to the dimes’ average mass. A simple mathematical manipulation (eqs 8 and 9) shows students that plots of MC versus NP and MC versus ND are expected to be linear (Figure 1). ΔMC ΔND ¼ PAM þ DAM ¼ PAM DAM > 0 ΔNP ΔNP
ð8Þ
ΔMC ΔNP ¼ DAM þ PAM ¼ DAM PAM < 0 ΔND ΔND
ð9Þ
rate of change for each must be equal to 1. Although the model using coins involves discrete data, the chemical situation being modeled involves a continuous, differentiable curve. Hence, we propose that the quantities (ΔNP/ΔND) and (ΔND/ΔNP) in a fixed coin collection are analogous to the quantities (d[solvent1]/ d[solvent2) and (d[solvent2]/d[solvent1) in binary solvent mixtures (vide infra) (Figure 2). Because the pennies’ average mass is greater than the dimes’ average mass, the rate of change of MC with respect to NP is positive ((ΔMC/ΔNP) > 0) and the rate of change of MC with respect to ND is negative ((ΔMC/ΔND) < 0). The intercept values of the plots in Figure 1 are related to 10 times the coins’ average mass (10 PAM and 10 DAM) as indicated by eqs 10 and 11, obtained after the integration and evaluation of eqs 8 and 9. Because there is a total of 10 coins in each collection, when NP = 0, MC = (10 DAM) and vice versa when ND = 0, MC = (10 PAM), and eqs 8 and 9 become
This is because their slopes are equal to (PAM DAM) and (DAM PAM), respectively. As the number of pennies increases by 1, the number of dimes must decrease by 1, and vice versa. So, the
MC ¼ ðPAM DAMÞNP þ 10DAM
ð10Þ
MC ¼ ðDAM PAMÞND þ 10PAM
ð11Þ
Alternatively, PAM and DAM values can be estimated from the slope and intercept values of MC/NP versus ND/NP (or MC/ND 934
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k1,benzene and k1,cyclohexane values can be estimated from the slope and intercept values of the kobsd versus [benzene] (or kobsd versus [cyclohexane]) plots. Alternatively, according to eqs 17 and 18, k1,benzene and k1,cyclohexane values can be estimated from the slopes and intercepts of kobsd/[benzene] versus [cyclohexane]/[benzene] or kobsd/[cyclohexane versus [benzene]/[cyclohexane] plots (Figure 3).
Table 2. Analogies between Coins and SolventC60 Exchange on (η2-C60)W(CO)5 Coin Analogy eq 7
(η2-C60)W(CO)5 System eq 6
eqs 10and 11 ΔNP ΔND orviceversa
eqs 14 and 15
Number of coins (NP and ND)
[solvent1] and [solvent2]
Coin average mass (PAM and DAM)
k1,solvent1 andk1,solvent2
Total mass of coins collection (MC)
kobsd
10 DAM and 10 PAM
[cyclohexane]0k1,cyclohexane and [benzene]0k1,benzene
eqs 12 and 13
eqs 17 and 18
d½cyclohexane d½benzene
orviceversa
vs NP/ND) linear plots (Figure 3) as indicated by eqs 12 and 13. MC ND ¼ DAM þ PAM NP NP
ð12Þ
MC NP ¼ PAM þ DAM ND ND
ð13Þ
’ CONNECTION WITH CHEMICAL KINETICS CONCEPTS: AN ACTUAL EXAMPLE “The Sunlight-Induced Photosynthesis of (η2-C60)M(CO)5 Complexes”10 and “[60]Fullerene Displacement from (DihaptoBuckminster-Fullerene) Pentacarbonyl Tungsten(0)”11 are educational activities published in this Journal that promote integration of physical chemistry (kinetics concepts) and inorganic chemistry (electronic structure and reactivity). Fine tuning the mechanistic description by including the role of the solvent in the ligandC60 exchange provides a chemical example of the coin analogy.11,12 In this example, the solventligand exchange on (η2-C60)W(CO)5 is performed in benzene/cyclohexane mixtures. Expressing eq 6 with solvent1 = benzene and solvent2 = cyclohexane provides an immediate connection with eq 7 for the coin analogy. Similarly, eqs 10 and 11 for the coin analogy are related to eqs 14 and 15 for the solventligand exchange on (η2C60)W(CO)5 in benzene/cyclohexane mixtures.
kobsd
d½cyclohexane ¼ k1, benzene þ k1, cyclohexane ½benzene d½benzene þ ½cyclohexane0 k1, cyclohexane
ð14Þ
d½benzene kobsd ¼ k1, cyclohexane þ k1, benzene ½cyclohexane d½cyclohexane þ ½benzene0 k1, benzene
ð15Þ
In these eqs, [cyclohexane]0 is the molar concentration of pure cyclohexane and [benzene]0 is the molar concentration of pure benzene. The ratios (d[cyclohexane]/d[benzene]) and (d[benzene]/d[cyclohexane]) values, 0.816(7) and 1.22(1), can be estimated from [cyclohexane] versus [benzene] and [benzene] versus [cyclohexane] plots (Figure 2), respectively. Thus, kobsd versus [benzene] and kobsd versus [cyclohexane] plots are expected to be linear with slope equal to {k1,benzene 0.816(7) k1,cyclohexane} and {k1,cyclohexane 1.22(1) k1,benzene}, respectively.
kobsd ½cyclohexane ¼ k1, benzene þ k1, cyclohexane ½benzene ½benzene
ð17Þ
kobsd ½benzene ¼ k1, cyclohexane þ k1, benzene ½cyclohexane ½cyclohexane
ð18Þ
’ DISCUSSION Because this activity is directed to upper-division undergraduate and graduate students, the instructor can involve students in discussions at various cognitive levels: (i) about the validity and reach of the analogy, (ii) about how the analogy connection with the kinetic model allows a meaningful assessment of the understanding of kinetics concepts, and (iii) about the experimental designs to test the validity of approximations or assumptions made in the activity. Table 2 contains the analogies between terms and equations. For example, in the activity, the quantities (ΔNP/ΔND) and (ΔND/ΔNP) are analogous to the quantities (d[cyclohexane]/ d[benzene]) and (d[benzene]/dcyclohexane]). Similar to plots of NP versus ND and ND versus NP, plots of [benzene] versus [cyclohexane] and [cyclohexane] versus [benzene] presented here are linear and the quantities (d[benzene]/d[cyclohexane]) and (d[cyclohexane]/d[benzene]) are constant values. Aspects related to the validity of the analogy can be included in the discussion. For example, the following questions “What is the physical-chemical meaning of [solvent] versus [solvent] linear plots?” and “Would the analogy be valid if the [solvent] versus [solvent] plots were nonlinear?” encouraged students’ participation. As a result of the discussion, some students proposed that the quantities (d[benzene]/d[cyclohexane]) and (d[cyclohexane]/d[benzene]) are equal to the slopes of [benzene] versus [cyclohexane] and [cyclohexane] versus [benzene] plots, respectively. Most students, graduate and undergraduate, correctly established the analogy between (ΔNP/ΔND) [or (ΔND/ΔNP)] and (d[benzene]/d[cyclohexane]) [or (d[cyclohexane]/d[benzene])]. Most undergraduate students stated that the analogy would still be valid even if [solvent] versus [solvent] plots were nonlinear. However, most graduate students proposed that nonlinear [solvent] versus [solvent] plots may reflect that volumes in solvent mixtures are not quantitatively additive and that the analogy in such cases may not be applicable. ’ CONCLUSION The activity provides opportunity for the assessment of conceptual learning via designed activities. It promotes graphical analysis as well as the integration of mathematical and kinetics concepts. Most importantly, it discourages rote learning and application of mathematical procedures without an underlying understanding of the chemical concepts involved. 935
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’ ASSOCIATED CONTENT
(14) Lide, D. R., Kehiaian, H. V. Handbook of Thermophysical and Thermochemical Data; CRC Press: Boca Raton, FL, 1994; pp 1525.
bS
Supporting Information Values of kobsd for the C60solventL exchange on W(CO)5C60 in benzene/cyclohexane mixtures at 74.2 °C. Plots in Figures 13 (inset) were constructed with information contained in this table. This material is available via the Internet at http://pubs.acs.org.
’ AUTHOR INFORMATION Corresponding Author
*E-mail: jose.cortes.fi
[email protected].
’ ACKNOWLEDGMENT The ideas presented in this article came from working on the project Electronic Structure, Molecular Structure, and Site of Bond Breaking in Facial and Meridional-(Dihapto-[60]fullerene)(Dihapto-Bidentate Ligand) Transition Metal(0) Complexes. The authors gratefully acknowledge the financial support to this project by The Donors of the Petroleum Research Fund, administered by the American Chemical Society (grant ACS-PRF-41267-B3). Helpful comments from reviewers are gratefully acknowledged. ’ ADDITIONAL NOTE a We use actual data from our laboratory (Table 1 of Supporting Information) for displacement of C60 Fullerene from (η2C60)W(CO)5. In the general formula LmM, M = W; L = C60 and CO; m = 6; solvent1 = benzene, solvent2 = cyclohexane).12 b
The concentration of pure solvents ([solvent]0) can be determined from their density and molar mass at the given temperature (= 1000 density/M). Densities at various temperatures for a variety of solvents are given in refs 13 and.14. The values [benzene]0 = 10.13(7) M and [cyclohexane]0 = 8.27(3) M reported in the text were obtained from the intercepts of [solvent] versus [solvent] linear plots. Error limits of quantities reported are given in parentheses as the uncertainties of the last(s) digit(s).
’ REFERENCES (1) Noggle, J. H. Physical Chemistry, 2nd ed.; Scott, Foreman and Co: Glenview, 1989; pp 528545. (2) Espenson, J. H. Chemical Kinetics and Reaction Mechanisms, 2nd ed.; McGraw-Hill: New York, 1995; pp 155157. (3) Espenson, J. H. Chemical Kinetics and Reaction Mechanisms, 2nd ed.; McGraw-Hill: New York, 1995; p 160. (4) Zhang, S.; Dobson, G. R.; Zang, V.; Bajaj, H. C.; van Eldik, R. Inorg. Chem. 1990, 29, 3477. (5) Bengali, A. A.; Stumbaugh, T. F. Dalton Trans. 2003, 354. (6) Wells, J. R.; House, P. G.; Weitz, E. J. Phys. Chem. 1994, 98, 8343. (7) Felix-Massa, T.; Cortes-Figueroa, J. E. J. Chem. Educ. 2010, 87, 426–428. (8) Zhang, S.; Dobson, G. R. Inorg. Chim. Acta 1991, 181, 103. (9) Bengali, A. A.; Charlton, S. B. J. Chem. Educ. 2000, 77, 1348. (10) Cortes-Figueroa, J. E. J. Chem. Educ. 2003, 80, 799–800. (11) Cortes-Figueroa, J. E.; Moore-Russo, D. A. J. Chem. Educ. 2006, 83, 1670–1673. (12) Cortes-Figueroa J. E. Dalton Trans., to be submitted for publication. (13) Singh, R. P.; Sinha, C. P.; Das, J. C.; Ghoshs, P. J. Chem. Eng. Data 1989, 34, 335–338. 936
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