In the Classroom
Secondary School Chemistry
W
Understanding Electrochemical Thermodynamics through Entropy Analysis Thomas H. Bindel 1 Pomona High School, 8101 West Pomona Drive, Arvada, CO 80005;
[email protected] I would like to present my thoughts on the teaching of thermodynamics and to describe a guided-discovery (1) activity in which entropy analysis (2–4) is applied to galvanic cells. The activity guides students into discovering the factors that directly determine electrochemical cell potential, while reinforcing the basic components and terminology of galvanic cells. While the activity is most appropriate for honors chemistry, advanced-placement chemistry, and first-year college chemistry courses, I have used it successfully in my first-year high-school chemistry classes for many years.2 I believe that thermodynamics can be taught effectively at the high-school level. The teaching and presentation of my unit on thermodynamics is approached in a nontraditional way.3 Entropy is defined in terms of energy dispersal (5–7), instead of the usual “disorder”. Reactions or processes are analyzed exclusively in terms of entropy and the second law of thermodynamics (2–4, 7–10). All too often, students are shown the Gibbs free energy relationship without an explanation of where it comes from or how it is rooted in the second law. This creates a shaky foundation upon which to build and leads to conceptual misunderstandings. Craig, a proponent of entropy analysis, has stated “the total entropy change is a more fundamental and more general index of spontaneous change than is ∆G ” (2). Atkins has indicated Table 1. Enthalpy of Formation and Entropy of Substances (3) Substance Ag(s)
∆Hf°/kJ mol ᎑1 S°/J K᎑1 mol ᎑1 0.0
42.6
Ag+(aq)
105.6
72.7
AgCl(s)
᎑127.1
96.2
᎑
Cl (aq)
᎑167.2
Cr3+(aq)
᎑255.4 a
Cr2O72᎑(aq) Cu(s) Cu2+(aq) Fe(s)
᎑1490.3 0.0
33.2 ᎑99.6 27.3 ᎑137.7
Fe3+(aq)
᎑48.5
᎑315.9
H+(aq)
0.0
0.0
H2O(ᐉ)
᎑285.8
69.9
0.0
32.0
2+
᎑220.8
᎑73.6
Pb(s)
0.0
64.8
2+
Pb (aq) Zn(s) 2+
Zn (aq) aThe
᎑1.7
10.5
0.0
41.6
᎑153.9
᎑112.1
change in Gibbs free energy was calculated for eq 6 using Gibbs free energies of formation. The enthalpy of formation for Cr3+ was then calculated from this ∆ rG.
Mn(s) + Cu2+(aq) → Mn2+(aq) + Cu(s)
(1)
→
+ Cu(s)
(2)
Zn(s) + Cu (aq) → Zn (aq) + Cu(s)
(3)
→
(4)
Pb(s) +
Cu2+(aq)
Pb2+(aq)
2+
Cu(s) +
261.9
᎑89.1
Mn (aq)
Set 1: Gathering Data
56.8
Fe2+(aq)
Mn(s)
Student Preparation Students should be somewhat familiar with the concepts of redox, half-reactions, galvanic or voltaic cells: anode, cathode, electrode polarity, and salt bridge. Prior to the activity, students calculate the change in the entropy of the universe (∆S univ ° )4 at 25 °C for eqs 1–6. (See Tables 1 and 2.)
᎑308
64.8 0
that Gibbs free energy disguises the entropy (11). This is not to say that Gibbs free energy should not be taught, but rather that an entropy basis should be established first. The topics I cover in the second-semester chemistry course are, in order, specific heat (1), enthalpy, entropy analysis and spontaneity, thermodynamic equilibrium (general equilibrium and Le Châtelier’s principle), kinetics, solubility equilibrium, and acid–base equilibrium. Within the entropy analysis and spontaneity unit, electrochemistry is introduced to a small extent, enough to allow students to successfully apply thermodynamics to galvanic cells. It has been my goal to support the thermodynamics unit with classroom activities. One such activity appeared in this Journal (8). I describe here a second, dealing with galvanic cells. A number of experiments using galvanic cells have appeared in this Journal : cell potential and metal activity (12), introduction to standard reduction potential table (13), galvanic cells using sodium silicate gels (14), and the Nernst equation (15). None of these provides an avenue for the student to discover the factors that relate to the cell potential.
2Ag+(aq)
2+
Cu2+(aq)
+ 2Ag(s)
Table 2. Thermodynamic Calculations at 25.O °C Eq
∆rH°/
∆S°surr /
∆rS°sys /
∆S°univ /
∆rG°/
᎑293.7
(kJ/mol rxn) (J/K mol rxn)a (J/K mol rxn) (J/K mol rxn)b (kJ/mol rxn)c.
1
᎑285.6
2
᎑66.5
3
᎑218.7
4 5
957.9
27.2
985.1
223
78.5
302
᎑90.0
733.5
᎑20.9
᎑146.4
491.0
᎑193.0
298.0
᎑88.85
᎑65.5
219.7
᎑33.3
186.4
᎑55.58
712.6
᎑212.5
6
᎑826.7
2774
᎑786.8
1987
᎑592.4
7
᎑365.1
1225
᎑213.9
1011
᎑301.4
1449
᎑165.8
1283
8
᎑432.0
9
᎑153.9
516.2
᎑32.2
484.0
10
᎑16.4
55.0
223.6
278.6
᎑382.5 ᎑144.3 ᎑83.06
from ∆S °surr = (᎑∆ rH ° × 1000 J/kJ)/T; T = 298.15 K. b∆S ° ° + ∆ rS °sys. univ = ∆S surr c∆ G ° = ᎑T∆S ° /(1000 J/kJ); T = 298.15 K. r univ aCalculated
JChemEd.chem.wisc.edu • Vol. 77 No. 8 August 2000 • Journal of Chemical Education
1031
In the Classroom
Set 2: The Importance of “n” Ag(s) + Cl᎑(aq) → e᎑ + AgCl(s) Ag+(aq) + e᎑ → Ag(s) __________________________ Ag+(aq) + Cl᎑ → AgCl(s)
(5)
Cr2O72᎑(aq) + 14H+(aq) + 3Cu(s) →
(6)
2Cr3+ + 7H2O(ᐉ) + 3Cu2+(aq)
Set 3: Predictions Zn(s) + 2 Ag+(aq) → 2Ag(s) + Zn2+(aq)
(7)
Mn(s) + 2 Ag (aq) → 2Ag(s) + Mn (aq)
(8)
Fe(s) + Cu2+(aq) → Fe2+(aq) + Cu(s)
(9)
Cu(s) + 2Fe3+(aq) → 2Fe2+(aq) + Cu2+(aq)
(10)
+
2+
Sample Calculations for Eq 1 ∆rH ° = νMn + ∆Hf°(Mn ) + νCu ∆Hf°(Cu) – [νMn∆Hf°(Mn) + νCu + ∆Hf°(Cu2+)] 2+
2
2
= (1 mol Mn2+/mol rxn)(᎑220.8 kJ/mol) + (1 mol Cu/mol rxn)(0.0 kJ/mol) – [(1 mol Mn/mol rxn)(0.0 kJ/ mol) + (1 mol Cu2+/mol rxn) (64.8 kJ/mol)] = ᎑285.6 kJ/mol rxn ∆rS sys ° = νMn +S°(Mn2+) + νCuS°(Cu) – [νMnS°(Mn) + νCu + S°(Cu2+)] 2
2
2+
= (1 mol Mn /mol rxn)(᎑73.6 J/K mol) + (1 mol Cu/mol rxn)( 33.2 J/K mol) – [(1 mol Mn /mol rxn)(32.0 J/K mol) + (1 mol Cu2+ /mol rxn)(᎑ 99.6 J/K-mol)] = 27.2 J/K mol rxn ∆S s°urr = (᎑∆rH ° × 1000 J/kJ)/298.15 K
Part II: Determination of the Faraday Constant The equation of the line obtained from part I should have the form E = k∆Suniv + 0, where k, a constant, is the slope of the line and has units of V/(J/K) or (V)(K)/J. Also note that the y-intercept for the equation from part I has a value close to zero. It should be close enough to zero for the student to replace it with zero. Alternatively, the student can force the intercept to zero and determine a new best-fit line. [slope = 1.47 × 10᎑3 (V )(K)/J .] The constant is really a collection of constants, as indicated by the equation k = T /(n F ), where T represents absolute temperature, n represents the number of moles of electrons involved in the reaction, and F represents the Faraday constant. The student substitutes into the equation the appropriate values and units for k, T, and n and solves for the value and
Table 3. Cathodes and Cell Potentials for Galvanic Cells Galvanic Cella
Cathode
Ecell / Vb
E°/ Vc
Cu
1.43
1.52
Mn(s) / Mn2+(aq) // Cu2+(aq) /Cu(s)
= (285.6 kJ/mol rxn)(1000 J/kJ)/298.15 K = 957.9 J/K mol rxn
Pb(s) / Pb (aq) // Cu (aq) /Cu(s)
Cu
0.48
0.47
Zn(s) / Zn2+(aq) // Cu2+(aq) /Cu(s)
Cu
1.07d
1.10
° + ∆rS sys ° = 985.1 J/K mol rxn ∆S univ ° = ∆S surr
Cu(s) / Cu2+(aq) // Ag+(aq) /Ag(s)
Ag
0.41
0.46
Ag(s) / Cl᎑(aq), AgCl // Ag+(aq) /Ag(s)
Ag
0.52
0.58
2+
2+
Cu(s)/Cu2+(aq)//Cr2O72᎑(aq),Cr3+(aq)/C
Activity Galvanic cells are constructed using nitrate and chloride salts (0.1 M), except for the Fe2+, Mn2+, and Cr3+ salts, in which sulfates (0.1 M in the metal ion concentration) were used. Laboratory materials, experiment construction, procedure, and student handout are supplied online.W There are two parts to the activity. The second part is for more advanced students upon completion of the first part. Answers are given in italics within brackets.
Part I The student makes 6 galvanic cells, one for each of the equations 1–6, then (i) measures the cell potential and (ii) determines the cathode. [See Table 3 for results.] For eqs 1–4, the student graphs the cell potential vs ∆S °univ and draws a best-fit line. (Best Fit Line: E = (1.45 × 1032
10 ᎑3)∆S°univ + 0.014 V; “E” is the measured cell potential; R2 = .99615). Next, the student plots the data for eqs 5 and 6 on the same graph. The silver (eq 5) and chromium (eq 6) cells lie outside the line of best fit and the student speculates as to why. [The number of moles of electrons transferred per mol rxn, n, is 1 for eq 5 and 6 for eq 6, which are different from eqs 1–4 where n is 2.] The student predicts the cell potential and cathode for the couples derived from eqs 7–10 through entropy analysis (calculation of ∆Suniv) and then either interpolation or extrapolation of the graph. [Eq 7: Extrapolation gives 1.48 V with Ag as the cathode. Eq 8: Extrapolation gives 1.87 V with Cu as the cathode. Eq 9: Interpolation gives 0.72 V with Cu as the cathode. Eq 10: Interpolation gives 0.42 V with an inert electrode, such as graphite, as the cathode.] The student constructs the cells, measures the potentials, and determines the cathodes (see Table 3).
C
0.78
0.99
Zn(s) / Zn2+(aq) // Ag+(aq) /Ag(s)
Ag
1.50
1.56
Mn(s) / Mn2+(aq) // Ag+(aq) /Ag(s)
Ag
1.83
1.98
Fe(s) / Fe2+(aq) // Cu2+(aq) /Cu(s)
Cu
0.62
0.78
0.36
0.43
Cu(s) / Cu2+(aq) // Fe2+(aq),Fe3+(aq) /C
Ce .
aThese are written in conventional terms: anode, metal solution, salt bridge, metal solution, cathode. bElectrodes were not cleaned between uses. Nitrates and chlorides were used, except in the case of Fe2+, Cr3+, and Mn2+, for which sulfates were used. “Porous ceramic cup” cells and mini-galvanic cells(18) gave similar cell potentials. cStandard cell potentials calculated from standard reduction potentials relative to NHE (17 ). d The zinc half-cell also contains 0.1 M Na SO . Without the Na SO , 2 4 2 4 the cell potential is 0.83 V. A behavior similar to this has been noted in the literature (14). ePlatinum
electrode gave the same cell potential.
Journal of Chemical Education • Vol. 77 No. 8 August 2000 • JChemEd.chem.wisc.edu
In the Classroom
units of the Faraday constant at 25 °C. [10.3 × 104 C/mol e᎑ for a slope of 1.45 × 10᎑3 ( V)(K)/J and 10.1 × 104 C/mol e᎑ for a slope of 1.47 × 10᎑3 ( V )(K)/J.] The actual value of the Faraday constant is 96,500 C/mol e᎑ (3 significant digits).6 Students calculate the percent error in their values for the Faraday constant. [7% and 5% error, respectively.] Optional. The student graphs E° vs ∆Su°niv and determines the line of best fit. [E° = 1.56 × 10᎑3 ∆S°univ + 7.3 × 10᎑4.] The student determines the value of the Faraday constant and the % error. [9.56 × 104 C/mol e᎑, 0.9% error.] Discussion
General Comments There is no need to discard the solutions after the activity; they can be and should be stored until the next use. The salt bridges may be stored in a large sealed container with a small quantity of water to prevent dehydration. All this serves many purposes: (i) minimizes the cost of materials; (ii) eliminates the waste of resources; (iii) reduces the impact to the environment when disposing of the materials; and (iv) saves time in making solutions and properly disposing of materials. High-input impedance digital multimeters should be used in measuring the Ecell values. Radio Shack Micronta Digital Multimeters (cat. No. 22-193) were used. Also, it should be noted that exact agreement with the E ° values is not expected for a couple of reasons: (i) standard state concentrations were not used, and (ii) E ° values are not obtained through a single measurement, but rather are determined by extrapolation. There may be a question of whether mini-galvanic cells (18) or cells constructed from porous ceramic cups7 would give better results than cells constructed with agar salt bridges. Both mini-galvanic cells and cells with porous ceramic cups gave similar results to the cells with agar salt bridges used in this activity. The use of mini-galvanic cells may eliminate some of the advantages noted above. The use of either minigalvanic cells or “porous-cup” cells could possibly lead to contamination or intermixing of the solutions in the half-cells. Mini-galvanic cells have the advantage that they require less solution and so would be less hazardous and less expensive. However, over the course of many class periods during the day and over many years of doing the activity, this advantage would be lost. Overall, I feel that the beakers and agar salt bridges are much easier for students to handle and there is less of a problem from contamination of the half-cells. If the availability or cost of manganese makes it difficult to obtain, then eq 7 may be substituted for eq 1 and eq 8 can be eliminated from the activity. If eq 7 is substituted for eq 1, then the best-line fit for the first four equations will be E = 0.00149 ∆Suniv + 0.0013 V
R 2 = .9973
Chromium salts are hazardous (chromium(III) sulfate is a suspected carcinogen and potassium dichromate is an alleged carcinogen [19]) and for this reason some instructors may choose not to use eq 6.
Part I A galvanic cell has the ability to store energy from a chemical system as electrical energy. The amount of stored electrical energy is equal to the absolute value of the Gibbs
free energy, |∆rG |, for reversible systems. The absolute value of the Gibbs free energy (see Table 2) for eqs 3–9 is less than the absolute value of the enthalpy of reaction, which is a direct result of the decrease in the entropy of the system (∆rS sys ° ). A portion of the enthalpy must be converted into enough thermal energy to offset the decrease in the entropy of the system. The amount of the enthalpy that must be converted or “wasted” is equal to T |∆rS s°ys |. As a consequence, the amount of electrical energy available from these chemical systems is equal to |∆ rH | – T |∆ rSsys|. For example, in the case of the cell constructed from eq 4, only 61% ((88.85/146.4) × 100) of the enthalpy is available as electrochemical energy. On the other hand, the absolute value of the Gibbs free energy for eqs 1, 2, and 10 is greater than the absolute value ° . Since the of the enthalpy, a result of an increase in ∆rS sys system is increasing in entropy, the surroundings can decrease in entropy to the same extent. The electrochemical cell converts some of the thermal energy of the surroundings into electrical energy, thereby adding to the energy available from the change in enthalpy of the reaction. The amount of electrical energy available from these chemical systems is equal to |∆ rH | + T∆ rSsys . The cell potential is determined primarily by two factors, the amount of energy available to do work (free energy) and the number of moles of electrons transferred, based on the chemical equation. The units of cell potential, V or J C᎑1, indicate an energy term (joule) per quantity of electrons (coulomb). The essence of this is captured in the familiar equation ∆ rG ° = ᎑nFE °. Solving for the potential gives E ° = ᎑∆ rG °/(nF ). This equation can also be expressed as E ° = T∆S°univ /(nF ). It is this last equation that forms the basis for the analysis of cell potential in terms of both ∆S °univ and n. In the first part of the activity, students realize that ∆S °univ directly relates to the cell potential. Later, from cells 5 and 6, they realize that possibly the number of moles of electrons transferred per mol of rxn, n, can also influence the cell potential. If the best-fit line from eqs 1–4 (E = (1.45 × 10᎑3)∆S °univ + 0.014 V and n = 2) is used, eqs 5 (n = 1) and 6 (n = 6) have predicted cell potentials of 0.28 and 2.89 V, respectively. The ratios of the predicted potentials to the measured cell potentials are 0.54 and 3.7, respectively. These roughly correspond to the ratios of the n’s 8 of 1:2 and 6:2, respectively. Through an in-class discussion, students are led to understand that cell potential (“voltage”) represents the amount of energy available when a quantity of electrons move within an electric potential, whether in an electrochemical cell or battery or in a household outlet. Students realize that multiplying the ∆S °univ by the temperature of 298 K (room temperature) converts the entropy into the negative of free energy at 25 °C, G, the energy capable of doing useful work. It is this energy that becomes available when “n” moles of electrons (determined from the chemical equation) move within an electric potential difference. That is, G is proportional to nE. Thus the more moles of electrons transferred for a given reaction, the less the electric potential, and vice versa. At the end of part I, students make predictions and test the predictions. The galvanic cells constructed from eqs 7 and 8 give good results (1% and 2% error, respectively). The results from cells corresponding to eqs 9 and 10 are not as good (16% and 14% error, respectively). Equation 10 represents an
JChemEd.chem.wisc.edu • Vol. 77 No. 8 August 2000 • Journal of Chemical Education
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In the Classroom
additional example of a galvanic cell with an inert electrode, graphite or platinum. It should be noted that the minigalvanic cell constructed from eq 10 produced a cell potential of 0.39 V, reducing the error to 7%.
Part II In the advanced part of the activity, students have an opportunity to derive a value for the Faraday constant, along with units. Students found errors within 7% of the accepted value. When the E ° values in Table 3 are used in the optional part of the activity, the value calculated for the Faraday constant is within 1% of the accepted value. Acknowledgments I would like to thank the following people: John C. Fochi and RoseMary Bindel for their assistance and invaluable support; Leah Phillips for her work on mini-galvanic cells; and Donald C. Zapien, from the department of chemistry at the University of Colorado at Denver, for conversations concerning the addition of sodium sulfate to galvanic cells and possible half-cells that would lead to different n’s. Lastly, I would like to thank the reviewers, especially #5, for suggestions that have led to a much improved manuscript. W
Supplemental Material
Supplemental material for this article is available in this issue of JCE Online. Notes 1. The author is known by his students as “Captain Carbon”. 2. The language and symbols of thermodynamics are simplified at the first-year high-school level. It is very important that entropy be introduced conceptually and dealt with from an “energy dispersal” point of view. A solid conceptual knowledge base needs to be established before any equations are handed to students. 3. In fact, when I learned this approach, the field of thermo-
1034
dynamics became more understandable and clear, and a certain “beauty” associated with it became evident. 4. There are many different symbols that can be used. For example, ∆Sθ is the same as ∆Ssurr, ∆Sσ is the same as ∆Ssys, and ∆Stot is the same as ∆Suniv. I use sys, surr, and univ because I feel high-school students will find them easier to use and remember. 5. “R” is the Pearson linear correlation coefficient. 6. A more precise value is 9.6485309 × 104 C mol᎑1 (16 ). 7. Similar in construction to a Daniell cell. 8. E (predicted) = ∆Suniv × 298/(2 F); E (measured) = ∆Suniv × 298/(n F ) E (predicted)/E (measured) = n/2.
Literature Cited 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.
Bindel, T. H.; Fochi, J. C. J. Chem. Educ. 1997, 74, 955. Craig, N. C. J. Chem. Educ. 1988, 65, 760. Craig, N. C. Entropy Analysis; VCH: New York, 1992. Craig, N. C. J. Chem. Educ. 1996, 73, 710. Lowe, J. P. J. Chem. Educ. 1988, 65, 403. Barón, M. J. Chem. Educ. 1989, 66, 1001. Atkins, P. W. The 2nd Law: Energy, Chaos, and Form; Scientific American Books: New York, 1994. Bindel, T. H. J. Chem. Educ. 1995, 72, 34. Brosnan, T. J. Chem. Educ. 1990, 67, 48. Craig, N. C. J. Chem. Educ. 1987, 64, 470. Atkins, P. W. Op. cit.; p 171. Lamba, R; Sharma, S.; Lloyd, B. W. J. Chem. Educ. 1997, 74, 1095. Tanis, D. O. J. Chem. Educ. 1990, 67, 602. Rapp, B. J. Chem. Educ. 1988, 65, 358. Hambly, G. F. J. Chem. Educ. 1985, 62, 875. Handbook of Chemistry and Physics, 76th ed.; Weast, R. C., Ed.; The Chemical Rubber Co.: Cleveland, OH, 1995. Zumdahl, S. S. Chemistry, 3rd ed.; D. C. Heath: Lexington, MA, 1993. Craig, N. C.; Ackermann, M. N.; Renfrow, W. B. J. Chem. Educ. 1989, 66, 85. Flinn Chemical Catalog Reference Manual; Flinn Scientific: Batavia, IL, 1994.
Journal of Chemical Education • Vol. 77 No. 8 August 2000 • JChemEd.chem.wisc.edu
