Galvanic Cells and the Determination of Equilibrium Constants

Mar 23, 2012 - ABSTRACT: Readily assembled mini-galvanic cells can be employed to compare their observed voltages with those predicted from the Nernst...
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Galvanic Cells and the Determination of Equilibrium Constants Jonathan L. Brosmer and Dennis G. Peters* Department of Chemistry, Indiana University, Bloomington, Indiana 47405, United States S Supporting Information *

ABSTRACT: Readily assembled mini-galvanic cells can be employed to compare their observed voltages with those predicted from the Nernst equation and to determine solubility products for silver halides and overall formation constants for metal−ammonia complexes. Results obtained by students in both an honors-level first-year course in general chemistry and an intermediate-level course for students majoring in chemistry, biochemistry, and biology agree well with data reported in the literature. In addition, these mini-galvanic cells can be used as concentration cells, together with a computer-generated calibration curve based on the Nernst equation, to determine the silver ion concentration in a dilute unknown solution of silver nitrate. KEYWORDS: First-Year Undergraduate/General, Upper-Division Undergraduate, Analytical Chemistry, Inorganic Chemistry, Laboratory Instruction, Electrochemistry, Electrolytic/Galvanic Cells/Potentials, Equilibrium, Oxidation/Reduction, Precipitation/Solubility

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ore than two decades ago, Craig and co-workers1 described a laboratory experiment involving the construction and use of mini-galvanic cells to compare experimentally measured cell voltages with those predicted theoretically with the aid of the Nernst equation. In addition, these mini-galvanic cells were proposed as a way for students to evaluate solubility products for silver halides and overall formation constants for complexes such as Ag(NH3)2+ and Cu(NH3)42+. However, no student-measured voltages for any of the mini-galvanic cells were presented, nor were solubility products or formation constants for complexes reported. Here we confirm the pedagogical value of these mini-galvanic cells by providing a summary and discussion of student results obtained from our honors introductory laboratory course for the period 1990−2000. An important feature of galvanic cells is that they can be employed to study equilibria under conditions where there is virtually no current flow and, therefore, no perturbation in the concentration of any of the solution-soluble species that comprise the individual half-cells. Thus, the conventional cell reaction written for a galvanic cell is actually a virtual process that never occurs under the conditions of a true potentiometric measurement of cell voltage; however, this virtual reaction does take place when the two electrodes are short-circuited by a conducting wire. Moreover, as is true for many galvanic cells employed in these experiments, the concentrations of potentialcontrolling species, for example, [Ag+] in saturated solutions of silver halides or [Zn2+] and [Cu2+] in equilibrium with their metal−ammonia complexes, Zn(NH3)42+ and Cu(NH3)42+, are so low that it would be difficult to determine them accurately by any reasonable procedure without perturbing the system. For these reasons, the construction and study of galvanic cells © 2012 American Chemical Society and Division of Chemical Education, Inc.

provide a powerful and reliable way to investigate chemical equilibria. Beside the report by Craig et al.,1 other articles pertain to this subject.2−6 Articles by Baca and Lewis2 and by Tanis3 describe assemblies with up to nine individual metal ion−metal halfcells, selected pairs of which can be connected with a salt bridge to create mini-galvanic cells, similar to those described by Craig et al.1 Using 12- or 24-well culture plates, Eggen et al.4 fabricated miniature galvanic cells to power a low-voltage lightemitting diode; later, the same authors5 prepared small-scale, low-cost galvanic cells consisting of Cu2+|Cu, Cl2|Cl−, and H+| H2 half-cells. Mills et al.6 describe galvanic cells, used to probe the validity of the Nernst equation, that involve the Ag+|Ag, Fe3+|Fe2+, I3−|I−, Cu2+|Cu, and quinone|hydroquinone halfreactions as cathodes, each being coupled to a AgCl|Ag anode; in addition, student-constructed galvanic cells were used to determine the formation constants for Ag(NH3)2+ and Ag(S2O3)23−, and concentration cells involving the Ag+|Ag couple with two different concentrations of Ag+ were studied. Work described here can be placed alongside electrochemistry experiments found in contemporary laboratory manuals for general chemistry. Typically, one finds experiments that deal with (a) establishing an activity series for a set of common metals (e.g., copper, iron, zinc, and tin) and (b) constructing and measuring the voltage of a Daniell cell (Zn|Zn2+||Cu2+|Cu) from large test tubes or beakers containing solutions of ZnSO4 and CuSO4 connected by a salt bridge fabricated either from a U-tube filled with an electrolyte−agar mixture or from a roll of filter paper saturated with an inert electrolyte. During the 1990s, the strategy presented by Craig et al.1 was used to develop experiments for an introductory honors-level Published: March 23, 2012 763

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solutions for galvanic cells used to determine [Ag+] in an unknown are described in the Supporting Information. For the electrodes, 5 cm lengths of Ag, Cu, and Ni wire (24 or 26 gauge) and strips (2 mm × 5 cm) of Cd, Pb, and Zn are used. After the experiment, all electrodes should be rinsed with distilled water, dried, and stored in separate containers for future use. For the construction of individual galvanic cells, the original procedure described by Craig et al.1 or in the protocol outlined in the Supporting Information can be followed. To measure cell voltages, a Beckman 31 or a Corning 340 pH meter, operated in the mV mode, is used.

laboratory course for 50 students. Although electrochemistry is no longer included in the syllabus for our current one-semester course in general chemistry, this experiment was resurrected in 2011 for a large-enrollment (160 students), intermediate-level course dealing with inorganic chemistry for students majoring in chemistry, biochemistry, and biology. For this latter course, six lectures are devoted to electrochemistry, featuring topics such as signal transduction in biological systems, fuel cells, electrolytic cells, writing and balancing redox reactions, assessing the strengths of oxidants and reductants, connecting equilibrium constants with cell voltages via the Nernst equation, and galvanic cells (both batteries and concentration cells); the last two topics are focal points of the experiments described here.

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PREDICTION AND MEASUREMENT OF CELL VOLTAGES Consider the cell Cu|Cu2+(0.100 M)||Ag+(0.100 M)|Ag. If this cell is constructed as prescribed by Craig et al.,1 the overall cell reaction is

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EXPERIMENT OVERVIEW Ordinarily, this set of experiments is conducted by students working in pairs, but individuals can accomplish the same work. As specified in the Supporting Information, students investigate the behavior of a few cells chosen from a longer list. During a single 3-h laboratory period, it is possible to complete all experimental work dealing with (a) simple galvanic cells and the measurement of their voltages and (b) cells used to determine solubility products or overall formation constants. For the final exercise, involving the use of galvanic (concentration) cells to determine the concentration of Ag+ in an unknown, a portion of a second 3-h laboratory period may be necessary to allow students to construct the galvanic cells, to perform the voltage measurements, and to verify the reliability of their results before processing and analyzing data. With the exception of pH measurements, laboratory work in electrochemistry, particularly at an introductory level, is uncommon. The simple experiments described here benefit students pedagogically: (i) the work reinforces the material on electrochemistry encountered in lectures and textbooks; (ii) the measured cell voltages agree well with those predicted from the Nernst equation; (iii) the experimentally determined equilibrium constants are in accord with literature values; and (iv) the determination of [Ag+] in an unknown, after a calibration graph based on the Nernst equation is constructed with known standards, is successful. Equipment and supplies for this experiment are inexpensive, and using the mini-galvanic cells conserves reagents and affords minimal chemical waste. This experiment can be expanded to include the determination of solubility products for other compounds such as AgSCN, Cu(IO3)2, PbI2, Pb(IO3)2, and PbSO4 and of formation constants for species such as Ag(SCN)2− and Ag(S2O3)23−. Moreover, it is possible to construct concentration cells for the determination of small concentrations of ionic species other than Ag+. In its earliest version, the set of experiments included mini-galvanic cells, for example, Pt| quinhydrone(s), HA (0.100 M)||Ag+(0.100 M)|Ag, designed to determine the dissociation constant (Ka) for a weak monoprotic acid in water (where HA denotes ClCH2CO2H or CH3CO2H).

Cu(s) + 2Ag+(aq) ⇌ Cu 2 +(aq) + 2Ag(s)

and, using the Nernst equation, Ecell (at 298 K) is predicted as follows: Ecell = Ecell° −

0.05915 [Cu 2 +] log = +0.433 V n [Ag+]2

Thus, the theoretical cell voltage (without a sign) is 0.433 V. Further details about this calculation can be found in the Supporting Information. A number of mini-galvanic cells was investigated in the introductory honors laboratory course. Table 1 identifies these Table 1. Comparison of Experimental Cell Voltages with Theoretical Values Calculated from the Nernst Equation Ecell (exptl)/ V

Cell 2+

+

Cu|Cu (0.100 M)||Ag (0.100 M)|Ag Zn|Zn2+(0.100 M)||Cu2+(0.100 M)|Cu Zn|Zn2+(0.100 M)||Ag+(0.100 M)|Ag Cd|Cd2+(0.100 M)||Ag+(0.100 M)|Ag Cd|Cd2+(0.100 M)||Cu2+(0.100 M)|Cu Zn|Zn2+(0.100 M)||Cd2+(0.100 M)|Cd Pb|Pb2+(0.100 M)||Ag+(0.100 M)|Ag Pb|Pb2+(0.100 M)||Cu2+(0.100 M)|Cu Zn|Zn2+(0.100 M)||Pb2+(0.100 M)|Pb Cd|Cd2+(0.100 M)||Pb2+(0.100 M)|Pb

0.428 ± 0.009 (N = 53) 1.099 ± 0.010 (N = 53) 1.526 ± 0.009 (N = 53) 1.132 ± 0.041 (N = 37) 0.713 ± 0.042 (N = 37) 0.392 ± 0.015 (N = 33) 0.896 ± 0.009 (N = 38) 0.474 ± 0.006 (N = 44) 0.627 ± 0.010 (N = 13) 0.228 ± 0.013 (N = 14)

Ecell (theory)/ V 0.433 1.100 1.533 1.173 0.740 0.360 0.896 0.463 0.637 0.277

cells and summarizes the results: the experimental mean cell voltage, Ecell (exptl), and the standard deviation, based on data obtained by N students in comparison with theoretical values calculated from the Nernst equation. Cells that involve cadmium metal routinely show the biggest discrepancy between observed and theoretical voltages, but the reason for this trend is unclear. Therefore, students and instructors might prefer to avoid mini-galvanic cells that involve a cadmium electrode and

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REAGENTS, SUPPLIES, AND INSTRUMENTATION Aqueous solutions (each 0.100 M, unless specified otherwise) needed to construct the various galvanic cells are as follows: CuSO4·5H2O, AgNO3, ZnSO4, Cd(NO3)2·4H2O, Pb(NO3)2, Ni(NO3)2·6H2O, KI (0.200 M), KBr (0.200 M), KCl (0.200 M), NH4NO3 (2.00 M), and NH4OH (7.00 M). Standard 764

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Table 2. Comparison of Experimental and Literature-Based Solubility Products for Silver Halides Solid AgCl AgBr AgI a

Ksp (lit)a

Ksp (exptl)

Cell −

−10

(3.1 ± 1.7) × 10 (N = 81) (3.1 ± 2.8) × 10−12 (N = 66) (2.8 ± 2.0) × 10−16 (N = 17)

+

Ag|AgCl(s), Cl (0.0500 M)||Ag (0.100 M)|Ag Ag|AgBr(s), Br−(0.0500 M)||Ag+(0.100 M)|Ag Ag|AgI(s), I−(0.0500 M)||Ag+(0.100 M)|Ag

1.76 × 10−10 5.14 × 10−13 9.50 × 10−17

Literature values are from ref 7.

Table 3. Comparison of Experimental and Literature-Based Overall Formation Constants for Metal−Ammonia Complexes

a

βn (exptl)

Complex

Cell

Zn(NH3)42+ Cu(NH3)42+ Cd(NH3)42+ Ag(NH3)2+

Zn|Zn(NH3)42+(0.0500 M), NH3 (3.30 M)||Ag+(0.100 M)|Ag Cu|Cu(NH3)42+(0.0500 M), NH3 (3.30 M)||Ag+(0.100 M)|Ag Cd|Cd(NH3)42+(0.0500 M), NH3 (3.30 M)||Ag+(0.100 M)|Ag Ag|Ag(NH3)2+(0.0500 M), NH3 (3.40 M)||Ag+(0.100 M)|Ag

(9.4 (2.0 (2.8 (1.0

± ± ± ±

1.1) 2.5) 3.0) 0.6)

× × × ×

108 (N = 76) 1013 (N = 57) 107 (N = 11) 107 (N = 27)

βn (lit)a,b 4.41 7.80 2.33 1.59

× × × ×

109 1012 107 107

Literature values are from ref 7. bSee ref 8, p 451, for a discussion of the meaning and significance of βn values.

cadmium(II)-containing solutions, especially because cadmium is known to be toxic.

and literature-based solubility products; for each Ksp, the standard deviation is based on data obtained by N students.

DETERMINATION OF SOLUBILITY PRODUCTS To determine the solubility product (Ksp) for a given compound, a galvanic cell is constructed for which the overall cell reaction corresponds to the formation of the precipitate from its component cation and anion, in other words, the reverse of the reaction that denotes dissolution of the chosen compound. A galvanic cell that can be constructed to determine the solubility product for silver bromide is as follows:

DETERMINATION OF OVERALL FORMATION CONSTANTS FOR COMPLEXES To determine the overall formation constant (βn) for a particular metal−ligand complex, a galvanic cell that involves the formation of the desired complex must be constructed. As an example, consider a determination of the overall formation constant (β4) for the tetraamminezinc(II), Zn(NH3)42+, species. A galvanic cell that can be used for this purpose is

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Zn|Zn(NH3)24 + (0.0500 M), NH3(3.30 M)||

Ag|AgBr(s), Br−(0.0500 M)||Ag+(0.100 M)|Ag

Ag+(0.100 M)|Ag

This shorthand denotation of the cell follows the IUPAC convention, where the left electrode is the true anode and the right electrode is the true cathode. Note that this cell is essentially a concentration cell, where the free silver ion concentration is different in each half-cell. The overall cell reaction is

The balanced overall cell reaction is Zn(s) + 2Ag+(aq) + 4NH3(aq) ⇌ Zn(NH3)24 + (aq) + 2Ag(s)

Ag+(aq) + Br−(aq) ⇌ AgBr(s)

and the Nernst equation is

which is the reverse of the reaction that represents the dissolution of AgBr(s). After this galvanic cell is constructed, the cell voltage (Ecell) can be measured, and the result can be expressed mathematically in terms of the Nernst equation: Ecell = Ecell° −

Ecell = Ecell° −

For purposes of illustration, suppose that Ecell was measured experimentally to be 1.880 V. Then, the concentrations of the three species can be substituted into this relation to reveal that E°cell = 1.839 V. To obtain β4 for the Zn(NH3)42+ species, one follows the steps presented in the Supporting Information to find that β4 = 2.14 × 109, which compares reasonably well with a literature value7 of 4.41 × 109. Table 3 provides a comparison of mean experimental formation constants for Zn(NH3)42+, Cu(NH3)42+, Cd(NH3)42+, and Ag(NH3)2+ (obtained by N students) with βn values found in the literature.7

0.05915 1 log n [Ag+][Br−]

where [Ag+] and [Br−] represent the concentrations in the cell. Using the experimentally measured Ecell, E°cell can be calculated and then the classic relation log K =

[Zn(NH3)24 + ] 0.05915 log n [Ag+]2 [NH3]4

nEcell° 0.05915

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DETERMINATION OF SILVER ION IN AN UNKNOWN A section of the Supporting Information describes another experiment for which a set of five galvanic cells is constructed. Each cell consists of two silver wire electrodes; one electrode is immersed in a different silver ion solution and the other electrode is always immersed in 0.100 M AgNO3 to give a concentration cell that can be represented as Ag|Ag+(c1)|| Ag+(0.100 M)|Ag, where c1 denotes the concentration of a certain silver ion-containing solution. By preparing a set of

can be used to obtain a value of K, which is (Ksp)−1 in this case. For purposes of illustration, suppose that Ecell is found experimentally to be 0.575 V. Then, E°cell and Ksp can be calculated to be 0.711 V and 9.54 × 10−13, respectively, the latter value being reasonably close to a literature-based7 Ksp of 5.14 × 10−13. More information about this cell appears in the Supporting Information. Results obtained by honors students for several compounds are compiled in Table 2, which shows the galvanic cells that were constructed, along with a comparison of student-obtained 765

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standard solutions containing different concentrations (c1) of silver ion in a medium of constant ionic strength, and by measuring the voltage of each cell, one can construct a calibration plot based on the Nernst equation from which [Ag+] in an unknown can be determined. A table of student results from one laboratory section appears in the Supporting Information.

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HAZARDS Because they act as skin irritants and are dangerous to inhale, concentrated (7 M) aqueous solutions of ammonia should be handled in fume hoods. Hazards arising from the use of other reagents are minimal; consulting appropriate MSDS information is recommended. Wearing of laboratory coats is always desirable, and the use of protective goggles and chemicalresistant gloves is mandatory. At the conclusion of the experiment, all metal-containing solutions must be collected in separate waste containers and disposed of according to EPA standards. If desired, particular attention can be given to the separate collection of cadmium(II) and lead(II), due to their toxicity.

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SUMMARY It should be evident that the mini-galvanic cells described above provide results that are relatively easy to obtain and that agree very well with theoretical or literature-based values.

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ASSOCIATED CONTENT

S Supporting Information *

Detailed description of the experimental procedure and theory; problems for the students. This material is available via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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REFERENCES

(1) Craig, N. C.; Ackermann, M. N.; Renfrow, W. B. J. Chem. Educ. 1989, 66, 85−86. (2) Baca, G.; Lewis, D. A. J. Chem. Educ. 1978, 55, 804−806. (3) Tanis, D. O. J. Chem. Educ. 1990, 67, 602−603. (4) Eggen, P.-O.; Grønneberg, T.; Kvittingen, L. J. Chem. Educ. 2006, 83, 1201−1203. (5) Eggen, P.-O.; Kvittingen, L.; Grønneberg, T. J. Chem. Educ. 2007, 84, 671−673. (6) Mills, K. V.; Herrick, R. S.; Guilmette, L. W.; Nestor, L. P.; Shafer, H.; Ditzler, M. A. J. Chem. Educ. 2008, 85, 1116−1119. (7) Sillén, L. G.; Martell, A. E. Stability Constants of Metal-Ion Complexes: Chemical Society: London, 1964; Spec. Publ. No. 17, pp 152−155, 286−288, 323−324, 338−339. (8) Skoog, D. A.; West, D. M.; Holler, F. J.; Crouch, S. R. Fundamentals of Analytical Chemistry, 8th ed.; Thomson−Brooks/ Cole: Belmont, CA, 2004, p 451.

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