Emf, Free Energy, pH,and Graphical Representation

Final Research Paper

CHEM 399

2 credit hours

April Kay Bobbie

Spring 2002

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Emf and Free Energy

The following paper was presented at the South Dakota Academy of Sciencemeeting in April, 2002: “Sets of Half-Reactions Which Yield Different Emf for an OverallReaction, But Equivalent Standard Gibbs Free Energy and Equilibrium Constants”.

INTRODUCTION

Standard electrode potentials (redox potentials, oxidation-reduction potentials)provide information about the behavior of chemical reactions. This project investigatedsets of half-reactions that produce the same cell reaction, different cell emf values(E°cell), but similar values of nE°cell, standard Gibbs free energy (∆G°) and equilibriumconstants (Keq). For mercury, chromium, iron, sulfur, chlorine, copper, and indium, setsof half-reactions were determined and cell emf values, nE°cell, equilibrium constants,and standard Gibbs free energy were calculated. E° values for half-reactions aretabulated in several sources. (Atkins 1998, Bard 1985, Latimer 1952) Where the NBSTables include ∆G°f for the species in a half-reaction, E° = -∆G°/nF, and this was usedas the preferred value. (Wagman 1982) Where the NBS Tables were incomplete, thesource of the E° used is referenced.

METHODS OF CALCULATION

For all calculations:F = 96485.3 C/molR = 8.31451 J/mol*KT = 298.15 K∆G° = -nFE°Keq = e nFE°/RT

RESULTS

Mercury: (Wagman 1982)2 Hg = Hg2

2+ + 2 e- E° = -0.7960 V2 Hg2+ + 2 e- = Hg2

2+ E° = +0.9110 V2 Hg + 2 Hg2+ = 2 Hg2

2+ E°cell = +0.115 VHg + Hg2+ = Hg2

2+ (n=1) n E°cell = +0.115 V

∆G° = -11.1 kJ mol-1 Keq = 87.7

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2 Hg = Hg22+ + 2 e- E° = -0.7960 V

Hg2+ + 2 e- = Hg E° = +0.8535 VHg + Hg2+ = Hg2

2+ E°cell = +0.0575 V(n=2) n E°cell = +0.115 V

∆G° = -11.1 kJ mol-1 Keq = 87.7

Chromium: (Atkins 1998)3 (Cr2+ + 2 e- = Cr) E° = -0.91 V2 (Cr = Cr3+ + 3 e- ) E° = +0.74 V3 Cr2+ = 2 Cr3+ + Cr E°cell = -0.17 V

(n=6) n E°cell = -1.02 V

∆G° = 98.4 kJ mol-1 K 100 kJ mol-1 Keq = 5.74x10-18

Cr3+ + 3 e- = Cr E° = -0.74 V3 (Cr2+ = Cr3+ + e-) E° = +0.41 V3 Cr2+ = 2 Cr3+ + Cr (n=3) E°cell = -0.33 V

n E°cell = -0.99 V∆G° = 95.5 kJ mol-1 K 100 kJ mol-1 Keq = 1.84x10-17

Iron: (Wagman 1982)3 (Fe2+ + 2 e- = Fe) E° = -0.409 V2 (Fe = Fe3+ + 3 e- ) E° = +0.049 V3 Fe2+ = 2Fe3+ + Fe E°cell = -0.360 V

(n=6) n E°cell = -2.16 V

∆G° = 208 kJ mol-1 Keq = 3.23x10-37

Fe3+ + 3 e- = Fe E° = +0.049 V3 (Fe2+ = Fe3+ + e-) E° = -0.769 V3 Fe2+ = 2Fe3+ + Fe (n=3) E°cell = -0.72 V

n E°cell = -2.16 V

∆G° = 208 kJ mol-1 Keq = 3.23x10-37

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Sulfur: (Wagman 1982)2 (S2

2- + 2 e- + 2 H+ = 2 HS-) E° = +0.287 V3 HS- = 3 H+ + 4 e- + S3

2- E° = -0.097 V2 S2

2- + 4 H+ = HS- + 3 H+ + S32- E°cell = +0.190 V

2 S22- + H+ = HS- + S3

2- (n=4) n E°cell = +0.760 V

∆G° = -73.3 kJ mol-1 Keq = 6.91x1012

S22- + 2 e- + 2 H+ = 2 HS- E° = +0.287 V

3 S22- = 2 e- + 2 S3

2- E° = +0.473 V4 S2

2- + 2 H+ = 2 HS- + 2 S32- E°cell = +0.760 V

2 S22- + H+ = HS- + S3

2- (n=1) n E°cell = +0.760 V

∆G° = -73.3 kJ mol-1 Keq = 6.91x1012

Chlorine: (Wagman 1982) in basic solutionClO3

- + H2O + 2 e- = ClO2- + 2 OH- E° = +0.271 V

2 (ClO2- = ClO2(g) + e-) E° = -1.071 V

ClO3- + ClO2

- + H2O = 2 ClO2(g) + 2 OH- E°cell = -0.800 V(n=2) n E°cell = -1.600 V

∆G° = 154.4 kJ mol-1 Keq = 1.07x1027

ClO3- + H2O + e- = ClO2(g) + 2 OH- E° = -0.530 V

ClO2- = ClO2(g) + e- E° = -1.071 V

ClO3- + ClO2

- + H2O = 2 ClO2(g) + 2 OH- E°cell = -1.601 V(n=1) n E°cell = -1.601 V

∆G° = 154.5 kJ mol-1 Keq = 1.12x1027

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Examples where the number of electrons is equal:Copper: (Wagman 1982)

2 (Cu+ + e- = Cu) E° = +0.520 VCu = Cu2+ + 2 e- E° = -0.340 V2 Cu+ = Cu2+ + Cu E°cell = +0.180 V

Cu2+ + 2 e- = Cu E° = +0.340 V2 (Cu+ = Cu2+ + e-) E° = -0.160 V2 Cu+ = Cu2+ + Cu E°cell = +0.180 V

n=2 n E°cell = +0.360 V

∆G° = -34.7 kJ mol-1 Keq = 1.22x106

Indium: (Wagman 1982)2 (In2+ + e- = In+) E° = -0.400 VIn+ = In3+ + 2 e- E° = +0.445 V2 In2+ = In3+ + In+ E°cell = +0.045 V

In3+ + 2 e- = In+ E° = -0.445 V2 (In2+ = In3+ + e-) E° = +0.490 V2 In2+ = In3+ + In+ E°cell = +0.045 V

n=2 n E°cell = +0.090 V

∆G° = -8.68 kJ mol-1 Keq = 33.1

DISCUSSION

For mercury, chromium, iron, sulfur, chlorine, copper, and indium, the calculated∆G° and Keq are equal for both sets of half-reactions. The cell reaction was the samefor each set of half-reactions. For all of the ions, except for copper and indium, thenumber of electrons transferred in each set of half-reactions was different. Regardlessof this difference, the standard Gibbs free energy, equilibrium constant, and nE°cell wereequivalent. Use of E° data from sources other than the NBS Tables tended to lead tomodest round-off error.

CONCLUSION

For each set of the two sets of half-reactions, the standard Gibbs free energyand equilibrium constant calculated were found to be equivalent. The same cellreaction is produced by both sets of half-reactions. Each set differs in the number ofelectrons transferred and the cell emf value, but agrees in their product, nE°cell. Using

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E° values based on the NBS Tables tended to give the most consistent results.

A paper addressing this same interesting quality of E° values was published byCooke and Willis, California State University, in the Journal of Chemical Eduction. Twogeneral chemistry students discovered the two different ways of calculating the samevalue for standard Gibbs free energy with a copper half-reaction. They realized thatboth of their answers were correct and recommend that textbooks describe chemicalreactions in terms of ∆G° rather than simply E° (Cooke and Willis).

Latimer Diagrams

Wendell Latimer, a chemical pioneer in the application of thermodynamics toinorganic solution chemistry, introduced this simple diagram in which the value of thestandard potential (in volts) is displayed above or below a horizontal line connecting thespecies of the element in different oxidation states. The Latimer diagram shows therelationships between various species of an element. “A species has a thermodynamictendency to disproportionate into its neighbors if the potential on the right of the speciesis higher than the potential on the left” (Shriver 300). The equation ∆G° = -nFE° allowscalculation of standard Gibbs free energy from the standard potential given in thediagram. Latimer diagrams summarize a large amount of information in a compact andclear style. All diagrams are from data for acidic solution, except for chlorine.

Mercury: (Wagman 1982)

0.9110 0.7960Hg2+---------------Hg2

2+---------------Hg |_________________________|

0.8535

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Chromium: (Atkins 1998)

-0.41 -0.91Cr3+---------------Cr2+---------------Cr |________________________|

-0.74

Iron: (Wagman 1982)

0.769 -0.409Fe3+---------------Fe2+---------------Fe|________________________|

-0.049

Sulfur: (Wagman 1982)

-0.287 0.473HS----------------S2

2----------------S32-

|_______________________|-0.097

Chlorine: (Wagman 1982) in basic solution

-0.530 1.071ClO3

---------------ClO2(g)--------------ClO2-

|__________________________|0.271

Copper: (Wagman 1982)

0.160 0.520Cu2+---------------Cu+---------------Cu |________________________|

0.340

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Indium: (Wagman 1982)

-0.490 -0.400In3+---------------In2+---------------In+

|_______________________|-0.445

Frost Diagrams

Arthur A. Frost developed a “new type of diagram” which plots the standard freeenergy versus the oxidation state for an element (Frost 2680). The slope of the linejoining any two points in a Frost diagram is equal to the standard potential of the coupleformed by the two species. The steeper the slope, the higher the standard reductionpotential for the couple. In the Frost diagram, the lowest lying species corresponds tothe most stable oxidation state of the element. Frost diagrams are most useful for the“quick qualitative impression they provide for trends in the chemical properties ofelements” (Shriver 302). The oxidizing agent in the couple with the more positive slopemay undergo reduction; the reducing agent in the couple with the less positive slopemay undergo oxidation. If an ion lies above the line connecting the two adjacentspecies, it is unstable. Any two species may comproportionate into an intermediatespecies which lies below the line joining the two species. For the following Frostdiagrams, the standard free energy was converted from kJ/mol to eV, using thisrelation: 1eV=96.485 kJ/mol. All diagrams are from data for acidic solution, except forchlorine.

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-2

-1.5

-1

-0.5

0

stan

dard

free

ene

rgy

(eV)

0 1 2 oxidation state

Hg

(Hg2)2+ Hg2+

MercuryFrost Diagram

element standard free energy (kJ/mol) standard free energy (eV) oxidation state calculated delta GHg 0 0.0000 0 0.0000 Hg22+ -153.52 -1.5911 1 -0.8519 Hg2+ -164.4 -1.7039 2 -1.7039 Cr 0 0.0000 0 0.0000 Cr2+ -175.6 -1.8200 2 -1.4800 Cr3+ -214.197 -2.2200 3 -2.2200 Fe 0 0.0000 0 0.0000 Fe2+ -78.9 -0.8177 2 -0.0244 Fe3+ -4.7 -0.0487 3 -0.0487 HS- 12.08 0.1252 -2 0.1252 S22- 79.5 0.8240 -1 0.5927 S32- 73.7 0.7638 -0.67 0.7638 ClO2- 168.438 1.7457 3 1.7457 ClO2(g) 271.738 2.8164 4 2.0163 ClO3- 220.647 2.2869 5 2.2869 Cu 0 0.0000 0 0.0000 Cu+ 49.98 0.5180 1 0.3394 Cu2+ 65.49 0.6788 2 0.6788 In+ -12.1 -0.1254 1 -0.1254 In2+ -50.7 -0.5255 2 -0.5706 In3+ -98 -1.0157 3 -1.0157 OH- -157.244 -1.6297 H2O -237.129 -2.4577

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-2.5

-2

-1.5

-1

-0.5

0

stan

dard

free

ene

rgy

(eV)

0 1 2 3 oxidation state

Cr

Cr2+Cr3+

ChromiumFrost Diagram

-1

-0.8

-0.6

-0.4

-0.2

0

stan

dard

free

ene

rgy

(eV)

0 1 2 3 oxidation state

Fe

Fe2+

Fe3+

IronFrost Diagram

0

0.2

0.4

0.6

0.8

1

stan

dard

free

ene

rgy

(eV)

-3 -2 -1 0 oxidation state

HS-

(S2)2-(S3)2-

SulfurFrost Diagram

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1.6 1.8

2 2.2 2.4 2.6 2.8

3

stan

dard

free

ene

rgy

(eV)

2 3 4 5 6 oxidation state

ClO2-

ClO2(g)

ClO3-

ChlorineFrost Diagram (base)

0

0.2

0.4

0.6

0.8

stan

dard

free

ene

rgy

(eV)

0 1 2 oxidation state

Cu

Cu+

Cu2+

CopperFrost Diagram

-1.2 -1

-0.8 -0.6 -0.4 -0.2

0

stan

dard

free

ene

rgy

(eV)

1 2 3 oxidation state

In+

In2+

In3+

IndiumFrost Diagram

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Pourbaix Diagram

A Pourbaix diagram is a plot of potential versus pH. The regions in a Pourbaixdiagram indicate the conditions of pH and potential under which each species isthermodynamically stable. The Pourbaix diagram of iron was created to include ironspecies found in natural water, at natural concentrations and pH levels. The normal pHrange found in natural waters is from 4-9. The Nernst equation, E=E°-(0.05916v/n)logQ, was used to calculate the potential, E. Two sloping lines for the boundaries ofthe stability field of water have also been included. Any species with a more positivepotential than the upper line will oxidize water to O2, just as any with potentials morenegative than the lower line will reduce water to H2.

pH E horizontal E - diagonal E - diagonal E O2/H2O E H2/H2O0 0.769 1.3138 0 1 0.769 1.25464 -0.05916 2 0.769 1.19548 -0.11832 3 0.769 1.13632 -0.17748 3 2 0.769 1.13632 -0.17748 4 0.59152 1.07716 -0.23664 5 0.41404 1.018 -0.2958 6 0.23656 0.95884 -0.35496 7 0.05908 0.89968 -0.41412 8 -0.1184 0.84052 -0.47328 9 -0.29588 -2 0.78136 -0.53244 9 -0.25127 0.78136 -0.53244 10 -0.31043 0.7222 -0.5916 11 -0.36959 0.66304 -0.65076 12 -0.42875 0.60388 -0.70992 13 -0.48791 0.54472 -0.76908 14 -0.54707 0.48556 -0.82824

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Fe(OH)2(s)

-2

-1

0

1

2

E (v

)

0 2 4 6 8 10 12 14 pH

Pourbaix DiagramIron

Fe3+

Fe2+ Fe(OH)3(s)

H2O/H2

O2/H2O

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References:Atkins, Peter W. Physical Chemistry 6th Ed. Freeman, New York. 1998.Bard, Allen J., Roger Parsons, and Joseph Jordan. Standard Potentials in

Aqueous Solution. Marcel Dekker, New York. 1985.Cooke, Ron C. and Grover C. Willis. “Which E° Is It?” Chemical

Principles Revisited. J. of Chem. Ed. 1996, 73, 450.Frost, Arthur A. ”Oxidation Potential-Free Energy Diagrams”. J. Am.

Chem. Soc. 1951, 73, 2680.Gray, Harry B., and Gilbert P. Haight, Jr. Basic Principles of Chemistry.

W.A. Benjamin, Inc., N.Y., 1967.Latimer, Wendell M. Oxidation Potentials 2nd Ed. Prentice-Hall,

Englewood Cliffs, N.J. 1952.Phillips, C.S.G., and R.J.P. Williams. Inorganic Chemistry I Principles and

Non-metals. Oxford University Press. 1965.Phillips, C.S.G., and R.J.P. Williams. Inorganic Chemistry II Metals.

Oxford University Press. 1966.Pourbaix, Marcel. Atlas of Electrochemical Equilibria in Aqueous

Solutions. National Association of Corrosion Engineers, TX, 1974.Shriver, D.F., Peter Atkins, and Cooper H. Langford. , D.F., Peter Atkins,

and Cooper H. Langford. Inorganic Chemistry 2nd Ed. W.H.Freeman and Company, N.Y., 1994.

Viste, Arlen. “Chemistry 341 Lecture Notes”. Fall, 1976.Wagman, Donald D., W.H. Evans, V.B. Parker, R.H. Schumm, Iva Halow,

S.M. Bailey, K.L. Churney, and R.L. Nuttall. The NBS Tables ofChemical Thermodynamic Properties. Journal of Physical andChemical Reference Data. 11: Supplement 2. American Instituteof Physics, Inc., N.Y., 1982.