Communication pubs.acs.org/jchemeduc
The Third Dimension in Pourbaix Diagrams: A Further Extension Pavel Anatolyevich Nikolaychuk* Department of Analytical and Physical Chemistry, Chelyabinsk State University, Chelyabinsk, 454001, Russian Federation ABSTRACT: Clarification is provided regarding the third axis in E−pH−M diagram. The other variants of three-dimensional Pourbaix diagram are proposed. KEYWORDS: Second-Year Undergraduate, Upper-Division Undergraduate, Physical Chemistry, Computer-Based Learning, Aqueous Solution Chemistry, Electrochemistry, Metals, Oxidation/Reduction, Thermodynamics
R
ecently Pesterfield et al.1 introduced the concept of 3D Pourbaix diagram. Two issues from this paper can be discussed further. The potential associated with a half-reaction a A + b B + m H + + ne− = c C + d D
and magnesium hydrides, which are present at the complete E− pH diagram for Mg,5 are not taken into account. The E−pH−ai diagram is not the only possible variant of applying the third axis in the Pourbaix plot. According to deBethune,6,7 the dependence of half-cell reaction electrode potential on temperature can be expressed as
(1)
in nonstandard conditions is determined by the Nernst equation2 E = Eo −
RT ln nF
aCc ·aDd aAa ·aBb ·a Hm+
⎛ dE o ⎞ o ⎟ + 0.5(T − 298.15)2 ETo = E298 + (T − 298.15) ·⎜ ⎝ dT ⎠298 ⎛ d2E o ⎞ ·⎜ 2 ⎟ ⎝ dT ⎠298
(2)
assuming −log aH+ = pH, the generalized equation that represents any line on an E−pH diagram can be derived E = Eo −
a c ·a d RT ln 10 RT ⋅m⋅ln 10 log Ca Db − pH n·F n·F aA ·aB
Therefore, the position and the slope of lines in the Pourbaix diagram depend on temperature. This way the E−pH−T diagrams can be introduced. For example, the E−pH−T diagram for H2O over the temperature range of liquid water is presented at Figure 2. The expressions for electrode potentials of corresponding half-cell reactions in form of eq 4 are listed in Table 2. The values of the first ((dEo/dT)298) and the second ((d2Eo/dT2)298) temperature coefficients were obtained from deBethune.6,7 Moreover, the standard Gibbs energy change associated with a half-cell reaction is linked with its standard electrode potential according to the equation
(3)
where ai is the thermodynamic activity of component i, other variables are the same as described by Bratsch.2 The activity units depend on the reference state of compounds. They are equal to molarities only for species in water solution.2 Thus, an E−pH−M diagram presented by Pesterfield et al.1 is only a particular case of E−pH−ai diagram in which varied parameters are the activities of ions in solution. It would be more correct in educational terms to change the titles of the third axis in diagrams of Pesterfield et al.1 from molarities of the species to their activities. Another case in which E−pH−ai diagram can be applied is if the half-reaction involves gases (in which fugacities may vary) or impure solid species (in which activities may vary). As the example, the E−pH−a diagram for Mg−H2O system is presented in Figure 1. Magnesium is the lightest structural metal, and therefore, its alloys are widely used in the components of modern cars and high-performance vehicles, in the aerospace industry, as orthopedic biomaterials, and as components in lenses.3 The corrosion resistance of such alloys has been widely studied.4 Therefore, the E−pH− aMg(s) diagram is the relevant example demonstrating how the electrochemical properties of magnesium depend on its activity in alloys. The expressions for electrode potentials of corresponding half-cell reactions are listed in Table 1. The activity of solid magnesium is variable; the activities of other species are set equal to unity. The values of standard Gibbs energies of formation of magnesium species are taken from Pesterfield et al.1 The compounds of monovalent magnesium © 2014 American Chemical Society and Division of Chemical Education, Inc.
(4)
ETo = −n·F ·Δr GTo
(5)
The standard Gibbs energies of formation of some metals and other compounds may depend on pressure8,9 2
3
A ·e a0·T + a1·T /2 + a2·T /3 + a3 / T Δ f G ( T , p) = (K 0 + K1·T + K 2·T 2) ·(n − 1) ·((1 + n·p ·(K 0 + K1·T + K 2·T 2))1 − 1/ n − 1)
(6)
where A, a0, a1, a2, a3, K0, K1, K2, and n are constants for the particular compound and phase. This results in dependence on pressure of standard Gibbs energy of reactions involving these compounds and consequently of corresponding standard electrode potentials. This way, the E−pH−p diagrams can be introduced. This can be important because the next generation Published: April 2, 2014 763
dx.doi.org/10.1021/ed400735g | J. Chem. Educ. 2014, 91, 763−765
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Communication
Figure 1. E−pH diagram for Mg−H2O system: (A) two-dimensional and (B) three-dimensional diagrams using individual curves and (C) three-dimensional diagram using surfaces. The activities of all species except Mg(s) are set equal to unity.
Figure 2. E−pH diagram for H2O at various temperatures: (A) twodimensional and (B) three-dimensional diagrams using individual curves and (C) three-dimensional diagram using surfaces. Temperature axis is calibrated in degrees Celsius for convenience. The activities of all species are set equal to unity.
of nuclear power plants will use supercritical water as working fluid10 and plotting Pourbaix diagrams at elevated pressures and
temperatures may be required to predict the corrosion properties of boilers and pipes in such power plants. In
Table 1. Basic Chemical and Electrochemical Equilibria in Mg−H2O System at 298.15 K Electrode Potentiala/V or solution pH
Half-Cell Reaction
Mg 2 +(aq) + 2e− = Mg(s) +
a
E = − 2.357 − 0.0295·log aMg(s) −
Mg(OH)2 (s) + 2H + 2e = Mg(s) + 2H 2O(l)
E = − 1.860 − 0.0591·pH − 0.0295·log aMg(s)
Mg(OH)2 (s) + 2H+ = Mg 2 +(aq) + 2H 2O(l)
pH = 8.394
Activities of all species except solid Mg are unity. 764
dx.doi.org/10.1021/ed400735g | J. Chem. Educ. 2014, 91, 763−765
Journal of Chemical Education
Communication
Table 2. Electrode Potentials of Hydrogen and Oxygen Electrodes with Respect to Temperature Electrode Potentiala/V
Half-Cell Reaction
2H+(aq) + 2e− = H 2(g)
E = 0.0000 + 0·(T − 298.15) + 0·(T − 298.15)2 − 1.9842·10−4 ·T ·pH
O2 (g) + 4H+(aq) + 4e− = 2H 2O(l)
E = 1.2291 − 8.456·10−4 ·(T − 298.15) + 2.7625·10−7 ·(T − 298.15)2 − 1.9842·10−4 ·T ·pH
a
The activities of all species are set equal to unity.
addition, E−pH−T and E−pH−p diagrams can be used in modeling of chemical and electrochemical processes in geothermal reservoirs.
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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) Pesterfield, L. L.; Maddox, J. B.; Crocker, M. S.; Schweitzer, G. K. Pourbaix (E−pH−M) Diagrams in Three Dimensions. J. Chem. Educ. 2012, 89 (7), 191−199. (2) Bratsch, S. G. Standard Electrode Potentials and Temperature Coefficients in Water at 298.15K. J. Phys. Chem. Ref. Data 1989, 18 (1), 1−21. (3) Polmear, I. J. Magnesium Alloys and Applications. Mater. Sci. Technol. 1994, 10 (1), 1−16. (4) Makar, G. L.; Kruger, J. Corrosion of Magnesium. Int. Mater. Rev. 1993, 38 (3), 138−153. (5) Perrault, G. G. The Potential−pH Diagram of the Magnesium− Water System. J. Electroanal. Chem. Interfacial Electrochem. 1974, 51, 107−119. (6) deBethune, A. J.; Licht, T. S.; Swendeman, N. The Temperature Coefficient of Electrode Potentials. The Isothermal and Thermal Coefficients − The Standard Ionic Entropy of Electrochemical Transport of the Hydrogen Ion. J. Electrochem. Soc. 1959, 106 (7), 616−625. (7) Salvi, G. R.; deBethune, A. J. The Temperature Coefficient of Electrode Potentials. II. The Second Isothermal Temperature Coefficient. J. Electrochem. Soc. 1961, 108 (7), 672−676. (8) Dinscale, A. T. SGTE Data for Pure Elements. CALPHAD: Comput. Coupling Phase Diagrams Thermochem. 1991, 15 (4), 317− 425. (9) Guillermet, A. F.; Gustafson, P.; Hillert, M. The Representation of Thermodynamic Properties at High Pressures. J. Phys. Chem. Solids 1985, 46 (12), 1427−1429. (10) Pioro I.; Mokry S.; Peiman W.; Saltanov E.; Grande L. Application of Supercritical Fluids in Power Engineering. In Proceedings of the 10th International Symposium on Supercritical Fluids (ISSF-2012); ISSF: San Francisco, CA, 2012; http://issf2012.com/ handouts/documents/109_004.pdf (accessed Mar 2014).
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