We can draw vertical lines on pE versus pH predominance diagrams, cutting the pre- dominance lines at known pH values. We can then investigate the speciation of the redox couple along each of these lines. We can write a mole balance on the element through which the electrons are transferred and couple the mole balance with the law of mass action for the half reaction. We can write two typical variations of the law of mass action: one isolating the reduced product on the LHS and one isolating the oxidized reactant on the LHS:
{ }ired = Kox.red{ }iox νi.ox{ } { }H+ nH e− nEνi.red1 (12.7)
{ } { }
{ } { }
. .
1
. ν ν
+ −
=
i red i ox
H E
ox red n n
ox red
i i
K H e
(12.8)
When species other than the oxidized and reduced species containing the element through which the electrons are transferred, protons, and electrons are present, they simply need to be included as appropriate in the law of mass action relation for the
half reaction. Let us apply Equations 12.7 and 12.8 along some vertical constant pH lines cutting some predominance lines.
Example 12.9 Draw two vertical lines (lines of constant pH) on the plot developed in Example 12.8. With the first line cut the H+–H2(g), Fe(OH)3(s)–Fe(II), and O2(g)–H2O predominance lines at pH 6.0. With the second, cut the Fe(III)–Fe(II) predominance line at pH2. Draw a third line cutting the sulfate–bisulfide predominance line of Figure A.9 at pH 10.0. Determine the abundances of the oxidized and reduced species as functions of pE along each of these vertical lines.
For the proton—hydrogen gas system, the proton activity is fixed at 10−6 M and the mole balance is trivial. We write a function with pH and pE as arguments. We can then use it at any value of pH. The value of pEH.H2° from Table A.5 is 0:
Then at pH 6.0 in Figure E12.9.1, we have plotted the predicted abundance of hydrogen gas, expressed as its equivalent partial pressure in atmospheres. We observe that the predicted partial pressure of hydrogen gas increases well above one atm as pE is lowered below the predominance line. Conversely, for each unit increase of pE above the line, the predicted partial pressure of hydrogen gas falls two orders of magnitude, in accord with the law of mass action statement. Certainly, the partial pressure of hydrogen gas in environmental systems would be limited to the range well below one atmosphere, suggesting that pE values below about 5.5 at pH 6.0 would be imaginary.
1×10–3–6.5 0.01 0.1 1 10
–6 –5.5
pE PH2(pH, pE)
–5 –4.5
Figure e12.9.1 a plot of predicted partial pressure of hydrogen gas (in atm) versus pE for redox equilibria in an aqueous solution at ph = 6.0.
For the oxygen–water system, the activity of water is taken as unity, again ren- dering the mole balance on oxygen to be trivial. From the law of mass action rela- tion, we write a function of pH and pE for the partial pressure of oxygen. The value of pEO2.H2O° from Table A.5 is 14.75:
In Figure E12.9.2, we plot the function versus pE values in the vicinity of the pre- dominance line. The predominance line passes through the point pH = 6 and pE = 14.75, so as is indicated in the plot, the partial pressure of 1 atm (equivalent to the unit activity of water) occurs at pE = 14.75. We note that PO2 would increase drastically at pE values beyond the predominance line and decreases four orders of magnitude with a decrease of pE from 14.75 to 13.75, in accord with the pre- diction of the law of mass action. In environmental systems, PO2 really cannot be much larger than 0.01 atm, thus at pH 6 pE values beyond about 14.25 would be imaginary.
For the solid ferric hydroxide–ferrous iron couple, the activity of the solid is unity and again the mole balance on dissolved iron would include only Fe(II) species. We have done those in Chapter 11, so let us focus upon the activity of free ferrous iron as a function of pE. We write the function for the activity of ferrous iron as a function of pH and pE:
In Figure E12.9.3, we plot this function along our vertical line at pH 6, varying pE. We have included the line for {Fe+2} = 1 M as a rough estimate of the maximum abundance of iron(II) in water. When we consider the complexes along with the free metal ion and
13.75 1×10–4 1×10–3 0.01 0.1 1 10
14 14.25 14.5 14.75 pE
PO2(pH, pE) 1
15
Figure e12.9.2 a plot of predicted partial pressure of oxygen (in atm) versus pE for redox equilibria in an aqueous solution of ph = 6.0.
that the activity coefficients would be significantly lower than unity, and we ignore the potential for formation of ferrous hydroxide solid, we guess that an activity around one molar might approximate the maximum {Fe+2} relative to dissolution in water. The significance is that the ferric hydroxide solid phase can exist at pE values well into ferrous iron predominance region. The predominance line drawn in Example 12.8 considered that the ferrous iron abundance would be 10−6 M, com- mensurate with pE ~ +4. We observe that the ferric hydroxide solid can exist at pE values five or more orders of magnitude below the predominance line. This predominance boundary is indeed rather wide.
The dissolved Fe+3–Fe+2 couple is rather simple. Here, we can employ the mole balance and let us suggest that the total dissolved iron (neglecting ionic strength effects) is 10−5 M. We write the mole balance as the sum of dissolved iron species and employ the redox equilibria much in the same manner as we would for a mono- protic acid to produce functions of pH and pE for Fe+3 and Fe+2:
We plot these two functions against pE in Figure E12.9.4. We may observe that the Fe+3–Fe+2 redox couple appears to behave in a manner relative to pE exactly as a monoprotic acid would behave relative to pH. For the Fe+3–Fe+2 system, a single
1×10–6 1×10–5 1×10–4 1×10–3 0.01 0.1 1 10 100
–3 –2 –1 0 1 2 3 4 5
pE Fe2(pH,pE)
1
Figure e12.9.3 a plot of predicted activity of fe+2 versus pE for an aqueous solution at ph = 6.0 in which ferric hydroxide solid is assumed to be present.
electron is transferred and thus the slopes of the relations are unity, in accord with the stoichiometry of the redox half reaction. We see that the predominance line sep- arating Fe+3 from Fe+2 is also wide, with both the reduced and the oxidized species able to exist in significant abundance at least a pE unit on the opposite side of the predominance line. In general, the magnitude of the slope of the log{i} versus pE trace will be equal to the number of electrons transferred.
For the sulfate–bisulfide couple, we have chosen pH 10 to minimize the signifi- cance of dissolved hydrogen sulfide. Within the predominance region below sulfate, we observe a vertical line at pH ~7. This is, of course, the predominance boundary between hydrogen sulfide and bisulfide. We specify a total sulfur abundance of 10−3 M.
We write the mole balance as the sum of sulfate and bisulfide and employ the redox equilibrium to write functions of sulfate and bisulfide activities with pH and pE as the arguments. The value of pESO4.HS° from Table A.5 is 4.25:
11 12 13
pE
14 15
1×10–8 1×10–7 1×10–6 1×10–5
Fe3(pH, pE) Fe2(pH, pE)
Figure e12.9.4 a plot of {fe+2} and {fe+3} versus pE for an aqueous solution containing 10−5 m fetot at ph 6.0.
–3 –2.8 –2.6 –2.4 –2.2 –2 pE
1×10–6 1×10–5
1×10–7 1×10–4 1×10–3 0.01
SO4(pH, pE) HS(pH, pE)
Figure e12.9.5 Plot of sulfate and bisulfide speciation versus pE at ph 6 for an aqueous solution containing 10–3 m total sulfur.
We plot these relations versus pE in Figure E12.9.5. Abundances of sulfate and bisulfide decrease dramatically as the pE value departs from the predominance line at pH 6. As the value of pE is decreased or increased one-half unit ({e–} changes by a factor of 3.16 or 1/ 3.16) , the abundances are decreased or increased four orders of magnitude, again, consistent with the stoichiometry of the half reaction. We would describe the predominance line separating the sulfate from bisulfide predominance region as very sharp.
12.4.3 determining system pe
Electrochemists and their engineering partners, the metallurgical engineers, tend to prefer using EH (and, of course, E °H) as their property for characterization of the electron availability in their targeted systems. They drive their processes using electrical currents and the volt is the most convenient unit for their characterization of electrical potential. Chemists tend to employ pE (and pE °) because the unit con- versions from volts to the seemingly dimensionless electron-normalized equilibrium constants are cumbersome and pE is a very convenient partner to pH. A third charac- terization of electron availability is the oxidation–reduction potential (ORP) with units of millivolts. In fact, the ORP electrode has been developed for physically measuring the electron availability of aqueous samples. In a manner similar to that used for pH and ion-specific ion electrodes, the ORP electrode employs an electro- chemical cell in order to sense the availability of electrons. The measurement is translated directly as an electrical potential. Of prime importance is the necessity to measure the ORP of samples prior to changes in sample character arising from the removal of the sample from its location within the system. When we can insert the electrode (or probe) into the aqueous solution involved with a process, we can obtain fairly accurate measurements. Also, when we can employ an ORP probe in a flow- through cell through which ground water is directed from a well, we can obtain fairly accurate measurements, as long as the path from the ground water source to the flow cell is tightly closed. We might even be able to insert an ORP probe into the sedi- ments below the sediment–water interface of a water body and obtain fairly accurate measurements. As analytical instruments, ORP electrodes must be calibrated and the calibration status continually verified and adjusted as necessary or output readings could be positively or negatively biased. Seemingly very simple, ORP measurement using an electrode is fraught with many possibilities for error. Then, as a backup for ORP measurements, or even as the primary means of determining electron
availability, whether we use EH, pE, or ORP as our unit, physical determination of the speciation of key redox species can yield accurate determinations of electron availability.
Environmental systems generally are not at true chemical equilibrium. This is especially true in regard to overall redox reactions. Electrons are flowing, via many intermediate reactions, and the species of interest are generally the original reactants and the final products of the redox process. While not at equilibrium, processes within environmental systems are often proceeding at steady or near-steady rates, from which quasi-equilibrium conditions arise. Thus, changes in conditions happen only slowly. Conversely, electron transfers occur rapidly. Given this rate disparity, from the presence (or absence) of oxidized and reduced reactants and products of specific half reactions, we can learn much about the availability of electrons. Consider the speciation of the various redox pairs of Example 12.9 in the vicinity of the pre- dominance line. When predominance lines are sharp, the presence of both oxidized and reduced species of a redox pair suggests that the electron availability must be at a level near the predominance line. Then, if we can measure the pH and by assay, determine the abundances of the oxidized and reduced species of a target redox pair, we can, with some degree of confidence through employment of equilibrium rela- tions, determine the electron availability.
Example 12.10 Examine the redox conditions within an operating digester at a typical wastewater renovation facility. Such digesters are completely mixed flow reactors (CMFRs) with hydraulic residence times in the range of 15–30 days. Most do not have cell recycle, so solids residence times and hydraulic residence times are equal. Digesters are operated most efficiently at ~36 °C, and our standard database is at 25 °C. Perhaps we can find the enthalpy of formation data available to adjust the equilibrium constants for temperature, but herein let us accept the errors associated with the standard temperature in favor of a focus upon the redox aspects of the pro- cess and use values specific to 25 °C from our database. Further, we know that the aqueous solution within this digester is not infinitely dilute with regard to electro- lytes, but again, to focus on the redox aspects, let us use the infinitely dilute assump- tion and accept the corresponding errors. Typically digesters operate efficiently in the pH range of 6.5–8.5. Other important information includes the composition of the gas phase above the digester liquid: YCO2 =~ 0.30; YCH4 =~ 0.60; and YH S2 =~ 0.01 with PTot ~ 1.0 atm. The aqueous solution within the digester typically contains ~0.01 and ~0.005 M total acetate and total propionate, respectively. At the pH levels of digesters, speciation will highly favor the conjugate bases. Sulfate abundance is typ- ically below limits of detection. Use the CO2(g)–CH4(g), CO2(g)–CH3COO– (Ac–), CO2(g)–CH3CH2COO– (Pr–), and CH3COO––CH4(g) redox couples to define the elec- tron availability within the digester over the specified range of pH for the stated con- ditions. Suggest which couples might be the most reliable.
We obtain the CO2(g)–CH4(g) half reaction and its pE° value from Table A.5, but must write (or obtain from other sources) the half reactions and compute pE° from Gibbs energy for the remaining redox couples:
We obtain the Gibbs energy of formation for acetate and carbon dioxide from Table A.1, and that for propanoic acid (HPr, ∆Gf° = −383.5 kJ / mol) from Dean (1992).
We might search other databases and secure a value for the propanoate ion, but herein we will choose to add the deprotonation of propanoic acid to produce pro- panoate (Pr–) to obtain the desired half reaction.
From our half reactions and Gibbs energy data, we obtain the equilibrium constants for the four selected half reactions, choosing to use the pE° format. For the carbon dioxide–acetate couple, we have the following:
For the carbon dioxide–propanoate couple, we have the following:
For the acetate–methane couple, we have the following:
We also know the abundances of species occupying both sides of the targeted redox couples:
We use these equilibrium constants, target specie abundances, and the equilibria for the corresponding half reactions to write functions for pE using pH as the master independent variable:
Each of these relations should give us an estimate of the electron availability within the digester as a function of the pH of the aqueous solution. We plot them versus pH in Figure 12.10.1 and obtain a visual result. We quickly observe close agreement bet- ween the results of the CO2(g)–CH3COO– and CO2(g)–CH3CH2COO– couples and reason- able agreement between results of these two couples and the CO2(g)–CH4(g) couple. The results from the CH3COO––CH4(g) couple are orders of magnitude apart from the others.
We might suggest that the CH3COO––CH4(g) is the anomaly. We can test our theory by examining the predicted abundance of sulfate based on the results from the four couples. With known vapor abundance of hydrogen sulfide and known pH we can predict aqueous hydrogen sulfide from Henry’s law and bisulfide abundances from acid dissociation. From bisulfide abundance we can predict, through the half reaction, the abundance of the sulfate ion. We write four corresponding MathCAD functions:
When we plot these functions in Figure E12.10.2, the resultant picture is worth the proverbial 1000 words. We immediately observe that the acetate–methane couple is the outlier as the predicted sulfate abundances are many orders of magnitude greater than is possible. From the result of the carbon dioxide–methane couple, we would predict that at pH 8.5 sulfate would be ~10−3 M, certainly not below limits of detection. Our conclusion is that the true electron availability is somewhere near the traces of the carbon dioxide–propanoate and carbon dioxide–acetate couples.
–86.5 pECO2.CH4(pH) pECO2.Ac(pH)
pEAc.CH4(pH) pECO2.Pr(pH)
–6 –4 –2 0 2
7 7.5
pH
8 8.5
Figure e12.10.1 a plot of pe predicted from Co2–Ch4, Co2–Ch3Coo–, Co2–Ch3Ch2Coo–, and Ch3Coo––Ch4 redox couples for typical conditions within the aqueous solution of an anaerobic digester.
6.5 SO4CO2.CH4(pH)
SO4CO2.Ac(pH)
SO4Ac.CH4(pH) SO4CO2.Pr(pH)
7 7.5
pH
8 8.5
1×10–9 1×10–3 1×103
1×10–15 1×109 1×1015 1×1021 1×1027 1×1033
Figure e12.10.2 Predicted sulfate activities from pE values derived from the Co2–Ch4, Co2– Ch3Coo–, Co2–Ch3Ch2Coo–, and Ch3Coo––Ch4 redox couples for typical conditions within the aqueous solution of an anaerobic digester.
When we dig a little deeper into the anaerobic digestion process, we find that we may divide the process into four major subprocesses, all occurring simultaneously in the CMFR that constitutes the digester:hydrolysis (breakdown of initial substrates); acidogenesis (conversion to long-chain carboxylic acids); acetogenesis (conversion to acetic acid); and methanogenesis (conversion to methane). The microbiologists inform us that of these four subprocesses, methanogenesis is the rate-limiting step due to the slow growth of methanogenic bacteria. This, of course, means that acetic acid, which is the product of the acetogenesis subprocess, will be in abundance well beyond the levels predicted by the acetate–methane redox equilibrium. Relative to the end product methane, all intermediate byproducts would tip the reactant side of Le Chatelier’s balance. Our reaction quotient, Q, would be well below unity, indicating a strong driving force for the reaction to pro- ceed, based on electron availability. Unfortunately, the microbes are the limiting factor in the process. The acetate–methane half reaction is far-removed from its equilibrium condition. We might choose to use the results from the carbon dioxide–
methane and carbon dioxide–propanoate redox couples as limiting cases for describing the pE in the targeted anaerobic digester.
In Example 12.10, we employed knowledge of the composition of a gas phase in contact with an aqueous phase together with general knowledge of the key compo- nents of the aqueous phase to estimate electron availability within the overall system.
We must be sure to understand that our result is but an overall estimate.
We may employ these principles in other systems involving fully dissolved com- ponents and involving an interface between an aqueous solution and a solid phase.
Estimating the pE of a ground water is one notable case we can investigate.
Particularly in the Midwestern United States, ground waters may contain measure- able quantities of iron. Certainly we know that at the pH values of ground waters (e.g., 6–8) the iron that is in the aqueous solution must be of the ferrous variety. Let us examine such a case.
Example 12.11 Consider that ground water may have pH between 6 and 8 and may contain total iron in the range of 1–10 ppmm. Also consider that total inorganic carbon would be in the range of 0.001–0.005 M. Lastly, ground water almost always con- tains sulfur mostly in the form of sulfate. In many instances, the sulfur can be in the reduced sulfide form. Let us suggest that, upon sampling, the field personnel would detect only a very slight odor of rotten eggs (associated with hydrogen sulfide gas) from the water when sampling. However, when tested using a field test kit, the sul- fide level was too low to be quantitated. Determine as closely as possible the electron availability of the ground water sampled.
Rather than a specific value or set of values correlated with pH, herein we seek to identify the region of the pE versus pH (or EH versus pH) diagram within which the pE must lie relative to the pH of the system. Most appropriately, we will pro- duce a plot. We must consider the potential existence of Fe(III) solids (e.g., Fe(OH)3(s), FeOOH(s), and Fe2O3(s)). For illustration herein, we will use Fe(OH)3(s).