Converting Reference Electrode Potentials

When reading the literature we often find that the reported reference electrode is different to the one we've used for an experiment. Converting between different reference electrodes isn't difficult but it is slightly error prone. For example, if a potential is measured as being at +0.1 V (vs SCE) and we want to know what potential this is against a saturated Ag/AgCl electrode, how can we do the conversion? At 25oC the SCE is +0.241 V (vs SHE) and the Ag/AgCl Sat. is +0.197 V (vs SHE). The simplest way forward is to draw out the different potential scales, as shown at the bottom of this article. This way we're less likely to make a mistake when adding and subtracting the relevant potentials. If you click 'convert' on the calculator below, the measured potential is marked up on the scale and we can see that for the above problem the converted potential is +0.144 V (vs Ag/AgCl Sat.) i.e. 0.1 V +0.241 V -0.197 V = +0.144 V.

Just be aware that the reported electrode values for different systems can vary a little in the literature. This seems to be especially true for mercury based electrodes. Consider the Mercury/Mercury Sulfate (Sat.) electrode, where Bard and Faulkner[1] give a value of +0.64 V vs. SHE and The Handbook of Analytical Chemistry[2] gives a value of +0.65 V vs. SHE. The value of +0.64 V vs. SHE seems to be consistent (< 2 mV) with recent experiments on the proton/hydrogen redox couple. These discrepancies for the Mercury/Mercury Sulfate (Sat.) electrode are not ideal, but if we compare these two texts for their reported values of the Hg/HgO (0.1 M NaOH), the numbers don't even seem to vaguely agree. The Analytical Handbook reports a value of +0.165 V (vs. NHE), whereas Bard and Faulkner* give a value of 0.926 V (vs. NHE)! Neither of these values actually seem to be completely accurate and if you are using a Hg/HgO reference electrode, I'd suggest you use this ACS Catalysis[3] paper to determine the 'best' potential for your electrode. For most other reference electrodes Ives and Janz[4] tends to be the primary literature source.

This variability in a standard electrode potential also arises in some other important cases. For example, let's consider the difference between the 'reversible hydrogen electrode' (RHE), the 'standard hydrogen electrode' (SHE) and the 'normal hydrogen electrode' (NHE). For all three electrodes, the potential is controlled by the proton/hydrogen redox couple: $$\mathrm{H^+} + e^- \rightleftharpoons \frac{1}{2}\mathrm{H_2}$$ The Nernst equation for this reaction is: $$ E = E^{\minuso} - \frac{RT}{F} \ln \frac{\left(p_{\mathrm{H_2}}/{p^{\minuso}}\right)^{1/2}}{a_{\mathrm{H^+}}}$$ here $a_{\mathrm{H^+}}$ is the activity of the proton, $p_{\mathrm{H_2}}$ is the hydrogen pressure and $p^{\minuso}$ is the standard pressure (1 bar), all other terms are as per usual. For all three electrodes (RHE, SHE and NHE) the hydrogen gas is defined as being at 1 bar. We can see from the Nernst equation above that the potential of the hydrogen electrode is sensitive to the hydrogen pressure. If a hydrogen electrode is experimentally set up in the lab, variables such as the local atmospheric pressure and the depth of the hydrogen bubbler influence the potential held on the electrode.[5] As an interesting aside, in 1982, when IUPAC changed the definition of the standard pressure from 1 atm (1.01325 bar) to 1 bar, the potential of the standard hydrogen electrode potential was itself also accordingly shifted by +0.17 mV. Although for all three electrodes, the hydrogen gas pressure is the same, these electrodes differ in their definitions of the solution phase proton composition.

It is often assumed that the SHE and NHE have the exact same potential i.e. that they are both zero by definition but, this is not quite correct. Although the Nernst equation is the same for these two electrodes, strictly the two reference potentials do differ in their definitions of the proton solution phase composition.[6] The SHE is a hypothetical situation where the proton is at unit activity ($a_{\mathrm{H^+}} = 1$). Conversely, in the NHE, the proton concentration is equal to 1 normal. A normal is a somewhat outdated measure of concentration and is, for a monoprotic acid, equal to its molar concentration. To make this difference a little clearer, we can express the proton activity in the Nernst equation on a concentration basis: $$ E = E^{\minuso} - \frac{RT}{F} \ln \frac{c^{\minuso}p_{\mathrm{H_2}}^{1/2}}{\gamma_\pm c_{\mathrm{H^+}} {p^{\minuso}}^{1/2}} $$ where \(c^{\minuso} \) is the standard concentration (1 mol dm-3) and $\gamma_\pm$ is the activity coefficient. But, here's the key point, a solution containing one molar protons does not have unit activity. The activity coefficient (\( \gamma_\pm \)) has, under these conditions, a value of approximately 0.8. Consequently, if we consider the Nernst equation for the reaction, then we can see at a one molar proton concentration, when we account for the non-unity activity coefficient, the potential of the NHE and SHE are expected to differ by approximately -5.7 mV. Furthermore, given that the activity coefficient varies not just as a function of the proton concentration but also depends on the identity of the acid,[7] the exact difference between the NHE and the SHE depends on the experimental system! In the calculator below a value of -5.7 mV is taken as the difference between the NHE and SHE, but as highlighted, this value should be taken with a pinch of salt.

The RHE is a markedly different case; the RHE does not have a fixed potential but it varies as a function of the pH of the solution. The idea behind the RHE is to measure the potential of the hydrogen/proton redox couple in the same electrolyte as being used in the experiment. The argument being that, doing this removes possible contaminations issues from the leaking of either chloride or sulfate into the electrochemical cell being studied. Furthermore, issues around unknown liquid/liquid junction potentials are removed(ish**). For the RHE, the solution phase proton concentration is not fixed and is simply the same as that used in the experimental electrochemical cell. Returning to the Nernst equation, the difference in the potential between the SHE and RHE can be expressed as: $$ E^{\minuso}_{RHE} = E^{\minuso}_{SHE} - \frac{RT}{F} \ln \frac{1}{a_{\mathrm{H^+}}}$$ The definition of pH is: $$ \mathrm{pH} = -\log a_{\mathrm{H^+}} $$ In some elementary texts, it's not uncommon to find the pH to be expressed on a concentration basis this is not correct doing so, is to assume that the proton activity coefficient is unity. We can substitute in the definition of pH into the expression for the difference in potential between the SHE and RHE electrode. Hence, we get: $$ E^{\minuso}_{RHE} = E^{\minuso}_{SHE} - \frac{RT}{F} \ln10 \cdot \mathrm{pH}$$ For instance in a pH 3.0 solution, at 298 K, the RHE has a potential of -0.177 V vs. SHE, whereas in a pH 7.0 solution the potential would be -0.414 V vs. SHE. Although, the construction of an RHE is a relatively straightforward task, there is an issue. We have a descent of information, the reference electrode potential is a combination of the standard hydrogen electrode potential and the pH of the system. Consequently, if we want to compare data measured against a RHE, to work reported against a SHE we need to accurately know the pH of the solution.

As a final point, although there is strictly a difference (as outlined above) between the definition of the SHE and NHE. These days when reading the literature it is advisable to assume that, unless stated otherwise, both the NHE and SHE refer to a hydrogen electrode with unit activity protons and that the historic differences have been set aside. This is the precisely the situation in Bard and Faulkner where the two electrodes are deemed to be equivalent.

Calculator (@25oC)

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Tabulated Values
Note: The pH values only influence the potential of the RHE.
† In this calculator the NHE has been given a value of -5.7 mV vs SHE; however, as discussed in the text the commonly adopted and used value is 0.0 V vs SHE.

[1] Bard, A. J., Faulkner, L. R. Electrochemical Methods: Fundamentals and Applications, 2nd Edition. United States: John Wiley & Sons (2000)
[2] Meites, Louis. Handbook of Analytical Chemistry, United States, McGraw-Hill Book Company (1963)
[3] Kawashima, Kenta, et al. "Accurate Potentials of Hg/HgO Electrodes: Practical Parameters for Reporting Alkaline Water Electrolysis Overpotentials." ACS Catalysis 13 (2023): 1893-1898
[4] David J. G., and George J. Janz. Reference Electrodes, Theory and Practice. New York: Academic (1961)
[5] Hills, G. J., and D. J. G. Ives. "The Hydrogen Electrode." Nature 163 (1949): 997-997
[6] Ramette, R. W. "Outmoded terminology: The normal hydrogen electrode." Journal of Chemical Education 64.10 (1987): 885
[7] McKay, H. A. C. "The activity coefficient of nitric acid, a partially ionized 1:1-electrolyte." Transactions of the Faraday Society 52 (1956): 1568-1573

* Actually, the value given in Bard and Faulkner (2nd edition) is the standard potential for the Hg/HgO redox couple. So it is the potential of the couple in the presence of unit activity protons and not 0.1 M NaOH.

** If we want to relate a measurement made against a RHE to a SHE, then we need an accurate measurement of the pH of the electrochemical cell. The issue is that most (all?) glass pH electrodes use a frit separating the outer solution from an inner reference electrode. So although the RHE does not have a liquid/liquid junction potential, the complementary pH measurement will.