8.5 Cell Potential
A cell potential is a measure of the electromotive force (emf) across two cells (or terminals) created by a separation of charge and is dependent on the the chemical environment such as the substances involved in the redox reaction.
8.5.1 Standard Cell Potential
The standard cell potential can be expressed as a voltage as follows
\[\begin{align*} E_{\mathrm{cell}}^{\circ} &= E_{\mathrm{red}}^{\circ} + E_{\mathrm{ox}}^{\circ} \\[1.5ex] \end{align*}\]
where E°red is the reduction potential of the reduction reaction whereas E°ox is the oxidation potential of the oxidation reaction. The oxidation potential is simply -(E°red) from the standard reduction potential table.
Another way to express the standard cell potential is
\[\begin{align*} E_{\mathrm{cell}}^{\circ} &= E_{\mathrm{red,cathode}}^{\circ} - E_{\mathrm{red,anode}}^{\circ} \\[1.5ex] \end{align*}\]
where E°cathode is the reduction potential of the reduction reaction (which takes place at the cathode) and E°anode is the reduction potential of the oxidation reaction (which takes place at the anode).
The reduction potentials can be found on a Standard Reduction Potential Table:
Note that all reactions are written as reductions. A pruned table is given below.
| Half-Reaction | E° (V) |
|---|---|
F2 + 2e– → 2F– |
2.866 |
Au3+ + 3e– → Au |
1.498 |
Cl2(g) + 2e– → 2Cl– |
1.35827 |
Br2(l) + 2e– → 2Br– |
1.066 |
Ag+ + e– → Ag |
0.7996 |
Fe3+ + e– → Fe2+ |
0.771 |
I2 + 2e– → 2I– |
0.5355 |
Cu2+ + 2e– → Cu |
0.3419 |
2H+ + 2e– → H2 |
0 |
Pb2+ + 2e– → Pb |
-0.1262 |
Ni2+ + 2e– → Ni |
-0.257 |
Fe2+ + 2e– → Fe |
-0.447 |
Zn2+ + 2e– → Zn |
-0.7618 |
Al3+ + 3e– → Al |
-1.662 |
Mg2+ + 2e– → Mg |
-2.372 |
Na+ + e– → Na |
-2.71 |
K+ + e– → K |
-2.931 |
Li+ + e– → Li |
-3.0401 |
Consider the galvanic cell previously mentioned in Figure 8.2. The cell notation is
\[\mathrm{Cu}(s) ~|~ \mathrm{Cu^{2+}}(aq) ~||~ \mathrm{Ag^+}(aq) ~|~ \mathrm{Ag}(s)\]
Looking up the standard reduction potentials for these half-reactions give
- Anode/Oxidation: Cu2+ + 2e– → Cu (0.3419 V)
- Cathode/Reduction: Ag+ + e– → Ag (0.7996 V)
The standard cell potential is therefore
\[\begin{align*} E_{\mathrm{cell}}^{\circ} &= E_{\mathrm{cathode}}^{\circ} - E_{\mathrm{anode}}^{\circ} \\[1.5ex] &= 0.7996~\mathrm{V} - 0.3419~\mathrm{V} \\[1.5ex] &= 0.46~\mathrm{V} \end{align*}\]
Electrons spontaneously flow from low-to-high potential.
Standard cell potentials, E°cell, are referenced at 298.15 K, though there exists a small, but negligible temperature dependence. The relationship between standard cell potential and temperature is nearly linear according to the work of deBethune.16,17 This temperature dependence is shown below.
\[\begin{align*} \Delta G^{\circ} &= -nFE_{\mathrm{cell}}^{\circ} \\[1.5ex] \Delta G^{\circ} &= \Delta H^{\circ} - T\Delta S^{\circ} \quad \therefore \\[2.5ex] \Delta E_{\mathrm{cell}}^{\circ} &= \dfrac{\Delta H^{\circ} - T\Delta S^{\circ}}{-nF} \end{align*}\]
Using standard thermodynamic values, we calculate the standard cell potential for two redox systems:
\[\mathrm{Cu}(s) ~|~ \mathrm{Cu^{2+}}(aq) ~||~ \mathrm{Ag^+}(aq) ~|~ \mathrm{Ag}(s)\]
\[\mathrm{Li}(s) ~|~ \mathrm{Li^+}(aq) ~||~ \mathrm{Cl_2}(g) ~|~ \mathrm{2Cl^-}(aq)\]Cu/Ag work
\[\mathrm{Cu}(s) + 2\mathrm{Ag^+}(aq) \longrightarrow \mathrm{Cu^{2+}}(aq) + 2\mathrm{Ag}(s)\]
| Substance | ΔH° (kJ mol–1) | S° (J mol–1 K–1) |
|---|---|---|
Cu(s) |
0 |
33.15 |
Cu2+(aq) |
64.77 |
-99.6 |
Ag(s) |
0 |
42.55 |
Ag+(aq) |
105.6 |
72.68 |
The thermodynamic values for the reaction are
\[\begin{align*} \Delta H^{\circ}_{\mathrm{rxn}} &= -146.43~\mathrm{kJ~mol^{-1}} \\[1.5ex] \Delta S^{\circ}_{\mathrm{rxn}} &= -0.19301~\mathrm{kJ~mol^{-1}~K^{-1}} \\[1.5ex] \Delta G^{\circ}_{\mathrm{rxn}} &= -88.88~\mathrm{kJ~mol^{-1}} \end{align*}\]
and the cell potentials are
\[\begin{alignat*}{3} \mathrm{Anode:}~~ &\mathrm{Cu}(s) \longrightarrow \mathrm{Cu^{2+}}(aq) + 2e^- \quad &&E^{\circ} = 0.3419~\mathrm{V}\\[1.5ex] \mathrm{Cathode:}~~ &\mathrm{Ag}(s) \longrightarrow \mathrm{Ag^+}(aq) + e^- \quad &&E^{\circ} = 0.799~\mathrm{V} \end{alignat*}\]
and the standard cell potential is
\[\begin{align*} E_{\mathrm{cell}}^{\circ} &= E_{\mathrm{cathode}}^{\circ} - E_{\mathrm{anode}}^{\circ} \\[1.5ex] &= 0.7996~\mathrm{V} - 0.3419~\mathrm{V}\\[1.5ex] &= 0.46~\mathrm{V} \end{align*}\]
or, determined with thermodynamic values, is
\[\begin{align*} \Delta E_{\mathrm{cell}}^{\circ} &= \dfrac{\Delta H^{\circ} - T\Delta S^{\circ}}{-nF} \\[1.5ex] &= \dfrac{-146.43~\mathrm{kJ~mol^{-1}} - \left ( 298.15~\mathrm{K} \times -0.19301~\mathrm{J~mol^{-1}~K^{-1}}\right ) }{-(2~\mathrm{mol~}e^-)(96.485~\mathrm{kJ~mol^{-1}~V^{-1}})} \\[1.5ex] &= 0.46~\mathrm{V} \end{align*}\]
Varying T in the above equation from 273.15 K to 373.15 K shows a very small change in cell potential (about 70 mV).
Li/Cl2 work
\[\mathrm{Li}(s) + \mathrm{Cl_2}(g) \longrightarrow \mathrm{Li^{+}}(aq) + 2\mathrm{Cl^-}(aq)\]
| Substance | ΔH° (kJ mol–1) | S° (J mol–1 K–1) |
|---|---|---|
Li(s) |
0 |
29.1 |
Li+(aq) |
-278.5 |
13.4 |
Cl2(g) |
0 |
223.1 |
2Cl–(aq) |
-167.2 |
56.5 |
The thermodynamic values for the reaction are
\[\begin{align*} \Delta H^{\circ}_{\mathrm{rxn}} &= -891.4~\mathrm{kJ~mol^{-1}} \\[1.5ex] \Delta S^{\circ}_{\mathrm{rxn}} &= -0.1415~\mathrm{kJ~mol^{-1}~K^{-1}} \\[1.5ex] \Delta G^{\circ}_{\mathrm{rxn}} &= -849.2~\mathrm{kJ~mol^{-1}} \end{align*}\]
and the cell potentials are
\[\begin{alignat*}{3} \mathrm{Anode:}~~ &\mathrm{Li}(s) \longrightarrow \mathrm{Li^{+}}(aq) + e^- \quad &&E^{\circ} = -3.0401~\mathrm{V}\\[1.5ex] \mathrm{Cathode:}~~ &\mathrm{Cl_2}(g) \longrightarrow \mathrm{2Cl^-}(aq) + e^- \quad &&E^{\circ} = 1.35827~\mathrm{V} \end{alignat*}\]
and the standard cell potential is
\[\begin{align*} E_{\mathrm{cell}}^{\circ} &= E_{\mathrm{cathode}}^{\circ} - E_{\mathrm{anode}}^{\circ} \\[1.5ex] &= 1.359~\mathrm{V} - -3.042~\mathrm{V}\\[1.5ex] &= 4.4~\mathrm{V} \end{align*}\]
or, determined with thermodynamic values, is
\[\begin{align*} \Delta E_{\mathrm{cell}}^{\circ} &= \dfrac{\Delta H^{\circ} - T\Delta S^{\circ}}{-nF} \\[1.5ex] &= \dfrac{-891.4~\mathrm{kJ~mol^{-1}} - \left ( 298.15~\mathrm{K} \times -0.1415~\mathrm{J~mol^{-1}~K^{-1}}\right ) }{-(2~\mathrm{mol~}e^-)(96.485~\mathrm{kJ~mol^{-1}~V^{-1}})} \\[1.5ex] &= 4.4~\mathrm{V} \end{align*}\]
Varying T in the above equation from 273.15 K to 373.15 K shows a very small change in cell potential (about 100 mV).
Figure 8.3: The small dependence of the standard cell potential on temperature
Given this small temperature dependence, we can safely use tabulated standard cell potentials at 298.15 K with little effect on the theoretical voltage.
8.5.2 Standard Hydrogen Electrode (SHE)
The standard reduction potentials lie on a relative scale. Notice that some reduction potentials are positive values, some are negative, and one is zero. In this case, the reduction reaction with a potential of 0 V is the reference electrode to which all other electrodes are relative to. The reference electrode, in this case, corresponds to the following reduction reaction
\[2\mathrm{H^+} + 2e^{-} \longrightarrow \mathrm{H_2}\] and is called the standard hydrogen electrode (SHE) which is shown below.
](files/ch17/she.svg)
Figure 8.4: Standard hydrogen electrode schematic Wikipedia
Scheme Outline:
- Platinum electrode
- Hydrogen gas
- 1 M H3O+ solution (pH = 0)
- Hydroseal (keeps oxygen out)
- Connection to another half-cell
The SHE operates as a half-cell, depending on what it is paired with. When acting as a reduction reaction, dissolved hydrogen ions (H+) receive two electrons and form H2 gas. The reduction potential is set to 0 V to act as the reference on our relative standard reduction potential scale. All other relative reduction potentials are determined by hooking up the half-cell to the SHE.