Chemistry · Ch 2 — Electrochemistry

Nernst Equation

2.3

Nernst Equation

Beyond standard conditions

Every standard electrode potential listed in Table 2.1 is measured with all species at unit concentration (1 M for ions in solution, 1 bar for gases) and 298 K.

Table 2.1Standard Electrode Potentials at 298 K

Ions are present as aqueous species and H₂O as liquid; gases and solids are shown by g and s.

Reaction (Oxidised form + $ne^-$ → Reduced form)	$E^\ominus$ /V
$F_2(g) + 2e^- \rightarrow 2F^-$	2.87
$Co^{3+} + e^- \rightarrow Co^{2+}$	1.81
$H_2O_2 + 2H^+ + 2e^- \rightarrow 2H_2O$	1.78
$MnO_4^- + 8H^+ + 5e^- \rightarrow Mn^{2+} + 4H_2O$	1.51
$Au^{3+} + 3e^- \rightarrow Au(s)$	1.40
$Cl_2(g) + 2e^- \rightarrow 2Cl^-$	1.36
$Cr_2O_7^{2-} + 14H^+ + 6e^- \rightarrow 2Cr^{3+} + 7H_2O$	1.33
$O_2(g) + 4H^+ + 4e^- \rightarrow 2H_2O$	1.23
$MnO_2(s) + 4H^+ + 2e^- \rightarrow Mn^{2+} + 2H_2O$	1.23
$Br_2 + 2e^- \rightarrow 2Br^-$	1.09
$NO_3^- + 4H^+ + 3e^- \rightarrow NO(g) + 2H_2O$	0.97
$2Hg^{2+} + 2e^- \rightarrow Hg_2^{2+}$	0.92
$Ag^+ + e^- \rightarrow Ag(s)$	0.80
$Fe^{3+} + e^- \rightarrow Fe^{2+}$	0.77
$O_2(g) + 2H^+ + 2e^- \rightarrow H_2O_2$	0.68
$I_2 + 2e^- \rightarrow 2I^-$	0.54
$Cu^+ + e^- \rightarrow Cu(s)$	0.52
$Cu^{2+} + 2e^- \rightarrow Cu(s)$	0.34
$AgCl(s) + e^- \rightarrow Ag(s) + Cl^-$	0.22
$AgBr(s) + e^- \rightarrow Ag(s) + Br^-$	0.10
$2H^+ + 2e^- \rightarrow H_2(g)$	0.00
$Pb^{2+} + 2e^- \rightarrow Pb(s)$	−0.13
$Sn^{2+} + 2e^- \rightarrow Sn(s)$	−0.14
$Ni^{2+} + 2e^- \rightarrow Ni(s)$	−0.25
$Fe^{2+} + 2e^- \rightarrow Fe(s)$	−0.44
$Cr^{3+} + 3e^- \rightarrow Cr(s)$	−0.74
$Zn^{2+} + 2e^- \rightarrow Zn(s)$	−0.76
$2H_2O + 2e^- \rightarrow H_2(g) + 2OH^-(aq)$	−0.83
$Al^{3+} + 3e^- \rightarrow Al(s)$	−1.66
$Mg^{2+} + 2e^- \rightarrow Mg(s)$	−2.36
$Na^+ + e^- \rightarrow Na(s)$	−2.71
$Ca^{2+} + 2e^- \rightarrow Ca(s)$	−2.87
$K^+ + e^- \rightarrow K(s)$	−2.93
$Li^+ + e^- \rightarrow Li(s)$	−3.05

Real cells almost never operate under these exact conditions — as a reaction proceeds, reactant concentrations fall and product concentrations rise, so the potential actually measured keeps changing. Nernst worked out how to correct the standard potential for any arbitrary concentration.

For a general single-electrode reduction

\text{M}^{n+}(aq) + ne^- \rightarrow \text{M}(s)

the electrode potential at any concentration, measured against the standard hydrogen electrode, is given by

E_{(\text{M}^{n+}/\text{M})} = E^\ominus_{(\text{M}^{n+}/\text{M})} - \frac{RT}{nF}\ln\frac{[\text{M}]}{[\text{M}^{n+}]}

Since M is a solid, its concentration (strictly, its activity) is taken as unity, which simplifies the expression to

E_{(\text{M}^{n+}/\text{M})} = E^\ominus_{(\text{M}^{n+}/\text{M})} - \frac{RT}{nF}\ln\frac{1}{[\text{M}^{n+}]}

where:

$E^\ominus_{(\text{M}^{n+}/\text{M})}$ — the standard electrode potential (already fixed for the couple)
$R$ — the gas constant, $8.314\ \text{J K}^{-1}\text{mol}^{-1}$
$F$ — the Faraday constant, $96487\ \text{C mol}^{-1}$
$T$ — the absolute temperature in kelvin
$n$ — the number of electrons transferred in the electrode reaction
$[\text{M}^{n+}]$ — the molar concentration of the metal ion in solution

This single relation is what is called the Nernst equation for an electrode.

Applying it to a full cell — the Daniell cell

Take the Daniell cell, whose two electrode reactions are governed independently by the relation above.

At the cathode (copper is deposited):

E_{(\text{Cu}^{2+}/\text{Cu})} = E^\ominus_{(\text{Cu}^{2+}/\text{Cu})} - \frac{RT}{2F}\ln\frac{1}{[\text{Cu}^{2+}(aq)]}

At the anode (zinc dissolves):

E_{(\text{Zn}^{2+}/\text{Zn})} = E^\ominus_{(\text{Zn}^{2+}/\text{Zn})} - \frac{RT}{2F}\ln\frac{1}{[\text{Zn}^{2+}(aq)]}

The overall cell potential is the cathode potential minus the anode potential, $E_{(\text{cell})} = E_{(\text{Cu}^{2+}/\text{Cu})} - E_{(\text{Zn}^{2+}/\text{Zn})}$ . Substituting both expressions,

E_{(\text{cell})} = E^\ominus_{(\text{Cu}^{2+}/\text{Cu})} - \frac{RT}{2F}\ln\frac{1}{[\text{Cu}^{2+}(aq)]} - E^\ominus_{(\text{Zn}^{2+}/\text{Zn})} + \frac{RT}{2F}\ln\frac{1}{[\text{Zn}^{2+}(aq)]}

and collecting the two logarithmic terms under the common $\dfrac{RT}{2F}$ factor,

E_{(\text{cell})} = E^\ominus_{(\text{Cu}^{2+}/\text{Cu})} - E^\ominus_{(\text{Zn}^{2+}/\text{Zn})} - \frac{RT}{2F}\left(\ln\frac{1}{[\text{Cu}^{2+}(aq)]} - \ln\frac{1}{[\text{Zn}^{2+}(aq)]}\right)

gives

E_{(\text{cell})} = E^\ominus_{(\text{cell})} - \frac{RT}{2F}\ln\frac{[\text{Zn}^{2+}]}{[\text{Cu}^{2+}]}

This confirms something intuitive: $E_{(\text{cell})}$ rises as $[\text{Cu}^{2+}]$ increases and falls as $[\text{Zn}^{2+}]$ increases — exactly the direction the spontaneous reaction is pushing the concentrations.

Converting the natural logarithm to base-10 and substituting the constants at $T = 298\ \text{K}$ collapses the coefficient $\dfrac{2.303RT}{F}$ to the familiar number $0.059\ \text{V}$ , giving

$E_{(\text{cell})} = E^\ominus_{(\text{cell})} - \frac{0.059}{2}\log\frac{[\text{Zn}^{2+}]}{[\text{Cu}^{2+}]}$

Both electrode expressions must use the same number of electrons, $n$ , even if the two half-reactions naturally involve different electron counts — the overall balanced cell reaction fixes a single common $n$ . As an illustration, for the cell

\text{Ni}(s)\,|\,\text{Ni}^{2+}(aq)\;\|\;\text{Ag}^+(aq)\,|\,\text{Ag}

the cell reaction is

\text{Ni}(s) + 2\text{Ag}^+(aq) \rightarrow \text{Ni}^{2+}(aq) + 2\text{Ag}(s)

and the Nernst equation is written as

E_{(\text{cell})} = E^\ominus_{(\text{cell})} - \frac{RT}{2F}\ln\frac{[\text{Ni}^{2+}]}{[\text{Ag}^+]^2}

where the silver-ion concentration appears raised to the power 2, matching its stoichiometric coefficient in the balanced reaction. …

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