Galvanic cells and standard EMF

Result

Galvanic cells and standard EMF

T-054Home CU-205Threads energy · equilibrium

Statement

Cell potential assembled from half-reactions.

Why it matters

A galvanic (voltaic) cell converts the free energy of a spontaneous redox reaction directly into usable electrical work, and this result is what lets that conversion be predicted and quantified from tabulated data alone, without ever having to build and test the cell. Knowing how to assemble a cell's overall potential from two independently tabulated half-reactions is the foundation for everything else in this unit: gibbs-emf-relation connects that potential to free energy and equilibrium constants, nernst-equation extends it to non-standard concentrations, and faraday-electrolysis uses the identical electron bookkeeping to run the process in reverse.

Hypotheses

Every half-reaction can be assigned a standard reduction potential relative to a common, universally agreed reference.Potentials cannot be measured for a single electrode in isolation — only the potential difference between two electrodes is physically measurable — so a reference point must be chosen by convention. The standard hydrogen electrode (SHE), \(2\text{H}^++2e^-\to\text{H}_2\), is defined to have \(E^\circ=0\,\text{V}\) exactly, at standard conditions, giving every other half-reaction's tabulated potential meaning only relative to this fixed zero. All species are at standard-state activity (commonly taken as \(1\,\text{mol/L}\) for solutes, 1 bar for gases, pure solids/liquids) and standard temperature (298 K).Departures from standard conditions change the cell potential according to the Nernst equation (nernst-equation); the standard potential \(E^\circ\) tabulated here is therefore a fixed reference value specific to standard conditions, not the potential of an arbitrary, as-assembled cell.

Proof

2\text{H}^+(aq) + 2e^- \to \text{H}_2(g), \qquad E^\circ \equiv 0\,\text{V (by convention)}

The standard hydrogen electrode is defined, by international convention, to have exactly zero standard reduction potential at standard conditions; every tabulated half-reaction potential is measured relative to this fixed reference by constructing a cell against the SHE and reading its voltage. A

\text{cathode: reduction occurs (electrons flow in); anode: oxidation occurs (electrons flow out).}

In a spontaneous galvanic cell, electrons flow externally from the electrode where oxidation occurs (the anode) to the electrode where reduction occurs (the cathode) — this assignment of anode/cathode by the direction of spontaneous electron flow is a fixed convention independent of which half-reaction happens to be tabulated as a reduction. A

E^\circ_{\text{cell}} = E^\circ_{\text{cathode}} - E^\circ_{\text{anode}}

Both half-reactions are compared on the identical SHE-referenced scale (Step 1); the overall cell potential is the difference between the reduction potential of the half-reaction that actually proceeds as reduction (cathode) and that of the half-reaction proceeding as oxidation (anode, still using its tabulated reduction potential, not a separately defined "oxidation potential"). A

\text{The spontaneous cell reaction pairs the half-reaction with the higher } E^\circ \text{ as cathode (reduction) against the lower as anode (oxidation).}

Since \(E^\circ_{\text{cell}}>0\) is required for spontaneity (linked to \(\Delta G^\circ<0\) via gibbs-emf-relation), Step 3 is maximised, and therefore made positive whenever physically possible, precisely by assigning the higher-potential half-reaction as the cathode; this determines, without further calculation, which of two given half-reactions is reduced and which is oxidised when they are combined. A

Result

E^\circ_{\text{cell}} = E^\circ_{\text{cathode}} - E^\circ_{\text{anode}}

Reading. A cell's standard potential is assembled purely by subtraction of two independently tabulated standard reduction potentials, with the higher one always playing the role of cathode in the spontaneous direction.

Scope. Valid at standard conditions only (Hypotheses); requires both half-reactions to be referenced to the same SHE-based scale and does not itself account for kinetic barriers that might make an otherwise spontaneous (positive \(E^\circ_{\text{cell}}\)) reaction proceed very slowly in practice.

Corollaries & converses

electrochemical-series is simply a ranked table of standard reduction potentials; this Result is exactly what such a table is for — predicting, by inspection alone, which of any two half-reactions in the table will act as cathode and which as anode when combined.
gibbs-emf-relation connects \(E^\circ_{\text{cell}}\), assembled here, directly to \(\Delta G^\circ\) and hence to the equilibrium constant of the overall cell reaction — this Result supplies the potential that quantity needs as an input.
Converse: a measured cell potential, together with one half-reaction's known standard potential, determines the other half-reaction's standard potential — the practical method by which most of the electrochemical series was originally measured, one couple at a time against the SHE or another known reference.

Fails without

Reverse the subtraction order of Step 3, computing anode minus cathode: the resulting sign of \(E^\circ_{\text{cell}}\) flips, and a genuinely spontaneous reaction is misidentified as non-spontaneous (or vice versa), even though every input potential was looked up correctly.
Apply the Result directly to a cell at non-standard concentrations without correction: the standard potentials tabulated and combined here hold only at standard-state activities (Hypotheses); the true, as-assembled cell potential at arbitrary concentrations requires the further correction supplied by nernst-equation.

Common errors

Reversing the subtraction order in Step 3 (computing anode minus cathode), which flips the sign and can turn a correctly spontaneous cell into an apparently non-spontaneous one.
Multiplying a half-reaction's tabulated \(E^\circ\) by a stoichiometric scaling factor when balancing electrons between the two half-reactions; \(E^\circ\) is an intensive quantity and must never be scaled, unlike \(\Delta G^\circ\) or \(\Delta H^\circ\), even though the half-reaction itself is scaled to balance electrons.
Assuming a large positive \(E^\circ_{\text{cell}}\) guarantees a fast reaction; spontaneity (thermodynamics, this Result) and rate (kinetics) are separate questions, exactly as in Hess's law's thermodynamics/kinetics distinction.

Discussion

Alessandro Volta constructed the first true galvanic cell (the voltaic pile) in 1800, well before a systematic, quantitative theory of electrode potentials existed; the modern convention of tabulating standard reduction potentials against the SHE reference developed over the following century as electrochemistry matured into a quantitative science. Luigi Galvani's earlier observations of "animal electricity" in frog-leg experiments, though based on a mistaken mechanism, gave the field its name and helped motivate Volta's subsequent, correctly interpreted work.

Every standard reduction potential in a modern electrochemical series table is, in effect, a difference measurement against the SHE, carried out under carefully controlled standard conditions; the entire self-consistent table is only as reliable as this shared zero point, which is why the SHE convention (Hypotheses) is treated as fixed and non-negotiable throughout electrochemistry.

Common misconception: that a half-reaction's tabulated potential is an "absolute" property of that couple, independent of any reference. Every value in the table is relative to the arbitrarily but universally chosen SHE zero (Step 1); only potential differences (cell potentials) are directly, physically measurable quantities.

Worked examples

\text{Zn}^{2+}+2e^-\to\text{Zn},\ E^\circ=-0.76\,\text{V}; \qquad \text{Cu}^{2+}+2e^-\to\text{Cu},\ E^\circ=+0.34\,\text{V}

Comparing the two tabulated standard reduction potentials, copper's is higher, so by Step 4 copper is reduced (cathode) and zinc is oxidised (anode) in the spontaneous cell — the classic Daniell cell. A

E^\circ_{\text{cell}} = E^\circ_{\text{cathode}}-E^\circ_{\text{anode}} = 0.34-(-0.76) = 1.10\,\text{V}

Applying Step 3 directly with copper as cathode and zinc as anode gives the standard cell potential of the Daniell cell, a widely cited benchmark value in electrochemistry. A

E^\circ_{\text{cell}}(\text{Daniell}) = +1.10\,\text{V}

Reading. A positive cell potential confirms the reaction is spontaneous as written, with zinc metal reducing aqueous copper ions while itself being oxidised to \(\text{Zn}^{2+}\).

Scope. The identical subtraction procedure applies to any pair of half-reactions drawn from the electrochemical series, regardless of how far apart their tabulated potentials lie.

Problems

Given \(\text{Ag}^++e^-\to\text{Ag}\), \(E^\circ=+0.80\,\text{V}\), and \(\text{Zn}^{2+}+2e^-\to\text{Zn}\), \(E^\circ=-0.76\,\text{V}\), determine the spontaneous cell reaction and its standard potential.
Solution
Silver's potential is higher, so silver is reduced (cathode) and zinc is oxidised (anode). Balancing electrons requires doubling the silver half-reaction (\(2\text{Ag}^++2e^-\to2\text{Ag}\)), but \(E^\circ\) is not scaled (Common errors). \(E^\circ_{\text{cell}}=0.80-(-0.76)=1.56\,\text{V}\). Overall reaction: \(\text{Zn}+2\text{Ag}^+\to\text{Zn}^{2+}+2\text{Ag}\).
A student computes a cell potential of \(-1.10\,\text{V}\) for the Daniell cell by subtracting cathode potential from anode potential in the wrong order. Explain what this negative sign would (incorrectly) imply, and identify the error.
Solution
A negative \(E^\circ_{\text{cell}}\) would (incorrectly) imply the reaction as written is non-spontaneous, i.e. that copper should be oxidised and zinc reduced — the reverse of what actually happens. The error is applying Step 3's subtraction in the wrong order (anode minus cathode instead of cathode minus anode), a frequent sign-error source (Common errors).
Using \(E^\circ(\text{F}_2/\text{F}^-)=+2.87\,\text{V}\) and \(E^\circ(\text{Li}^+/\text{Li})=-3.04\,\text{V}\), find the standard potential of a cell combining these two half-reactions, and state which electrode is the cathode.
Solution
Fluorine's reduction potential is far higher, so fluorine is reduced (cathode) and lithium metal is oxidised (anode). \(E^\circ_{\text{cell}} = 2.87-(-3.04) = 5.91\,\text{V}\), an unusually large standard cell potential reflecting fluorine's position as the strongest common oxidiser and lithium's as one of the strongest common reducers in the electrochemical series.