Galvanic cells: half-cell potentials, cell diagrams, and standard reduction potentials

Intuition Beginner

A galvanic cell converts chemical energy into electrical energy by physically separating a redox reaction into two half-reactions. Each half-reaction occurs in its own compartment -- a half-cell -- containing an electrode dipped in an electrolyte. A wire connects the electrodes so electrons can travel through the external circuit, and a salt bridge (or porous barrier) allows ions to migrate between the half-cells, maintaining charge balance.

The voltage produced by a galvanic cell is its cell potential $E_{\text{cell}}$, measured in volts. A positive cell potential means the reaction is spontaneous: it will produce electrical current when the circuit is closed. The magnitude of $E_{\text{cell}}$ depends on which species are being oxidised and reduced.

Every half-cell has a characteristic standard reduction potential $E^{\circ }$, a voltage measured against the standard hydrogen electrode (SHE). The SHE is assigned $E^{\circ }=0.00\text{V}$ by international convention. A species with a more positive $E^{\circ }$ has a greater tendency to be reduced; a species with a more negative $E^{\circ }$ is more easily oxidised.

Visual Beginner

A galvanic cell is represented by a cell diagram -- a compact notation that specifies the composition of each half-cell and the direction of electron flow.

The general form is:

$$\text{anode}\mid \text{anode electrolyte}\parallel \text{cathode electrolyte}\mid \text{cathode}$$

The single vertical bar $|$ separates phases within a half-cell. The double bar $\parallel$ represents the salt bridge or porous barrier. The anode (oxidation) is written on the left, the cathode (reduction) on the right.

For the Daniell cell: $\text{Zn}(s)\mid {\text{Zn}}^{2+}(aq)\parallel {\text{Cu}}^{2+}(aq)\mid \text{Cu}(s)$.

Worked example Beginner

Calculate $E_{\text{cell}}^{\circ }$ and determine spontaneity for a cell constructed from ${\text{Ag}}^{+}/\text{Ag}$ and ${\text{Cu}}^{2+}/\text{Cu}$ half-cells.

Standard reduction potentials: $E^{\circ }({\text{Ag}}^{+}/\text{Ag})=+0.80\text{V}$, $E^{\circ }({\text{Cu}}^{2+}/\text{Cu})=+0.34\text{V}$.

Step 1. Identify anode and cathode. The species with the more negative $E^{\circ }$ is oxidised at the anode. Copper has the less positive reduction potential, so copper is oxidised. Silver is reduced at the cathode.

Step 2. Calculate cell potential.

$$E_{\text{cell}}^{\circ }=E_{\text{cathode}}^{\circ }-E_{\text{anode}}^{\circ }=0.80-0.34=0.46\text{V}$$

Step 3. Write half-reactions. Anode (oxidation): $\text{Cu}(s)\to {\text{Cu}}^{2+}(aq)+2e^{-}$. Cathode (reduction): ${\text{Ag}}^{+}(aq)+e^{-}\to \text{Ag}(s)$. To balance electrons, multiply the reduction by 2: $2{\text{Ag}}^{+}(aq)+2e^{-}\to 2\text{Ag}(s)$.

Step 4. Overall reaction. $\text{Cu}(s)+2{\text{Ag}}^{+}(aq)\to {\text{Cu}}^{2+}(aq)+2\text{Ag}(s)$.

Step 5. Cell diagram. $\text{Cu}(s)\mid {\text{Cu}}^{2+}(aq)\parallel {\text{Ag}}^{+}(aq)\mid \text{Ag}(s)$.

Step 6. Spontaneity. $E_{\text{cell}}^{\circ }=+0.46\text{V}>0$, so the reaction is spontaneous under standard conditions. $\Delta G^{\circ }=-nF{E}^{\circ }=-2\times 96485\times 0.46=-88,766\text{J/mol}\approx -88.8\text{kJ/mol}$.

Check your understanding Beginner

Formal definition Intermediate+

Standard reduction potentials and the electrochemical series

A standard reduction potential $E^{\circ }$ is the potential of a half-cell measured relative to the SHE under standard conditions: $25{}^{\circ }\text{C}$, $1\text{M}$ concentrations for all solutes, $1\text{atm}$ pressure for all gases, and pure solid or liquid phases. The SHE half-reaction is:

$$2{\text{H}}^{+}(aq,1\text{M})+2e^{-}\rightleftharpoons {\text{H}}_2(g,1\text{atm}),E^{\circ }=0.00\text{V}$$

Selected standard reduction potentials at $25{}^{\circ }\text{C}$:

Half-reaction	$E^{\circ }$ (V)
${\text{F}}_2+2e^{-}\to 2{\text{F}}^{-}$	+2.87
${\text{MnO}}_4^{-}+8{\text{H}}^{+}+5e^{-}\to {\text{Mn}}^{2+}+4{\text{H}}_2\text{O}$	+1.51
${\text{Cl}}_2+2e^{-}\to 2{\text{Cl}}^{-}$	+1.36
${\text{Ag}}^{+}+e^{-}\to \text{Ag}$	+0.80
${\text{Cu}}^{2+}+2e^{-}\to \text{Cu}$	+0.34
$2{\text{H}}^{+}+2e^{-}\to {\text{H}}_2$	0.00
${\text{Zn}}^{2+}+2e^{-}\to \text{Zn}$	-0.76
${\text{Al}}^{3+}+3e^{-}\to \text{Al}$	-1.66
${\text{Na}}^{+}+e^{-}\to \text{Na}$	-2.71
${\text{Li}}^{+}+e^{-}\to \text{Li}$	-3.04

Species near the top are strong oxidising agents (they readily gain electrons). Species near the bottom are strong reducing agents (they readily lose electrons). This ordering is the electrochemical series.

Cell diagrams: notation rules

A cell diagram encodes the composition and structure of a galvanic cell in a single line. The rules are:

1. The anode (oxidation) is written on the left, the cathode (reduction) on the right.
2. A single vertical bar $|$ separates different phases within a half-cell (e.g., electrode and solution).
3. A double vertical bar $\parallel$ represents the salt bridge or porous barrier between half-cells.
4. Concentrations (or pressures) are written in parentheses after each species when non-standard.
5. Inert electrodes (Pt, graphite) are included when neither half-reaction species is a conducting solid.

Example: The cell $\text{Pt}(s)\mid {\text{Fe}}^{2+}(aq),{\text{Fe}}^{3+}(aq)\parallel {\text{MnO}}_4^{-}(aq),{\text{Mn}}^{2+}(aq),{\text{H}}^{+}(aq)\mid \text{Pt}(s)$ involves two solution-phase redox couples, each requiring an inert platinum electrode.

When two soluble species share a half-cell (both dissolved), they are separated by a comma rather than a bar, since they are in the same phase.

Calculating cell potential and spontaneity

For any galvanic cell:

$$E_{\text{cell}}^{\circ }=E_{\text{cathode}}^{\circ }-E_{\text{anode}}^{\circ }$$

where both values are taken directly from the reduction potential table. The anode half-reaction is reversed (oxidation), but its $E^{\circ }$ value is not sign-flipped; the subtraction handles the sign.

The spontaneity criterion is:

• $E_{\text{cell}}^{\circ }>0$: reaction is spontaneous under standard conditions ($\Delta G^{\circ }<0$).
• $E_{\text{cell}}^{\circ }=0$: system is at equilibrium.
• $E_{\text{cell}}^{\circ }<0$: reaction is non-spontaneous under standard conditions ($\Delta G^{\circ }>0$).

The relationship between cell potential and Gibbs energy is:

$$\Delta G^{\circ }=-nF{E}_{\text{cell}}^{\circ }$$

where $n$ is the number of electrons transferred and $F=96485\text{C/mol}$.

The standard hydrogen electrode

The SHE consists of a platinum electrode coated with platinum black (finely divided Pt to maximise surface area), immersed in $1\text{M}$ ${\text{H}}^{+}$ solution, with ${\text{H}}_2$ gas at $1\text{atm}$ bubbling over the electrode. The platinum provides an inert conducting surface for the electron transfer. The half-cell potential of any unknown couple is measured by constructing a galvanic cell pairing the unknown half-cell with the SHE and reading the cell potential.

If the unknown species is more easily reduced than ${\text{H}}^{+}$, the measured $E_{\text{cell}}$ is positive and the unknown $E^{\circ }$ equals this positive value. If ${\text{H}}^{+}$ is more easily reduced, the SHE is the cathode and the sign convention still assigns $E^{\circ }$ to the unknown couple as the measured value with the appropriate sign.

Common reference electrodes

The SHE is the primary reference but is cumbersome in practice. Three secondary references are common:

Saturated calomel electrode (SCE): ${\text{Hg}}_2{\text{Cl}}_2(s)+2e^{-}\rightleftharpoons 2\text{Hg}(l)+2{\text{Cl}}^{-}(aq)$, $E=+0.241\text{V}$ vs SHE. Contains $\text{Hg}/{\text{Hg}}_2{\text{Cl}}_2$ paste in saturated KCl. Robust and easy to construct.

Silver-silver chloride electrode: $\text{AgCl}(s)+e^{-}\rightleftharpoons \text{Ag}(s)+{\text{Cl}}^{-}(aq)$, $E=+0.197\text{V}$ vs SHE (in saturated KCl). Miniature, used in pH electrodes and biomedical sensors.

Normal hydrogen electrode (NHE): Functionally identical to the SHE but with historical variations in the definition of standard conditions. In modern usage, SHE and NHE are treated as synonymous.

To convert a potential measured against one reference to another, simply add the reference electrode offset: $E_{\text{vs SHE}}=E_{\text{vs SCE}}+0.241$.

Counterexamples and common errors

• Never multiply $E^{\circ }$ by stoichiometric coefficients. $E^{\circ }$ is an intensive property. Doubling a half-reaction doubles $\Delta G^{\circ }$ and $n$ but leaves $E^{\circ }$ unchanged because $E^{\circ }=-\Delta G^{\circ }/(nF)$.
• When combining two half-reactions with different $n$ values, do not average $E^{\circ }$ values. Convert each to $\Delta G^{\circ }=-nF{E}^{\circ }$, add the $\Delta G^{\circ }$ values, and convert back: $E_{\text{combined}}^{\circ }=-\Delta G_{\text{total}}^{\circ }/(n_{\text{total}}F)$.
• The stronger reducing agent is the one that is more easily oxidised, not the one with the more positive $E^{\circ }$. Lithium ($E^{\circ }=-3.04\text{V}$) is a far stronger reducing agent than silver ($E^{\circ }=+0.80\text{V}$) because lithium gives up electrons far more readily.

Key result Intermediate+

The electrochemical series predicts redox spontaneity

Any species in the electrochemical series will spontaneously oxidise any species below it (under standard conditions). This single rule predicts the direction of every two-couple redox reaction.

Worked example: will ${\text{Cl}}_2$ oxidise ${\text{Br}}^{-}$ to ${\text{Br}}_2$?

$E^{\circ }({\text{Cl}}_2/{\text{Cl}}^{-})=+1.36\text{V}$, $E^{\circ }({\text{Br}}_2/{\text{Br}}^{-})=+1.07\text{V}$.

Chlorine has the more positive $E^{\circ }$ (stronger oxidising agent), so it is reduced. Bromide is oxidised. $E_{\text{cell}}^{\circ }=1.36-1.07=0.29\text{V}>0$. Yes, the reaction is spontaneous.

Will ${\text{I}}_2$ oxidise ${\text{Br}}^{-}$ to ${\text{Br}}_2$?

$E^{\circ }({\text{I}}_2/{\text{I}}^{-})=+0.54\text{V}$, $E^{\circ }({\text{Br}}_2/{\text{Br}}^{-})=+1.07\text{V}$.

Iodine has the less positive $E^{\circ }$. It is a weaker oxidising agent than bromine, so it cannot oxidise bromide. $E_{\text{cell}}^{\circ }=0.54-1.07=-0.53\text{V}<0$. Non-spontaneous.

Cell potential and the equilibrium constant

The standard cell potential determines the equilibrium constant:

$$E^{\circ }=\frac{RT}{nF}\ln K$$

At $25{}^{\circ }\text{C}$: $E^{\circ }=\frac{0.0592}{n}{\log }_{10}K$.

Large positive $E^{\circ }$ corresponds to large $K$: the reaction goes nearly to completion. Even modest cell potentials give enormous equilibrium constants. For $n=2$ and $E^{\circ }=0.46\text{V}$ (the Ag/Cu cell): $K=10^{0.46\times 2/0.0592}=10^{15.5}\approx 3.4\times 10^{15}$.

Exercises Intermediate+

Interfacial potentials and liquid junction potentials Master

Origin of the cell potential at the atomic level

The cell potential arises from the difference in electrochemical potential of electrons between the two electrodes. When a metal electrode is immersed in a solution containing its ions, an equilibrium establishes at the interface:

$$\text{M}(s)\rightleftharpoons {\text{M}}^{n+}(aq)+n{e}^{-}$$

If the metal tends to lose ions (oxidise), a small negative charge builds on the electrode and a layer of cations accumulates in solution, creating an electrical double layer. The potential difference across this layer is the electrode potential. The magnitude depends on the metal-ion couple, the ion concentration, and temperature.

For two half-cells connected by a salt bridge, the measured cell potential is the difference between the two interfacial potentials. The salt bridge minimises (but does not entirely eliminate) the liquid junction potential that develops where two different electrolytes meet.

Liquid junction potentials

At the interface between two electrolyte solutions of different composition, ions diffuse across the boundary at rates proportional to their mobilities. If the cation diffuses faster than the anion (or vice versa), a small charge separation develops, creating a potential difference called the liquid junction potential $E_j$. Typical values range from 1 to 50 mV.

The Henderson equation estimates $E_j$ for a junction between solutions 1 and 2:

$$E_j=\frac{RT}{F}\frac{(u_{+}-u_{-})(c_2-c_1)}{(u_{+}+u_{-})(c_2-c_1)}\cdot \frac{\sum |z_i|u_i(c_{i,2}-c_{i,1})}{\sum |z_i|u_i(c_{i,2}-c_{i,1})}$$

where $u_i$ are ionic mobilities and $c_{i,1}$, $c_{i,2}$ are concentrations of ion $i$ in each solution. Salt bridges use concentrated KCl or ${\text{KNO}}_3$ because ${\text{K}}^{+}$ and ${\text{Cl}}^{-}$ have nearly equal mobilities ($u_{{\text{K}}^{+}}\approx u_{{\text{Cl}}^{-}}$), so their junction potentials nearly cancel.

The full cell potential equation

For a real cell, the measured potential includes the electrode potentials and the junction potential:

$$E_{\text{measured}}=E_{\text{cathode}}-E_{\text{anode}}+E_j$$

In introductory treatments, $E_j$ is neglected. For precision potentiometry (measurements to better than $\pm 1\text{mV}$), the junction potential must be estimated and corrected. The use of a saturated KCl salt bridge reduces $E_j$ to typically less than 1 mV, which is adequate for most analytical purposes.

The standard hydrogen electrode: construction and conventions

The SHE is defined operationally. A platinised platinum electrode is immersed in a solution of ${\text{H}}^{+}$ at unit activity ($a_{{\text{H}}^{+}}=1$, which corresponds to approximately 1.2 M HCl) while hydrogen gas at 1 bar pressure bubbles over the electrode surface. The platinum black provides a high-surface-area catalytic surface for the reaction $2{\text{H}}^{+}+2e^{-}\rightleftharpoons {\text{H}}_2$.

The standard reduction potential of the SHE is defined as exactly $0.000\text{V}$ at all temperatures. This is a convention, not a measurement. All measured $E^{\circ }$ values are differences relative to this zero.

The absolute potential of the SHE (relative to vacuum or to a point at infinity) has been estimated by indirect methods to be approximately $+4.44\text{V}$, but this quantity is never needed in electrochemistry because only potential differences are measurable.

Reference electrode thermodynamics

The saturated calomel electrode (SCE) potential derives from the ${\text{Hg}}_2{\text{Cl}}_2/\text{Hg}$ couple:

$${\text{Hg}}_2{\text{Cl}}_2(s)+2e^{-}\rightleftharpoons 2\text{Hg}(l)+2{\text{Cl}}^{-}(aq)$$

The Nernst equation gives: $E=E^{\circ }({\text{Hg}}_2{\text{Cl}}_2/\text{Hg})-(RT/2F)\ln [{\text{Cl}}^{-}{]}^2$. In saturated KCl ($[{\text{Cl}}^{-}]\approx 4.2\text{M}$), the chloride activity is constant, so $E$ is fixed at $+0.241\text{V}$ vs SHE at $25{}^{\circ }\text{C}$.

The Ag/AgCl reference: $E=E^{\circ }(\text{AgCl/Ag})-(RT/F)\ln [{\text{Cl}}^{-}]$. In saturated KCl, $E=+0.197\text{V}$ vs SHE. In 3.5 M KCl, $E=+0.205\text{V}$ vs SHE. The dependence on $[{\text{Cl}}^{-}]$ means the reference potential shifts slightly with KCl concentration and temperature.

Formal potentials and conditional potentials

The formal potential $E^{\circ \prime }$ is the measured potential of a half-cell at specified conditions of ionic strength, pH, and temperature, incorporating all activity-coefficient and complexation effects. It replaces $E^{\circ }$ in the Nernst equation when concentrations (rather than activities) are used:

$$E=E^{\circ \prime }-\frac{RT}{nF}\ln Q_c$$

where $Q_c$ is the concentration quotient. Tabulated formal potentials (e.g., in Harris's Quantitative Chemical Analysis or Bard & Faulkner's appendices) allow accurate predictions without explicit activity corrections. For example, $E^{\circ \prime }({\text{Fe}}^{3+}/{\text{Fe}}^{2+})=+0.70\text{V}$ in 1 M HCl but $+0.68\text{V}$ in 1 M ${\text{H}}_2{\text{SO}}_4$, because ${\text{Cl}}^{-}$ and ${\text{SO}}_4^{2-}$ complex ${\text{Fe}}^{3+}$ to different degrees.

Worked example: converting between reference electrodes

A potentiometric measurement gives $E=+0.532\text{V}$ vs SCE. What is the potential vs SHE and vs Ag/AgCl (sat'd KCl)?

$E_{\text{vs SHE}}=E_{\text{vs SCE}}+0.241=0.532+0.241=+0.773\text{V}$.

$E_{\text{vs Ag/AgCl}}=E_{\text{vs SHE}}-0.197=0.773-0.197=+0.576\text{V}$.

Three-digit precision in the conversion requires knowing the exact reference electrode potential at the measurement temperature (both SCE and Ag/AgCl have temperature coefficients of about $-0.5\text{mV/K}$).

Connections Master

• Electrochemistry fundamentals 14.11.01 provides the Nernst equation and the $\Delta G=-nFE$ framework that this unit applies to specific galvanic cell constructions. The half-cell potential table and cell diagram notation here are the practical tools that make the Nernst equation usable.

• Thermodynamics 14.06.01 underpins the spontaneity criterion. The sign of $E_{\text{cell}}^{\circ }$ is the electrochemical expression of $\Delta G^{\circ }<0$ at constant $T$ and $P$. The equilibrium constant from $E^{\circ }$ is derived from $\Delta G^{\circ }=-RT\ln K$ combined with $\Delta G^{\circ }=-nF{E}^{\circ }$.

• Electrolysis 14.11.03 is the reverse of galvanic-cell operation. Where galvanic cells produce electricity from spontaneous reactions, electrolytic cells consume electricity to drive non-spontaneous reactions. The half-cell potential framework is shared, but the roles of anode and cathode signs are reversed.

• Batteries and fuel cells 14.11.04 are engineered galvanic cells. The lead-acid battery, the lithium-ion cell, and the hydrogen fuel cell are all specific galvanic cells whose voltages are predicted by the standard reduction potentials developed in this unit.

• Acid-base chemistry 14.10.01 connects through pH-dependent half-cell potentials and the role of ${\text{H}}^{+}$ in many half-reactions. The Nernst equation for pH-dependent couples reduces to $E=E^{\circ }-0.0592\text{pH}$ (for the water couples), which is the bridge to the Pourbaix diagram.

• Corrosion science applies the electrochemical series to predict whether a metal will corrode in a given environment. The galvanic series in seawater (a practical version of the electrochemical series for engineering alloys) determines galvanic corrosion rates when dissimilar metals are in contact.

• Analytical chemistry uses potentiometric measurements with reference electrodes for quantitative ion determination. The ion-selective electrode is a half-cell whose potential follows the Nernst equation with respect to a single ion activity.

Historical notes Master

The galvanic cell traces its origin to Luigi Galvani's 1791 experiments with frog legs and Alessandro Volta's 1800 construction of the voltaic pile, the first device to produce a sustained electric current. Volta's pile stacked alternating zinc and copper discs separated by brine-soaked cardboard, producing about 1 V per cell pair. The pile launched electrochemistry as a discipline and provided the current that enabled the isolation of sodium, potassium, calcium, and other electropositive metals by Humphry Davy within a few years.

John Frederick Daniell developed the Daniell cell in 1836 as a improved voltaic cell with a porous barrier separating the $\text{Zn}/{\text{Zn}}^{2+}$ and $\text{Cu}/{\text{Cu}}^{2+}$ compartments. The Daniell cell's stable voltage ($\sim 1.1\text{V}$) made it the standard voltage reference for telegraphy throughout the nineteenth century.

Walther Nernst derived the thermodynamic relationship between cell potential and composition in 1889, providing the quantitative framework that connects measured voltages to Gibbs energies. The same year, Svante Arrhenius published his theory of ionic dissociation, and the two works together established the ionic basis of electrochemistry.

The systematic tabulation of standard reduction potentials was begun by Wendell Latimer in his 1938 monograph The Oxidation States of the Elements and Their Potentials in Aqueous Solution (2nd edition 1952). Latimer introduced the diagrammatic notation that bears his name (see 14.11.01) and compiled the first comprehensive table of $E^{\circ }$ values. The IUPAC revision by Bard, Parsons, and Jordan (1985) updated and standardised these values.

The standard hydrogen electrode convention was formalised by IUPAC in the 1950s, resolving decades of confusion between the "normal hydrogen electrode" (with various definitions) and a reproducible standard. The modern definition specifies $a_{{\text{H}}^{+}}=1$ (unit activity, not unit molarity), $P_{{\text{H}}_2}=1\text{bar}$ (updated from 1 atm), and $E^{\circ }=0.000\text{V}$ at all temperatures.

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