Electrochemical Potentials & Concentration Cells

SECTION 1

Historical Context & Motivation

The study of electrochemistry traces its origins to the late eighteenth century, when scientists first observed that chemical reactions could produce electrical current and, conversely, that electrical energy could drive chemical transformations. The concept of an electrochemical potential — the thermodynamic quantity that governs the tendency of a species to gain or lose electrons — emerged gradually from experiments with voltaic piles, galvanic cells, and careful measurements of electrode behavior. Understanding how voltage arises from differences in chemical environment, particularly from concentration gradients, became central to physical chemistry, analytical chemistry, and later to biological science and materials engineering.

1800

Volta's Pile

Alessandro Volta constructed the first true battery — the voltaic pile — demonstrating that a sustained electromotive force could arise from the contact of dissimilar metals separated by brine-soaked cloth, establishing the foundation of electrochemistry as an experimental science.

1834

Faraday's Laws of Electrolysis

Michael Faraday quantified the relationship between the amount of substance transformed at an electrode and the total charge passed, introducing the concept of the Faraday constant (F ≈ 96 485 C mol⁻¹) and unifying chemistry with electricity on a quantitative basis.

1889

The Nernst Equation

Walther Nernst derived the equation relating cell potential to the activities of reactants and products, providing the theoretical framework for understanding how concentration affects electromotive force and earning him the Nobel Prize in Chemistry in 1920.

1953

Biological Membrane Potentials

Hodgkin and Huxley published their Nobel-Prize-winning model of the nerve action potential, demonstrating that concentration-cell principles govern ion transport across biological membranes, thereby bridging electrochemistry with physiology.

A persistent question drove much of this work: if two half-cells employ the same electrode material and the same overall reaction, can a net voltage still arise? The answer is yes — whenever the concentrations (or more precisely, activities) of the electroactive species differ between the two compartments, thermodynamics dictates a spontaneous tendency to equalize those concentrations, and this tendency manifests as a measurable cell potential. This type of device is the concentration cell, and understanding it requires a firm grasp of electrochemical potentials and the Nernst equation.

SECTION 2

Core Principles & Definitions

Before diving into concentration cells, it is essential to establish the thermodynamic underpinning of electrode potentials. Every galvanic cell converts chemical free energy into electrical work. The Gibbs free energy change of the cell reaction, ΔG, relates to the cell's electromotive force (EMF), E, through the expression ΔG = −nFE, where n is the number of moles of electrons transferred per mole of reaction and F is the Faraday constant. When all species are in their standard states — solutes at unit activity, gases at 1 bar, pure solids and liquids — the cell potential is denoted E° and is called the standard cell potential. Deviations from standard conditions are handled by the Nernst equation, which introduces the reaction quotient Q.

Electrochemical Potential (μ̃)

The electrochemical potential of a charged species i is μ̃_i = μ_i + z_iFφ, combining the chemical potential μ with the electrical energy of the ion in a phase at inner potential φ. It is the master variable governing ion transfer equilibria.

Standard Electrode Potential (E°)

Each half-reaction has a standard reduction potential measured relative to the Standard Hydrogen Electrode (SHE). A positive E° indicates a strong tendency for reduction; a negative value indicates a tendency to be oxidized relative to H⁺/H₂.

Nernst Equation

E = E° − (RT/nF) ln Q relates the cell potential to the reaction quotient Q, capturing how departures from unit activity shift the driving force of the reaction. At 25 °C this becomes E = E° − (0.02569 V / n) ln Q.

Concentration Cell

A galvanic cell in which both half-cells involve the same electrode and same electrolyte at different concentrations. Because the electrode materials are identical, E° = 0, and the entire EMF originates from the concentration gradient, making the cell a direct probe of activity ratios.

✦ KEY TAKEAWAY

Think of a concentration cell as two reservoirs of water at different heights connected by a pipe. No pump (no E°) is needed — the water flows spontaneously from high to low. Similarly, ions flow from the compartment of higher activity to the one of lower activity, and the resulting current can do electrical work. The height difference is analogous to the Nernst potential, and the volume of water that could flow maps onto the Gibbs free energy available to drive the process.

SECTION 3

Visual Explanation — Anatomy of a Concentration Cell

A copper concentration cell. Both electrodes are metallic Cu immersed in CuSO₄ solutions of different concentrations. Oxidation occurs at the dilute side (anode), dissolution raising [Cu²⁺]; reduction occurs at the concentrated side (cathode), depositing Cu and lowering [Cu²⁺]. Electrons flow through the external wire from anode to cathode, and the salt bridge maintains electroneutrality.

The diagram above captures the essential architecture of a concentration cell. Notice that both electrodes are the same metal — in this case, copper — and both electrolyte solutions contain the same ion, Cu²⁺. The driving force does not come from a difference in electrode chemistry (as it would in a Daniell cell, for example) but entirely from the inequality in Cu²⁺ activity between the two compartments. Thermodynamics favors equalizing the two concentrations: the dilute side acts as the anode (Cu dissolves, raising its Cu²⁺ concentration), while the concentrated side acts as the cathode (Cu²⁺ plates out, lowering its concentration). The net effect is a spontaneous flow of electrons from the anode to the cathode through the external circuit, producing a measurable EMF that persists until the two solutions reach the same activity.

💡 Why Activities, Not Concentrations?

In dilute solutions, molarity is a reasonable approximation for activity. At higher ionic strengths, however, ion–ion interactions cause the effective concentration (activity) to deviate significantly from the molar concentration. The Nernst equation rigorously requires activities, a = γ × m (activity coefficient times molality). For an introductory treatment, we often assume ideal behavior (γ ≈ 1) and substitute molarities, but keep in mind that at high concentrations this simplification introduces error.

SECTION 4

Mathematical Framework

The quantitative heart of electrochemistry is the connection between Gibbs free energy and electrical work. For a cell operating reversibly (infinitesimally small current), the maximum non-expansion work equals −ΔG. Because the work done in moving n moles of electrons through a potential difference E is nFE, we obtain the fundamental relation:

GIBBS–EMF RELATION

ΔG = −nFE

n = moles of electrons transferred; F = 96 485 C mol⁻¹ (Faraday constant); E = cell EMF in volts. A positive E corresponds to a negative (spontaneous) ΔG.

When the system is not at standard conditions, the Gibbs energy depends on the reaction quotient Q through ΔG = ΔG° + RT ln Q. Substituting the Gibbs–EMF relation on both sides (ΔG = −nFE and ΔG° = −nFE°) and dividing by −nF yields the Nernst equation:

NERNST EQUATION (GENERAL)

E = E° − (RT / nF) ln Q

R = 8.314 J mol⁻¹ K⁻¹; T = absolute temperature in K; Q = reaction quotient expressed in activities. At 25 °C the prefactor RT/F = 0.02569 V, giving E = E° − (0.02569 / n) ln Q.

For a concentration cell the two half-cells have identical electrode reactions. The standard potentials cancel (E° = E°_cathode − E°_anode = 0), and the entire cell potential is determined by the activity ratio:

CONCENTRATION CELL EMF

E = (RT / nF) ln (a₂ / a₁)

a₂ = activity of the electroactive ion in the cathode (concentrated) compartment; a₁ = activity in the anode (dilute) compartment. Because a₂ > a₁ the logarithm is positive and E > 0, confirming spontaneity. For dilute solutions, replace a with molar concentration c.

BASE-10 FORM (25 °C)

E = (0.05916 V / n) log₁₀(c₂ / c₁)

This form follows from converting ln to log₁₀ using ln x = 2.303 log₁₀ x. The factor 2.303 × 0.02569 V = 0.05916 V at 25 °C. Many general chemistry texts quote this version.

🌡️ Temperature Dependence

Because the factor RT/nF appears explicitly, the EMF of a concentration cell is directly proportional to absolute temperature — a property exploited in thermometric titrations and in measuring the entropy change of cell reactions via the temperature coefficient (∂E/∂T)_P = ΔS / nF.

SECTION 5

How EMF Varies with Concentration Ratio

A key insight from the Nernst equation applied to concentration cells is the logarithmic dependence of EMF on the concentration ratio. This means that each tenfold change in the ratio c₂/c₁ adds the same increment of voltage: 0.05916/n volts at 25 °C. For a divalent ion like Cu²⁺ (n = 2), each decade of concentration ratio contributes roughly 29.6 mV. This logarithmic sensitivity is the basis of the ion-selective electrode (ISE), including the ubiquitous pH electrode, which is conceptually a concentration cell responding to H⁺ activity.

The EMF of a Cu²⁺/Cu concentration cell increases linearly with log₁₀(c₂/c₁). Each decade of concentration ratio contributes ≈ 29.6 mV when n = 2. For a monovalent ion (n = 1), the slope doubles to ≈ 59.2 mV per decade — the well-known Nernstian slope used in pH measurements.

As the graph illustrates, the relationship between EMF and the logarithm of the concentration ratio is strictly linear. This linearity is not an approximation — it follows directly from the Nernst equation. The y-intercept is zero because when c₂ = c₁ the logarithm vanishes and no driving force exists. Practically, this means that even modest concentration differences produce measurable voltages. A hundred-fold ratio (log = 2) yields about 59 mV for n = 2, well within the range of a decent voltmeter. Conversely, by measuring the EMF one can determine the activity ratio — which is the operating principle behind potentiometric analysis.

EMF values for various concentration ratios at 25 °C

c₂ / c₁	log₁₀(c₂/c₁)	E (mV), n = 1	E (mV), n = 2
1	0	0	0
10	1	59.2	29.6
100	2	118.3	59.2
1 000	3	177.5	88.7
10 000	4	236.6	118.3

SECTION 6

Worked Example — Silver Concentration Cell

Consider a concentration cell constructed from two silver electrodes, one dipped in 0.0100 M AgNO₃ and the other in 0.500 M AgNO₃, at 25 °C. Calculate the EMF of the cell and identify the anode and cathode. Assume activity coefficients of unity (ideal dilute solution).

Silver Concentration Cell at 25 °C

Step 1 — Write the Half-ReactionsBoth half-cells involve the same reaction: Ag⁺(aq) + e⁻ → Ag(s), with E° = +0.7996 V. Because both half-cells share this reaction, E°_cell = E°_cathode − E°_anode = 0.7996 − 0.7996 = 0 V.

E° = 0 V

Step 2 — Identify Anode and CathodeThe cathode is the half-cell where reduction is favored — the compartment with higher Ag⁺ concentration (0.500 M) because there are more Ag⁺ ions available to be reduced. The anode is the dilute compartment (0.0100 M), where Ag will dissolve to increase Ag⁺ concentration.

Cathode: 0.500 M; Anode: 0.0100 M

Step 3 — Write the Nernst Equation for a Concentration CellWith E° = 0 and n = 1 (one electron transferred per Ag⁺ reduced), the equation simplifies to E = (RT / nF) ln(c₂ / c₁). At 25 °C using the base-10 form: E = (0.05916 V / 1) × log₁₀(0.500 / 0.0100).

Step 4 — Evaluate the Logarithmc₂ / c₁ = 0.500 / 0.0100 = 50.0. Therefore log₁₀(50.0) = 1.699.

log₁₀(50.0) = 1.699

Step 5 — Calculate the EMFE = 0.05916 V × 1.699 = 0.1005 V ≈ 100.5 mV. This positive value confirms the assignment of electrodes: electrons flow from the dilute (anode) to the concentrated (cathode) compartment through the external circuit. As the cell operates, the dilute solution becomes more concentrated and the concentrated solution becomes more dilute until equilibrium is reached and E → 0.

E = 0.1005 V (100.5 mV)

SECTION 7

Strengths, Limitations & Practical Context

Concentration cells occupy a unique niche in electrochemistry. They serve primarily as measurement tools rather than energy sources, because their EMF is inherently small (millivolt range) and decays to zero as the system reaches equilibrium. Nevertheless, their elegance and directness make them invaluable for extracting thermodynamic data.

Strengths and limitations of concentration cells

Strengths	Limitations
Directly measures activity ratios (and hence activity coefficients) without requiring any absolute calibration beyond the Nernst equation.	Very low EMF values make precise measurement challenging; requires high-input-impedance voltmeters.
Provides a clean route to thermodynamic quantities (ΔG, ΔS, ΔH) via temperature-dependent EMF measurements.	Assumes reversible behavior; irreversible junction potentials at the salt bridge introduce systematic errors.
Conceptually simple — same electrode on both sides eliminates many variables.	Cell potential decays over time as concentrations equalize; cannot deliver sustained current.
Foundation of ion-selective electrodes (ISEs), pH meters, and biological membrane-potential theory.	Ideal-dilute assumption (γ = 1) breaks down at moderate to high ionic strength, requiring activity coefficient corrections.

✦ KEY TAKEAWAY

Concentration cells are the electrochemical equivalent of a perfectly calibrated thermometer: they are not designed to do heavy work (boil water), but to make exquisitely precise measurements of a single thermodynamic variable — the activity of an ion. Every time you dip a pH electrode into a solution, you are exploiting the same principle: a thin glass membrane separating two solutions of different H⁺ activity generates a Nernstian potential that your meter reads as pH.

SECTION 8

Connection to Advanced Electrochemical Theory

The introductory treatment of concentration cells rests on ideal assumptions — unit activity coefficients, no liquid junction potential, and purely thermodynamic (zero-current) measurements. In advanced electrochemistry, each of these assumptions is relaxed, leading to richer and more realistic models. The table below previews how the core ideas in this lesson extend into more sophisticated territory.

From introductory to advanced electrochemistry

Introductory Treatment	Advanced Extension
Replace activity with concentration (γ = 1).	Debye–Hückel theory and extended models provide γ as a function of ionic strength; cells with/without transference distinguish between mean ionic activity coefficients.
Neglect the liquid junction potential (salt bridge assumed ideal).	The Henderson equation and the Planck integration model quantify the junction potential; cells without transference eliminate it entirely.
Measure open-circuit (equilibrium) EMF only.	Under finite current, overpotentials arise from charge-transfer kinetics (Butler–Volmer equation) and mass transport limitations, studied via voltammetry and impedance spectroscopy.
Single aqueous electrolyte at 25 °C.	Non-aqueous electrolytes, molten salts, and solid-state ion conductors (e.g., YSZ in oxygen sensors) extend the concentration-cell concept to extreme conditions.

As you progress through physical chemistry, you will encounter cells with transference and cells without transference as distinct experimental designs. In a cell with transference (a single electrolyte, no salt bridge), the EMF includes a contribution from the differing mobilities of cation and anion — this is the transference number correction. In a cell without transference, a common-ion salt bridge or a special cell design eliminates this complication, allowing a direct determination of the mean ionic activity coefficient γ±. These refinements illustrate the remarkable depth that can be extracted from what is, at its heart, a simple concentration gradient.

SECTION 9

Practice Problems

PROBLEM 1 — CONCEPTUAL

In a concentration cell, why does E° equal zero? Explain the physical reasoning, not just the mathematical cancellation.

PROBLEM 2 — BASIC CALCULATION

A Zn²⁺/Zn concentration cell has [Zn²⁺] = 0.0050 M in the anode compartment and [Zn²⁺] = 0.50 M in the cathode compartment at 25 °C. Calculate the cell EMF assuming ideal behavior.

PROBLEM 3 — INTERMEDIATE

A concentration cell is constructed using two Ag/Ag⁺ half-cells. The measured EMF at 25 °C is 0.085 V. If the concentration in one compartment is 0.100 M, calculate the concentration in the other compartment. Identify which compartment is the anode.

PROBLEM 4 — APPLIED

A pH glass electrode functions as a concentration cell for H⁺ ions with n = 1. At 25 °C, the electrode reads an EMF of +0.296 V when the internal reference solution has [H⁺] = 1.00 M (pH 0). Determine the pH of the external test solution and explain why the Nernstian slope for pH is 59.16 mV per pH unit.

PROBLEM 5 — CRITICAL THINKING

Suppose you measure the EMF of a Cu²⁺/Cu concentration cell at several temperatures and obtain the temperature coefficient (∂E/∂T)_P = +4.2 × 10⁻⁵ V K⁻¹. From this datum, calculate ΔS for the cell reaction and comment on why ΔS is nonzero even though the cell reaction net involves no change in the number of moles of any chemical species.

SUMMARY

Lesson Summary

This lesson introduced the foundational ideas behind electrochemical potentials and concentration cells. The Gibbs–EMF relation (ΔG = −nFE) connects the free energy of a cell reaction to its electromotive force, and the Nernst equation (E = E° − (RT/nF) ln Q) quantifies how deviations from standard conditions alter the cell potential. In a concentration cell, both electrodes are identical, so E° = 0 and the entire driving force arises from the activity (concentration) ratio of the electroactive species between the two compartments. The cell spontaneously transfers material from the high-activity side (cathode, where reduction occurs) to the low-activity side (anode, where oxidation occurs) until equilibrium is reached.

Practically, the logarithmic dependence of EMF on concentration ratio — with a slope of 0.05916 V / n per decade at 25 °C — underpins technologies such as the pH electrode and other ion-selective electrodes. Strengths of concentration cells include their conceptual simplicity and their ability to measure activity coefficients directly; limitations include small EMF magnitudes and sensitivity to liquid junction potentials. Advanced treatments address these issues through the Debye–Hückel theory, the Henderson equation, and cells with versus without transference — topics that build naturally on the principles established here.