Concentration Cell

A battery with no chemical reaction

Almost every battery you have ever used runs on a chemistry mismatch. In a Daniell cell, zinc dissolves and copper plates out because copper "wants" the electrons more strongly than zinc does — that imbalance in standard electrode potentials is worth 1.10 V. A concentration cell throws that idea away entirely. It uses the same metal on both electrodes and the same salt in both beakers. There is no favorable redox couple, no chemistry gradient at all. Wire up the standard electrode potentials and they cancel perfectly: E°cell = E°cathode − E°anode = 0 V. By the textbook rule, this cell should be dead.

Yet hook a voltmeter across it and you read a real, persistent voltage — usually tens of millivolts. The only difference between the two halves is how much ion is dissolved on each side. That difference alone is a source of free energy, and a concentration cell is the device that converts it into electric current. It is the cleanest possible demonstration that concentration is energy: even when the chemistry is identical, a gradient does work.

The classic copper setup

The canonical example uses two strips of pure copper, each dipped in copper(II) sulfate, joined by a salt bridge:

Cu(s) | Cu²⁺ (0.001 M) ‖ Cu²⁺ (0.10 M) | Cu(s)

Both half-reactions are the same couple, Cu²⁺ + 2e⁻ ⇌ Cu(s), with E° = +0.34 V on each side. The system has one goal: equalize the two concentrations. To do that, the dilute side must gain ions and the concentrated side must lose them. So:

• Dilute half-cell (anode, −): Cu(s) → Cu²⁺ + 2e⁻. The electrode dissolves, pumping ions into the thin solution to raise its concentration.
• Concentrated half-cell (cathode, +): Cu²⁺ + 2e⁻ → Cu(s). Ions plate out onto the electrode, draining the rich solution to lower its concentration.

Electrons therefore flow through the external wire from the dilute electrode to the concentrated electrode, while the salt bridge carries counter-ions to keep each compartment electrically neutral. The two concentrations march toward each other — 0.001 M creeping up, 0.10 M creeping down — and the moment they meet, the gradient vanishes and the voltage falls to exactly zero. The battery has discharged not by consuming a reactant, but by erasing a difference.

The numbers: the Nernst equation with E° = 0

All of this is quantified by the Nernst equation, which corrects a standard potential for non-standard concentrations:

E = E° − (RT / nF)·ln Q

Here R = 8.314 J·mol⁻¹·K⁻¹, T is temperature in kelvin, n is the number of electrons transferred, F = 96 485 C·mol⁻¹ is the Faraday constant, and Q is the reaction quotient. At 25 °C (298.15 K) the prefactor (RT/F)·ln(10) collapses to the famous 0.0592 V, so the equation is usually written in base-10 form:

E = E° − (0.0592 / n)·log Q

For a concentration cell the two half-reactions are reverses of each other, so E° = 0 and the reaction quotient is simply the ratio of the two ion concentrations. Writing it with the cathode (concentrated) on top:

E = (0.0592 / n)·log( C_cathode / C_anode )

That single expression contains everything. The voltage is positive only because the cathode is the concentrated side; flip the labels and the log goes negative, which just means you mislabeled the anode and cathode. For our copper example, n = 2 and the ratio is 0.10/0.001 = 100:

E = (0.0592 / 2)·log(100) = 0.0296 × 2 = 0.0592 V ≈ 59 mV

Two practical rules of thumb fall out of this. First, every factor-of-10 in the concentration ratio is worth 0.0592/n volts — 59 mV for a singly-charged ion, only 30 mV for a doubly-charged one. Second, because the dependence is logarithmic, even a thousand-fold gradient buys you under 200 mV for n = 1. Concentration cells are intrinsically low-voltage devices; their value is in measurement and biology, not in powering a flashlight.

Where the energy comes from: entropy of mixing

If there is no chemical reaction, what is the cell actually spending? The answer is entropy. A concentrated solution is a low-entropy arrangement: many ions crowded into a small space. Diluting it lets those ions spread out, and spreading out increases the number of accessible microstates — exactly the same statistical push that makes a drop of ink bloom through still water or a gas fill its container. The free-energy change for moving one mole of ions from concentration C₂ down to C₁ is purely entropic:

ΔG = RT·ln(C₁ / C₂) < 0 (since C₁ < C₂)

That negative ΔG is the fuel. Through the bridge ΔG = −nFE, so the cell simply converts this entropy of mixing into electrical work instead of letting it dissipate as aimless diffusion. This is why a concentration cell still obeys the second law: total entropy rises as the gradient flattens, and the cell goes dead precisely at equilibrium, when the two chemical potentials are equal, Q = 1, ΔG = 0, and E = 0. A concentration cell is, in a very literal sense, a machine for extracting work from the spontaneous tendency of things to mix.

How it compares to a normal galvanic cell

The concentration cell is best understood as a stripped-down special case of the broader galvanic cell family. The table below puts the two side by side, and adds a worked voltage for several common ions.

Property	Ordinary galvanic cell (Daniell)	Concentration cell
Electrode materials	Different (Zn vs Cu)	Identical (Cu vs Cu)
Electrolytes	Different salts	Same salt, different concentration
E°cell	+1.10 V (large)	0 V (cancels exactly)
Source of voltage	Favorable redox couple	Concentration gradient only
Driving force	Enthalpy + entropy of reaction	Entropy of mixing
Typical voltage	≈ 1–2 V	≈ 10–200 mV
Dead when	A reactant is exhausted	Concentrations equalize (Q = 1)
Anode	More easily oxidized metal	Dilute half-cell

And the voltage for a clean 10:1 and 100:1 gradient, by ion charge, all at 25 °C:

Ion (couple)	n	E for 10:1 ratio	E for 100:1 ratio
Ag⁺ / Ag	1	59.2 mV	118.4 mV
H⁺ (pH-type)	1	59.2 mV	118.4 mV
Cu²⁺ / Cu	2	29.6 mV	59.2 mV
Zn²⁺ / Zn	2	29.6 mV	59.2 mV
Al³⁺ / Al	3	19.7 mV	39.5 mV

Why it matters: meters, membranes, and rust

Concentration cells are everywhere once you know the signature — a voltage that tracks a logarithm of concentration.

• pH meters and ion-selective electrodes. The glass electrode is a concentration cell: a thin glass membrane separates a known internal H⁺ concentration from the unknown sample, and the cell voltage reports the H⁺ ratio. Because every pH unit is a 10× change in [H⁺], the meter sees a clean ~59 mV per pH unit — the Nernst slope. The same physics drives sensors for Na⁺, K⁺, Ca²⁺, F⁻ and nitrate. (See the Nernst equation and pH scale.)
• Biological membrane potentials. Living cells keep K⁺ high inside and Na⁺ high outside, and these gradients are concentration cells across the membrane. The Nernst equation predicts the equilibrium potential of each ion (≈ −90 mV for K⁺, ≈ +60 mV for Na⁺), and their weighted combination — the Goldman equation — gives the neuron's −70 mV resting potential and the spike of the action potential. Every nerve impulse and heartbeat is a concentration cell being charged and discharged.
• Differential-aeration corrosion. When part of a steel surface sees more oxygen than another part (under a water droplet, a gasket, or at the waterline), the two regions form a concentration cell with respect to dissolved O₂. The oxygen-starved region becomes the anode and corrodes preferentially — this is why pitting and crevice corrosion concentrate exactly where you can't see them. (Compare redox reactions.)
• Reference electrodes and oxygen sensors. The automotive zirconia lambda sensor is an oxygen concentration cell comparing exhaust O₂ to ambient air; its sharp voltage swing near the stoichiometric point tells the engine computer how to trim the fuel mixture.

Common misconceptions

• "E° = 0 means no voltage." E° is the voltage at standard (equal) concentrations only. The Nernst correction makes the actual E nonzero whenever the concentrations differ.
• "The concentrated side is the anode." Backwards — the dilute side is the anode. The system raises the dilute concentration by dissolving that electrode.
• "There's a chemical reaction driving it." The same species is just shuttled from high to low concentration; the net "reaction" is dilution. The driver is entropy, not chemistry.
• "Bigger gradient, proportionally bigger voltage." The dependence is logarithmic. Going from 10:1 to 100:1 only doubles the voltage, not multiplies it tenfold.
• "It never runs out." It absolutely does — at exactly the point where the two concentrations become equal, where ΔG = 0 and the cell sits at equilibrium.
• "Charge doesn't matter." The n in 0.0592/n halves the voltage for a 2+ ion and thirds it for a 3+ ion at the same ratio.