Electrochemistry: Galvanic Cells, the Nernst Equation and Conductance

The Daniell cell converts the spontaneous reaction Zn(s) + Cu2+(aq) to Zn2+(aq) + Cu(s) into electrical energy, generating exactly 1.1 volts when both ion concentrations sit at unity, a device called a galvanic (or voltaic) cell. Apply an external opposing voltage smaller than 1.1 V, and the reaction keeps proceeding in the same direction, just slightly slowed. Raise that external voltage to exactly 1.1 V, and the reaction stops entirely, current drops to zero. Push the external voltage past 1.1 V, and something genuinely striking happens: the entire reaction reverses direction, zinc now deposits while copper dissolves, precisely because it now takes electrical energy to force a non-spontaneous reaction to occur. This reversed setup, using electrical energy to drive a chemical reaction rather than the other way around, is called an electrolytic cell, and both kinds of cell share the exact same physical hardware, only the direction of electron flow differs.

Every galvanic cell splits into two half-cells, each a metallic electrode dipped in its own electrolyte, connected externally through a wire and internally through a salt bridge that lets ions migrate without letting the two solutions mix. At each electrode-electrolyte interface, a genuine tug-of-war plays out: metal ions in solution tend to deposit onto the electrode, while metal atoms on the electrode tend to dissolve into solution, and the resulting equilibrium separation of charge produces a measurable electrode potential. In a galvanic cell, oxidation always happens at the anode, which develops a negative potential, while reduction always happens at the cathode, which develops a positive potential, and electrons flow from the negative anode to the positive cathode through the external wire. By strict convention, cells are always written with the anode on the left and cathode on the right, a vertical line marking each electrode-electrolyte interface and a double vertical line marking the salt bridge, so Cu(s)|Cu2+(aq)||Ag+(aq)|Ag(s) directly tells you copper is oxidised and silver is reduced.

No individual electrode's potential can actually be measured in isolation, only the difference between two electrodes connected in a cell, which creates a genuine problem: without some fixed reference, every measured value would only be relative to whatever the other electrode happened to be. Chemists solved this by arbitrarily assigning one specific electrode, the standard hydrogen electrode (a platinum electrode coated in platinum black, dipped in an acidic solution with hydrogen gas bubbled through at 1 bar and H+ at 1 M), a standard electrode potential of exactly 0.00 volts at every temperature. Every other electrode's standard potential is then just the measured emf of a cell built from that electrode paired with the standard hydrogen electrode. Copper measured against hydrogen gives +0.34 V, meaning Cu2+ is reduced more easily than H+; zinc measured against hydrogen gives -0.76 V, meaning H+ is reduced more easily than Zn2+, meaning zinc itself is the one that ends up oxidised instead.

Arranged from most positive at the top to most negative at the bottom, the standard electrode potential table doubles as a ranked list of oxidising and reducing strength. Fluorine sits at the very top, +2.87 V, meaning F2 has the greatest tendency of any species on the table to be reduced, making it the strongest oxidising agent, while fluoride ion is correspondingly the weakest possible reducing agent. Lithium sits at the very bottom, -3.05 V, meaning Li+ has the least tendency to be reduced, making lithium metal the strongest reducing agent available, readily giving up its electron to almost anything. Moving down the table, oxidising power steadily decreases on the left side of each half-reaction while reducing power steadily increases on the right, and this single, ordered list lets chemists extract an enormous amount of practical information: predicting reaction feasibility, calculating equilibrium constants, and even determining pH, all without running a single additional experiment.

Standard electrode potentials assume every species sits at exactly unit concentration, but real solutions rarely cooperate. Nernst worked out exactly how electrode potential shifts away from its standard value as concentration changes, E = E° - (RT/nF) ln(1/[Mn+]) for a simple reduction, and applying this to a full cell like the Daniell cell, combining both electrodes' individual expressions, gives E(cell) = E°(cell) - (RT/nF) ln([Zn2+]/[Cu2+]), or at 298 K converted to base-10 logarithms, E(cell) = E°(cell) - (0.059/n) log([Zn2+]/[Cu2+]). This confirms something genuinely intuitive: cell potential increases as the concentration of the species being reduced (Cu2+) rises, and decreases as the concentration of the species being oxidised (Zn2+) rises, exactly matching how a reaction naturally runs down as products accumulate and reactants deplete. For any general reaction, aA + bB to cC + dD, the same logic extends directly to E(cell) = E°(cell) - (RT/nF) ln Q, where Q is the reaction quotient built from products over reactants.

Close a Daniell cell's circuit and let it run, and something predictable happens: Zn2+ concentration climbs, Cu2+ concentration falls, and the voltmeter reading steadily drops. Eventually, the concentrations stop changing and the voltmeter reads exactly zero, the unmistakable signature of equilibrium. Setting E(cell) to zero in the Nernst equation, then substituting the equilibrium ratio [Zn2+]/[Cu2+] for the reaction's actual equilibrium constant Kc, produces E°(cell) = (0.059/n) log Kc at 298 K, an equation letting chemists calculate a genuinely difficult-to-measure equilibrium constant purely from an easily-measured cell voltage. For the Daniell cell's own E°(cell) of 1.1 V with n = 2, this works out to log Kc = 37.29, or Kc around 2 x 10^37, an equilibrium constant so enormous it confirms the reaction proceeds essentially to completion, something no direct concentration measurement could ever practically capture.

Electrical work done reversibly by a galvanic cell equals its Gibbs energy decrease, giving the direct relation DrG = -nFE(cell), where n is the number of moles of electrons transferred and F is Faraday's constant. Under standard conditions this becomes DrG° = -nFE°(cell), a single equation quietly connecting three ideas usually met in three separate chapters: electrochemistry's cell potential, thermodynamics' Gibbs energy, and equilibrium's own equilibrium constant, since DrG° also equals -RT ln K from earlier work. For the Daniell cell, with n = 2, F = 96487 C/mol, and E°(cell) = 1.1 V, this gives DrG° = -2 x 1.1 x 96487, roughly -212 kJ/mol, a substantial negative Gibbs energy confirming the reaction is genuinely, strongly spontaneous, exactly consistent with the enormous equilibrium constant calculated the same way from the same starting E° value.

Just as a wire's resistance depends on its length and cross-sectional area, so does an electrolyte solution's, and the constant of proportionality, resistivity, has a genuinely useful inverse: conductivity, symbol kappa, describing how well a solution carries current per unit volume. But raw conductivity alone can't fairly compare two different solutions, since a more concentrated solution simply has more ions available to carry current regardless of how good each individual ion actually is at conducting; molar conductivity, symbol Λm, solves this by dividing conductivity by concentration, Λm = kappa/c, giving the conductance contributed specifically by one mole of dissolved electrolyte. This is measured practically using a conductivity cell with two platinum electrodes, calibrated first against a known-conductivity solution (typically KCl) to determine the cell's own geometric cell constant, before that same cell constant is applied to calculate an unknown solution's conductivity from its measured resistance.

Molar conductivity always increases with dilution, but the reason differs sharply between strong and weak electrolytes. Strong electrolytes, already fully dissociated at any concentration, show only a slow, gentle rise, following Λm = Λ°m - A(square root of c), a straight line when plotted against the square root of concentration, since dilution's only real effect here is reducing the ionic interference that slightly slows ion movement at higher concentration. Weak electrolytes, only partially dissociated to begin with, show a genuinely steep rise instead, since dilution actively shifts the dissociation equilibrium further toward more ions being produced, a fundamentally different mechanism, meaning a weak electrolyte's limiting molar conductivity (Λ°m, its value at infinite dilution) can never be found by simply extrapolating a straight-line graph the way it can for a strong electrolyte, since the curve near zero concentration becomes far too steep to extrapolate reliably.

Kohlrausch noticed something remarkable studying limiting molar conductivities: the difference between, say, KCl and NaCl's Λ°m values stays essentially constant no matter which halide anion is paired with those same two cations, roughly 23.4 S cm2/mol every time. This regularity led directly to the law of independent migration of ions: at infinite dilution, each ion migrates and conducts entirely independently of whatever other ion it happens to be paired with, so an electrolyte's limiting molar conductivity is simply the sum of its cation's and anion's own individual contributions, Λ°m = n+ lambda°+ + n- lambda°-. This is genuinely powerful for weak electrolytes specifically, since their own Λ°m can't be measured by direct extrapolation; acetic acid's Λ°m can instead be calculated indirectly from the easily-measured, fully-dissociated strong electrolytes HCl, NaCl, and sodium acetate combined algebraically, sidestepping the extrapolation problem entirely.