Watch a galvanic cell (such as the Daniell cell) in real time: electrons flow through the external circuit, ions migrate across the salt bridge, and the electrodes are oxidized and reduced. Visualize the standard EMF E°cell and how the Nernst equation E = E°−(0.0592/n)logQ shifts the potential with concentration.
Standard EMF. Computed as the difference of the standard reduction potentials E° (vs the standard hydrogen electrode, SHE). E°cell > 0 means spontaneous (a galvanic cell).
Nernst equation. R: gas constant, T: absolute temperature [K], n: electrons transferred, F: Faraday constant (96485 C/mol), Q: reaction quotient.
$$Q = \frac{[\text{Anode}^{z+}]}{[\text{Cathode}^{z+}]}, \qquad \Delta G = -nFE$$
The reaction quotient Q is the anode-product ion concentration divided by the cathode-reactant ion concentration. As discharge proceeds, Q rises and E falls; at Q→K, E→0 (equilibrium).
What is cell potential (EMF)?
🙋
So "cell EMF" is basically the battery's voltage, right? I learned a Daniell cell is about 1.1 V — where does that number come from?
🎓
Roughly speaking, the EMF is the difference in how badly each electrode "wants electrons." Every metal has a standard reduction potential E°: copper is +0.34 V, zinc is −0.76 V. Subtract the anode (zinc) from the cathode (copper): 0.34 − (−0.76) = 1.10 V. That's the standard EMF E°cell. Press the "Standard conditions" preset and you'll see exactly 1.10 V appear.
🙋
Got it! But in the diagram on the left the yellow "−" markers flow from zinc to copper. Why do the electrons leave from the zinc side?
🎓
Because zinc gives up electrons more readily than copper. Zinc dissolves as Zn → Zn²⁺ + 2e⁻, leaving electrons behind (oxidation = anode). Those electrons travel through the external wire to the copper side, where copper plates out as Cu²⁺ + 2e⁻ → Cu (reduction = cathode). Ions move through the salt bridge to keep each solution electrically neutral. Watch the green ions head toward the cathode and the pink ones toward the anode.
🙋
So changing the concentrations changes the voltage too? With the "Dilute cathode" preset the voltage dropped…
🎓
That's where the Nernst equation comes in: E = E° − (0.0592/n)·logQ, with Q = [anode ion]/[cathode ion]. Making the cathode Cu²⁺ dilute raises Q, so the logQ term lowers the voltage. As discharge proceeds, Cu²⁺ is consumed and Zn²⁺ builds up, Q keeps growing, and finally at Q→K the EMF reaches E→0 — the battery is "dead" (at equilibrium). Try the "Near equilibrium" preset and watch the operating point slide toward the lower right of the curve.
Physical model and key equations
Standard EMF: $E^{\circ}_{cell} = E^{\circ}_{cathode} - E^{\circ}_{anode}$, the difference of the standard reduction potentials E° (referenced to the standard hydrogen electrode, SHE). When E°cell > 0 the reaction proceeds spontaneously and the system acts as a galvanic cell that can drive current through an external circuit.
Nernst equation: $E = E^{\circ}_{cell} - \dfrac{RT}{nF}\ln Q$. At 25°C, $\dfrac{RT}{F}\ln 10 = 0.0592\,\mathrm{V}$, so $E = E^{\circ}_{cell} - \dfrac{0.0592}{n}\log_{10} Q$. Here n is the number of electrons transferred and Q the reaction quotient.
Free energy and equilibrium: $\Delta G = -nFE$. At equilibrium $E=0$ and $\Delta G=0$, so $\log K = \dfrac{nE^{\circ}_{cell}}{0.0592}$. For the Daniell cell (n=2, E°=1.10 V), $K\approx1.5\times10^{37}$ — enormous, meaning the reaction runs essentially to completion.
Real-world applications
Primary and secondary battery design: The nominal voltage of dry cells, lithium-ion batteries, and lead-acid batteries is set by the difference of the positive and negative electrode potentials. Because the change in active-material concentration during discharge (rising Q) shows up as the voltage drop along the discharge curve, the Nernst equation underpins capacity design and state-of-charge (SOC) estimation.
Corrosion and protection: When dissimilar metals contact, the more active metal becomes the anode and dissolves preferentially (galvanic corrosion). A larger potential difference means a stronger driving force, while sacrificial-anode cathodic protection (zinc or magnesium) deliberately exploits this same principle.
Electrochemical sensors: pH meters, ion-selective electrodes (ISE), and dissolved-oxygen probes use the fact that the target ion concentration appears as a potential following the Nernst equation. The 59 mV/decade slope (monovalent ion, 25°C) is the basis for calibration.
Electrolysis and electroplating: Driving a non-spontaneous reaction (E°cell < 0) requires an external power supply plus overpotential. The minimum required voltage follows from the cell potential, with overpotential and resistive losses added for the real applied voltage.
Common misconceptions and pitfalls
First, the misconception that "E°cell is the sum of the electrode potentials." It's actually a difference: E°cell = E°(cathode) − E°(anode), with both kept as standard reduction potentials. You can instead flip the anode to an oxidation potential and add, but that invites sign errors. This tool stays consistent with reduction potentials — compare the "Anode E°" and "Cathode E°" stat cards.
Next, the mistaken idea that "doubling the concentration doubles the voltage." In reality it's logarithmic: Q must change tenfold to move the potential by just 0.0592/n V (about 30 mV for two electrons). Lower the cathode concentration from 1 M to 0.001 M (a factor of 1000) in the simulator and E drops only ~0.09 V. The concentration effect is surprisingly small.
Finally, "E→0 means the cell is broken" is wrong. E→0 means the reaction has reached equilibrium (Q→K) and can no longer deliver net current. Thermodynamically it's the natural endpoint where ΔG=0. In a rechargeable secondary cell, an external supply pushes this equilibrium back in reverse. Use the "Near equilibrium" preset to see where the operating point lands on the Nernst curve.
Frequently asked questions
E°cell = E°(cathode) − E°(anode). Keep both electrodes as standard reduction potentials E° (vs SHE), then subtract the anode's reduction potential from the cathode's. For a Daniell cell that is 0.34 − (−0.76) = 1.10 V. A positive E°cell means a spontaneous galvanic cell; a negative value means electrolysis (an external power supply is needed).
It is RT/F × ln10. With R = 8.314 J/(mol·K), T = 298.15 K (25°C), and F = 96485 C/mol, this equals about 0.05916 V. So at 25°C, E = E° − (0.0592/n)logQ. The coefficient changes with temperature, so this tool recomputes RT/nF in real time from the temperature slider.
Q is the concentration ratio at any instant; K is that ratio at equilibrium. As discharge proceeds Q rises and E falls, and the moment Q = K, E = 0 (equilibrium, ΔG = 0). The equilibrium constant follows from logK = nE°cell/0.0592; for the Daniell cell K ≈ 1.5×10³⁷, so the reaction runs essentially to completion.
At the anode, oxidation (the metal releases electrons and dissolves) leaves a surplus of electrons; at the cathode, reduction (metal ions accept electrons and plate out) consumes them. Electrons therefore travel anode → cathode through the external wire. Meanwhile ions move through the salt bridge to keep the solutions electrically neutral and close the circuit.
Choose the galvanic cell combination under "Electrode pair" (the default is the Zn/Cu Daniell cell). The anode (the more active metal) dissolves and the cathode (the nobler metal) plates out.
Set the anode-side and cathode-side ion concentrations (M) and the temperature. The tool automatically computes E°cell = E°(cathode) − E°(anode) and the cell potential from the Nernst equation E = E° − (0.0592/n)logQ.
Watch the left diagram as electrons flow anode → cathode through the external circuit and ions migrate across the salt bridge in real time. On the right Nernst curve, the operating point (red) moves with log Q.
Worked Example
Daniell cell at standard conditions: anode Zn (E° = −0.76 V), cathode Cu (E° = +0.34 V), [Zn²⁺] = [Cu²⁺] = 1 M, 25°C. Then E°cell = 0.34 − (−0.76) = 1.10 V; with Q = 1, logQ = 0, the cell potential is E = 1.10 V. Diluting the cathode Cu²⁺ to 0.001 M gives Q = 1000, logQ = 3, so E = 1.10 − (0.0592/2)×3 ≈ 1.01 V — a drop of about 0.09 V.
Practical Notes
Standard reduction potentials E° are literature values (25°C, vs SHE, unit activity). In real solutions, activity coefficients and complexation change the effective concentration, so at high concentration the Nernst equation should use activities instead of concentrations.
This tool computes the thermodynamic EMF (open-circuit voltage). Once current actually flows, overpotentials (activation, concentration, and resistive polarization) make the terminal voltage lower than the EMF.
For a monovalent ion such as Ag⁺ at a silver electrode, the half-reaction is Ag⁺ + e⁻ → Ag, but to match the overall electron count n it is treated here as 2Ag⁺ + 2e⁻ → 2Ag. Note that the exponent in Q also depends on the ionic charge.
🎬 Watch it in motion
Electric Dipole | the field two opposite charges weave #Shorts
