Nernst Equation
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Direct answer
Electrode and cell potentials drift from their standard values the moment concentrations depart from 1 M, and the Nernst equation prices the drift: Ecell = E°cell − (2.303 RT/nF) log Q, which at 298 K collapses to Ecell = E°cell − (0.059/n) log Q, with n the electrons transferred and Q the reaction quotient of the cell reaction as written. For the Daniell cell (Zn + Cu2+ → Zn2+ + Cu, E° = 1.1 V), the working form is E = 1.1 − (0.059/2) log([Zn2+]/[Cu2+]): product zinc ions piled high or copper ions depleted both drag the voltage down. The equation's two limit points anchor the chapter — when E = 0 the battery is dead and Q = K, giving E°cell = (0.059/n) log K, and a concentration cell built from identical electrodes in different solutions gives E = (0.059/n) log(c2/c1) with E° = 0.
What you must remember
- Working equation at 298 K: Ecell = E°cell − (0.059/n) log Q; n counts electrons in the balanced cell reaction, not per electrode.
- Daniell template: E = 1.1 − (0.059/2) log([Zn2+]/[Cu2+]); raising [Cu2+] raises E, raising [Zn2+] lowers it.
- General electrode form: for M^n+ + ne− → M, E = E° + (0.059/n) log[M^n+] — more concentrated ion, higher reduction potential.
- Dead battery condition: E = 0 means equilibrium, so Q = K and log K = nE°/0.059 — the bridge from electrochemistry to equilibrium constants.
- Concentration cells: same electrodes, E° = 0, so E = (0.059/n) log(c2/c1); current flows until concentrations equalise.
- Gibbs link: ΔG = −nFE and ΔG° = −nFE°, so spontaneous cells (E positive, ΔG negative) are the exam's sign-consistency check.
- Gas electrodes: partial pressures enter Q — for the hydrogen electrode at pressure p, E = −0.059 × pH, the basis of pH measurement by potentiometry.
Pricing the Daniell cell in the exam's favourite way
Set [Zn2+] = 0.1 M and [Cu2+] = 0.001 M: E = 1.1 − 0.0295 × log(0.1/0.001) = 1.1 − 0.0295 × log(100) = 1.1 − 0.059 = 1.041 V. Notice every discipline the equation demands: n = 2 from zinc losing and copper gaining two electrons each; Q written with products over reactants using only the aqueous species (solid zinc and copper sit out, activity 1); log of the ratio taken before multiplying by 0.059/n.
Flip the assignment to [Zn2+] = 0.001 and [Cu2+] = 0.1 and the log term turns negative, delivering 1.159 V — higher than standard, the physical statement that copper-rich, zinc-poor solutions push the reaction harder. When the paper asks instead for the equilibrium constant of the Daniell cell: log K = nE°/0.059 = 2 × 1.1/0.059 ≈ 37.3, K ≈ 2 × 10^37, a number so large it explains why the cell runs to completion in our perception.
Where candidates lose the mark
Three slips recur in NEET grading. Wrong n: candidates read one electron at each electrode and write 1; the balanced reaction Zn + Cu2+ → Zn2+ + Cu transfers two, and the 0.059 halves accordingly. Inverted Q: products over reactants, with solids, pure liquids and water omitted — an option set always includes the value from the flipped ratio. Sign drift on spontaneity: E positive means ΔG = −nFE negative and the cell as written is spontaneous; a negative E simply means run the cell backward, not "no reaction". The concentration-cell question catches those who insert an E° — identical electrodes make it zero by definition, so the entire voltage is the 0.059 log term. Finally, remember the 0.059 constant is a 298 K special case of 2.303RT/F; a paper that hands you another temperature expects the full form, and integer-type questions have used exactly that variation.
Frequently asked questions
What is the Nernst equation for the Daniell cell at 298 K?
E = 1.1 V − (0.059/2) log([Zn2+]/[Cu2+]), zinc ion in the numerator as the cell reaction's product.
What happens to cell potential when the cell reaches equilibrium?
E falls to zero — the driving force is exhausted — and Q becomes the equilibrium constant K, linked to the standard potential by log K = nE°/0.059.
Why does a concentration cell have zero standard potential?
Both electrodes are chemically identical, so their standard reduction potentials cancel; the measurable voltage comes purely from the concentration ratio.
How does the hydrogen electrode measure pH?
For Pt|H2(1 atm)|H+, E = −0.059 × pH at 298 K, so the electrode potential reads off hydrogen-ion concentration directly.
How are Gibbs energy and cell potential related?
ΔG = −nFEcell, so a positive cell potential means negative Gibbs energy and a spontaneous cell reaction — the sign pairing NEET tests in assertion-reason form.