Electrochemistry Notes: Half-Reactions, Potentials, and Cell Theory

Two-Compartment Electrochemical Cells and the Half-Reaction Strategy

The instructor emphasizes that students are not electricians, but electricity and delta G concepts are essential for understanding electrochemical cells.
Volts (voltage) come from the redox chemistry, not from a mysterious force; a voltage is set by the tendency of a redox pair to gain or lose electrons.
In a galvanic/voltaic cell, there are two compartments: the anode compartment and the cathode compartment. They are physically separated to study each half-reaction independently and to simplify the analysis of the overall cell.
- Separation allows studying different cathode/anode combinations without rebuilding whole batteries each time.
- Electrons flow from the anode to the cathode through an external circuit, creating electrical energy that can be tapped.
The half-reaction framework:
- Oxidation is one half-reaction.
- Reduction is the other half-reaction.
- The full redox reaction (redox couple) is the combination of the two half-reactions.
The standard reference half-reaction is the reduction at the cathode; the other (anode) half-reaction is treated as the reference and often assigned a reference potential, sometimes set to zero in the context of a given cell diagram.
The cell voltage (E_cell) is determined by the potentials of the two half-reactions; the overall potential is the contribution from both halves, while the reference half is effectively set to zero in the context of the chosen reference.
- Example discussion: the hydrogen half-reaction is often used as a reference, with the hydrogen reference electrode having a potential of 0 V by convention.

Key Concepts: Half-Reactions, Reduction Potentials, and Spontaneity

Redox chemistry decomposes into two half-reactions:
- Oxidation: loss of electrons.
- Reduction: gain of electrons.
The term “reduction potential” on a table corresponds to the tendency of a species to gain electrons (the reduction form).
The overall cell voltage is determined by combining these half-reaction potentials.
Hydrogen as a reference electrode:
- The reference half-reaction is taken as the hydrogen redox couple, H⁺/H₂, with a reduction potential defined as 0 V.
- Any other half-reaction can be compared against this reference to determine its tendency to be reduced.
Relationship between voltage and energy ( spontaneity):
- The voltage is related to the free energy change, ΔG, by the equation
- oxed{ \Delta G = -n F E_{cell}}
- Here, n is the number of moles of electrons transferred in the balanced redox reaction, and F is the Faraday constant.
- If ΔG < 0 (negative), then Ecell > 0 (spontaneous in the forward direction). If ΔG > 0, then Ecell < 0 (non-spontaneous in that direction).
The instructor notes that a negative cell voltage (Ecell < 0) means the reaction is not spontaneous in the given direction; flipping the reaction direction makes it spontaneous (the sign of Ecell changes).
Example interpretation:
- For a half-reaction like Mn²⁺ + e⁻ → Mn(s), the reduction potential is typically negative for many metals (not necessarily the numbers shown in the lecture), which would imply non-spontaneity in the reduction direction without coupling to a more favorable reduction half-reaction.
- A second half-reaction with a positive reduction potential can drive the overall spontaneous cell when paired correctly.

Important Table and Values Mentioned in the Lecture

Reduction potential table reference points (examples mentioned):
- Ag⁺ + e⁻ → Ag(s) has a positive reduction potential (about +0.80 V to +0.84 V depending on the table); this makes Ag⁺ reduction favorable.
- H⁺/H₂ is used as the zero reference potential (
  E^\b0{H^+/H2} = 0
  ).
When the reaction is written in the reverse direction (oxidation instead of reduction), the sign of the potential flips.
- For example, the oxidation of Ag(s) → Ag⁺ + e⁻ would have a reduction potential of about +0.80 V for the reverse process, which is equivalent to about −0.80 V for the oxidation direction.
The cathode vs anode potentials: the species at the top of the standard table (with positive reduction potentials) tend to be reduced (cathode), and those with negative reduction potentials tend to be oxidized (anode).
Note on a concrete example discussed in class: zinc versus silver and the corresponding half-reactions lead to a net positive cell potential when the more favorable reduction half is used at the cathode and the less favorable one is reversed at the anode. The teacher uses the phrase that zinc “wants to lose electrons” (oxidation tendency) and silver “doesn’t want to be oxidized” (favorable reduction).
A value mentioned for a large equilibrium constant (for a favorable spontaneous reaction) is approximately
- $K \approx 6.9 \times 10^{12}.$
- A large K suggests the products are highly favored at equilibrium for the forward reaction.
The Faraday constant:
- $F = 96485\ \mathrm{C \, mol^{-1}}.$
The number of electrons transferred, n, depends on the balanced redox equation. It is not fixed for all reactions; it can be 1, 2, 3, or 4 in different systems. The example discussed uses n = 2 for the zinc/copper-type couple, but the instructor notes that n can vary and must be determined from the balanced half-reactions.
Q and standard state considerations:
- The reaction quotient Q reflects concentrations (or activities) of reactants and products.
- In standard state, the concentrations are set to 1 M, and Q = 1, which makes E = E° (the standard cell potential).
- If concentrations differ from 1 M, then Q ≠ 1 and the cell potential departs from E° according to the Nernst equation (see below).
Comment on misalignments between references and tables:
- The instructor notes you should be able to locate the corresponding half-reactions on a table and determine their reduction potentials; some species (e.g., Mg and Sr) may not be on every table, requiring lookup of the correct half-reaction and its reduction potential.

Key Equations and How to Use Them

Standard cell potential (from half-cells):
- E^\b0{cell} = E^\b0{cathode} - E^\b0_{anode}}.
Relationship between ΔG° and E°cell:
- \Delta G^\circ = -n F E^\b0_{cell}.
General Nernst equation (temperature-dependent):
- E = E^\b0_{cell}} - \frac{R T}{n F} \ln Q.
- At 25°C (298 K), this simplifies to
- E = E^\b0{cell} - \frac{0.05916}{n} \log{10} Q.
Reaction quotient for a general reaction:
- For a reaction aA + bB ⇌ cC + dD with activities aA, aB, aC, aD, the reaction quotient is
- $Q = \frac{a<em>C^{c} a</em>D^{d}}{a<em>A^{a} a</em>B^{b}}.$
Faraday constant:
- $F = 96485\ \mathrm{C \, mol^{-1}}.$
Specific notes about n:
- The number n is the total number of electrons transferred in the balanced redox reaction; it can be 1, 2, 3, or 4 (not always 2). When balancing a redox reaction with multiple metals, you may need to multiply half-reactions by appropriate coefficients to balance electrons (e.g., 3 Mg -> 3 Mg^{2+} + 6 e⁻ and 3 Cu^{2+} + 6 e⁻ -> 3 Cu).
Temperature and unit caution:
- Be careful with units when combining energy terms (e.g., kJ vs J). Energies in ΔG or ΔH are often in kJ, while E and F use joules; ensure unit consistency when calculating quantities like ΔG from E° or E from Q.
Practical lab note from the lecture:
- Students will measure half-reactions in the lab; the measured potentials help assess the spontaneity of each half-reaction and the overall cell behavior.

How to Apply These Concepts to a Zinc-Copper (or Zinc-Silver) Cell (Conceptual Outline)

Write the two half-reactions in reduction form (as reduction potentials are tabulated):
- Example (Copper side): Cu^{2+} + 2 e⁻ → Cu(s) with E° ≈ +0.34 V (table value varies by source).
- Example (Zinc side): Zn^{2+} + 2 e⁻ → Zn(s) with E° ≈ -0.76 V (table value varies by source).
Determine which half-reaction will be reduced at the cathode (the more positive reduction potential) and which will be oxidized at the anode (the less positive reduction potential).
Compute the standard cell potential:
- E^\b0{cell} = E^\b0{cathode} - E^\b0_{anode}}.
- Alternatively, if you flip an electrode’s role (oxidation vs reduction), the sign of its potential changes accordingly.
Determine n (the total number of electrons transferred) from the balanced overall equation; balance the electron transfer to ensure electron conservation (e.g., for Zn + Cu^{2+} → Zn^{2+} + Cu, n = 2).
Use the Faraday constant to relate E°cell to ΔG°:
- \Delta G^\ = -n F E^\b0_{cell}.
If the reaction is not at standard concentrations, apply the Nernst equation to find the actual cell potential at the given conditions:
- E = E^\b0_{cell} - \frac{R T}{n F} \ln Q.
Recognize that a large equilibrium constant (K) implies the reaction strongly favors products and typically corresponds to a large positive E°cell and a negative ΔG°, indicating spontaneity in the forward direction.
Remember that the standard hydrogen electrode is the zero reference, so any potential can be measured relative to this reference.

Conceptual Takeaways and Practical Implications

Separation of compartments aids analysis and design by allowing independent study of oxidation and reduction processes.
The sign and magnitude of E°cell reflect the thermodynamic tendency for the overall redox reaction to proceed spontaneously in the forward direction.
The sign of ΔG and E°cell are linked: negative ΔG corresponds to a positive E°cell for the cell as written (spontaneous).
The number of electrons transferred (n) is crucial for calculating ΔG and for converting between E and ΔG; miscounting n leads to incorrect energy estimates.
Concentration effects (via Q in the Nernst equation) shift the actual cell potential away from E°cell; hence lab measurements at non-standard conditions may differ from tabulated standard potentials.
The Faraday constant is a key bridge between electrochemistry and energy: it converts moles of electrons to charge, enabling energy calculations for electrochemical processes.

Quick Reference: Summary of Symbols and Common Values

E°{cell} = E°{cathode} − E°_{anode}
ΔG° = − n F E°_{cell}
E = E°_{cell} − (R T / n F) ln Q
At 25°C: E = E°_{cell} − (0.05916 / n) log10 Q
Q = (activities of products)^(coefficients) / (activities of reactants)^(coefficients)
F = 96485 C/mol
n = number of electrons transferred (varies by reaction; e.g., n = 2 for Zn^{2+}/Zn and Cu^{2+}/Cu in the classic Zn-Cu cell)
E(H⁺/H₂) = 0 V (standard hydrogen electrode as reference)
K (equilibrium constant) can be very large for spontaneous reactions (e.g., ~6.9 × 10^{12} in an example discussed)