Electrochemistry: Galvanic Cells, Electrode Potentials, and Nernst Relations

Overview of Electrochemical Cells

• Convert energy forms: chemical energy to electrical energy in galvanic (Voltaic) cells; electrical energy to chemical energy in electrolytic (electrolysis) cells.
• Key components: electrodes (anode and cathode) immersed in electrolytes, and an external circuit for current flow.
• Basic terms:
  • Electrochemical cell: device that drives or uses redox reactions to convert energy forms.
  • Electrical energy: energy carried by moving electrons in the external circuit (current).
  • Chemical energy: energy stored in chemical bonds and species in the cell.

Anode and Cathode: Definitions

• Anode: electrode at which oxidation takes place (loss of electrons).
• Cathode: electrode at which reduction takes place (gain of electrons).
• In galvanic cells, electrons flow from anode to cathode through the external circuit; current is the conventional flow from cathode to anode.
• In galvanic cells, reactions should occur in separate containers to allow current generation (prevent direct mixing of oxidant and reductant).

Galvanic (Voltaic) Cells: Functioning and Examples

• Functioning concept: a spontaneous redox reaction generates electrical energy.
• Example: Daniell cell (Zn | Zn^{2+} || Cu^{2+} | Cu)
  • Oxidation (anode): $ext{Zn (s)} <br /> ightarrow ext{Zn}^{2+} (aq) + 2e^-$
  • Reduction (cathode): $ext{Cu}^{2+} (aq) + 2e^- <br /> ightarrow ext{Cu (s)}$
  • Overall cell reaction: $ext{Zn (s)} + ext{Cu}^{2+} (aq) <br /> ightarrow ext{Zn}^{2+} (aq) + ext{Cu (s)}$
  • Anode is negative; cathode is positive in galvanic cells.
• Cell notation (cell diagram): example for Daniell cell
  • $ext{Zn (s)} | ext{Zn}^{2+} (aq) || ext{Cu}^{2+} (aq) | ext{Cu (s)}$
• Electrode potential (E): potential difference between an electrode and its solution; absolute potentials are not directly measurable due to interfacial effects.
• Factors influencing electrode potential:
  • Type of electrode/metal used
  • Concentration of ions in solution
  • Temperature
  • Presence of other ions (activity vs concentration effects)

Half-Cells and Electrode Potentials

• Types of half-cells (as discussed in the slides):
  1) Metal-Metal ion half-cell (e.g., Fe^{2+}/Fe, Cu^{2+}/Cu) where reduction or oxidation occurs at the metal interface.
  2) Gaseous electrode (e.g., H2/ H^+) with an inert electrode (commonly Pt) for the electron transfer. 3) Inert electrode half-cells (Pt, graphite) used for redox couples where the electrode itself does not participate chemically. 4) Metal-Metal insoluble salt half-cell (e.g., Ag/AgCl with AgCl(s) ⇌ Ag^+ + Cl^−; K{sp} relations apply).
• Important concepts:
  • Each half-reaction occurs at its own half-cell; the overall cell combines the oxidation half-reaction (anode) with the reduction half-reaction (cathode).
  • Cell diagrams show the interphase between phases with a vertical line and the phase boundary; a double vertical line indicates the salt bridge or junction between the solutions.
  • The direction of the reaction and electron flow is determined by the standard reduction potentials; a more positive reduction potential tends to be the cathodic reaction.

Electrode Potentials and Nernst Equation

• Electrode potential depends on:
  • Type of metal/ion (E{oxidation} vs E{reduction})
  • Ion concentrations, temperature, and interfacial conditions.
• Standard electrode potentials (E°):
  • Defined for standard states (1 M, 1 atm, 25°C) for each half-reaction.
  • Absolute electrode potentials are not directly measurable; E° serves as a reference.
• Relating electrode potentials to cell emf:
  • For a galvanic cell, the cell emf is the difference between cathode and anode potentials: $E<em>{ ext{cell}} = E</em>{ ext{cathode}} - E<em>{ ext{anode}} = E^ ext{°}</em>{ ext{cell}} - ext{(non-standard effects)}$
• Nernst equation (at 25°C approximation):
  • General form: E{ ext{cell}} = E^ ext{°}{ ext{cell}} - rac{0.05916}{n}
    abla ext{log} Q
  • Here, $Q$ is the reaction quotient for the overall cell reaction, and $n$ is the number of electrons transferred.
• Relation between ΔG°, E°, and K (thermodynamics):
  • $riangle G^ ext{°} = -n F E^ ext{°}_{ ext{cell}}$
  • $riangle G^ ext{°} = -RT <br /> abla ext{ln} K$
  • Therefore, E^ ext{°}_{ ext{cell}} = rac{RT}{nF}
    abla ext{ln} K = rac{0.05916}{n}
    abla ext{log} K ext{ at 25°C}
• The Nernst equation can be used to compute E_{ ext{cell}} at any condition from E° and Q.

Standard Hydrogen Electrode (SHE) and E°

• The SHE is defined with E° = 0 for the half-reaction:
  • $ext{2 H}^+ (aq) + 2e^- <br /> ightarrow ext{H}_2 (g)$
• It serves as the reference electrode for reporting standard electrode potentials.

Concentration Cells and Salt Bridge

• Concentration cells: same redox couple but different concentrations lead to a measurable emf.
  • Example: If two half-cells use the same couple but with different ion activities, the emf arises from concentration differences.
  • General form for a concentration cell: E{ ext{cell}} = rac{0.05916}{n} abla ext{log} rac{a ext{oxidant, high}}{a_ ext{oxidant, low}} (at 25°C, base-10 log form)
• Salt bridge: completes the circuit, maintains electrical neutrality, and prevents liquid-liquid junction potentials.
  • Prepared with a strong electrolyte (e.g., KCl, NH4NO3) in a gel (agar or gelatin).
  • Important properties of salts in the bridge: high ionic mobility, inertness, and not participating in redox, to avoid side reactions.
  • Precautions: avoid salts that would precipitate sparingly soluble salts with ions in the half-cells (e.g., Ag^+, Pb^{2+}, Hg_2^{2+}) which could form insoluble salts and disrupt the cell.

4 Types of Half-Cells (Summary)

• Metal-metal ion: e.g., Fe^{2+}/Fe or Cu^{2+}/Cu with one electrode being a metal and the other ion in solution.
• Gaseous electrode: H_2/H^+ with an inert conductor (Pt) to enable electron transfer.
• Inert electrode: electrodes like Pt or C that participate only as electron conductors, not as reactants.
• Metal-metal insoluble salt: e.g., Ag/AgCl; involves dissolution of the solid salt to provide Ag^+ in solution with a chloride ion activity governed by solubility product (K_{sp}).

Worked Outline: Building and Analyzing a Cell

• Step 1: Identify oxidation and reduction half-reactions and their E° values.
• Step 2: Compute E°cell = E°cathode - E°anode.
• Step 3: Write the overall balanced cell reaction.
• Step 4: Determine Q from activities/ions (solutions conc.).
• Step 5: Use Nernst equation to find Ecell at non-standard conditions: E{ ext{cell}} = E^ ext{°}{ ext{cell}} - rac{0.05916}{n} ext{log} Q.
• Step 6: If ΔG is required, use $riangle G = -n F E_{ ext{cell}}.$

Relationship Between E°, ΔG°, and Keq

• At standard conditions: $riangle G^ ext{°} = -n F E^ ext{°}_{ ext{cell}}.$
• At equilibrium: $riangle G = 0 <br /> ightarrow K_{ ext{eq}} = e^{- riangle G^ ext{°}/(RT)}.$
• Connect to electrode potentials: E^ ext{°}_{ ext{cell}} = rac{RT}{nF} ext{ln} K = rac{0.05916}{n} ext{log} K ext{ at 25°C}.

Example Calculation Template (Zn | Zn^{2+} || Cu^{2+} | Cu)

• Standard potentials (typical values):
  • $E^ ext{°}_{ ext{Cu}^{2+}/ ext{Cu}} <br /> = +0.34 ext{ V}$
  • $E^ ext{°}_{ ext{Zn}^{2+}/ ext{Zn}} <br /> = -0.76 ext{ V}$
• E°cell:
  • $E^ ext{°}<em>{ ext{cell}} = E^ ext{°}</em>{ ext{cathode}} - E^ ext{°}_{ ext{anode}} = 0.34 - (-0.76) = 1.10 ext{ V}$
• Example condition: [Zn^{2+}] = 1.0×10^{-2} M, [Cu^{2+}] = 1.0×10^{-1} M, n = 2
  • Q for the overall reaction: Q = rac{[ ext{Zn}^{2+}]}{[ ext{Cu}^{2+}]} = rac{1.0 imes10^{-2}}{1.0 imes10^{-1}} = 0.10
  • E_{ ext{cell}} = 1.10 - rac{0.05916}{2} ext{log}(0.10) = 1.10 - 0.02958(-1)
    = 1.1296 ext{ V (approximately 1.13 V)}
• Note: If Q > 1, Ecell decreases; if Q < 1, Ecell increases relative to E°cell.

Additional Concepts and Formulas

• Ecell interpretation: an intensive property; the potential difference depends on the system, not on the size of the container.
• Cell diagrams and interphases:
  • Anode side: species undergoing oxidation.
  • Cathode side: species undergoing reduction.
  • A single vertical line represents a phase boundary; a double vertical line represents a salt bridge/junction.
• Concentration cells and temperature effects: variations with temperature (T) also affect Ecell through ΔG and E° relationships; thermodynamics framework includes ΔH and ΔS terms for temperature dependence.
• Practical thermodynamics connections:
  • ΔG° = ΔH° - T ΔS° and the effect of temperature on cell emf via the fundamental relation ΔG° = -n F E°cell.
  • E°cell relates to K via E°cell = (RT/nF) ln K.

Quick Reference: Key Equations (LaTeX)

• Cell emf (Nernst form):
  E{ ext{cell}} = E^ ext{°}{ ext{cell}} - rac{RT}{nF}
  abla ext{ln} Q = E^ ext{°}_{ ext{cell}} - rac{0.05916}{n} ext{log} Q ext{ at 25°C}

• Gibbs relation for cell emf and electrons transferred:
  $riangle G = -n F E_{ ext{cell}}$

• Standard relation to equilibrium constant:
  $riangle G^ ext{°} = -n F E^ ext{°}<em>{ ext{cell}} = -RT ext{ln} K$ E^ ext{°}{ ext{cell}} = rac{RT}{nF} ext{ln} K = rac{0.05916}{n} ext{log} K ext{ at 25°C}

• Insoluble salt (K_{sp}) example (AgCl):

  • $ext{AgCl (s)} <br /> ightleftharpoons ext{Ag}^+ (aq) + ext{Cl}^- (aq), ext{ with } K_{sp} = [ ext{Ag}^+][ ext{Cl}^-]$
  • In a half-cell, Ag/AgCl potential depends on [Ag^+] and [Cl^-] via the Nernst equation.

Notes for Exam Preparation

• Be comfortable identifying anode vs cathode from standard reduction potentials.
• Be able to write and balance half-reactions, then assemble the full cell reaction and the cell diagram.
• Be able to apply the Nernst equation for any given Q and n, and to interpret how changes in concentrations alter Ecell.
• Remember the special role of the Standard Hydrogen Electrode and how to use it to locate E° values for other couples.
• Understand the purpose and proper use of a salt bridge in maintaining neutrality and avoiding junction potentials.
• Know how to relate E°cell to Keq and ΔG°, especially for quick conversion between equilibrium constants and cell potentials.
• Practice problems: compute Ecell for Daniell-type cells, concentration cells, and cells involving insoluble salts using K_{sp} and Nernst equations.