Cell Potentials & EMF

Charge Flow in Galvanic Cells

• Electrons always move from anode → cathode.
  • Anode = site of oxidation (loss of e⁻)
  • Cathode = site of reduction (gain of e⁻)
• Question addressed: How do we know which species is oxidized or reduced? → Use reduction potentials.

Reduction Potentials (General)

• Reduction potential (E)
  • Electrical potential (measured in volts, V) assigned to a half-reaction written as a reduction.
  • Specifies the intrinsic “tendency” of a species to gain electrons.
• Reference electrode: Standard Hydrogen Electrode (SHE)
  • Conventional value: $E_{\text{SHE}} = 0\;\text{V}$
• Interpretation of sign/magnitude
  • More positive $E_{\text{red}}$ → stronger drive to be reduced.
  • Less positive (or negative) $E_{\text{red}}$ → species is more easily oxidized (acts as anode in a galvanic cell).

Standard Conditions

• All tabulated "standard" values assume:
  • Temperature: $25\;^\circ\text{C} \;(298\;\text{K})$
  • Pressure: $1\;\text{atm}$
  • Concentrations: $1\;\text{M}$ for all aqueous species.

Choosing Anode vs. Cathode with $E_{\text{red}}$

• Galvanic cell (spontaneous)
  • Electrode with more positive $E_{\text{red}}$ → cathode
  • Electrode with less positive $E_{\text{red}}$ → anode
  • Overall cell free energy: \Delta G < 0
• Electrolytic cell (non-spontaneous, driven by external voltage)
  • External source forces the electrode with the more positive $E_{\text{red}}$ to be oxidized → becomes anode.
  • Electrode with the less positive $E_{\text{red}}$ is forced to be reduced → cathode.
  • \Delta G > 0

Converting Reduction ↔ Oxidation Potentials

• Oxidation potential = negative of the reduction potential for the reverse reaction.
  • Reverse the half-reaction and change the sign.
  • Example (from transcript)
  • Reduction: $\text{Ti}^{+} + e^- \rightarrow \text{Ti} \quad\; E_{\text{red}} = -0.34\;\text{V}$
  • Oxidation: $\text{Ti} \rightarrow \text{Ti}^{+} + e^- \quad\; E_{\text{ox}} = +0.34\;\text{V}$
• MCAT & most tables supply only $E_{\text{red}}$; convert as needed.

Example 1 – Ag⁺ / Ti⁺ Half-Cells (Galvanic)

• Given half-reactions (all as reductions):
  • $\text{Ag}^+ + e^- \rightarrow \text{Ag} \quad E_{\text{red}} = +0.80\;\text{V}$
  • $\text{Ti}^+ + e^- \rightarrow \text{Ti} \quad E_{\text{red}} = -0.34\;\text{V}$
• Analysis
  • Ag⁺ has the more positive reduction potential → will be reduced (cathode).
  • Ti(s) will be oxidized to $\text{Ti}^+$ (anode).
• Net ionic equation (spontaneous):
  $\text{Ag}^+ + \text{Ti(s)} \;\rightarrow\; \text{Ti}^+ + \text{Ag(s)}$

Electromotive Force (EMF)

• Definition: Voltage difference between two half-cells under standard conditions.
• Formula (standard):
  $E<em>{\text{cell}}^{\circ} = E</em>{\text{red,cathode}}^{\circ} - E_{\text{red,anode}}^{\circ}$
• Do not multiply potentials by stoichiometric coefficients—potential depends on identity, not quantity, of species.

Example 2 – Sm³⁺ vs. $\text{RhCl}_{6}^{3-}$

• Given $E_{\text{red}}$ values
  • $\text{Sm}^{3+} + 3e^- \rightarrow \text{Sm} \quad E_{\text{red}} = -2.41\;\text{V}$
  • $\text{RhCl}<em>{6}^{3-} + 3e^- \rightarrow \text{Rh} + 6\text{Cl}^- \quad E</em>{\text{red}} = +0.44\;\text{V}$
• Half-reaction roles as written in transcript’s equation
  • $\text{Sm}^{3+}$ is reduced.
  • $\text{Rh}$ (within $\text{RhCl}_{6}^{3-}$ framework) is oxidized.
• EMF calculation: $E_{\text{cell}}^{\circ} = (-2.41\;\text{V}) - (+0.44\;\text{V}) = -2.85\;\text{V}$
  • Negative EMF → electrolytic under stated orientation.
  • Reversing the reaction (making it galvanic) would give $+2.85\;\text{V}$ and proceed spontaneously in the opposite direction.

Practical & Conceptual Notes

• Lead–acid and nickel–cadmium batteries (mentioned) employ the same sign conventions—tables usually list reduction potentials, even when oxidation half-reactions are discussed.
• In calculations of $\Delta G$ for electrochemical cells: $\Delta G^{\circ} = -nF E_{\text{cell}}^{\circ}$
  • $n$ = moles of electrons transferred
  • $F$ = Faraday constant $\approx 96,485\;\text{C\,mol}^{-1}$
• Ethical / safety implication: Knowing EMF and directionality guides safe battery charging/discharging and electrolysis operations.
• Philosophical note: The arbitrary zero point (SHE) illustrates convention-driven measurement systems—absolute potentials aren’t measurable, only differences.