Free energy and cell potential Knowledge-Based Study

Study notes

1. In electrochemistry, electrical energy is directly related to the thermodynamic driving force of a redox reaction. The maximum electrical work that an electrochemical cell can perform corresponds to the change in Gibbs free energy ($\Delta G$).

   The relationship between Gibbs free energy and cell potential is defined by the fundamental thermodynamic equation:

   $$\Delta G = -nFE_{\text{cell}}$$

   Under standard state conditions (solutes at 1 M, gases at 1 bar or 1 atm, and a specified temperature, typically 298.15 K), this relationship is expressed as:

   $$\Delta G^\circ = -nFE^\circ_{\text{cell}}$$

   The components of this equation are defined as follows:

   • $n$: The number of moles of electrons transferred in the balanced redox equation (a dimensionless quantity representing mole equivalents).
   • $F$: The Faraday constant, representing the electric charge per mole of electrons, approximately equal to $96485\text{ C/mol}$ (or $\text{J}/(\text{V}\cdot\text{mol})$).
   • $E_{\text{cell}}$: The cell potential, measured in volts ($\text{V}$), where $1\text{ V} = 1\text{ J/C}$.

   The negative sign in the equation $\Delta G = -nFE_{\text{cell}}$ establishes the relationship between thermodynamic spontaneity and electrochemical cell potential:

   • Spontaneous processes (Galvanic / Voltaic Cells): A spontaneous redox reaction does electrical work on the surroundings. For a thermodynamically favorable process, $\Delta G < 0$, which requires that $E_{\text{cell}} > 0$.
   • Non-spontaneous processes (Electrolytic Cells): A non-spontaneous redox reaction requires external electrical energy to proceed. In this case, $\Delta G > 0$, which corresponds to $E_{\text{cell}} < 0$.
   • Equilibrium state: When a cell reaction reaches dynamic chemical equilibrium, the cell cannot produce electrical work (it is fully discharged). At equilibrium, $\Delta G = 0$ and therefore $E_{\text{cell}} = 0\text{ V}$.

   Standard cell potential ($E^\circ_{\text{cell}}$) provides a direct measure of the equilibrium constant ($K$) of a chemical system. Standard Gibbs free energy is related to the equilibrium constant by:

   $$\Delta G^\circ = -RT \ln K$$

   Setting this equal to $\Delta G^\circ = -nFE^\circ_{\text{cell}}$ gives:

   $$-nFE^\circ_{\text{cell}} = -RT \ln K \implies E^\circ_{\text{cell}} = \frac{RT}{nF} \ln K$$

   At standard temperature (298.15 K), substituting the values for the universal gas constant ($R = 8.314\text{ J}/(\text{mol}\cdot\text{K})$) and Faraday constant simplifies this relationship to:

   $$E^\circ_{\text{cell}} = \frac{0.0592\text{ V}}{n} \log_{10} K$$

   A positive standard potential ($E^\circ_{\text{cell}} > 0$) leads to $K > 1$, meaning that products dominate at equilibrium under standard conditions.

   A precise understanding of free energy and cell potential requires distinguishing between intensive and extensive quantities:

   • Intensive vs. Extensive Properties: Cell potential ($E^\circ_{\text{cell}}$) is an intensive property; multiplying a half-reaction by a stoichiometric coefficient does not alter its potential. In contrast, Gibbs free energy ($\Delta G^\circ$) is extensive. Multiplying a balanced equation doubles $n$, which proportionally doubles $\Delta G^\circ$ while leaving $E^\circ_{\text{cell}}$ unchanged.
   • Energy Units: The product $-nFE_{\text{cell}}$ yields energy in Joules ($\text{J}$). Because standard thermodynamic tables often report values in kilojoules ($\text{kJ}$), remember to divide by 1000 when comparing values.
   • Standard State vs. Dynamic State: Do not confuse $E_{\text{cell}}$ with $E^\circ_{\text{cell}}$. At equilibrium, $E_{\text{cell}} = 0$, but $E^\circ_{\text{cell}}$ remains a fixed constant characteristic of the reaction at a given temperature.