Cell Potential, \(\Delta_{r}G^\circ\), and the Equilibrium Constant \(K\)
A galvanic (voltaic) cell converts a spontaneous redox reaction into electrical work, measured by the cell potential \(E_{\text{cell}}\). The same driving force can be expressed thermodynamically using the standard Gibbs energy change \(\Delta_{r}G^\circ\) and the equilibrium constant \(K\). A cell potential, \(\Delta_{r}G^\circ\) and equilibrium constant \(K\) calculation typically reports \(\Delta_{r}G^\circ\) and/or \(K\) from a given \(E^\circ_{\text{cell}}\).
Core definitions and essential formulas
\[
\Delta_{r}G = -\,n F E_{\text{cell}}
\]
\[
\Delta_{r}G^{\circ} = -\,R T \ln K
\]
\[
\ln K = \frac{n F E^{\circ}_{\text{cell}}}{R T}
\]
\(n\) is the number of moles of electrons transferred in the balanced overall reaction, \(F \approx 96485~\mathrm{C\cdot mol^{-1}}\) is the Faraday constant, \(R \approx 8.314~\mathrm{J\cdot mol^{-1}\cdot K^{-1}}\) is the gas constant, and \(T\) is absolute temperature in kelvin. \(E_{\text{cell}}\) is in volts, and \(\Delta_{r}G\) or \(\Delta_{r}G^\circ\) is in \(\mathrm{J\cdot mol^{-1}}\). \(K\) is dimensionless and applies to the balanced overall reaction under standard conditions when using \(E^\circ_{\text{cell}}\).
How to interpret results
Larger positive \(E^\circ_{\text{cell}}\) gives more negative \(\Delta_{r}G^\circ\) and a larger \(K\), meaning products are strongly favored at equilibrium. If \(E^\circ_{\text{cell}} = 0\), then \(\Delta_{r}G^\circ = 0\) and \(K = 1\). If \(E^\circ_{\text{cell}} < 0\), then \(\Delta_{r}G^\circ > 0\) and \(K < 1\), so reactants are favored (the reverse reaction is spontaneous).
- Unit mix-up: using Celsius instead of kelvin for \(T\).
- Wrong \(n\): using coefficients instead of the electron count from the balanced redox equation.
- Base-10 confusion: using \(\log_{10}\) where \(\ln\) is required.
- Nonstandard conditions: treating \(E^\circ_{\text{cell}}\) as valid when concentrations or pressures are not standard.
Micro example: With \(n=2\), \(E^\circ_{\text{cell}}=1.10~\mathrm{V}\), \(T=298~\mathrm{K}\),
\(\ln K=\dfrac{2\times 96485\times 1.10}{8.314\times 298}\approx 85.6\), so \(K \approx 1.5\times 10^{37}\).
Use these relationships to connect electrochemistry outputs (\(E^\circ_{\text{cell}}\)) to thermodynamic favorability and equilibrium position (\(\Delta_{r}G^\circ\), \(K\)). Do not use them to predict actual cell voltage away from standard conditions; a common next step is applying the Nernst equation to include reaction quotient effects.
