Nernst Equation
Electrochemical context
The Nernst equation links electrode potential to concentration/activity ratios. It enables realistic cell-voltage analysis beyond standard-state conditions.
\[ E = E^\circ - \frac{RT}{nF}\ln Q \]
It is fundamental for galvanic cells, electrochemical sensors, corrosion analysis, and redox system modeling.
- Concentration and temperature impact on voltage
- Back-calculation of reaction quotient from measured potential
- Comparison of standard vs operating conditions
Context page with explicit cell-voltage focus: Cell Voltage (Non-Standard)
Formulas (MathJax)
\[ E = E^\circ - \frac{RT}{nF}\ln Q \]
\[ \ln Q = \frac{(E^\circ-E)nF}{RT},\qquad Q=\exp\left(\frac{(E^\circ-E)nF}{RT}\right) \]
\[ E_{\mathrm{cell}} = E_{\mathrm{cathode}} - E_{\mathrm{anode}} \]
\[ E_{\mathrm{cell}} = \left(E^\circ_{\mathrm{C}} - \frac{RT}{n_{\mathrm{C}}F}\ln Q_{\mathrm{C}}\right) - \left(E^\circ_{\mathrm{A}} - \frac{RT}{n_{\mathrm{A}}F}\ln Q_{\mathrm{A}}\right) \]
\[ E_{298\,K}=E^\circ-\frac{0.025693}{n}\ln Q \]
Symbol legend
- \(E\): electrode potential under operating conditions [V]
- \(E^\circ\): standard electrode potential [V]
- \(E_{\mathrm{cell}}\): cell voltage [V]
- \(E_{\mathrm{cathode}}, E_{\mathrm{anode}}\): half-cell potentials [V]
- \(R\): gas constant \(8.314\,\mathrm{J\,mol^{-1}\,K^{-1}}\)
- \(T\): absolute temperature [K]
- \(n\): number of transferred electrons
- \(F\): Faraday constant \(96485\,\mathrm{C\,mol^{-1}}\)
- \(Q\): reaction quotient (activity/concentration form)
Detailed examples
Example 1:
\(E^\circ=1.10\,V\), \(n=2\), \(T=298.15\,K\), \(Q=10\)
\(E\approx1.070\,V\)
\(E^\circ=1.10\,V\), \(n=2\), \(T=298.15\,K\), \(Q=10\)
\(E\approx1.070\,V\)
Example 2:
Same system, but \(Q=100\)
Potential decreases further because \(\ln Q\) grows.
Same system, but \(Q=100\)
Potential decreases further because \(\ln Q\) grows.
Example 3:
Given \(E=1.07\,V\), \(E^\circ=1.10\,V\)
Back-calculation yields \(Q\approx 10\) for \(n=2\), \(T=298\,K\).
Given \(E=1.07\,V\), \(E^\circ=1.10\,V\)
Back-calculation yields \(Q\approx 10\) for \(n=2\), \(T=298\,K\).
Example 4 (Zn/Cu galvanic cell):
\(E^\circ_{\mathrm{C}}=0.34\,V\), \(E^\circ_{\mathrm{A}}=-0.76\,V\), \(Q_C=Q_A=1\)
\(E_{\mathrm{cell}}\approx1.10\,V\)
\(E^\circ_{\mathrm{C}}=0.34\,V\), \(E^\circ_{\mathrm{A}}=-0.76\,V\), \(Q_C=Q_A=1\)
\(E_{\mathrm{cell}}\approx1.10\,V\)
Practical note:
For high-precision work, activities should replace pure concentrations, especially at higher ionic strengths.
For high-precision work, activities should replace pure concentrations, especially at higher ionic strengths.
Deeper insight
Temperature effect on E
As temperature increases, \(RT/(nF)\) increases, so the concentration term has a stronger influence on potential.
Practical applications
The Nernst equation is used in battery diagnostics, potentiometric sensing, corrosion engineering, and electrochemical analytics for consistent interpretation of measured voltages.
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