Nernst equation Knowledge-Based Study

Study notes

1. Cell Potentials Beyond Standard States

   In electrochemistry, standard cell potentials ($E^{\circ}_{\text{cell}}$) are measured under strictly defined standard-state conditions:

   • Solute concentrations of exactly 1.0 M
   • Gas partial pressures of 1.0 atm (or 1.0 bar)
   • A specified temperature, conventionally 298.15 K (25 °C)

   However, real-world electrochemical systems—such as operating batteries, biological nerve impulses, and corrosion processes—rarely function at standard concentrations. As a galvanic cell discharges, reactants are consumed and products accumulate, continuously altering concentration levels and driving the cell potential away from its standard value.

   Linking Free Energy to Cell Potential

   The dependence of electrical potential on concentration is rooted in chemical thermodynamics. The change in Gibbs free energy ($\Delta G$) under non-standard conditions is related to the standard free energy change ($\Delta G^{\circ}$) and the reaction quotient ($Q$) by the relation:

   $$\Delta G = \Delta G^{\circ} + RT \ln Q$$

   For an electrochemical process delivering electrical work, free energy change relates directly to cell potential via:

   $$\Delta G = -nFE_{\text{cell}} \quad \text{and} \quad \Delta G^{\circ} = -nFE^{\circ}_{\text{cell}}$$

   Where:

   • n is the number of moles of electrons transferred in the balanced redox equation.
   • F is the Faraday constant (approximately 96485 C/mol e⁻).
   • R is the universal gas constant (8.314 J/(mol·K)).
   • T is the absolute temperature in Kelvin.
   • Q is the reaction quotient.

   Substituting these definitions yields the fundamental Nernst Equation:

   $$E_{\text{cell}} = E^{\circ}_{\text{cell}} - \frac{RT}{nF} \ln Q$$

   Simplified Practical Form

   Electrochemical measurements are frequently performed at 298.15 K (25 °C). By substituting the numerical values for $R$, $T$, and $F$, and converting the natural logarithm to a base-10 logarithm ($\ln Q = 2.3026 \log_{10} Q$), we obtain a convenient simplified formula:

   $$\frac{2.3026 \times RT}{F} = \frac{2.3026 \times 8.314 \times 298.15}{96485} \approx 0.0592 \text{ V}$$

   Thus, at 298.15 K, the Nernst equation is commonly written as:

   $$E_{\text{cell}} = E^{\circ}_{\text{cell}} - \frac{0.0592}{n} \log Q$$

   This equation indicates that:

   • When $Q < 1$ (reactants predominate), $\log Q < 0$, and $E_{\text{cell}} > E^{\circ}_{\text{cell}}$.
   • When $Q = 1$ (standard concentrations), $\log Q = 0$, and $E_{\text{cell}} = E^{\circ}_{\text{cell}}$.
   • When $Q > 1$ (products predominate), $\log Q > 0$, and $E_{\text{cell}} < E^{\circ}_{\text{cell}}$.

   Applying the Equation to a Daniell Cell

   Consider a zinc-copper cell operating at 25 °C:

   $$\text{Zn}(s) + \text{Cu}^{2+}(aq) \rightarrow \text{Zn}^{2+}(aq) + \text{Cu}(s) \quad (E^{\circ}_{\text{cell}} = 1.10 \text{ V})$$

   Suppose $[\text{Cu}^{2+}] = 0.010 \text{ M}$ and $[\text{Zn}^{2+}] = 1.0 \text{ M}$.

   1. Determine $n$: The overall reaction involves the transfer of 2 electrons ($n = 2$).
   2. Write the expression for $Q$: Pure solids do not enter the reaction quotient:
      $$Q = \frac{[\text{Zn}^{2+}]}{[\text{Cu}^{2+}]} = \frac{1.0}{0.010} = 100 = 10^2$$
   3. Compute $E_{\text{cell}}$:
      $$E_{\text{cell}} = 1.10 - \frac{0.0592}{2} \log(10^2)$$
      $$E_{\text{cell}} = 1.10 - 0.0296 \times 2 = 1.10 - 0.0592 = 1.0408 \text{ V} \approx 1.04 \text{ V}$$

   Because product concentration is higher than reactant concentration relative to standard state, the driving force decreases.

   Special Cases of the Nernst Equation

   1. Concentration Cells

   A concentration cell consists of two identical half-cells containing the same electrodes and ions, differing only in solute concentration. Because the electrodes and solutions share the same chemical species, $E^{\circ}_{\text{cell}} = 0$. The cell potential is generated exclusively by the concentration gradient:

   $$E_{\text{cell}} = -\frac{0.0592}{n} \log \left(\frac{[\text{ion}]_{\text{dilute}}}{[\text{ion}]_{\text{concentrated}}}\right)$$

   2. Relationship to the Equilibrium Constant

   When a cell completely discharges, the system reaches chemical equilibrium: $E_{\text{cell}} = 0$ and $Q = K$. Substituting these values into the Nernst equation gives:

   $$0 = E^{\circ}_{\text{cell}} - \frac{0.0592}{n} \log K \implies E^{\circ}_{\text{cell}} = \frac{0.0592}{n} \log K$$

   This fundamental relationship allows experimental standard cell potentials to be used to calculate equilibrium constants for redox reactions.