The Nernst equation is a powerful tool in electrochemistry, connecting cell potential to the concentrations of reactants and products. It helps us understand how electrochemical cells behave under non-standard conditions, crucial for predicting spontaneous redox reactions and their driving forces.

This equation bridges the gap between thermodynamics and electrochemistry, allowing us to calculate cell potentials, equilibrium constants, and even determine unknown concentrations. It's a key concept in understanding real-world applications of electrochemical principles in batteries, corrosion, and analytical chemistry.

Deriving the Nernst equation

Relationship between Gibbs free energy and cell potential

• The Nernst equation relates the reduction potential of an electrochemical reaction to the standard electrode potential and the activities of the electrochemical species in the reaction
• It is derived from the change in Gibbs free energy (ΔG) for the electrochemical reaction at any moment in time
  • ΔG is related to the cell potential (Ecell) and the reaction quotient (Q)
  • The relationship is expressed as ΔG = -nFEcell, where n is the number of moles of electrons transferred in the cell reaction and F is Faraday's constant

Nernst equation expression and components

• The Nernst equation is expressed as: $Ecell = E°cell - (RT/nF)lnQ$
  • $E°cell$ is the standard cell potential
  • $R$ is the gas constant (8.314 J mol⁻¹ K⁻¹)
  • $T$ is the absolute temperature (in Kelvin)
  • $n$ is the number of moles of electrons transferred in the cell reaction
  • $F$ is Faraday's constant (96,485 C mol⁻¹)
  • $Q$ is the reaction quotient, which is the ratio of the product of the concentrations of the products raised to their stoichiometric coefficients divided by the product of the concentrations of the reactants raised to their stoichiometric coefficients

Relationship between standard cell potential and equilibrium constant

• At equilibrium, ΔG = 0 and Ecell = 0
  • This means that $E°cell = (RT/nF)lnK$, where K is the equilibrium constant
  • This relationship connects the standard cell potential to the thermodynamic equilibrium constant
• The equilibrium constant (K) is related to the standard Gibbs free energy change (ΔG°) by the equation $ΔG° = -RTlnK$
  • This equation can be combined with the relationship between ΔG and Ecell to derive the Nernst equation at equilibrium

Cell potential under non-standard conditions

Calculating cell potential using the Nernst equation

• The Nernst equation allows for the calculation of the cell potential (Ecell) under non-standard conditions by accounting for the concentrations or partial pressures of the electrochemical species involved in the reaction
• The reaction quotient (Q) in the Nernst equation is calculated using the actual concentrations or partial pressures of the species, raised to their respective stoichiometric coefficients
  • For example, in the reaction aA + bB ⇌ cC + dD, $Q = ([C]^c [D]^d) / ([A]^a [B]^b)$, where [A], [B], [C], and [D] are the concentrations or partial pressures of the respective species

Comparing non-standard cell potential to standard cell potential

• When the concentrations of the species are not equal to 1 M or the partial pressures are not equal to 1 atm, the cell potential will differ from the standard cell potential (E°cell)
• The sign of the $(RT/nF)lnQ$ term in the Nernst equation determines whether the non-standard cell potential is higher or lower than the standard cell potential
  • If Q < 1, then lnQ < 0, and Ecell > E°cell
  • If Q > 1, then lnQ > 0, and Ecell < E°cell

Concentration cell example

• A concentration cell is an electrochemical cell where the two half-cells contain the same redox couple but at different concentrations
• The cell potential of a concentration cell can be calculated using the Nernst equation
  • For example, in a concentration cell with the redox couple Cu²⁺/Cu, where [Cu²⁺]₁ = 0.1 M and [Cu²⁺]₂ = 0.01 M, the cell potential is calculated as: $Ecell = (RT/nF)ln([Cu²⁺]₁/[Cu²⁺]₂)$

Predicting spontaneous redox reactions

Standard reduction potential and spontaneity

• The standard reduction potential (E°) is a measure of the tendency of a chemical species to be reduced and is used to predict the direction of spontaneous redox reactions
• In a redox reaction, the species with the more positive standard reduction potential will spontaneously be reduced, while the species with the more negative standard reduction potential will spontaneously be oxidized
  • The species with the higher reduction potential will have a greater tendency to gain electrons and be reduced
  • The species with the lower reduction potential will have a greater tendency to lose electrons and be oxidized

Calculating standard cell potential

• The standard cell potential (E°cell) can be calculated by subtracting the standard reduction potential of the anode (oxidation half-reaction) from the standard reduction potential of the cathode (reduction half-reaction)
  • $E°cell = E°cathode - E°anode$
• If E°cell is positive, the redox reaction is spontaneous in the forward direction under standard conditions
• If E°cell is negative, the redox reaction is spontaneous in the reverse direction under standard conditions

Examples of spontaneous redox reactions

• In the reaction between zinc and copper(II) ions, Zn(s) + Cu²⁺(aq) → Zn²⁺(aq) + Cu(s), the standard reduction potentials are:
  • Zn²⁺(aq) + 2e⁻ → Zn(s), E° = -0.76 V
  • Cu²⁺(aq) + 2e⁻ → Cu(s), E° = +0.34 V
  • E°cell = E°cathode - E°anode = 0.34 V - (-0.76 V) = 1.10 V > 0, so the reaction is spontaneous in the forward direction
• In the reaction between silver and iron(III) ions, Ag(s) + Fe³⁺(aq) → Ag⁺(aq) + Fe²⁺(aq), the standard reduction potentials are:
  • Ag⁺(aq) + e⁻ → Ag(s), E° = +0.80 V
  • Fe³⁺(aq) + e⁻ → Fe²⁺(aq), E° = +0.77 V
  • E°cell = E°cathode - E°anode = 0.80 V - 0.77 V = 0.03 V > 0, so the reaction is spontaneous in the forward direction, but the driving force is relatively small

Concentration of electroactive species

Using the Nernst equation to calculate concentration

• The Nernst equation can be used to calculate the concentration of an electroactive species in an electrochemical cell if the cell potential, standard cell potential, and the concentrations of the other species are known
• By rearranging the Nernst equation, the concentration of the unknown species can be solved for
  • $[unknown species] = exp((nF/RT)(E°cell - Ecell)) × [other species]$

Applications in analytical chemistry

• This application is useful in analytical chemistry, where the concentration of an analyte can be determined by measuring the cell potential of an electrochemical cell
• The concentration of the electroactive species can also be determined using the Nernst equation in conjunction with other analytical techniques, such as potentiometric titrations
  • In a potentiometric titration, the potential of an electrochemical cell is measured as a function of the volume of titrant added, and the endpoint is determined by the inflection point in the potential curve

Ion-selective electrodes

• The Nernst equation is the basis for the operation of ion-selective electrodes, which are used to measure the concentration of specific ions in solution by measuring the potential difference between the electrode and a reference electrode
• Ion-selective electrodes have a membrane that is selectively permeable to a specific ion, and the potential difference across the membrane is proportional to the logarithm of the ion concentration
  • For example, a pH electrode is an ion-selective electrode that measures the concentration of hydrogen ions (H⁺) in solution
• The potential of an ion-selective electrode is given by a modified form of the Nernst equation
  • $E = E° + (RT/nF)ln[ion]$, where [ion] is the concentration of the ion that the electrode is selective for