Summarizing the Nernst Equation

Key Concepts and Summary

Electrical work (w_ele) is the negative of the product of the total charge (Q) and the cell potential (E_cell). The total charge can be calculated as the number of moles of electrons (n) times the Faraday constant (F = 96,485 C/mol e⁻). Electrical work is the maximum work that the system can produce and so is equal to the change in free energy.

Thus, anything that can be done with or to a free energy change can also be done to or with a cell potential. The Nernst equation relates the cell potential at nonstandard conditions to the logarithm of the reaction quotient. Concentration cells exploit this relationship and produce a positive cell potential using half-cells that differ only in the concentration of their solutes.

Key Equations

• \({E}_{\text{cell}}^{°}=\phantom{\rule{0.2em}{0ex}}\cfrac{RT}{nF}\phantom{\rule{0.4em}{0ex}}\text{ln}\phantom{\rule{0.3em}{0ex}}K\)

• \({E}_{\text{cell}}^{°}=\phantom{\rule{0.2em}{0ex}}\cfrac{0.02\text{57 V}}{n}\phantom{\rule{0.2em}{0ex}}\text{ln}\phantom{\rule{0.2em}{0ex}}K=\phantom{\rule{0.2em}{0ex}}\cfrac{0.0\text{592 V}}{n}\phantom{\rule{0.2em}{0ex}}\text{log}\phantom{\rule{0.2em}{0ex}}K\phantom{\rule{5em}{0ex}}(\text{at 298.15}\phantom{\rule{0.2em}{0ex}}K)\)

• \({E}_{\text{cell}}={E}_{\text{cell}}^{°}-\phantom{\rule{0.2em}{0ex}}\cfrac{RT}{nF}\phantom{\rule{0.4em}{0ex}}\text{ln}\phantom{\rule{0.3em}{0ex}}Q\phantom{\rule{5em}{0ex}}\text{(Nernst equation)}\)

• \({E}_{\text{cell}}={E}_{\text{cell}}^{°}-\phantom{\rule{0.2em}{0ex}}\cfrac{0.02\text{57 V}}{n}\phantom{\rule{0.2em}{0ex}}\text{ln}\phantom{\rule{0.2em}{0ex}}Q={E}_{\text{cell}}^{°}-\phantom{\rule{0.2em}{0ex}}\cfrac{0.05\text{92 V}}{n}\phantom{\rule{0.2em}{0ex}}\text{log}\phantom{\rule{0.2em}{0ex}}Q\phantom{\rule{5em}{0ex}}(\text{at 298.15}\phantom{\rule{0.2em}{0ex}}K)\)

• ΔG = −nFE_cell

• \(\text{Δ}{G}^{°}=\text{−}nF{E}_{\text{cell}}^{°}\)

• \({w}_{\text{ele}}={w}_{\text{max}}=\text{−}nF{E}_{\text{cell}}\)

Glossary

concentration cell

galvanic cell in which the two half-cells are the same except for the concentration of the solutes; spontaneous when the overall reaction is the dilution of the solute

electrical work (w_ele)

negative of total charge times the cell potential; equal to w_max for the system, and so equals the free energy change (ΔG)

Faraday’s constant (F)

charge on 1 mol of electrons; F = 96,485 C/mol e⁻

Nernst equation

equation that relates the logarithm of the reaction quotient (Q) to nonstandard cell potentials; can be used to relate equilibrium constants to standard cell potentials