Nernst equation derivation from "summary" of Physical Chemistry by Robert G. Mortimer
To derive the Nernst equation, we begin with the fundamental equation for the Gibbs energy change of a cell reaction. This equation relates the standard Gibbs energy change to the equilibrium constant of the cell reaction. By using the relationship between the standard Gibbs energy change and the cell potential, we can derive the Nernst equation. The standard Gibbs energy change of a cell reaction is related to the standard cell potential by the equation ΔG° = -nFΔE°, where n is the number of moles of electrons transferred in the reaction and F is Faraday's constant. This equation allows us to calculate the standard Gibbs energy change of a cell reaction from the standard cell potential. The equilibrium constant of a cell reaction, K, is related to the standard Gibbs energy change by the equation ΔG° = -RTlnK, where R is the gas constant and T is the temperature in Kelvin. By equating the expressions for ΔG°, we can derive the relationship between the standard cell potential and the equilibrium constant: ΔE° = (RT/nF)lnK. The Nernst equation is an extension of this relationship to non-standard conditions. By considering the effect of concentrations on the cell potential, we can derive the Nernst equation: E = E° - (RT/nF)ln(Q), where Q is the reaction quotient for the cell reaction under non-standard conditions. In summary, the Nernst equation allows us to calculate the cell potential under non-standard conditions by taking into account the effect of concentrations on the equilibrium constant. By deriving the Nernst equation from the fundamental equations for the Gibbs energy change of a cell reaction, we can understand the behavior of electrochemical cells in various situations.
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