Introduction to electrochemical science using Legendre transforms

The Legendre transform is a mathematical method that is very useful for finding basic equations that describe electrical surface phenomena. This lecture describes how the Legendre transform facilitates the transition between the internal energy and the electrochemical Gibbs free energy of classical electrochemistry. A rigorous summary of the electrochemical potential from well-defined principles of thermodynamics is provided, and this strategy is effective at efficiently integrating thermodynamics and electrostatics. In addition, the Lippmann equation, Gibbs adsorption equation, entropy equation, and capacitance equation were deduced from the fundamental equation of a system with electric and surface work.