{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,20]],"date-time":"2026-10-20T16:32:24Z","timestamp":1792513944718,"version":"4.5.0"},"reference-count":46,"publisher":"The Electrochemical Society","issue":"11","license":[{"start":{"date-parts":[[2024,11,27]],"date-time":"2024-11-27T00:00:00Z","timestamp":1732665600000},"content-version":"vor","delay-in-days":26,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0\/"},{"start":{"date-parts":[[2024,11,27]],"date-time":"2024-11-27T00:00:00Z","timestamp":1732665600000},"content-version":"tdm","delay-in-days":26,"URL":"https:\/\/iopscience.iop.org\/info\/page\/text-and-data-mining"}],"content-domain":{"domain":["iopscience.iop.org"],"crossmark-restriction":false},"short-container-title":["J. Electrochem. Soc."],"published-print":{"date-parts":[[2024,11,1]]},"abstract":"We revisit the classical derivation of the Butler-Volmer equation to include the effect of the electrode metal. If the metal is replaced by one with a different work function, keeping other conditions in the electrode constant, the chemical potential of electrons \n \n\n<\/jats:tex-math>\n \n \n \n \u03bc<\/mml:mi>\n <\/mml:mrow>\n \n e<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:inline-formula> and the Galvani potential \n \n\n<\/jats:tex-math>\n \n \u03c6<\/mml:mi>\n <\/mml:math>\n <\/jats:inline-formula> change in a complementary manner. Changes in \n \n\n<\/jats:tex-math>\n \n \n \n \u03bc<\/mml:mi>\n <\/mml:mrow>\n \n e<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:inline-formula> and \n \n\n<\/jats:tex-math>\n \n \u03c6<\/mml:mi>\n <\/mml:math>\n <\/jats:inline-formula> each impact the free energies of activation of the forward and backward electron transfer reactions, so we modify the classical expressions which relate them to applied voltage E<\/jats:italic> by including also the effect of \n \n\n<\/jats:tex-math>\n \n \n \n \u03bc<\/mml:mi>\n <\/mml:mrow>\n \n e<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n .<\/mml:mo>\n <\/mml:math>\n <\/jats:inline-formula> Inserting these expressions in an Eyring-Polyani or Arrhenius type equation in the traditional way, we obtain a modified Butler-Volmer equation which expresses current density as a function of both \n \n\n<\/jats:tex-math>\n \n E<\/mml:mi>\n <\/mml:math>\n <\/jats:inline-formula> and \n \n\n<\/jats:tex-math>\n \n \u0394<\/mml:mi>\n \n \n \u03bc<\/mml:mi>\n <\/mml:mrow>\n \n e<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n .<\/mml:mo>\n <\/mml:math>\n <\/jats:inline-formula> The exchange current density \n \n\n<\/jats:tex-math>\n \n \n \n j<\/mml:mi>\n <\/mml:mrow>\n \n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:inline-formula> appears as an exponential function of \n \n\n<\/jats:tex-math>\n \n \u0394<\/mml:mi>\n \n \n \u03bc<\/mml:mi>\n <\/mml:mrow>\n \n e<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n .<\/mml:mo>\n <\/mml:math>\n <\/jats:inline-formula> For the work function \n \n\n<\/jats:tex-math>\n \n \u03a6<\/mml:mi>\n <\/mml:math>\n <\/jats:inline-formula> of the metal, the approximation \n \n\n<\/jats:tex-math>\n \n \u0394<\/mml:mi>\n \n \n \u03bc<\/mml:mi>\n <\/mml:mrow>\n \n e<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n \u2248<\/mml:mo>\n \u2212<\/mml:mo>\n F<\/mml:mi>\n \u0394<\/mml:mi>\n \u03a6<\/mml:mi>\n <\/mml:math>\n <\/jats:inline-formula> yields a linear relationship between \n \n\n<\/jats:tex-math>\n \n ln<\/mml:mi>\n \n \n \n j<\/mml:mi>\n <\/mml:mrow>\n \n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:inline-formula> and \n \n\n<\/jats:tex-math>\n \n \u03a6<\/mml:mi>\n .<\/mml:mo>\n <\/mml:math>\n <\/jats:inline-formula> The linear increase in \n \n\n<\/jats:tex-math>\n \n ln<\/mml:mi>\n \n \n \n j<\/mml:mi>\n <\/mml:mrow>\n \n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:inline-formula> with \n \n\n<\/jats:tex-math>\n \n \u03a6<\/mml:mi>\n <\/mml:math>\n <\/jats:inline-formula> has long been reported. We show two experimental examples: the aqueous Fe2+<\/jats:sup>\/Fe3+<\/jats:sup> couple with positive slope and the hydrogen evolution reaction (HER) with parallel lines for the d and sp metals, both with positive slopes.<\/jats:p>","DOI":"10.1149\/1945-7111\/ad8f01","type":"journal-article","created":{"date-parts":[[2024,11,5]],"date-time":"2024-11-05T22:53:40Z","timestamp":1730847220000},"page":"116503","update-policy":"https:\/\/doi.org\/10.1088\/crossmark-policy","source":"Crossref","is-referenced-by-count":12,"title":["Editors\u2019 Choice\u2014The Butler-Volmer Equation Revisited: Effect of Metal Work Function on Electron Transfer Kinetics"],"prefix":"10.1149","volume":"171","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3006-8837","authenticated-orcid":true,"given":"D. 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Published on behalf of The Electrochemical Society by IOP Publishing Limited","name":"copyright_information","label":"Copyright Information"},{"value":"2024-07-31","name":"date_received","label":"Date Received","group":{"name":"publication_dates","label":"Publication dates"}},{"name":"date_accepted","label":"Date Accepted","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2024-11-27","name":"date_epub","label":"Online publication date","group":{"name":"publication_dates","label":"Publication dates"}}]}}