Editors’ Choice—The Butler-Volmer Equation Revisited: Effect of Metal Work Function on Electron Transfer Kinetics

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>μ</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>e</mml:mi> </mml:mrow> </mml:msub> </mml:math> and the Galvani potential <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>φ</mml:mi> </mml:math> change in a complementary manner. Changes in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>μ</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>e</mml:mi> </mml:mrow> </mml:msub> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>φ</mml:mi> </mml:math> 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 by including also the effect of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>μ</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>e</mml:mi> </mml:mrow> </mml:msub> <mml:mo>.</mml:mo> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>E</mml:mi> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:msub> <mml:mrow> <mml:mi>μ</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>e</mml:mi> </mml:mrow> </mml:msub> <mml:mo>.</mml:mo> </mml:math> The exchange current density <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>j</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> </mml:math> appears as an exponential function of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:msub> <mml:mrow> <mml:mi>μ</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>e</mml:mi> </mml:mrow> </mml:msub> <mml:mo>.</mml:mo> </mml:math> For the work function <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="normal">Φ</mml:mi> </mml:math> of the metal, the approximation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:msub> <mml:mrow> <mml:mi>μ</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>e</mml:mi> </mml:mrow> </mml:msub> <mml:mo>≈</mml:mo> <mml:mo>−</mml:mo> <mml:mi>F</mml:mi> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mi mathvariant="normal">Φ</mml:mi> </mml:math> yields a linear relationship between <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>ln</mml:mi> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>j</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="normal">Φ</mml:mi> <mml:mo>.</mml:mo> </mml:math> The linear increase in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>ln</mml:mi> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>j</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="normal">Φ</mml:mi> </mml:math> has long been reported. We show two experimental examples: the aqueous Fe 2+ /Fe 3+ couple with positive slope and the hydrogen evolution reaction (HER) with parallel lines for the d and sp metals, both with positive slopes.