A Theory of the Excess Chemical Potential of Metals in Alloys as Related to Electrochemical Measurements

Abstract

+P AV^-T AS^ (at constant P and T).The excess internal energy A t/ E is evaluated from the structure-dependent internal energy of the metals. T ZlS E is taken as zero (to a good approximation in lithium-magnesium). P A V E is found to be negligible at atmospheric pressure. A calculation is made for the alloy Li-Mg at about 600 °C for the excess chemical potentialWe summarize here the main results for the cohesive energy in the pseudopotential theory of metals 3 (per atom at density n):where Ni is the number of lithium ions, which is found to be of the right magnitude compared to two sets of experimental results. TheoryThe electromotive force in a galvanic cell, involving no concentration gradients, is proportional to the difference between the chemical potentials of the two metallic electrodes 2 . The chemical potential f.1 of a metal is the Gibbs free energy per atom(1)More accurately, ju t = (dG/dNi)r,p for a mixture of particle species i(i = l,2, 3) where temperature T and pressure P are kept constant. In Eq.(1), the unknown quantity is the internal energy U of the metal 3 , since the volumes V are measured or interpolated linearly for alloys (to a good approximation), whereas the entropy can be approximated by that of an ideal mixture 4 . Thus, the possibility of predicting the Gibbs free energy of a metallic electrode depends largely on the prediction of the internal or cohesive energy of the metal. Fortunately, considerable progress has been made on this last problem with the pseudopotential theory of metals 3 which, although still poor for defects in metals, provides good results for the cohesive energy of metals and the relative stability of crystal structures. 6 . The sixth term is written in the Aschcroft pseudopotential approximation 3 with a core of radius R c for the electron-ion potential*. This term is referred to as the Hartree term. The seventh term, which depends strongly on the structure of the metal, represents the Coulomb direct and indirect interaction between the ions of density n and pair correlation function g{r)~. | k T is the average kinetic energy of the ions. Equation (2) is characterized by strong separation between the conduction electron gas (the first four terms) and the ionic gas (the last two terms) of valency Z. The remaining two terms represent interaction terms between ions and electrons. r s is the radius of the sphere occupied by one electron r s = (3/4 Tine) 1 / 3 .Finally, we remark that Eq. (2) looks deceptively simple because it depends still on two unknowns: (1) u e ff(r) or the effective interaction between two ions which depends among other things on the order of perturbation theory in the electron-ion potential (usually a pseudopotential) and the screening used to represent the electron gas; (2) g(r) or the radial distribution function 4?rr 2^( r) Avhich is usually determined by x-ray diffraction. Method of Analysis and ResultsWe shall proceed in three steps: (1) obtain the cohesive energy U of the pure metal (lithium), (2) obtain the cohesive energy of the alloy ...