Nernst Equation for Lithium ion battery

Question

For electrochemical cells we know $$ \Delta G = \Delta G^{\ominus} + RT\ln Q=- RT\ln Keq + RT\ln Q=-nFE $$ So for lithium ion batteries the reactions are Cathode $$ \ce{CoO2 + Li+ + e- <=> LiCoO2} $$ Anode $$ \ce{LiC6 <=> C6 + Li+ + e^-}$$ Overall $$\ce{ LiC6 + CoO2 <=> C6 + LiCoO2} $$ Is it safe to say the cobalt oxide is a solid as is the Lithium and can be ignored in the reaction quotient? As in a Zinc copper battery i do not think the mass of the zinc and copper affect the cell potential only the aqueous concentrations. For a lithium ion battery the cell potential is a function of the state of charge and temperature. but what are the concentrations in the reaction quotient for a lithium ion battery as most of the products and reactants are solids, is it not accurate to ignore them due to intercalation and are not exactly solids? What is nernst equation for lithium ion batteries

Answer 1

The Nernst equation for the anode is :$$E_a = E°_a + \frac{RT}{F}ln{\frac{[Li^+]_e [C_6]_s}{[LiC_6]_s}}$$ where the concentrations $[C_6]_s$ and $[LiC_6]_s$ are defined in the solid phase of the anode, and not in the electrolyte. The concentration $[Li^+]_e$ is defined in the electrolyte. When the anode is working, the concentration $[LiC_6]_s$ decreases in the solid phase and the concentration $[C_6]_s$ increases.

The same reasoning may be done on the Nernst equation at the cathode : $$E_c = E°_c + \frac{RT}{F}· ln{\frac{[Li^+]_e[CoO_2]_{s'}}{[LiCoO_2]_{s'}}}$$ where $[CoO_2]_{s'}$ is the concentration of $CoO_2$ in the solid phase of the cathode, and $[LiCoO_2]_{s'}$ is the concentration of $LiCoO_2$ in the solid phase of the cathode.

The overall voltage of the battery $\Delta E$ can be written : $$\Delta E = E°_c - E°_a + \frac{RT}{F} · ln\frac{[LiC_6]_s [CoO_2]_{s'}}{[C_6]_s [LiCoO_2]_{s'}}$$ As a consequence, both $[LiC_6]_s$ and $ [CoO_2]_{s'}$ decrease when the battery is working. So the overall voltage decreases.