# Nernst Equation for Lithium ion battery

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

• Imagine what the equation for Zn/Cu battery would be, if there were used respective amalgams instead of pure metals. Feb 23 '20 at 7:26

## 1 Answer

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.

• After discharge and charge the voltage jumps or drops immediately due to the activation and ohmic overpotentials. But the change due to the mass transport loss/concentration loss takes time. This change is due to the concentration diffusion but the Li+ ions concentration at the electrodes are not in the overall voltage. Are you saying the solid concentration is changing with time while no current is flowing? Feb 27 '20 at 0:28