Relativity and the Lead-Acid Battery

The energies of the solid reactants in the lead-acid battery are calculated ab initio using two different basis sets at nonrelativistic, scalar-relativistic, and fully relativistic levels, and using several exchange-correlation potentials. The average calculated standard voltage is 2.13 V, compared with the experimental value of 2.11 V. All calculations agree in that 1.7–1.8 V of this standard voltage arise from relativistic effects, mainly from ${\mathrm{PbO}}_2$ but also from ${\mathrm{PbSO}}_4$.

Figure 1

The relativistic shifts in energies of formation $E_f$ (per formula unit), calculated at DFT–PBEsol-D level. The nonrelativistic energy, $\mathrm{NR}=E_f(\mathrm{NR})$, is chosen as reference; ${\Delta }_{\mathrm{SR}}=E_f(\mathrm{SR})-E_f(\mathrm{NR})$ and ${\Delta }_{\mathrm{FR}}=E_f(\mathrm{FR})-E_f(\mathrm{NR})$. Note the smallness of the effect on ${\mathrm{SO}}_3(g)$.

Figure 2

Total and partial DOS ($\mathrm{\text{states}}/\mathrm{eV}/\mathrm{\text{cell}}$) of Pb in $\beta \mathrm{\text{−}}{\mathrm{PbO}}_2$ calculated at DFT-VWN level (the Fermi energy $E_F$ is set to 0). Note the pronounced relativistic shift in both occupied and unoccupied Pb $6s$ states. (NR, nonrelativistic; SR, scalar relativistic; FR, fully relativistic). Gray (orange), Pb $6s$; black (green), Pb $6p$.

Figure 3

Effect of relativity on the EMF of model reaction, $M(s)+M{\mathrm{O}}_2(s)\to 2M\mathrm{O}(s)$, calculated with four different methods ($A–D$). Experimental values for $M=\mathrm{Pb}$ ($+0.82\mathrm{V}$) and $M=\mathrm{Sn}$ ($-0.085\mathrm{V}$) are indicated with the horizontal dashed lines and dotted lines, respectively. (A) BAND (VWN), (B) BAND (PBEsol), (C) BAND (PBEsol-D), and (D) FPLO (PW92). The level C has empirical dispersion corrections.

Figure 4

Relativistic (FR) and nonrelativistic (NR) Hartree-Fock orbital energies for the tin and lead atoms in their $n{s}^{2}n{p}^2$ ground state. Note the relativistic stabilization of the $ns$ level (lowest in the figure), leading to higher oxidative power for ${\mathrm{Pb}}^{\mathrm{IV}}$ than for ${\mathrm{Sn}}^{\mathrm{IV}}$. Data from Desclaux [33].