During discharge of a lead storage cell the density of sulphuric acid in the cell:

Discharging reaction 

$\mathrm{Pb}\left(\mathrm{s}\right)+{\mathrm{PbO}}_2\left(\mathrm{s}\right)+2{\mathrm{H}}_2{\mathrm{SO}}_4\to 2{\mathrm{PbSO}}_4+2{\mathrm{H}}_2\mathrm{O}$

Mechanism:

$$\begin{gathered} \mathrm{Pb}+{\mathrm{SO}}_4^{-2}\to {\mathrm{PbSO}}_4+2{\mathrm{e}}^{-} \\ {\mathrm{PbO}}_2+4{\mathrm{H}}^{+}{\mathrm{SO}}_4^{-2}+2{\mathrm{e}}^{-}\to {\mathrm{PbSO}}_4+2{\mathrm{H}}_2\mathrm{O} \end{gathered}$$

In this process the ${\mathrm{H}}_2{\mathrm{SO}}_4$ is consumed and residual water is left behind or evaporated.So,density of sulphuric acid decreases.