The anode half-reaction occurring during the discharge of a lead storage battery is:
(a) \[Pb(s) + S{O_2} + {O_2} \to PbS{O_4}(s)\]
(b) \[Pb(s) + S{O_4}^{2 - }(aq.) \to PbS{O_4}(s) + 2{e^ - }\]
(c) \[Pb{O_2}(s) + 4{H^ + }(aq.) + 2{e^ - } + S{O_4}^{2 - }(aq.) \to PbS{O_4}(s) + 2{H_2}O\]
(d) \[P{b^{2 + }}(aq.) + S{O_4}^{2 - }(aq.) \to PbS{O_4}(s)\]

Hint: An electrochemical cell consists of a negative (anode) and a positive (cathode) electrode respectively which are immersed in an electrolyte. The oxidation occurs at the anode whereas the reduction occurs at the cathode.

Complete Step by Step Solution:
Lead storage battery also called the rechargeable battery or also called an electrochemical cell or Lead-acid batteries. The lead storage battery can convert chemical energy by reversible chemical reactions. The lead storage battery can be recharged by passing the direct electric current again and again.

In the lead storage battery, the lead (\[Pb\]) electrode behaves as an anode and the lead dioxide (\[Pb{O_2}\]) electrode behaves as a cathode. While the \[{H_2}S{O_4}\] is act as an electrolyte.

The chemical reaction that takes place inside the lead-storage battery can be represented below.
At anode: \[Pb(s) + S{O_4}^{2 - }(aq.) \to PbS{O_4}(s) + 2{e^ - }\]
At cathode: \[Pb{O_2}(s) + 4{H^ + }(aq.) + S{O_4}^{2 - }(aq.) + 2{e^ - } \to PbS{O_4}(s) + 2{H_2}O(l)\]

The complete chemical reaction can be shown as:
\[Pb(s) + Pb{O_2}(s) + 2{H_2}S{O_4}(aq.) \to 2PbS{O_4}(s) + 2{H_2}O(l)\]
When the lead storage battery gets discharged, the electrolyte undergoes dissociation into \[{H_2}\] and \[S{O_4}\] . The dissociated \[{H_2}\] and \[S{O_4}\] further combined with oxygen that is produced on the positive electrode to form water(\[{H_2}O\]), and hence this cause the reduction in the amount of electrolyte.
Hence, from above we can conclude that option C will be the correct answer.

Note: The lead storage batteries are very important for their application in cars and heavy vehicles because they are capable of being recharged. The lead acid -batteries upon discharge lose much of their dissolved acid.