To solve this problem, we need to identify the reagent that converts$\\$ethyl propanoate (CH$_3$CH$_2$COO$^-$Na$^+)$ to ethane (CH$_3$CH$_3$) and sodium carbonate (Na$_2$CO$_3$).$\\$This reaction is a decarboxylation reaction, where a carboxylate salt is heated$\\$with a reagent to remove the carboxyl group as carbon dioxide (CO$_2$).$\\$The reaction given is:$\\$CH$_3$CH$_2$COO$^-$Na$^+ + ? \rightarrow$CH$_3$CH$_3 +$Na$_2$CO$_3 \\$The reagent that typically facilitates this type of reaction is calcium oxide (CaO).$\\$This is known as the soda lime decarboxylation reaction, where soda lime is a mixture$\\$of NaOH and CaO. The reaction proceeds as follows:$\\$1. The carboxylate ion (CH$_3$CH$_2$COO$^-)$ reacts with NaOH to form sodium carbonate (Na$_2$CO$_3$) and$\\$ ethane (CH$_3$CH$_3$).$\\$2. CaO acts as a catalyst and helps in the removal of CO$_2$ from the carboxylate ion.$\\$Therefore, the missing reagent is CaO, which corresponds to Option 3.$\\$Thus, the correct answer is Option 3: CaO.