Give the reactions for the following conversions:
$1$. Polyvinyl chloride from ethyne
$2$. Methane from propyne
$3$. $2,3-$Dimethylbutane from propene

(N/A) $1$. Polyvinyl chloride from ethyne:
$nHC \equiv CH$ $\xrightarrow{HCl, HgCl_2} nCH_2=CHCl$ $\xrightarrow{\text{Polymerisation}} -(CH_2-CHCl)_n-$
$2$. Methane from propyne:
$CH_3C \equiv CH$ $\xrightarrow{\text{Alkaline } KMnO_4, \Delta} CH_3COOH$ $\xrightarrow{NaOH, CaO, \Delta} CH_4 + Na_2CO_3$
$3$. $2,3-$Dimethylbutane from propene:
$CH_3CH=CH_2$ $\xrightarrow{HBr} CH_3CH(Br)CH_3$ $\xrightarrow{2Na, \text{dry ether}} CH_3-CH(CH_3)-CH(CH_3)-CH_3 + 2NaBr$