Write the reactions for the conversion of methanol into ethanol,propan-$2$-ol,and $2$-methylpropan-$2$-ol.

(N/A) The conversion of methanol $(CH_3OH)$ to higher alcohols involves the formation of aldehydes followed by Grignard reaction:
$1$. Conversion to Ethanol:
$CH_3OH$ $\xrightarrow{Cu, \Delta} HCHO$ $\xrightarrow{CH_3MgBr} CH_3CH_2OMgBr$ $\xrightarrow{H_3O^+} CH_3CH_2OH$
$2$. Conversion to Propan-$2$-ol:
$CH_3CH_2OH$ $\xrightarrow{Cu, \Delta} CH_3CHO$ $\xrightarrow{CH_3MgBr} CH_3CH(OMgBr)CH_3$ $\xrightarrow{H_3O^+} CH_3CH(OH)CH_3$
$3$. Conversion to $2$-methylpropan-$2$-ol:
$CH_3CH(OH)CH_3$ $\xrightarrow{Cu, \Delta} CH_3COCH_3$ $\xrightarrow{CH_3MgBr} (CH_3)_3COMgBr$ $\xrightarrow{H_3O^+} (CH_3)_3COH$