The conversion of $2$ -methylpropan-$1$-ol to $2$ -methylpropan-$2$-ol is a rearrangement reaction. In these reactions, the carbon skeleton of a molecule is rearranged to give a structural isomer of the original molecule. Here, the alcohol group of $2$-methylpropan-$1$-ol moves from one atom to another atom in the same molecule to give $2$-methylpropan-$2$-ol.