Reaction mechanism for how to convert methanol to ethanol? ?

Method 1

First lets react the methanol with PCl3 or HCl …(in case of HCl) the H+ gonna attack the OH bond to give a final result of Chloromethane
Do either Wurtz Reaction(Na with dry ether ) or Frankland Reaction (Zn) with 2 moles of Chorlomethan to get a final product of Ethane
Reaction of Ethane with Cl2 in presence of Sun light will give Chloroethane
Reaction of Chloroethane with Aq KOH will take up Cl to add OH to give a final product Ethanol

Method 2{Shorter}

First React Methanol with PCC to give a final product of HCHO
Reaction of Grignard Reagent (RMgX) take R=CH3 so reaction of CH3MgX in “acidic” medium will directly give you Ethanol

Method 3

CH3-OH + HBr ------> CH3Br + H2O
2CH3Br + (alc) KOH----->CH2=CH2
Reaction of of CH2=CH2 with H2SO4 ————→ CH3CH2OH