What is the equation to convert ethanol to propanoic acid ? And what conditions needed for the conversion ?

If we start from ethanol, then we can use ##"PBr"_3## in pyridine to turn ethanol into ethyl bromide, which can then be converted to a Grignard reagent by reacting with magnesium solid in dry ether, followed by a reaction with carbon dioxide.

1. ##\mathbf("PBr"_3)## acts as an electrophile for the alcohol to attack, while pyridine acts as a proton acceptor. A ##""^(-)"O"-"PBr"_2## (bromophosphite) group leaves when the bromide that left from ##"PBr"_3## attacks again as a nucleophile (Organic Chemistry, Bruice, Ch. 10.2).
2. Formation of a Grignard reagent; you should know this if you have taken second-semester organic chemistry for a few weeks. It must be dry to prevent deactivation of the nucleophile.
3. Grignard reagents can react in an ##"S"_N2## reaction with ##\mathbf("CO"_2)##.
4. A bit of aqueous acid finishes up the reaction by protonating the carboxylate and hydrolyzing the ##"MgBr"^(+)##, neutralizing any remaining Grignard reagent as well by (easily) protonating the ethyl anion.