My son's 98 Mercury Sable smoking black badly from exhaust after we added coolant

blown head gasket maybe?

It is not clear whether you mean from the tailpipe or in the engine compartment. If in the engine compartment, l would guess that some coolant was spilled in the process and it is now burning off ot the hot surfaces, particularly the exhaust manifold. If that is the case, it will subside.

Coolant does not cause black smoke, if u spilled it, it would steam off(white smoke) black smoke(out of the exhaust) is caused by carbon deposits inside the engine being burned off, or if the engine is running excessively rich it will smoke black. There is a SLIGHT chance that if its coming out of the exhaust that u have some coolant leaking into a cylinder or two(blown head gasket) causing the engine to produce excessive heat & burn the carbon out of the combustion chamber & off piston top. If the problem does not go away, take it to a mechanic, OR if u cannot afford that, chances r u have a freind that may be mechanically inclined.

Coolant temperature sensor may be bad causing computer to trigger full rich as if engine was cold.Any trouble codes in the computer?If check engine light is on have a diagnostic check done to find the faulty sensor.

Was the car super hot and died out,Then later on you will filled it back up, then started the engine .If so the vehicle could have flooded out and was burning off the extra fuel.If the car is not running hot now , then take it down the road and mess the pedal to the floor so it will blow the smute out.