1993 Toyota Corolla emits smoke

Need a lot more than that to go on. Is the smoke white, blue, or black? Do I assume it's coming out of the tail pipe or is it coming from under the hood or somewhere else? Are you losing coolant or oil? When does this occur? For just a few minutes after starting the engine? All the time? Only after the engine is hot?

White smoke from the exhaust is a sign of burning engine coolant. That is caused by a leaking cylinder head gasket. Your mechanic can perform a chemical test at the radiator that will confirm a leaking head gasket about 95 percent of the time. You can also add a small bottle of dark purple dye to the coolant, then search a day or two later with a black light. The dye will show up as a bright yellow stain that you can follow back to the source. If the head gasket is leaking, you'll find the dye inside the tail pipe.

The "white compartment" is probably the coolant reservoir. A leaking head gasket will send combustion gases into the cooling system where they'll escape from that reservoir. You could see that as smoke too.