I have white smoke coming out from the exhaust of my audi a6 2003 and the coolant tank is empty

First you need to find out what is causing the smoke. What colour is the smoke?
White Exhaust smoke is usually coolant getting into the cylinders.
Blue Exhaust smoke is usually Oil getting into the cylinders and burning with the fuel.
Black Exhaust smoke is caused by the fuel to air mixture being too rich. This could be your Air Filter so check it is clean.
A common cause of Blue or White smoke is a blown head gasket. This can also show up as oil in the coolant tank and/or a 'mayonnaise' like buildup inside the Oil filler hole.

he causes of white exhaust smoke can vary; however, it is common to see white exhaust smoke when first starting a car, especially on cooler days. This is generally steam caused by condensation. As the engine warms up and the condensation dissipates the white exhaust smoke (steam) is no longer seen. If excessive white exhaust smoke is present well after the engine warms up, it is necessary to have the car inspected for possible internal coolant leaks. Indicators of an internal coolant leak include billowing white exhaust smoke accompanied by a sweet odor or a low coolant reservoir level. An internal coolant leak can also contaminate the engine oil giving it a frothy, milky appearance. Even small amounts of coolant entering the combustion chamber will produce white exhaust smoke.
One of the main causes of white exhaust smoke and coolant loss is a cracked or warped cylinder head, a cracked engine block, or head gasket failure caused by overheating. A cracked head may allow coolant to leak into one or more cylinders or into the combustion chamber of the engine. Dirty coolant, a poorly maintained cooling system, a low coolant level, or a non-functioning cooling fan can cause engine overheating. In addition, engine wear can eventually cause the gaskets to lose their capacity to seal properly allowing internal coolant loss. Intake manifold gasket and head gasket failures are two of the most common sources of internal coolant loss caused by engine wear.
Never remove the radiator cap or coolant reservoir cap while the engine is hot or running as it can cause serious injury; always allow the car to cool down completely first. Checking for a low coolant level in the reservoir is the first step in determining if coolant loss is causing the white exhaust smoke. If the coolant reservoir is at the proper level but excessive white exhaust smoke is present, a cooling system pressure check is required to determine where, if any, coolant leaks are located.

What is year--make--model?
Are you losing coolant? The white smoke may be coolant in the combustion cylinder, possibly caused by a head gasket problem. If head gasket problem, fix that first.

With idle issues, clean/check anything that has to do with idle air system. Any applicable trouble codes?

You have a leaking head gasket, there are several additives you might try to seal it, but be prepared for lots of white smoke and an overheated motor if you leave it as is.

Sounds like you have a leaking head gasket or damage to the engine from when it overheated.
A leaking head gasket can cause compression to leak into the cooling system and pull coolant into the cylinders which is burned out the exhaust. It does not have to leak into the oil.
A shop would check for exhaust gas in the cooling system and run a compression test on the cylinders.

I was going to say that white smoke wouldn't be oil but I see that you figured that out. Look for loose coolant hose clamps on or near the reservoir or possibly a small crack in the reservoir itself. As a general rule for gasoline engines: White smoke = coolant, Blue smoke = oil, Gray smoke = fuel, Black smoke = "Uh oh, that can't be good." :-)

White exhaust smoke from a gasoline engine usually means coolant leaking into the engine block.
Is the coolant in the radiator full ?

If the smoke has a sweet smell to it then it is coolant from an internal leak inside the engine like a head gasket or other source. To confirm that you have a leaking head gasket have a shop problem the coolant recovery/surge tank with a gas analyzer for the presence of hydrocarbons in the coolant tank, this indicates for sure that a head gasket is the cause of the white smoke.

You have coolant burning. Your 2.7T is a hot-running engine and the plastic tank sits above the left side exhaust manifold. The heat from the manifold is known to cook the tank and make it brittle enough to crack and leak. I think if you look at the tank you'll see it's leaking slightly on the manifold, causing the coolant smoke you're seeing. You can get a new tank from www.GermanAutoParts.com in Albany NY - great place to deal with (I always order parts for jobs from them) and replacement is simple.