As I said, an estimate.
Okay, let's go with your figures and see what it comes to. Let's start with 2.2x10^10 and double it just for the hell of it.
5000 trillion tons of atmo / 2.2x10^10 = 0.0000044
0.0000044 of the mass of the atmosphere. Still tiny. That's 44 parts per 10 million, or to put it better 4.4 parts-per-million.
That means if I put 250,000 dots on a (really big) page, then put one dot by itself, the one dot represents your figures doubled.
My past research has indicated that human contribution to greenhouse gasses comprised on the close order of 1 part in 1,250 of all greenhouse gasses (water vapor being the big one, accounting for about 94% of all greenhouse-effect gasses, caused by ocean evaporation mainly.)
AGW just isn't very credible when viewed from that perspective.
No, seen from that perspective, the A in AGW is questionable. However, looking at CO2, the increase has be en 36% over the past century, about 3/4 of which is attributable to human activity. That amounts to a 27% increase that can be attributed to the A in AGW. So, why look at CO2, and not H20? Well, the amount of carbon dioxide is constant, while the amount of water vapor is not. The warming caused by the carbon dioxide, in fact, also increases the content of water vapor, as warm air holds more of the stuff than cold air does. Here, we have a feedback loop.
Still, you are absolutely correct that the difference is very small. Carbon dioxide is a small component of the atmosphere, so that 27% increase doesn't translate to a lot in absolute terms.
So, why can't human contribution to the equation be dismissed?
The fact is that the warming of the Earth is also very small. Scientists tell us that the average temperature of the Earth, averaging winter, summer, tropics, arctic, everything, is about 86% F, and that that has increased from about 85% in the past century. Now, that's not a lot.
If you stop to consider that the 1 degree increase is really 1 degree out of well over 500 degrees absolute, since zero degrees F is really just an arbitrary temperature. Absolute zero is about -451 degrees.
So, what is the point of all that? The increase in temperature is 1 degree out of over 500, or less than two tenths of a percent.
So, not much increase in CO2 has resulted in a very small increase in average temperature.
Still, you may be right that human activities are not a part of the equation. Most scientists think they are, but there is no absolute proof of that.
Further, if it is 86 degrees or 85 degrees, outside, most of us wouldn't notice the difference, so what does it matter?
The average temperature overall translates to greater differences in some places than others. Some places, like Northern Europe for example, may actually be getting cooler.
The problem is the average temperature on the mountaintops (see also, melting glaciers) and that in the Arctic (melting ice caps).
Is global warming an impending disaster? We don't know. Will it result in benefits in some places? probably. Will it negatively impact some places? Without a doubt.
What is the final result likely to be in any given place?
Now, there is the prize question, the one we should be researching instead of emitting tons of hot air into the atmosphere trying to either deny global warming, or use it to impose taxes that may not be necessary.