The formula above is what the IPCC uses to calculate how long CO2 remains in the atmosphere.
The first thing I am going to do is re-arrange the formula and to get rid of the exp() and replace it with powers of 2 so the formula now becomes:

0.22 + 0.26 * 2^(-t/120) + 0.34 * 2^(-t/13) + 0.19 * 2^(-t/0.82)

The 2^(-t/120) is a half life formula (see here). this means you could describe the above formula as: 22% of CO2 remains permanently in the atmosphere, 26% has a half life of 120 years, 34% has a half life of 13 years and 19% has a half life of 301 days.

You might think that because the formula is full of half-life expressions it behaves the same as a typical half-life formula. You would be wrong. The above formula has some strange properties.

The first one is that you can not calculate how much of a given quantity of CO2 will disappear in a given time period unless you know when all of its components entered the atmosphere.

For example.

Scenario 1
In 1995 1.7 Gt of CO2 is released leaving 1Gt remaining in 2012
in 2000 3.1 Gt of CO2 is released leaving 2Gt remaining in 2012
in 2005 4.3 Gt of CO2 is released leaving 3Gt remaining in 2012
in 2010 4.9 Gt of CO2 is released leaving 4Gt remaining in 2012
Total remaining in 2012 = 1+2+3+4 = 10GT

Scenario 2
In 1995 6.8 Gt of CO2 is released leaving 4Gt remaining in 2012
in 2000 4.7 Gt of CO2 is released leaving 3Gt remaining in 2012
in 2005 2.8 Gt of CO2 is released leaving 2Gt remaining in 2012
in 2010 1.2 Gt of CO2 is released leaving 1Gt remaining in 2012
Total remaining in 2012 = 1+2+3+4 = 10Gt

The question is: How much of the 10Gt in 2012 will be left in 2022?
For scenario 1 the answer is: 0.9 + 1.7 + 2.6 + 3.1 = 8.3Gt
For scenario 2 the answer is: 3.6 + 2.6 + 1.7 + 0.8 = 8.9Gt

Obviously all CO2 was not created equal

A second property is the time it takes CO2 to half and then half again. If you add 4Gt of CO2 today it will take 31 years before there is just 2Gt remaining. It will take a further 343 years for the 2Gt to reduce to 1Gt. However, if you add another 2Gt in 31 years it will take that 2Gt just 31 years to half.

Clearly the IPCC's equations do not make sense.