All components would have the same temperature. There are cases where you can have different particle types with different temperatures in the same place (fusion reactors have different electron and ion temperatures, for example), but atoms in the atmosphere collide with each other far too often (billions of times per second) to sustain any difference.

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As you are already are likely aware, some gases affect heating/cooling a lot more than others, a popular example is greenhouse gases, however because the atmosphere is mixed togeter and the molecules interact a lot, there is not much difference in the temperature of a volume that is recieving the same amount of thermal energy.

As was already said, all components of the atmosphere will have the same temperature. As the air cools or warms, this will remain true because the air is pretty well mixed.

However, you may be interested to know that temperature is not the same as heat. The "specific heat capacity" of a material or gas is the amount of heat required to raise its temperature by one degree. Earth's atmosphere is almost entirely made up of nitrogen, oxygen, and argon, with trace amounts of other gasses including carbon dioxide. A gas with a low specific heat capacity, like argon, doesn't need to lose/gain as much heat to change temperature when compared with a gas with a high specific heat capacity, like carbon dioxide.

Finally – say you add a particular amount of heat to the atmosphere. The proportion of that heat that goes into different gasses depends on both the specific heat capacity and the proportion of each gas in the atmosphere. A gas like carbon dioxide may have a high specific heat capacity, but because of its very low proportion in the atmosphere (<0.1%), it won't take a very significant proportion of the heat. In this way, I think the major atmospheric gasses all have about the same influence on changes in temperature (i.e., their proportions and specific heat capacities all very roughly balance out).

You should rephrase your question. Air is well mixed, which means that all of the molecules in an air sample should be the same temperature. I think what you’re really wanting to know is which molecule takes the least amount of energy to raise its temperature by itself.

If you start with 1 gram of oxygen, 1 gram of nitrogen, and 1 gram of argon heat by 100 Celsius, then how much additional energy does each have…?

Oxygen specific heat is 0.92 J/gK. 100 Celsius of change is the same as 100 Kelvin of change so (0.92 J/gK)(1 gram)(100 K) = 92 J

Nitrogen specific heat is 1.04 J/gK. 100 Celsius of change is the same as 100 Kelvin of change so (1.04 J/gK)(1 gram)(100 K) = 104 J

Argon specific heat is 0.52 J/gK. 100 Celsius of change is the same as 100 Kelvin of change so (0.52 J/gK)(1 gram)(100 K) = 52 J

From all of this, if you had an air mixture of 1 gram Oxygen, 1 gram Nitrogen, and 1 gram Argon and wanted a temperature change of 100 Celsius, you would need 92+104+52 = 258 Joules to heat it up.

As you can see of these three gases, nitrogen takes the most energy to increase its temperature. With respect to what’s in earth’s atmosphere and ignoring non-naturally occurring industrial pollutants, carbon dioxide, methane, and water vapor take a lot more energy to heat up and must lose more energy to cool down.

An example of electron temperature being different in a more common, familiar technology is a fluorescent lamp in which the electron temperature is much higher than the gas temperature. That's part of the explanation of how it can efficiently produce UV to then excite the phosphor.

I call it a common, familiar technology, but it is rapidly on the way out now that LEDs offer better performance at low cost.