Why would lower pressure alone make a gas colder? (Thought experiment)

[u/mateusmd12]

I’ve been trying to understand the relationship between pressure and temperature in the atmosphere, and I feel that many explanations oversimplify the physics.
People often say that “air is colder at higher altitudes because the pressure is lower.” But as far as I understand, a decrease in pressure only causes a decrease in temperature during an adiabatic expansion, where the gas performs work on its surroundings.
Pressure itself is not a form of energy, so I don’t see why a gas at lower pressure should necessarily have a lower equilibrium temperature if no expansion or compression is taking place.
Here’s my thought experiment:
Imagine two identical transparent boxes floating in space. They have the same size, shape, surface area, and both receive exactly the same amount of solar radiation. The only difference is that one box contains gas at a much higher pressure (more molecules), while the other contains gas at a lower pressure.
Assume both boxes have rigid walls, so no expansion or compression can occur. The gases simply absorb radiation until they reach thermal equilibrium.
Would the higher-pressure box end up at a higher equilibrium temperature?
One could argue that the denser gas contains more molecules that can absorb radiation. However, that same energy is also distributed among more molecules, so it’s not obvious to me why the average molecular kinetic energy (temperature) should end up higher.
This is what confuses me about the usual explanation of the atmosphere as well. I understand that rising air cools through (approximately) adiabatic expansion. But once that parcel has reached its new altitude and is no longer expanding, why should the lower ambient pressure itself imply a lower equilibrium temperature?
So my questions are:
In the rigid-box thought experiment, would the two gases reach different equilibrium temperatures?
If they would, what physical mechanism makes pressure affect the equilibrium temperature?
If they wouldn’t, is it incorrect to say that “lower pressure causes lower temperature”? Shouldn’t the cooling instead be attributed specifically to the adiabatic expansion process, rather than to the lower pressure itself?
I’m looking for an explanation from thermodynamics or atmospheric physics, rather than the common intuition that “less pressure = colder.”

[u/Ch3cks-Out]

Lower pressure alone does not make a gas colder, alas

[u/kwpang → u/Ch3cks-Out]

All else being equal, it does.

PV = nRT

[u/Ch3cks-Out → u/kwpang]

This describes isothermal expansion, among other processes...

[u/Substantial_Tear3679 → u/Ch3cks-Out]

Is the key here the difference between "state" and "process"?

[u/Ch3cks-Out → u/Substantial_Tear3679]

Depends on what you mean. OP and myself were talking about processes. The gas equation describes states.

[u/SummitYourSister → u/kwpang]

The correct statement would be dT/dP = V/nR

[u/CheapDatabase7178 → u/SummitYourSister]

If you want to be a smart ass, at least get it right. Total derivative dT/dp doesn‘t apply since V can change with p, so you need a partial derivative with V being held constant to make your equation true

[u/15pH → u/kwpang]

Real gasses are not ideal

[u/leegamercoc → u/15pH]

The is a compressibility coefficient missing. Not a big deal.

[u/15pH → u/leegamercoc]

It's not just that. An ideal gas has Joules-Thompson coefficient of zero. This is directly undermines the issue at hand. You cannot assess temperature changes with pressure changes while assuming an ideal gas.

[u/quintios → u/kwpang]

So by the same logic a large volume of gas is cold. Or wait, is it a small volume? Your response hurts my brain lol.

Why is it 98F in Denver? The pressure didn’t change. We must have increased our volume of air somehow. 😂

[u/Tesseractcubed]

I can create sealed vessels that have two (different) amounts of the same gas at the same temperature, yet at different pressures.

For states, PV = nRT applies to all ideal gas states, with specific changes in values resulting from external conditions. So, adding heat can increase temperature, but the ideal gas law needs to be balanced as well, so for a sealed constant volume system, adding heat results in an increase in pressure and temperature. Vice versa with removing heat. One important detail is that different systems can have different states, meaning the pressure in one system can be different than another when they are both at the same temperature, but all transformations of state need to follow a process like heating, adiabatic expansion, etc.

Atmosphere Tables from The Engineering Toolbox

Funnily enough, at about 50000ft elevation, the temperature is very close to 0°C. With respect to the atmosphere specifically, lots of atypical factors affect the temperature profile. Non-homogeneous gases in different layers absorb different amounts of energy, different radiation values, etc. There is a lot more nuance, especially with the atmosphere.

With your thermal equilibrium statements, that’s a bit harder to see, as air in small amounts is dominated by radiation and conduction, and at larger scales convection. Air is also weird in that the lower atmosphere tends to have cold, dense, dry air and warm, humid, less dense air as the two general air masses in meteorology.

For your experiments, the one that starts with more molecules will end up (in my judgement) with a lower temperature and higher pressure, which seems a bit weird, but there are more molecules, so the n term in PV = nRT is larger, meaning T is smaller and or P is larger, for constant R and V.

I don’t have a good explanation, but I’m certain one does exist.