# P = nRT/V vs. P=F/A paradox?

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So when Day After Tomorrow came out, a lot of people were saying online that a temperature decrease would, per the ideal gas law, decrease pressure, and that the movie failed to portray the effects of this drawing air out of enclosed surfaces. This summer after noticing the office's alcohol based sanitizer containers dripping from overpressurization the topic has come to mind again.

However, pressure is also proportional to the weight of the air above a given surface. So if the surface area of Earth remains the same, would that not mean the average pressure has to remain the same? Would therefore the volume of the atmosphere contract? (I would assume thermal expansion of liquid/solid interior of the Earth is at a less drastic rate.)

Conversely, real-life climate change is known to increase average global temperatures, which I would presume increases the total volume of the atmosphere. However, it's also known to be more pronounced at the poles than the tropics (I often see this pointed out but I haven't come across anything pointing out why) so would that mean the shape of the atmosphere as well?

This also got me thinking; for the ideal gas law; or some hypothetical differential equivalent thereof; to work at every point in space, that would mean there has to be more volume per unit space where the temperature's higher. Doesn't this suggest the present shape of the atmosphere is actually different than the shape of the Earth? That it would bulge more at the equator than the Earth does, and shrink more at the poles than the Earth does? If climate change expands the atmosphere more rapidly at the poles than the tropics, will this create an atmospheric shape more similar to that of the Earth? (Ie. Larger but more "to scale" than before?)

Edited by ScienceNostalgia101

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A temperature decrease causing a pressure drop from PV=nRT assumes V and n are constant (or only have small changes) which are probably not good assumptions.

If water condenses out, n will drop. The atmosphere does not have a fixed boundary, so V may effectively be lower. You would have to know the size of those effects to draw a conclusion about P

5 minutes ago, ScienceNostalgia101 said:

So when Day After Tomorrow came out, a lot of people were saying online that a temperature decrease would, per the ideal gas law, decrease pressure, and that the movie failed to portray the effects of this drawing air out of enclosed surfaces. This summer after noticing the office's alcohol based sanitizer containers dripping from overpressurization the topic has come to mind again.

However, pressure is also proportional to the weight of the air above a given surface. So if the surface area of Earth remains the same, would that not mean the average pressure has to remain the same? Would therefore the volume of the atmosphere contract?

What if there is less atmosphere?

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3 minutes ago, swansont said:

A temperature decrease causing a pressure drop from PV=nRT assumes V and n are constant (or only have small changes) which are probably not good assumptions.

If water condenses out, n will drop. The atmosphere does not have a fixed boundary, so V may effectively be lower. You would have to know the size of those effects to draw a conclusion about P

What if there is less atmosphere?

Ah, the condensation part makes sense, then.

So is phase change of compounds/elements involved the only means by which "n" changes? Or is it also possible for an expanding/contracting/otherwise changing atmosphere to shoot more matter out into space and/or attract more matter from outer space?

And another thing I forgot to consider earlier on, as you mentioned the lack of a fixed boundary... Earth's atmosphere is exponentially decreasing with distance, rather than having any arbitrary "threshold" where Earth ends and space begins. So that raises the question... what would decreasing V be equivalent to in that context? An exponential function with a different base? Different rate of change? Both? Neither?

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15 minutes ago, ScienceNostalgia101 said:

So is phase change of compounds/elements involved the only means by which "n" changes? Or is it also possible for an expanding/contracting/otherwise changing atmosphere to shoot more matter out into space and/or attract more matter from outer space?

We don’t rapidly lose much gas; what we lose most easily is H2 and He. We don’t have a lot to start with. And colder means less energy, so that reduces the chance of loss.

15 minutes ago, ScienceNostalgia101 said:

And another thing I forgot to consider earlier on, as you mentioned the lack of a fixed boundary... Earth's atmosphere is exponentially decreasing with distance, rather than having any arbitrary "threshold" where Earth ends and space begins. So that raises the question... what would decreasing V be equivalent to in that context? An exponential function with a different base? Different rate of change? Both? Neither?

Ideal gas law ignores gravity, so it may be hard to reconcile.

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Ah, so it's just an approximation, then. That makes more sense of it.

Thanks again.

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