Does hot air have high pressure, and does compreesed air have high pressure? I have heard a lot about pressure.  Warm air is less dense and has low pressure, but I thought warm air has high pressure. I know that compressed air is going to be hot, so wouldn't it have high pressure? 
We know that the air molecules in cold air move slower than in warm air, but there are more of them packed closer together. So the chances of colliding are more significant. Is that why cold air has high pressure? 
This is a counter-argument because the molecules in hot air are farther apart but move faster so they would also have a big chance of colliding, right? 
I would appreciate it if the gentleman or women who reply to this could answer all of my questions if you can, but any reply would be appreciated. 
Thanks, Gary
 A: The answer is none of the above.  You can have warm air that is lower pressure, higher pressure, or equal pressure to cold air.  However, if you want to focus on a single body of air (such as the air in a room, or air in a balloon), we can talk about how changes affect pressure.
The key equation, as I.E.P. mentioned in the comments is $PV=nRT$ where:


*

*P is the pressure of the gas

*V is the volume of the gas

*n is the number of mols of gas

*R is the ideal gas constant

*T is the temperature (in kelvin)


So we can see from this equation that if you take a fixed amount of gas (n doesn't change), and you increase its temperature by heating it, either pressure or volume (or both) must change.  If you do that heating in an enclosed chamber, the pressure will increase.  This would lead you to the idea that warm air is high pressure, but that's not quite a true statement.  What is true is that if you heat air in an enclosed space, you will increase its pressure.  On the other hand, if you heat the air in an open container with no lid, the pressure wont increase at all: the volume will increase.
Just because air is compressed does not automatically make it "hot."  It just means it has a high pressure.  If you have experience with compressors, this might not be intuitive because, in practice, you find that compressors heat the air quite a bit (especially if you're filling air tanks for divers!).  However, that is really an issue of inefficiency.  The piston based compressors take an initial volume of air and compress it really quickly into a small volume.  This is going to either raise the pressure, the temperature,  or both.  In practice, it raises both.  That's not a desirable thing -- it wastes money to generate all that heat.
Finally, to your model of collisions, if you have a hot gas and a cold gas at the same pressure, what you will find is that the hot gas will have fewer collisions, but they will be more energetic (higher velocity).  The cold gas will have more collisions, but they will have less energy.  They're two different approaches to cause the same pressure.
A: Sorry for the necro-answer, but this question bugged me awhile and I wanted to post the thoughts that helped me understand the issue.  Because I couldn't understand how the two could be decoupled at all, and thought Temperature=Pressure.
Imagine you've got a large room - and it's empty except for 10 hydrogen atoms.  The temperature of the system is how much energy/speed those atoms have.  So if they're moving fast, the temperature is hot; if they're moving sluggishly, the temperature is cold.  Now, that's not the same thing as pressure.  Pressure is how much force is being put on the sides of the room, pushing outwards.  Well, there's only 10 atoms - it's going to be low pressure even if those atoms are relatively hot, because it's only 10 atoms exerting pressure against the sides of the container.
Or another thought experiment: imagine you've got a container of an ideal gas.  You can't heat the gas up without making it expand; you can't change the volume of the gas without changing its temperature - so it might seem like Temperature=Pressure.  But imagine dividing the container in two, letting one half of the gas out, and then letting the gas reexpand to fill the container.  So the gas cools and depressurizes during this expansion... but what happens when you heat the gas back to its original temperature?  The pressure goes up... but not all the way up to what it was before!  Because the Ideal Gas Law (Pressure x Volume = Constant x MoleculeCount x Temperature) - one of those variables was changed: MoleculeCount.  It's been halved.  Which means, if you heat the temperature back up to what it was before, but it only has half the mass, the pressure is going to be half of what it was before - even though the temperature is exactly the same!
So if you want a low-pressure, high temperature system?  Make it low mass.  If you want a high-pressure, low-temperature system?  Pack in lots of molecules.
Short Story: Temperature is highly related to Pressure... but only if the mass doesn't change.  Change the mass, change the ratio between Temperature and Pressure.
