h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

Sorry, I just realised that I had been thinking wrong, and that if the density of air *does* decrease with altitude, then the actual thickness of the air *would* be greater than you would expect .

But anyway, my question still stands why is the pressure the same as the weight of the air column?

"But anyway, my question still stands why is the pressure the same as the weight of the air column?"

Because that's what's pressing down. Actually I suspect things get more complicated when you bear in mind that the air isn't still, but basically it's the weight of the air that causes air pressure.

When you're calculating, bear in mind that the column won't be 1 square inch the whole way up. The Earth's curvature means that vertical lines will diverge.

Yes, it is possible to weigh air. If you take the weight of your box of air, you then need to weigh an identical box with no air in it (ie, containing a vacuum) and subtract the one from the other. Make sure the boxes are heavy enough that the one without air doesn't float away.

In this country, 'pound' is something you do with a pestle and mortar.

B

"Is that true? The weight of the air is [mass of the air]*[acceleration due to gravity]. 14.7 pounds-force = 65.38 Newtons"

Good instinct there - go for the SI units, they make MUCH more sense.

"implying that there is a mass of air [...] of about 10 000 kg per sq-metre of the earth's surface area."

Correct.

"Density of the air is reckoned at about 1.3 kg per cubic metre (according to one website)."

WHOAH! Density of air WHERE? At what temperature?

Go back to that ideal gas equation:

PV = nRT

Rearrange it to give the density of the gas, i.e. the NUMBER of molecules per unit VOLUME.

n/V = P/RT

So the density is DIRECTLY proportional to the pressure (i.e. the higher the pressure, the higher the density), but INVERSELY proportional to the temperature (the higher the temp, the lower the density).

Now, air temperature doesn't really vary *that* much (by less than 100 degrees in any air a human can breathe), but the pressure varies a LOT.

For instance, average air pressure in a ski resort a mile up is only 83% of the pressure at sea level. What does that mean? It means there is physically less air up there - fewer molecules per m3. You notice this very quickly if you try to run up the stairs in a ski resort - you get out of breath MUCH quicker, because each breath is bringing in less oxygen.

That density (and hence the pressure) drops off VERY quickly as you increase in altitude. At an altitude of only 32 km, air pressure is down to less than 1% of its value at sea level. So that column of air very, VERY rarefied at the top.

"This implies the atmosphere is about 7.7 km deep?"

That assumes a constant density. But the density is not constant...

"Does the density of air increase significantly at any particular altitude? Or is my maths wrong?"

Your maths is right, but it makes a wrong assumption - constant density. And no, there is no particular altitude where the density suddenly goes up - it's a smooth progression.

"By "weight of air" do (they) mean the *measured* weight of air, per square inch..."

Yes. But do try to get away from this business of using imperial measurements. It really does make more sense in SI. (Engineer talking...)

"If you had a closed piston, with the volume of the piston in vacuum, and the piston being pushed out by a balance of somekind (ie, a spring), then if the end of the piston was 1sq-inch, then you would measure 14.6 pounds-force on the end of the piston, because you'd measure 14.6 psi * 1 si = 14.6 p.

Absolutely correct.

"But that's not really the same as weight, is it?"

Yes, it is. Ask yourself - what is weight?

Weight is the FORCE exerted on a body because of two things:
(1) its mass
(2) the gravity field it's in.

The problem with your imperial units is that it's very easy to confuse the units of weight and mass. This is not a problem in SI.

Consider your question:

"...if the end of the piston was 1 m2, then you would measure 100,000 N on the end of the piston, because you'd measure 100,000 N/m2 (i.e. atmospheric pressure, more or less) * 1 m2 = 100,000 N."

And it would never occur to you to say that that 100,000 N wasn't the same as weight, would it?

"Or is it? How do you measure the mass of a volume of gas?"

"If you put a section of gas in a box, could you weigh the box? (ie, and then subtract the weight of the box, etc..)"

Yup.

"Does the added downward velocity due to gravity increase the pressure experienced on the bottom of the box in order to allow you to measure the mass of the gas?"

Whoah, again hang on. What "velocity"?

There is no "velocity", because there is no movement.

What you've got is an additional net FORCE, downwards due to the mass of the gas.

Here's the tricky part... the *pressure* experienced on the bottom of the box is exactly the same as the pressure experienced on the top of the box.

To be honest, that's giving me pause. Hmm. Must think about that one.

"Is the problem related to bouyancy? Probably... ???"

Sort of. Buoyancy is all about *relative* density.

Is any of this helping?

SoRB

>>"You notice this very quickly if you try to run up the stairs in a ski resort - you get out of breath MUCH quicker, because each breath is bringing in less oxygen."

Well, there are other factors involved there as well, but they'd be getting off-topic and more into biology.

I think factor number one is that it's very hard to run up stairs with skis on.

B

"there are other factors involved"

Of course - but the fundamental fact remains that if I breath in a lungful of air in Manchester, I get about 25% more oxygen into my lungs than I do if I take exactly the same size lungful of air in Val Thorens.

SoRB

When suddenly *starting* heavy exercise (uphill sprinting, stair-running, etc) at altitude, especially when not acclimatised, it can take longer before an increased breathing rate kicks in, since breathing rate is not really controlled by oxygen levels, but by CO2 levels/blood acidity, which themselves are affected by altitude.

The result can be that you breathe more slowly than you might do at sea-level, and *before* you feel noticably out of breath, you can find your leg muscles tiring and fading due to oxygen running low.

Of course, the fact that there is less oxygen is an important factor whatever the breathing rate, but the fact that many people will under-breathe compared with the rate they'd breathe at at sea level can have a great effect, particularly when initiating heavy work, though acclimatisation and/or deliberate breathing during exercise can at least correct for breathing-rate issues to some extent.

Is that why you go down instead of up Potholer?

Ok, ok, my calculation over simplified things a lot... I explained in my third post that I'd mistakenly believed it would provide an overestimate rather than an underestimate, so the numbers can be forgotten about. Probably shouldn't have done it so late at night... I started the calculation thinking I'd get something obviously nonsense, but it turned out a bit too good NB: The fact that the column wouldn't have parrallel sides would of course make the real answer even shorter than my 7km.

""That assumes a constant density. But the density is not constant...""

My calculation only gives a factor of about 4 to play with?... ie, 7km cf. ~30km?? Is that enough?

"""...if the end of the piston was 1 m2, then you would measure 100,000 N on the end of the piston, because you'd measure 100,000 N/m2 (i.e. atmospheric pressure, more or less) * 1 m2 = 100,000 N."""

That's exactly the same calculation as I did above, and it still doesn't make sense.

"""And it would never occur to you to say that that 100,000 N wasn't the same as weight, would it?"""

Yes, it would occur to me to say 100,000 N wasn't the same as weight. Any arbitrary force != weight.

Weight is measured by the reaction force of a mass contacting another object in a gravitational field? If you put the piston horizontally, you should still measure the same force on the end of the piston (from the air pressure)? Then you definately *wouldn't* say it was weight, would you?

How is it possible that air molecules at the top of the column of air will have any significant impact on the measuring apparatus at the bottom? If you tried to measure a box of air, and then measured a box of air that was twice as tall, how would the measuring apparatus be aware of the air at the top of the box, and "know" to double it's reading?

""Whoah, again hang on. What "velocity"?

There is no "velocity", because there is no movement.""

I meant the velocity of the air molecules. Air pressure is clearly derived from the transfer of momentum of the air molecules to the measuring apparatus. Since gravity points downwards, the air molecules should impact the bottom of a box quicker (ie, with more momentum) than when they do at the top, right? (?) Thus, there is a greater transfer of momentum *downwards* so, if the box is sitting at rest on a measuring apparatus, there will be a greater downwards force upon the scales than if you'd had an empty box? But I don't believe that this would be enough to allow you to claim that you were measuring the weight of the air; see my previous point about 'how do the scales even know about the air high above it...'

cool, thanks for that additional information Potholer. So if I deliberately try to breathe harder when I'm not acclimitized, that might help?

>>"So if I deliberately try to breathe harder when I'm not acclimitized, that might help?"

Certainly, at reasonable altitudes (~200m), I've found when starting activities like sprinting up slopes, it really helps to start deep breathing when starting to run - it avoids getting ~10s into exercise and then finding legs turning to jelly with no warning.
Likewise, in my limited skiing experience, I found I tended not to breathe fast enough on starting descents, and rapidly tired. Maintaining a deliberate breathing rhythm made a definite difference.

In sustained exercise like carrying heavy packs uphill, I'm not aware of deliberately breathing, but it may be that I'm synchronising breathing with footfalls, or breathing partly to try and dump heat, or maybe having muscles running pretty much constantly at their aerobic limit, feedback mechanisms to control breathing just work better.

200 m eh? So if I were 1 km up, that wouldn't work?

Cheers for spotting the typo - should have been 2000m.

"Yes, it would occur to me to say 100,000 N wasn't the same as weight."

What do you measure weight in?

Newtons. Therefore ANYTHING you can measure in Newtons is analogous to weight.

"Any arbitrary force != weight."

Well... yeah. That's kind of the point. Weight is a measure of FORCE, not MASS. This is one of those really difficult to understand but incredibly basic concepts which, once you get it, makes a lot of things clearer. Until you do get it, however, you'll struggle with many things to do with forces.

"Weight is measured by the reaction force of a mass contacting another object in a gravitational field?"

Well, yeah. Or in ANY accelerating frame of reference.

Consider: there is NO experiment you can do that can distinguish, from inside a sealed room, whether you're on the outer surface of the earth, on the INNER surface of a large rotating cylinder in deep space, or inside a spacecraft accelerating in deep space at exactly 9.81m/s^2.

What that means is that in all of those situations - which from the outside look completely different - your WEIGHT is exactly the same. It's just a force, measured in Newtons.

"If you put the piston horizontally, you should still measure the same force on the end of the piston (from the air pressure)? Then you definately *wouldn't* say it was weight, would you?"

The force you apply to the piston is not weight. But the AMOUNT of force you have to apply to move it is equal to (or rather, very slightly more than) the WEIGHT of the air. This is no different than the fact that if you want to lift a 100N weight, you have to apply a force upwards of 100N plus a bit.

The orientation of the piston doesn't matter because the weight of the air is manifested as *pressure*, which acts in all directions equally.

"How is it possible that air molecules at the top of the column of air will have any significant impact on the measuring apparatus at the bottom?"

You might just as well as why, when you stand on a bathroom scale, does it measure the weight of your whole body and not just your feet. I mean, how is it possible that your head could have an impact on the measuring apparatus five feet or so away?

"If you tried to measure a box of air, and then measured a box of air that was twice as tall, how would the measuring apparatus be aware of the air at the top of the box, and "know" to double it's reading?"

If you tried to measure my weight, then got my girlfriend to stand on my shoulders, how would the bathroom scales "know" to increase their reading?

""Whoah, again hang on. What "velocity"?

There is no "velocity", because there is no movement.""

I meant the velocity of the air molecules. Air pressure is clearly derived from the transfer of momentum of the air molecules to the measuring apparatus."

Yes!!!! BUT - pressure acts equally in all directions. The NET velocity, the AVERAGE of all the velocities of those gas molecules is zero.

But you're right - at a very, very basic level, the transfer of momentum from the moving air molecules to the container is what pressure IS. You have had a very deep insight there.

"Since gravity points downwards, the air molecules should impact the bottom of a box quicker (ie, with more momentum) than when they do at the top, right? (?)"

Yeah, I've been thinking about that one. And in theory, I suppose you're right. But the difference it makes is *tiny*. Not worth thinking about. Certainly far too small to measure.

"Thus, there is a greater transfer of momentum *downwards* so, if the box is sitting at rest on a measuring apparatus, there will be a greater downwards force upon the scales than if you'd had an empty box? But I don't believe that this would be enough to allow you to claim that you were measuring the weight of the air; see my previous point about 'how do the scales even know about the air high above it.."

The facts would appear to contradict you.

And here is the crucial thing about science - you say you don't "believe" that this would be enough to allow you to "claim" you're measuring the weight of the air. But the universe doesn't care what you believe, and the fact of the matter is an evacuated box demonstrably weighs less than one full of air.

If a box full of air weighs more than one that is empty, what do you propose we are measuring, if not the weight of the air?

If you can think of a better theory, I'm all ears. But your theory must fit the facts, and the fact is the air filled box weighs more than the empty box.

SoRB

Hi Potholer,

actually I didn't know it was a typo, honestly! I was just asking an honest question.

>>"actually I didn't know it was a typo, honestly! I was just asking an honest question."

That's OK - whatever the reason, it was good it was spotted earlier rather than later.

I think the issue is that I've been considering weight as being mass*[acceleration due to gravity] whereas you're clearly defining it in terms of mass*[acceleration due to any arbitrary force]. Admittedly Einstein would suggest that they can be equivalent. I think that's not quite fair. I mean, yes an air molecule is unaware that it's weight is a superposition of gravitational forces and the forces from collisions with other air molecules, but from our point of view, viewing the system separately, it's pretty easy to make the distinction, is it not? In describing the weight of the air, I was describing it from the reference of an external observer, not the air itself.

""If a box full of air weighs more than one that is empty, what do you propose we are measuring, if not the weight of the air?""

An inflated balloon would appear to weigh more than an uninflated balloon and the same amount of air, because it would be more dense, and less bouyant in the surrounding air. Behind my previous discussions, I would have considered this to be an *apparent* weight. I don't believe that a box full of air and an empty box would measure differently, in air (assuming they're the same size box). Bouyancy occurs because of displaced volume, and the volumes of the boxes should be equal. In a vacuum I'm happy to accept that the boxes would measure differently. What I don't understand is how the difference between the two could be equal to the weight of the air (as in, mass of the air * acceleration due to gravity). I mean, going back to the root question, I've never denied that the weight of the air is important; clearly if it had no weight it'd just fly off into space and we'd be living in a vacuum. What I don't understand is how the apparent air pressure (and force from that pressure) is identically equivalent to the weight of air.

""You might just as well as why, when you stand on a bathroom scale, does it measure the weight of your whole body and not just your feet.""

This is surely related: I can understand that it's straightforward to measure your entire body weight, because your head, for example, is constantly pressing down against your body. You head is, in effect, in constant collision with your lower body (and your lower body in constant collision with your bathroom scales). Although air molecules at the top of a box will collide with some air molecules lower down the box, they spend a significant part of their time in free space. You and I both said that the gravitational component of the pressure on the bottom of the box is insignificant. If it is, how on earth could you measure their weight?

""But the universe doesn't care what you believe""

Thankyou for pointing that out. All of these years, I've been thinking that actually, it is *I* that control every element of the universe. Now you've told me that I've been wrong all this time; the laws of physics are defined separate from myself. Wow. Sorry, I'll need to spend some time trying to come to terms with that. You'll have to excuse me if you don't hear from me for a while.

Sorry, that *is* wrong. When I talked about boxes, I was assuming they'd be hung from a balance. If you rested them on bathroom scales, for example, then it might be different.

In that case, I may be able to accept that an empty box (containing a vacuum) and a full box (of air) would be observed to weigh differently. Again, however, I don't see how the difference is equal to the weight of the air. If the gravitational pressure on the bottom of the box is insignificant, how could it be?

Generally, when calculating molecular properites, gravity is completely neglected b/c it is such a small fraction of the intermolecular forces present. For a "box of air" this is still true - gravity can be completely neglected when describing the collisions of the gas molecules with themselves.

I guess what is happening is that despite its (relatiively) small effect, when this small affect is summer over the number of gas molecules in the box of air (10^23 =
100000000000000000000000)
the result is a measurable macroscopic effect.

Maybe a useful analogy would be to imagine a box full of super-bouncy balls. Assuming there's no loss to friction (as would approximately be the case in the box of air), if the bouncy balls are put into a highly agitated state, they can bounce like crazy inside the box, in all directions, etc. But they will be accelerated towards the bottom, and impact more force on the bottom (than the other sides) even though they're motion is energetic enough such that they're hitting the other walls.

Then consider this. Using the analogy above, you wouldn't expect the weight of the box to change if the balls were all at rest on the bottom, or bouncing like crazy on the inside, would you?