PhD., University of Maryland
Jonathan is a published author and recently completed a book on physics and applied mathematics.
Atmospheric pressure is the pressure caused by the mass of our gaseous atmosphere. It can be measured using mercury in the equation atmospheric pressure = density of mercury x acceleration due to gravity x height of column of mercury. Atmospheric pressure can be measured in atm, torr, mm Hg, psi, Pa, etc.
So let's talk about atmospheric pressure, the pressure associated with the atmosphere of air we've got all around us. Well air is a fluid, it's something that can change its shape and size without too much pushing on it. So what that means is that we're sitting here immersed in air and that means we got support the weight of all the air above us and that means that we get pressure. Alright so we're ready right? Liquid pressure, fluid pressure here we go, density of the fluid times g times the height. Okay what do we use for the density of air, I mean we know what the density is down here but the density of air gets smaller as we go up so what should we use in this formula and what about h? What should we use for the height of the atmosphere? What do you think? I don't know and actually those 2 things are related to each other. So this is kind of a problem, we can't use this formula, so we need to do something else and what we're going to do is we're just going to measure it directly. So we're going to take a tube full of mercury we're going to immerse it in a bath of mercury and then we're going to bring the tube up like that okay.
Now there wasn't any air inside the tube, so what that means is that the mercury is going to fall down a little bit and there's nothing there. So the pressure in here is 0, nothing to support. Out here on the other hand, we've got the whole atmosphere pushing down. So that means that the difference in pressure between the atmospheric pressure and 0 is supporting this height of a column of mercury. And now we can use density times g times h because the density of the mercury is about constant and we can measure the height. So the atmospheric pressure is given by the density of mercury times 9.8 meters per second squared times however high this column of mercury is. Alright, so this atmospheric pressure is proportional to the height of the column of mercury. And that leads to one of the earliest measurements one of the earliest units for pressure the millimeter of mercury.
Now when we do a standard measurement what we find is the atmospheric pressure is just around 760 millimeters of mercury. Now we've actually got a definition for what we mean by the standard atmosphere and so let's look over here. One standard atmosphere is equal to 760 torr. This torr is the unit that was supposed to be like a millimeter of mercury but is named after a physicist named Torr Charlie who did a lot of work on pressure in 1600s. So if you actually look at what really that is in terms of millimeters of mercury it's 763.3 so it's close but not exactly the same thing. Well for a standard unit in the United States people often use PSI pounds per square inch. One atmosphere in PSI is approximately 14.7 PSI. So what does that mean?
Well that means that for every square inch of your face so inch, inch something like that I've got 14.7 pounds pushing in every single square inch. That's a lot how come I don't just collapse, well I'm breathing in air which is at that same pressure. So that air pushes out, the air on the outside pushes in and the skin doesn't have to support very much. The same ideas used by scuba divers who dive down deep and they'll breath in pressurized air, so that there're taking in a higher pressure and then their skin doesn't have to support that much of a pressure difference between the water of the ocean and their internal, their lungs and their blood and everything like that. So anyway that's just kind of a little side note.
To write it in terms of SI units, because of course we got always use SI units, it'll be written as 101,325 Pascal's so that's one atmosphere and again this is a standard definition. When somebody says a standard atmosphere that's what they mean 101,325 no at any given day the actual pressure of the atmosphere might be a little bit higher or a little bit lower. But this is just kind of the standard atmosphere. Alright so let's go ahead and use this idea to solve a problem. So when I'm sucking through a straw, essentially what I'm doing is the same thing that we saw over here with this mercury column. I'm removing some of the air out of the top of the straw so that now the pressure exerted inside my mouth is less than the pressure that's exerted by the atmosphere. So that difference in pressure makes a column of water or soda or whatever I'm drinking rise up in the straw. So now I've got a question for you, what's the maximum height of water that you can support in a straw? So if you suck as hard as you possibly can how high up can that column get? Could it get infinitely high?
Well here's the idea, just like before you've got your column of water and really what's supporting that is the atmospheric pressure. So the best you can possibly do is like what we saw over with the mercury barometer. 0 pressure in the top, now you can't actually do that, you can try though but that would be kind of like the idealized best you can possibly even imagine doing 0 pressure up there. And then you got atmospheric pressure down here and that's got to support this column of water. So what we're going to do is we're going to say p atmospheric equals density of water times acceleration due to gravity times the height. And that's that height that we're after right? Well now it's just real easy, I just got to solve and plug in numbers long as everything is in SI units I'm good to go. So h will be p atmosphere 101,325 over density of water 10 to the third, g 9.8 and when you work all that out, you'll find that it's about 10.34. What would the unit be? Well everything is in SI units, so everything is in SI units, it's a height so it's meters. So that means that the maximum height of a column of water that you can support on earth is 10.34 meters.
You want to support bigger than that, you got to push harder you can't just let the atmosphere do your work for you. Alright well of course if it's a denser liquid like mercury the maximum height you can support would be 763.3 millimeters, which is a little bit less about 75% 76.33% of a meter okay so it's much less than that and that's because mercury is a lot denser than water. Alright so let's go ahead and look at some of the other types of problems that you might be asked to solve associated with this. Some of them are kind of weird there're not hard but you might not have thought of what to do okay. So question, what is the total weight of the atmosphere? That's kind of a strange question right? You look at that question and you're like what? How on earth would I know that? alright, weight of the atmosphere, remember we said that the pressure associated with the atmosphere was really due to its weight. So I know the pressure, so what about the weight? Well the weight is going to, that's the force right, so the weight is going to be pressure times area.
Well I know the pressure that's atmospheric, what's the area? Well the atmosphere acts over the whole planet. So this pressure is acting over the surface area of the whole planet. Well what is the surface area of the whole planet? Well it's 4 pi times the radius of the whole planet squared, 4 pi r squared like we know from Geometry right? So now all I got to do is plug in numbers, the radius of the earth is about 6,370 kilometers alright. So we're going to write this out 101,325 again I'm going to ignore units because I know what the units are going to be at the end as long as everything is in SI units I'm fine. So p atmosphere 4 pi is 12.54 alright and then we've got the radius of the, it's got to be in SI units. I said 6,370 kilometers, I'm not allowed to use kilometers. I got to say 6.37 times 10 to the 6 meters and I'll square that. And when I plug all those numbers into the calculator I end up with a weight of 5.17 times to the 19 alright now what's the unit? It's a weight, so that's force and in SI units that's given in Newtons, that's a pretty huge weight.
But that's the whole atmosphere of the earth and notice how easy it really is. It just follows directly from what does weight mean, what does pressure mean and what does area mean? And then just kind of okay multiply the numbers out, alright so that's the way that that goes. Alright let's look at this last one what is the scale height of the atmosphere assuming constant density. Alright so this will take a minute just to kind of figure out what I mean by that. Remember over at the beginning we wanted to use density times acceleration due to gravity times height to determine the pressure at the surface of the earth of the whole atmosphere. But we said there were a couple of problems, the density wasn't constant right and we didn't know what to use by the height or for the height. So in this problem what it's asking us to do is to assume constant density and just take the density at sea level. And then determine what the height, the appropriate height would be that would give us the pressure that we measure.
Alright, so let's go ahead and do that because it'll be an interesting answer. So we'll say p atmosphere equals density gh I want h, so we'll have h equals p atmosphere 101,325 over the density of air at sea level is about 1.3 kilograms per cubic meter. So I'll have 1.3, 9.8 and when you work all that out you end up with 7.9 kilometers. Now that's pretty crazy 7.9 kilometers is really not very high up at all okay. It's actually less than the flying height of most airplanes. Okay at least once they go long distances and so there're got to be air above that and so that indicates to us directly that the density of the air has got to change because if it was just constant the whole way up atmosphere ends way too low. Alright that's atmospheric pressure.
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