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## Physics library

### Course: Physics library>Unit 9

Lesson 1: Density and Pressure

# Finding height of fluid in a barometer

Using our understanding of fluid pressure to figure out the height of a column of mercury. Created by Sal Khan.

## Want to join the conversation?

• I thought standard measurement was 760 mm Hg= 1 atm. Here you got 770 mm Hg?
• Yes it is 760mm. The mistake is coming from the atm conversion to Pa.
1atm is 101325Pa not 103000Pa.
• I don't understand why the mercury should be going up the test tube. I understand there is no atmosphere acting on it, but there is also no force dragging it up. Please help! :|
• The atmosphere outside the whole system is pushing down on the mercury in the bowl, and that pressure is distributed evenly throughout the mercury in the bowl. As there is no atmosphere in the test tube (its a vacuum), there is no pressure inside the test tube to push down on the mercury below it. The mercury therefore gets pushed up the test tube due to the pressure being exerted on the whole bowl of mercury, and the lack of pressure pushing back from within the tube. It is only because of the vacuum that the mercury goes up the tube, if that was just a regular test tube you pulled out of a drawer (with no vacuum), this wouldn't work. This is at least how I picture it.
• Q 1: Will the vacuum of the test tube try to raise / suck the Hg into it? If so, do we have to compensate for an added force upward? I can see that if it were not a vacuum, it would be pushing the Hg out of the tube.
Q 2: Since we are dealing with a vacuum in the tube, do we have to deal with the Hg trying to evaporate and creating a gas in the available vacuum? I think this would yield some inaccurate measurements since you can't measure the pressure contribution of the gas via height.
• Q1: This is an incredibly common misconception. A complete vacuum cannot exert any pressure whatsoever. It has no density, so it's pressure (given by P = (rho)*g*h) must be zero. It is the difference in pressures that would drive the mercury to be "sucked into" the space occupied by the vacuum. If the pressure on one side is 0 atm, and the other is 1 atm, then higher pressure (in this case 1 atm) will generate a force to push the mercury to the lower-pressure side (0 atm).

Q2: This is pretty interesting. However, if you consult a phase diagram for mercury, you'll notice that mercury is does some weird things at low and high pressures. So, the short answer is yes, it would create a problem, but no, we don't worry about that here.
• So does the cross-sectional area of the test tube not matter when determining how high the mercury will rise? I would think this would have to be accounted for.
• Nope, it does not matter. The mercury will stop rising when its weight inside the tube equals the force being exerted by the atmospheric pressure on the same area (the cross sectional area of the test tube). So, when weight of liquid inside tube equals force of atmosphere inside tube:

Weight of liquid in tube = density*Area*Height*g
Force of atmosphere = AtmPressure*Area of tube

If you equal the two you get: density*Area*Height*g = AtmPressure*Area
The Areas are the same so they cancel each other out, leaving you with the equation Sal used: density*h*g = AtmPressure

Hope it helped.
• Since the portion above the mercury is a vacuum, how is it that the mercury stops climbing up the tube? The pressure of 1 atmosphere must exist at the top and push upwards, correct?
• To understand why it stop you need to understand what a vacuum is. Try to think in terms of forces. First "Vacuums" don't suck things up, or cause things to move by exerting forces because they are vacuums - they are a space of nothing. The reason that things seem to be "sucked" up or "pulled" by the vacuum is because other forces are pushing said things in towards the vacuum. Why are they pushing things towards the vacuum, because there is nothing pushing back from the vacuum. Now to your question, if nothing is pushing back then why doesn't it go all the way up? Nothing is pushing back rather gravity is pulling down. Make a free-body-diagram on the column of mercury, you will find that the at that height up the tube, the pressure dude to gravity pulling the column of mercury down is equal to the pressure pushing it up due to the atmosphere at the bottom of the tube. I hope this helps (^-^)/ Forgive me if I'm wrong!
• I Didnt Understand the 1atm at France Paris?

I know that 1 atm is an approx. value, But in France does it have the correct measurment?

(Thank You!)
• The standard atmosphere (atm) is equal to 101325 Pascal. This value was meant to represent the mean atmospheric pressure at mean sea level at the latitude of Paris.
• Does this also mean that the height of the mercury in the bowl from its deepest point to the point where the pressure was equal is also going to be 0.77 meters, since the pressure is the same, the density and the gravitational force?
• No, the 1 atmosphere of pressure that was used to calulate the height of the mercury was at the surface of the mercury in the bowl so the height is measured from there.

If you used the pressure at the deepest part of the bowl you would get a height of 0.77 + the depth of the bowl.
• If both water faucets upstairs and downstairs were turned fully on, will more water come out of the downstairs faucet? Why?
• This is a great question! I'm not sure entirely what the answer would be, and maybe someone with more knowledge of fluid mechanics could correct me, but I don't see how more water would come out of a downstairs faucet as I'd imagine the entire system is under relatively constant pressure. If there was a difference, I think it would be minimal.
(1 vote)
• why is there a pressure of 1 atm pushing up on the base of the column of mercury
()
• To make more sense conceptually, Imagine we are in a complete vacuum room and the part of the dish with mercury exposed to the atmosphere is completely covered by a weightless cover, but there is a hole in the middle where there is a completely empty vacuum tube that perfectly fits and seals the hole in the weightless cover. If we apply with our hands the pressure to the weightless cover downward onto the mercury bowl the mercury will go up the tube, because the mercury cannot be compressed and that the pressure exerted on a fluid is evenly distributed on every portion of the fluid and to the walls of its container, but because one "wall" is not a wall but rather a hole, the pressure pushing up on that imaginary "wall" (which does not exist cause its just a hole to the vacuum tube) is the pressure being applied by the hands on the weightless cover. if we apply a 1atm with our hands then the mercury will travel up the tube until it reaches a height in which the pressure due to gravity pulling the column of mercury down is equal to 1atm at the opening of the vacuum tube. Hope that helps (^-^)/ forgive me if im wrong! thanks!
• Does the Atmospheric pressure acting on our body at inside home and outside home is different? I mean that when we are inside the is only a few meters of air column up to the ceiling. So, the pressure must be smaller that outside pressure!
• yes unless your room is airtight perhaps a vacuum

## Video transcript

In the last video, we learned that the pressure at some depth in a fluid is equal to the density of the fluid times how deep we are in the fluid, or how high is the column of fluid above us times gravity. Let's see if we can use that to solve a fairly typical problem that you'll see in your physics class, or even on an AP physics test. Let's say that I have a bowl. And in that bowl, I have mercury, and then I also have this kind of inverted test tube that I stick in the middle of-- this is the side view of the bowl, and I'll draw everything shortly. Let's say my test tube looks something like this. Let's say I have no air in this test tube-- there's a vacuum here-- but the outside of the bowl, this whole area out here, this is exposed to the air. We are actually on Earth, or actually in Paris, France, at sea level, because that's what an atmosphere is defined as-- the atmospheric pressure. Essentially, the way you could think about it-- the weight of all of the air above us is pushing down on the surface of this bowl at one atmosphere. An atmosphere is just the pressure of all of the air above you at sea level in Paris, France. And in the bowl, I have mercury. Let's say that that mercury-- there's no air in here, and it is actually going to go up this column a little bit. We're going to do the math as far as-- one, we'll see why it's going up, and then we'll do the math to figure out how high up does it go. Say the mercury goes up some distance-- this is all still mercury. And this is actually how a barometer works; this is something that measures pressure. Over here at this part, above the mercury, but still within our little test tube, we have a vacuum-- there is no air. Vacuum is one of my favorite words, because it has two u's in a row. We have this set up, and so my question to you is-- how high is this column of mercury going to go? First of all, let's just have the intuition as to why this thing is going up to begin with. We have all this pressure from all of the air above us-- I know it's a little un-intuitive for us, because we're used to all of that pressure on our shoulders all of the time, so we don't really imagine it, but there is literally the weight of the atmosphere above us. That's going to be pushing down on the surface of the mercury on the outside of the test tube. Since there's no pressure here, the mercury is going to go upwards here. This state that I've drawn is a static state-- we have assumed that all the motion has stopped. So let's try to solve this problem. Oh, and there are a couple of things we have to know before we do this problem. It's mercury, and we know the specific gravity-- I'm using terminology, because a lot of these problems, the hardest part is the terminology-- of mercury is 13.6. That's often a daunting statement on a test-- you know how to do all the math, and all of a sudden you go, what is specific gravity? All specific gravity is, is the ratio of how dense that substance is to water. All that means is that mercury is 13.6 times as dense as water. Hopefully, after the last video-- because I told you to-- you should have memorized the density of water. It's 1,000 kilograms per meter cubed, so the density of mercury-- let's write that down, and that's the rho, or little p, depending on how you want to do it-- is going to be equal to 13.6 times the density of water, or times 1,000 kilograms per meter cubed. Let's go back to the problem. What we want to know is how high this column of mercury is. We know that the pressure-- let's consider this point right here, which is essentially the base of this column of mercury. What we're saying is the pressure on the base of this column of mercury right here, or the pressure at this point down, has to be the same thing as the pressure up, because the mercury isn't moving-- we're in a static state. We learned several videos ago that the pressure in is equal to the pressure out on a liquid system. Essentially, I have one atmosphere pushing down here on the outside of the surface, so I must have one atmosphere pushing up here. The pressure pushing up at this point right here-- we could imagine that we have that aluminum foil there again, and just imagine where the pressure is hitting-- is one atmosphere, so the pressure down right here must be one atmosphere. What's creating the pressure down right there? It's essentially this column of water, or it's this formula, which we learned in the last video. What we now know is that the density of the mercury, times the height of the column of water, times the acceleration of gravity on Earth-- which is where we are-- has to equal one atmosphere, because it has to offset the atmosphere that's pushing on the outside and pushing up here. The density of mercury is this: 13.6 thousand, so 13,600 kilogram meters per meter cubed. That's the density times the height-- we don't know what the height is, that's going to be in meters-- times the acceleration of gravity, which is 9.8 meters per second squared. It's going to be equal to one atmosphere. Now you're saying-- Sal, this is strange. I've never seen this atmosphere before-- we've talked a lot about it, but how does an atmosphere relate to pascals or newtons? This is something else you should memorize: one atmosphere is equal to 103,000 pascals, and that also equals 103,000 newtons per meter squared. One atmosphere is how much we're pushing down out here. So it's how much we're pushing up here, and that's going to be equal to the amount of pressure at this point from this column of mercury. One atmosphere is exactly this much, which equals 103,000 newtons per meters squared. If we divide both sides by 13,609.8, we get that the height is equal to 103,000 newtons per meter cubed, over 13,600 kilograms per meter cubed times 9.8 meters per second squared. Make sure you always have the units right-- that's the hardest thing about these problems, just to know that an atmosphere is 103,000 pascals, which is also the same as newtons per meter squared. Let's just do the math, so let me type this in-- 103,000 divided by 13,600 divided by 9.8 equals 0.77. We were dealing with newtons, so height is equal to 0.77 meters. And you should see that the units actually work, because we have a meters cubed in the denominator up here, we have a meters cubed in the denominator down here, and then we have kilogram meters per second squared here. We have newtons up here, but what's a newton? A newton is a kilogram meter squared per second, so when you divide you have kilogram meters squared per second squared, and here you have kilogram meter per second squared. When you do all the division of the units, all you're left with is meters, so we have 0.77 meters, or roughly 77 centimeters-- is how high this column of mercury is. And you can make a barometer out of it-- you can say, let me make a little notch on this test tube, and that represents one atmosphere. You can go around and figure out how many atmospheres different parts of the globe are. Anyway, I've run out of time. See you in the next video.