Pressure Gauge: Barometer: Study with Video Lessons, Practice Problems & Examples

Pressure gauges, like barometers, measure atmospheric pressure using the height of a liquid column. The fundamental equation is $P_b=P_t+\rho gh$, where P_b is the pressure at the bottom, P_t is the pressure at the top (zero in a vacuum), ρ is the liquid density, g is the acceleration due to gravity, and h is the height of the liquid column. Mercury is commonly used due to its high density, allowing for a more compact design. Atmospheric pressure varies with altitude, making barometers essential for measuring local conditions.

Pressure Gauges: Barometer

Hey everyone. In this video, we're going to start talking about devices used to measure pressure, which are called pressure gauges and specifically we're going to be talking about the barometer. So let's go ahead and jump right in. A pressure gauge uses height differences to calculate pressure. Whenever you have height differences and densities of materials like liquids, you're going to use the Pbottom=Ptop+ρgh. Nothing new here. Let me just show you how the barometer works. Usually, it kind of looks like this: you have some container and that container is filled with usually mercury or some other heavy liquid like this. And you have a column which is really just a glass tube like this. The idea here is that on one side of it, the bottom of the container is actually exposed or is open to the ambient air around it. The idea here is that the sort of molecules of air that you have in the ambient atmosphere are going to push down on this liquid and they're going to force it up this column or up the glass tube. Now on the top side of it, you have a little tiny cap where there's a vacuum. So, in other words, what happens is you have no pressure over here. So basically, the atmosphere forces it down and it sort of goes up the tube and then it reaches some kind of a balance when you've equalized the pressure. The height of this column is going to be really important here. So remember, when we have pressures and heights, we're going to go ahead and use our Pbottom=Ptop+ρgh. Now the Pbottom is going to be right over here, and because you're touching air, this Pbottom is really just going to be Pair. So this Pbottom is just Pair like this. And the pressure at the top, is going to be right over here. This is going to be Ptop. Now because this touches the vacuum, what do you think this pressure turns out to be? This one right over here. It's basically going to be 0. Again, whenever you're touching zero pressure or a vacuum, your Ptop is going to go away. So that's going to be 0. So what this equation sort of works out to is that the air pressure is equal just to ρgh. And this is the equation that you're going to use, the sort of simplified equation for barometers. Basically, what this equation tells us is if you want to calculate the air pressure, which is really just going to be the atmospheric pressure, then all you need is you need the density, and you're going to use the density of some kind of a known material. So you're going to use a known density like this. And if you have the g, which is just a constant depending on where you are, then all you have to do is just measure the height of the column itself using a ruler or sometimes these things have actually built-in notches or whatever to show you how much is in the container. So you measure this, and basically, what this does is this is a device that is used to measure the air pressure around it. Now you might be wondering, isn't that always just going to be 1 atmosphere? And the answer is not really. It's 1 atmosphere if you're at sea level, that's standard atmospheric pressure, But if you take this barometer and you go up to a mountain where the air pressure is different, that's what you use this for. You use this h to basically measure what the ambient air pressure is around. That's why it's called a pressure gauge. The last thing I want to mention here is that the classic barometer, invented by Torricelli, uses mercury because it's 13.6 times denser than water. So here's what's going on here. If you have water, in which the density is a thousand, then basically what happens here is if you have Patmosphere=1atm, then by this equation, what you would work out is that the hcolumn would actually have to be 10.3 meters high. So basically what happens is if you set this equal to 1 atmosphere and you use the density of water, which is a 1,000, if you solve for that, then you have these three variables that are known. Your h, the height of the column would have to be 10.3 meters high. So one way to make this a little bit more compact is you basically want to increase the density to the highest that you possibly can so that the height can go down. If you use mercury, which is one of the densest liquids that we know of, that's sort of liquid at room temperature, then what happens is this height goes down and it's like roughly about a meter or a little bit less than a meter, and it makes this instrument a lot more compact. So that's why historically, we use mercury in these things. It's because it's a really, really, really dense liquid. So you don't need that much of it basically to push against the air pressure that's on the outside. So it's kind of like a more conceptual point here, but if you plug in these numbers, basically what you'll get is you'll get h. So you can go ahead and try that for yourself. That's basically how barometers work. Let's go ahead and take a look at some examples, so I can show you some questions that might be asked.

A classic barometer (shown below) is built with a 1.0-m tall glass tube and filled with mercury (13,600 kg/m³). Calculate the atmospheric pressure, in ATM, surrounding the barometer if the column of liquid is 76 cm high. (Use g=9.8 m/s².)

• A. 0 Pa
• B. 1.01 Pa
• C. 1.01×10⁵ Pa
• D. 1.01×10⁷ Pa

Here’s what students ask on this topic:

How does a barometer measure atmospheric pressure?

A barometer measures atmospheric pressure by using the height of a liquid column, typically mercury, in a glass tube. The fundamental equation used is:

$P{b}_{}=P_{t_{}}+\rho gh$

where $P{b}_{}$ is the pressure at the bottom, $P_{t_{}}$ is the pressure at the top (zero in a vacuum), $\rho$ is the liquid density, $g$ is the acceleration due to gravity, and $h$ is the height of the liquid column. The atmospheric pressure forces the liquid up the tube, and the height of the liquid column is used to calculate the pressure.

Why is mercury used in barometers instead of water?

Mercury is used in barometers instead of water because it has a much higher density. Mercury's density is approximately 13.6 times that of water. This high density allows the barometer to be more compact. For example, if water were used, the height of the column would need to be about 10.3 meters to measure standard atmospheric pressure. In contrast, with mercury, the column height is only about 0.76 meters. This makes the instrument more practical and easier to use.

What is the equation used to calculate atmospheric pressure with a barometer?

The equation used to calculate atmospheric pressure with a barometer is:

$P{b}_{}=P_{t_{}}+\rho gh$

In a barometer, the pressure at the top $P_{t_{}}$ is zero because it is a vacuum. Therefore, the equation simplifies to:

$P{a}_{}=\rho gh$

where $P{a}_{}$ is the atmospheric pressure, $\rho$ is the density of the liquid, $g$ is the acceleration due to gravity, and $h$ is the height of the liquid column.

How does altitude affect the reading of a barometer?

Altitude affects the reading of a barometer because atmospheric pressure decreases with increasing altitude. At higher altitudes, there is less air above the barometer, resulting in lower atmospheric pressure. This causes the height of the liquid column in the barometer to decrease. Conversely, at lower altitudes, the atmospheric pressure is higher, causing the liquid column to rise. Therefore, barometers are essential for measuring local atmospheric pressure, which can vary significantly with changes in altitude.

How do you calculate the density of an unknown liquid using a barometer?

To calculate the density of an unknown liquid using a barometer, you can rearrange the barometer equation:

$P{a}_{}=\rho gh$

to solve for density $\rho$:

$\rho =\frac{P}{a_{}}$

Here, $P{a}_{}$ is the atmospheric pressure, $g$ is the acceleration due to gravity, and $h$ is the height of the liquid column. By measuring the height of the liquid column and knowing the atmospheric pressure and gravitational constant, you can calculate the density of the unknown liquid.