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Measuring atmospheric pressure

 One way of measuring atmospheric pressure is by means of Boyles's law. However, some concepts need to be introduced first.

Absolute, atmospheric and manometric pressure

These three types of pressure can be related as,

$p_{abs}=p_{man}+p_{atm}$        Eq. (01)

where $p_{abs}$ stands for absolute pressure, $p_{man}$ for manometric pressure and $p_{atm}$ for atmospheric pressure. All these types of pressure can be schematically represented in U-tube as follows,

Fig. 01 At point 1 and 2 the pressure is the same. At both sides manometric pressure is zero at the interface but at this very same position atmospheric pressure is different from zero.

Understanding hydrostatic pressure in a U-tube

Hydrostatic pressure is usually presented as a single pressure but in fact it is a difference pressure. Here is its mathematical representation,

$\Delta p=p_{bottom}-p_{top}=Hg\rho$


and it can be graphically represented as follows,

Fig. 02 Sketch to understand hydrostatic pressure in a tank

As you may recall from the previous section, the manometric pressure at the top is zero! (because the it is a tank open to the air). Notice that the hydrostatic pressure is only related to the pressure exerted vertically only. That is why you only use the fluid height $H$.

$p_{top}=0$

However, the pressure at the bottom is not zero and is the one usually called hydrostatic pressure. Then,


$\Delta p=p_{bottom}=Hg\rho$

Now, how do you interpret the hydrostatic pressure in a U-tube as the one shown in Fig. 01?

Since, both sides in the U-tube are open to the air, it does not make sense to talk about hydrostatic pressure since it is zero at both sides. 

However, hydrostatic pressure comes into play when one of the inlets is closed and pressurized. For example, as in the figure below

Fig. 03 Recognizing the hidrostatic pressure changes in a U-tube with one of the sides sealed.

As you can see in the figure above the gas in the enclosure becomes compressed as more water is poored into the U-tube. The pressure at the open side is still zero but the pressure at the interface in the sealed side becomes different from zero. 

Equation (01) and the sketch in Fig. 03 can be used along with the Boyle's law to determine the atmospheric pressure.

Using Boyle's law to estimate the atmospheric pressure

Boyle's law is tipically written as,

$p_1V_1=p_2V_2$        Eq. (02)

where $p$ is for pressure and $V$ for volume. Subscripts 1 and 2 refer to two different states in a system featured by changes in pressure and volume.

Let us take from Fig. 03 these two different states where we can easily identify changes in pressure and volume.

Fig. 04 Two sketches showing the two states expressed in Boyle's law. You should notice that the volume changing is that of the gas enclosed in the right leg from $V_1$ to $V_2$.

However, the qestion remains: how do we estimate the atmospheric pressure from the explanation above?

If we consider the aboslute pressure at the interface in the sealed leag in state 1  along with the absolute pressure at the interface in the sealed leg in state 1 (Fig. 04) the next two expressions follow. For state 1,

$p_{abs1}=p_{man1}+p_{atm}$

$p_{abs1}=p_{atm}$        Eq. (03)

because $p_{man1}=0$ at the interface since it is at the same height that the open leg. For state 2,

$p_{abs2}=p_{man2}+p_{atm}$        Eq. (04)

Notice that $p_{man2}$ is different from zero and unknown!

Next, using Boyle's law and volumes $V_1$ and $V_2$, as shown in Fig. 04, we obtain,

$p_{abs1}V_1=p_{abs2}V_2$

$p_{atm}V_1=\left( p_{man2}+p_{atm}\right) V_2$        Eq. (05)

At this point, $p_{man2}$ can be easily determined from the hydrostatic pressure formula stated previously in Fig. 03. All you need is to measure the height $H$ and find the density of the liquid at the corresponding temperature.

Then, in Eq. (05) the atmospheric pressure $p_{atm}$ is the only unknown. This can be easily isolated to be,

$p_{atm}=p_{man2}\dfrac{V_2}{V_1-V_2}$        Eq. (06)

Volumes $V_1$ and $V_2$ easily determined with the formula for volume of a cylinder. Since you know the diameter of the tube all you need to do is to measure the distance from the interface to the seal (in the sealed leg of course).

Since this experiment may involve different sources of error, it is recommended to repeat the experiment several times to reduce the error or at least to give a margin of it.

An experimental aid

Here is a template you can use to record your results. This is an automated sheet built to work in SI units. You will need to load some data as the fluid properties and your city atmospheric pressure.

Boyle's law experiment template for recording data

Please, make a copy of  the file and modify it as you need. I hope this could be useful.

Any question? Write in the comments and I shall try to help.

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