Mercury barometer with a small amount of air

In summary, the pressure in a closed space is given by the pressure of the atmospheric pressure plus the pressure of the column of mercury. If the barometer is faulty, the pressure of the column of mercury will be measured rather than the actual pressure.
  • #1
plain stupid
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Homework Statement
We have a faulty mercury barometer with a small amount of air trapped at the very top, above the mercury. For an atmospheric pressure of 768 mmHg, our barometer shows 761 mmHg, and for an outside pressure of 740 mmHg, it shows 736 mmHg. What's the length of the barometer tube?
Relevant Equations
p = ρgh
$$\rho_{Hg} gh_{actual} = \rho_{Hg} gh_{measured} + \rho_{air} gh_{air}$$

Note: by "actual", I mean "theoretical", i.e. what the barometer would measure were there no air inside it. By "measured" I mean "as measured by the faulty barometer, i.e. with some air introduced".

I believe this assumption should hold, because those are the only things in that tube. The pressures should be in an equilibrium, meaning the pressure from the outside (actual pressure) should correspond to the pressure exerted by the column of mercury + the tiny bit of air. I'm assuming here that because air is a gas (or multiple gasses) that it should spread out evenly, so its "height" should correspond to the height of the remaining part of the tube. I don't think I should assume its density is known, because it should change (since its mass remains the same, but its volume varies).

I can cross out the gravitational constant from both sides, and replace the density of air with its mass over its volume, assuming the tube's cross-sectional area is constant throughout:

$$\rho_{Hg} h_{actual} = \rho_{Hg} h_{measured} + \frac{m_{air}}{A_{tube} h_{air}} h_{air}$$

simplifying yields

$$\rho_{Hg} \left(h_{actual} - h_{measured} \right) = \frac{m_{air}}{A_{tube}}$$

Now, I'm pretty sure this ratio of air mass and barometer cross-sectional area should be constant, and so should be the density of mercury, so I'm not at all sure what I've arrived at here. What am I even looking for here? I've been looking at this for a couple of days now, but am not sure what I'm missing.

Also, it's not clear what's exactly given (known) aside from the obvious, but I think we could assume the density of mercury could be read from a table, and even the average density of air, if that could be of help.

I'm not really sure how to proceed from here, even though the problem statement suggests there should somehow be two equations with two unknowns...

Any hints? Thank you!
 
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  • #2
Your first equation is wrong. Air pressure in a closed space isn't just a question of its weight, e.g. what would happen if the air were heated?
Also, your use of "actual" is a bit misleading. The measured height is the actual height. You mean the height it would be if there were no air. Maybe "theoretical"?
 
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  • #3
haruspex said:
Your first equation is wrong. Air pressure in a closed space isn't just a question of its weight, e.g. what would happen if the air were heated?
Also, your use of "actual" is a bit misleading. The measured height is the actual height. You mean the height it would be if there were no air. Maybe "theoretical"?
I don't think there are any indications that the air is heated or that the temperature changes (I assumed an equilibrium state). Do you think this is actually pertinent considering how relatively basic this problem is?

Yeah, by "actual" I did mean "theoretical", but if the barometer is faulty, then we can reasonably talk about an actual pressure (in terms of the millimeters of mercury) vs. a measured one (on the faulty barometer). I'll try to edit my post to clarify what I meant regardless.
 
  • #4
Should I try to play with my setup and get the result? I.e. am I on the right way (perhaps extracting that height of air and cancelling those terms in particular wasn't the right idea, but I should continue playing with something similar)?
 
  • #5
plain stupid said:
I don't think there are any indications that the air is heated
You are missing the point. The way you have calculated the pressure the air exerts is as though the tube continues up into space and is open at the top. It would be minute.
Pressure in a closed volume is given by PV=nRT.
 
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  • #6
Once the vacuum is broken, the column may reach a very low level.
If we imagine the air bubble as being big enough, it could completely fill up the space within the tube while being at atmospheric pressure, for which condition the height of the column of mercury would be zero.

If we then apply a vacuum pump at the top end of the vertical tube, the level of the column would increasely go up as mass of air is being removed from the cavity and its absolute pressure decreases.

When a high degree of vacuum is reached inside the top cavity of the tube, for an atmospheric pressure of 768 mm Hg, our barometer would show 768 mm above the open free surface of mercury.
 
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  • #7
haruspex said:
You are missing the point. The way you have calculated the pressure the air exerts is as though the tube continues up into space and is open at the top. It would be minute.
Pressure in a closed volume is given by PV=nRT.

Okay, I see. Thank you very much! I'm sorry for not realizing the hint in your first reply, and having you spell it out for me.

Taking that into account, I get the right solution. I'll show a few steps below just in case someone else is interested.

$$\rho_{Hg}gh_{theoretical} = \rho_{Hg} gh_{measured} + \frac{nRT}{V_{air}}$$

$$\rho_{Hg}g(h_{theoretical}-h_{measured}) = \frac{1}{h_{air}} \frac{nRT}{A_{tube}}$$

Simplifying (##C## is a constant and doesn't change, assuming the temperature of air is the same for both cases, ##R## is a constant, and the number of particles of air is conserved)...

$$\rho_{Hg} \cdot g \cdot (h_{theoretical}-h_{measured}) \cdot h_{air} = C$$

The above has to be evaluated for the two cases (measurements), and from those we get the ratio between two heights of air columns.

$$\frac{h_{air,1}}{h_{air,2}} = ratio$$

There's also a second equation we can easily come up with, since the height of the whole tube doesn't change:

$$h_{measured,1} + h_{air,1} = h_{measured,2} + h_{air,2}$$

These two equations are enough to get the solution.
 
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  • #8
plain stupid said:
Okay, I see. Thank you very much! I'm sorry for not realizing the hint in your first reply, and having you spell it out for me.

Taking that into account, I get the right solution. I'll show a few steps below just in case someone else is interested.

$$\rho_{Hg}gh_{theoretical} = \rho_{Hg} gh_{measured} + \frac{nRT}{V_{air}}$$

$$\rho_{Hg}g(h_{theoretical}-h_{measured}) = \frac{1}{h_{air}} \frac{nRT}{A_{tube}}$$

Simplifying (##C## is a constant and doesn't change, assuming the temperature of air is the same for both cases, ##R## is a constant, and the number of particles of air is conserved)...

$$\rho_{Hg} \cdot g \cdot (h_{theoretical}-h_{measured}) \cdot h_{air} = C$$

The above has to be evaluated for the two cases (measurements), and from those we get the ratio between two heights of air columns.

$$h_{air,1} \cdot h_{air,2} = ratio$$

There's also a second equation we can easily come up with, since the height of the whole tube doesn't change:

$$h_{measured,1} + h_{air,1} = h_{measured,2} + h_{air,2}$$

These two equations are enough to get the solution.
Very good. Thanks for posting your solution.
 

Related to Mercury barometer with a small amount of air

1. How does a mercury barometer with a small amount of air work?

A mercury barometer with a small amount of air works by measuring the atmospheric pressure exerted on the mercury in the tube. As the air pressure increases, it pushes down on the mercury, causing it to rise in the tube. The height of the mercury column can then be used to determine the air pressure.

2. What is the purpose of the small amount of air in the mercury barometer?

The small amount of air in the mercury barometer serves as a cushion for the mercury column. It helps to prevent the mercury from fluctuating too quickly and provides a more stable reading of the atmospheric pressure.

3. How accurate is a mercury barometer with a small amount of air?

A mercury barometer with a small amount of air is highly accurate in measuring atmospheric pressure. It is able to detect even small changes in pressure and can provide precise readings.

4. What are the advantages of using a mercury barometer with a small amount of air?

One advantage of using a mercury barometer with a small amount of air is its high accuracy in measuring atmospheric pressure. It is also a simple and relatively inexpensive instrument to use. Additionally, the use of mercury makes it resistant to changes in temperature, ensuring consistent readings.

5. Are there any safety concerns when using a mercury barometer with a small amount of air?

Yes, there are safety concerns when using a mercury barometer with a small amount of air. Mercury is a toxic substance and should be handled with caution. It is important to properly dispose of any broken or unused mercury barometers to prevent harm to both humans and the environment.

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