- #1

plain stupid

- 18

- 1

- 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!

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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