# A glass tube filled with mercury

• ValeForce46
In summary, the force balance equation for a mercury column is as follows:-The weight of the mercury-The atmospheric pressure-The buoyant force-The pressure of the air in the closed section
ValeForce46
Homework Statement
A glass tube with a section of ##A= 10^{-4} m^2## and length ##l=1.14 m## is closed at one end and open at the other end. Inside the tube there's a mercury's column ##l_2=0.3 m##. When the tube is placed horizontally, the columns of air on the right and left of the column of mercury have the same length ##l_1=l_3=0.42 m##. The tube is now placed vertically with the open end to the top. Determine the length of the column of air ##l_1'## of the closed end. How long the length ##l_1''## of air would be if the tube was closed before it was placed vertically? Assume temperature constant during every process, air is a perfect gas and ##ρ_{Hg}=13.6 ρ_{air}##.
Relevant Equations
Boyle's law: ##P*V=constant##

I guess I have to find out something when the tube is on the horizontal.
What force should I consider? Is the pressure of the air inside still ##p_{atm}##?
When it is on the vertical, the equation is something like this?
##P_{Hg}+p_{atm}*A=p_{atm}*A+ρ_{air}*A*(l_1-l_1')*g## with ##P_{Hg}=ρ_{Hg}*A*l_2*g## weight of the column of mercury.
Or maybe I shouldn't face this problem considering the force...?

Please try to justify your equation. (I fail to see how the weight of air in the closed section comes into it.)

That was just an attempt, I don't even think that is right.
I considered as downward forces the weight of the column of mercury ##P_{Hg}## and the atmospheric pressure times the section ##p_{atm}*A##.
I considered as upward forces the bouyant force ##ρ_{air}∗A∗(l1−l′1)∗g ## and the pressure of the air in the closed section. Is it right to consider the bouyant force? What about the pressure of the closed section?
haruspex said:
I fail to see how the weight of air in the closed section comes into it.
Really, I didn't even considered it...

ValeForce46 said:
Is it right to consider the bouyant force?
no. (This is where you seemed to be doing something with weight of air.)
ValeForce46 said:
What about the pressure of the closed section?
Indeed, what about it? What law tells you how the shrinkage in its length leads to greater pressure?

haruspex said:
What law tells you how the shrinkage in its length leads to greater pressure?
Boyle's law. My main dubt is: can I consider the initial pressure of the closed section ##p_{atm}##? Why?
In such case, it would be ##p_{atm}*A*l_1=p_{f}*A*(l_1-l_1')##
But I still need something else. Where should I look for it? Do I have to adjust the equation of forces (when the tube is on the vertical)?

ValeForce46 said:
can I consider the initial pressure of the closed section ##p_{atm}##? Why?
Yes, because while horizontal any difference from atmospheric pressure would have caused the mercury to move.
ValeForce46 said:
In such case, it would be ##p_{atm}*A*l_1=p_{f}*A*(l_1-l_1')##
Doesn’t look right to me.
Always worth checking trivial special cases. What does your equation give if l1 doesn't change?

Last edited:
haruspex said:
Always worth checking trivial special cases. What does your equation give if l1 doesn't change?
It gives ##p_{atm}=p_f##. Why is this wrong? If ##l_1'## doesn't change, the air would take the same volume.
Anyways, how should I apply Boyle's law?

EDIT:
well, now I got it: ##p_{atm}*A*l_1=p_f*A*l_1'##.
What is the next step?

Last edited:
ValeForce46 said:
It gives ##p_{atm}=p_f##. Why is this wrong? If ##l_1'## doesn't change, the air would take the same volume.
Anyways, how should I apply Boyle's law?

EDIT:
well, now I got it: ##p_{atm}*A*l_1=p_f*A*l_1'##.
What is the next step?
Write the force balance equation for the mercury using pf.

Is this right?
##ρ_{Hg}*A*l_2*g+p_{atm}*A=p_f*A##
The first term is the weight of mercury. (On the homework statement I did a mistake ##ρ_{Hg}=13.6*ρ_{H2O}##)
If yes, then ##l_1'=0.3 m##.

ValeForce46 said:
Is this right?
##ρ_{Hg}*A*l_2*g+p_{atm}*A=p_f*A##
The first term is the weight of mercury. (On the homework statement I did a mistake ##ρ_{Hg}=13.6*ρ_{H2O}##)
If yes, then ##l_1'=0.3 m##.
Correct.
The easiest way is to start with atmospheric pressure as 0.76m of Hg. The 0.3m in the tube is going to increase the pressure to 1.06m, i.e. in the ratio 106/76. So the volume of air is reduced in that ratio: (76/106)0.42=0.3.

ValeForce46
For the second part (the tube is completely closed before it is on the vertical), do I have to repeat the reasoning I did before?
Now, when the air under the mercury is compressed, the air above the mercury expands.
Boyle's Law:
##p_{atm}*A*l_1=p_1*A*(l_1-x)## for the air under the mercury
##p_{atm}*A*l_3=p_3*A*(l_3+x)## for the air above the mercury
and by the force balance equation:
##ρ_{Hg}∗A∗l_2∗g+p_3*A=p_1*A##
3 equation and 3 unknows ##p_1##, ##p_3## and ## x##. Then I can find ##l_1''=l_1-x##
Let me know if this is right please.
Just one last thing, why can I consider the length of mercury constant? I did it for the first part of the problem too.

ValeForce46 said:
Let me know if this is right please.
Looks good.
ValeForce46 said:
why can I consider the length of mercury constant?
Why would it change (by more than a minuscule amount)?

ValeForce46
...Yes, actually there's no reason to change, in this problem. Thanks a lot

ValeForce46 said:
...Yes, actually there's no reason to change, in this problem. Thanks a lot
Yes, Mercury is highly incompressible. Jupiter however...
(sorry!)

collinsmark and ValeForce46

## 1. What is a glass tube filled with mercury?

A glass tube filled with mercury is a cylindrical container made of glass and filled with the liquid metal mercury. It is often used in laboratory settings for experiments and measurements.

## 2. How is a glass tube filled with mercury used in science?

In science, a glass tube filled with mercury is commonly used in experiments involving pressure, temperature, and density measurements. It can also be used as a conductive medium in electrical circuits.

## 3. How does a glass tube filled with mercury work?

The glass tube filled with mercury works by utilizing the properties of the liquid metal mercury. Mercury is a good conductor of electricity and its density changes with temperature, making it useful for various scientific measurements.

## 4. Is a glass tube filled with mercury safe?

Mercury is a toxic substance and should be handled with caution. However, a glass tube filled with mercury is relatively safe as long as it is not broken or heated. Proper safety precautions should always be taken when working with mercury.

## 5. Can a glass tube filled with mercury be recycled?

Yes, a glass tube filled with mercury can be recycled. However, it must be done by a professional recycling company due to the toxic nature of mercury. It is important to properly dispose of mercury-containing items to avoid environmental contamination.

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