Heated gas pushing a mercury column out of a cylinder

[highschoboy004]

Homework Statement

A vertical cylinder contains a definite amount of ideal gas, sealed on top with a massless and frictionless piston. On top of the piston there is a mercury column of height h = 76cm. Let atmospheric pressure be p0 = 100000 N/m2, cross-section area of piston is S = 20cm2, initially the piston is 26 cm above the bottom of the cylinder. We then heat the gas so that it expands and push all of the mercury out of the cylinder. Calculate the amount of work done by the gas from the beginning until all of the mercury has been pushed out.

Relevant Equations

nRT=pv; even integrals...

Firstly I figure out that there is a point where the temperature of the gas is the hottest and beyond that no more amount of heat is necessary, descibed as following inequation:

 

[jbriggs444]

then heat the gas so that it expands and push all of the mercury out of the cylinder. Calculate the amount of work done by the gas from the beginning until all of the mercury has been pushed out.
[...]
Firstly I figure out that there is a point where the temperature of the gas is the hottest and beyond that no more amount of heat is necessary, descibed as following inequation:

You are not asked for the amount of heat required. Only for the work done.

I do not think that you are expected to solve for the work expended in increasing the kinetic energy of the mercury as it spurts out at increasing speed due to the decreasing weight of the remaining column of mercury.

One could, perhaps, consider removing heat from the gas as needed so that the mercury is expelled at a low rate throughout the scenario.

 

[highschoboy004]

Sorry, I was interrupted. I agree with supposing that the mercury be expelled at a low rate, it help simplify the problem. Initially, I divide the process into two phase:
The first one being heating up the gas to the hottest temperature: a certain amount of mercury has been expelled. Can I suppose that the work is equal to the weight of expelled mercury?
The second phase is after the gas has been heated up to hottest temp, it pushes the remaining mercury out without having to supply more heat, is this process adiabatic or isothermal?

 

[scottdave]

I think there is not enough information to consider change in kinetic energy. So we could assume it is "slow enough". You could think of an imaginary level trough at the top, long enough to hold all of the mercury. Once mercury is in this trough, no more potential energy is added. One question - is the 76 cm height starting at the piston, or from the bottom of the tube?

 

[Steve4Physics]

As highlighted in the other posts, the information in the question is ambiguous/unclear. Is that the exact and complete question? Is there a diagram that comes with the question?

We are not told the initial distance between the top of the mercury and the top of the tube. If necessary, assume this distance is zero (and explicitly state this assumption in your answer).

We are not told the thickness of the piston, so presumably it is not needed.

Can I suppose that the work is equal to the weight of expelled mercury?

No. Work (measured in J) and weight (measured in N) can't be equated.

 

[highschoboy004]

Sorry, I forgot to supply the diagram:

I think I should add that the cylinder-piston system is insulated and heat is supplied from within the chamber.

 

[highschoboy004]

I think there is not enough information to consider change in kinetic energy. So we could assume it is "slow enough". You could think of an imaginary level trough at the top, long enough to hold all of the mercury. Once mercury is in this trough, no more potential energy is added. One question - is the 76 cm height starting at the piston, or from the bottom of the tube?

If this trough can hold all of the mercury, then I think that during the first phase, mercury is pushed up at a constant and slow rate, therefore the amount of work done by the gas is equal to the weight of the mercury times height displacement?

 

[Steve4Physics]

If this trough can hold all of the mercury, then I think that during the first phase,

Is there more than 1 phase?

mercury is pushed up at a constant and slow rate, therefore the amount of work done by the gas is equal to the weight of the mercury times height displacement?

By "height displacement" do you mean the 76cm?

If every part of the mercury were very slowly raised by 76cm (e.g. if you lifted the whole apparatus) then the work done on the mercury would be equal to "the weight of the mercury times height displacement".

But not every part of the mercury is being raised by 76cm. In fact the only part raised by 76cm is the part at the bottom. The part of the mercury initially near the top, gets raised by only a small amount.

Hint: it helps to think in terms of centre-of-gravity and gravitational potential energy.

Also, don't forget the effect of the atmosphere. If the 'experiment' were done in a vacuum (atmospheric pressure = 0) would that affect the work done by the gas?

 

[Lnewqban]

Initially, I divide the process into two phase:
The first one being heating up the gas to the hottest temperature: a certain amount of mercury has been expelled.

What is your reasoning to believe that there is that hottest temperature?

The second phase is after the gas has been heated up to hottest temp, it pushes the remaining mercury out without having to supply more heat,

What would make the gas do additional work on the remaining mass of mercury after reaching that temperature?

is this process adiabatic or isothermal?

May I ask what subject is this problem related to?
If I assume that you are referring to the process of simultaneously heating and expanding that gas, then the adiabatic possibility is out of question, as energy is flowing into it in form of heat.

Please, see:
https://en.wikipedia.org/wiki/Adiabatic_process

 

[Steve4Physics]

The first one being heating up the gas to the hottest temperature: a certain amount of mercury has been expelled.

There is no 'hottest temperature' (unless you haven't given us the full question?). You can make the gas as hot as you like. As the gas temperature increases, its volume increases. You can push all the mercury out by making the gas hot enough.

You are asked to “calculate the amount of work done by the gas from the beginning until all of the mercury has been pushed out”.

Think about what the work actually does. From conservation of energy, we know that the work done by the gas will be the sum of:

a) the gravitational potential energy gained by the mercury;

b) the energy needed to ‘push back’ the atmosphere to make space for the increased gas volume.

(We’re assuming the process is slow enough so that the amount of kinetic energy gained by the mercury is negligible.)

We need to calculate and add a) and b). Job done. Exactly how the gas is heated is irrelevant.

 

[highschoboy004]

Think about what the work actually does. From conservation of energy, we know that the work done by the gas will be the sum of:

a) the gravitational potential energy gained by the mercury;

b) the energy needed to ‘push back’ the atmosphere to make space for the increased gas volume.

(We’re assuming the process is slow enough so that the amount of kinetic energy gained by the mercury is negligible.)

We need to calculate and add a) and b). Job done. Exactly how the gas is heated is irrelevant.

I agree, maybe I was just complicating the problems from the start by adding those phases and hottest temp. Thank you all a lot.