Thermodynamics Questions

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Sowieso
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Thermodynamics Questions

Postby Sowieso » Sun Jan 23, 2011 11:32 pm UTC

Couple of HW questions I could use tips on:

A helium balloon is used to lift a load of 119 N. The weight of the envelope of the balloon is 45.5 N and the volume of the helium when the balloon is fully inflated is 29.1 m3. The temperature of the air is 0°C and the atmospheric pressure is 1.00 atm. The balloon is inflated with a sufficient amount of helium gas that the net upward force on the balloon and its load is 27.0 N. Neglect any effects due to the changes of temperature as the altitude changes.

(a) How many moles of helium gas are contained in the balloon?
(b) At what altitude will the balloon be fully inflated?
(c) Does the balloon ever reach the altitude at which it is fully inflated?
(d) If the answer to Part (c) is "Yes," what is the maximum altitude attained by the balloon? (If the answer to Part (c) is "No", enter no.)

Alright so for part (a)
Bouyant force [F(b)] - envelope weight - load = 27.0 N
F(b) = 191.5 N
F(b) = p(density of air)*V*g = 1.29 kg/m^3 * V * 9.81 m/s^2
V = F(b)/(p*g)
V = 15.1 m^3 *(1000L/1 m^3) = 15100 L

PV=nRT
n = PV/RT = (1 atm)*(15100 L) / (.08206)*(273.15 K)
n = 674 mol
but this isn't right. Why?






Second question:
A cylinder is filled with 0.10 mol of an ideal gas at standard temperature and pressure, and a 1.4-kg piston seals the gas in the cylinder (see figure) with a frictionless seal. The trapped column of gas is 2.5-m high. The piston and cylinder are surrounded by air, also at standard temperature and pressure. The piston is released from rest and starts to fall. The motion of the piston ceases after the oscillations stop with the piston and the trapped air in thermal equilibrium with the surrounding air.

(a) Find the height of the gas column.
(b) Suppose that the piston is pushed down below its equilibrium position by a small amount and then released. Assuming that the temperature of the gas remains constant, find the frequency of vibration of the piston.

For (a), I would have thought use ideal gas law to find volume, but you don't know anything about the dimensions of the cylinder so volume wouldn't help me find height. I'm really not sure where to start with this one. Hints are appreciated.

Thanks for the help!

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jmorgan3
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Re: Thermodynamics Questions

Postby jmorgan3 » Sun Jan 23, 2011 11:44 pm UTC

First Problem:
You're screwing up either here
Sowieso wrote:F(b) = p(density of air)*V*g

or here (pick only one):
Sowieso wrote:Bouyant force [F(b)] - envelope weight - load = 27.0 N

depending on how you want to define bouyant force.

Second problem:
Just write the volume in each case as cross-sectional area (A) times the relevant height. Work through the problem symbolically, and the area should cancel out.
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Sowieso
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Re: Thermodynamics Questions

Postby Sowieso » Mon Jan 24, 2011 12:30 am UTC

jmorgan3 wrote:First Problem:
You're screwing up either here
Sowieso wrote:F(b) = p(density of air)*V*g

or here (pick only one):
Sowieso wrote:Bouyant force [F(b)] - envelope weight - load = 27.0 N

depending on how you want to define bouyant force.

Hm. Perhaps I misunderstand the concept of bouyant force; it's not the displaced air acting upwards on the balloon?

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jmorgan3
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Re: Thermodynamics Questions

Postby jmorgan3 » Mon Jan 24, 2011 12:55 am UTC

If you define the buoyant force as the integrated pressure force over the surface of the object, then the buoyant force is equal in magnitude to the weight of the displaced fluid. In that case, the problem is here:
Sowieso wrote:Bouyant force [F(b)] - envelope weight - load = 27.0 N

There's a weight you're forgetting.
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Sowieso
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Re: Thermodynamics Questions

Postby Sowieso » Mon Jan 24, 2011 1:07 am UTC

jmorgan3 wrote:If you define the buoyant force as the integrated pressure force over the surface of the object, then the buoyant force is equal in magnitude to the weight of the displaced fluid. In that case, the problem is here:
Sowieso wrote:Bouyant force [F(b)] - envelope weight - load = 27.0 N

There's a weight you're forgetting.


Bouyant force of the air against the weight of the helium, yea! Thanks a bunch. From once I got the number of moles I was able to find the pressure on the balloon when the volume was maximized, and from the pressure attain the altitude. Now I just have to figure out how high it WILL go.
And I have to figure out the oscillations part on number 2; any tips?

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ChicagoPianoTuner
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Re: Thermodynamics Questions

Postby ChicagoPianoTuner » Mon Jan 24, 2011 11:28 pm UTC

jmorgan3 wrote:Just write the volume in each case as cross-sectional area (A) times the relevant height. Work through the problem symbolically, and the area should cancel out.

Pretty sure the area doesn't cancel out, but you can indeed solve for it given the initial conditions: you know the moles of gas at STP, height of gas (before piston is dropped). That's enough to find the area.

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ChicagoPianoTuner
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Re: Thermodynamics Questions

Postby ChicagoPianoTuner » Mon Jan 24, 2011 11:35 pm UTC

To find the frequency, try to use Newton's second law, where the force due to the pressure of the gas in the cylinder depends on how far it is depressed. When you do, you'll be left with a familiar differential equation that you can solve to find the frequency.

Edited to say that this method should work, but I'm having trouble getting it to work out - the differential equation is messy, and maybe not solvable? I haven't tried to solve any differential equations for quite some time.
Last edited by ChicagoPianoTuner on Mon Jan 24, 2011 11:54 pm UTC, edited 1 time in total.

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jmorgan3
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Re: Thermodynamics Questions

Postby jmorgan3 » Mon Jan 24, 2011 11:42 pm UTC

ChicagoPianoTuner wrote:
jmorgan3 wrote:Just write the volume in each case as cross-sectional area (A) times the relevant height. Work through the problem symbolically, and the area should cancel out.

Pretty sure the area doesn't cancel out, but you can indeed solve for it given the initial conditions: you know the moles of gas at STP, height of gas (before piston is dropped). That's enough to find the area.

Correct. I misread the problem initially.
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ChicagoPianoTuner
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Re: Thermodynamics Questions

Postby ChicagoPianoTuner » Tue Jan 25, 2011 4:09 am UTC

Just figured it out. Need to use PV^gamma=constant. Taylor expand and then use N2L. Post again if you're still having trouble. Tricky problem!

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jmorgan3
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Re: Thermodynamics Questions

Postby jmorgan3 » Tue Jan 25, 2011 5:33 am UTC

ChicagoPianoTuner wrote:Just figured it out. Need to use PV^gamma=constant. Taylor expand and then use N2L. Post again if you're still having trouble. Tricky problem!

That would be isentropic; the problem says isothermal.
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ChicagoPianoTuner
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Re: Thermodynamics Questions

Postby ChicagoPianoTuner » Tue Jan 25, 2011 1:42 pm UTC

PV^gamma=constant holds for adiabatic compression or expansion of ideal gases. Adiabatic means no energy enters or leaves the system by heat (Q=0). At least I'm pretty sure that's accurate. I'm also pretty sure it describes this problem.

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jmorgan3
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Re: Thermodynamics Questions

Postby jmorgan3 » Wed Jan 26, 2011 12:54 am UTC

The problem says "Assuming that the temperature of the gas remains constant." When you're doing work on a fluid, adiabatic != isothermal.
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ChicagoPianoTuner
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Re: Thermodynamics Questions

Postby ChicagoPianoTuner » Wed Jan 26, 2011 2:13 pm UTC

I agree with you that the problem does not explicitly state that the expansion/compression is adiabatic, but my point was that it is not unreasonable to make the assumption that it is, given a small change in volume. I believe it is unsolvable otherwise (in closed form), though I would love to be proven wrong if you have worked it out.

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jmorgan3
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Re: Thermodynamics Questions

Postby jmorgan3 » Wed Jan 26, 2011 6:34 pm UTC

If the problem didn't state that the expansion/compression was isothermal, then the best assumption would be that the process is isentropic (adiabatic and reversible), and P*V^gamma would be constant. However, isothermal compression/expansion of a gas cannot be adiabatic. There is a pretty obvious way to get a similar expression relating P and V for the isothermal condition that I will PM you.
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