Calculating Piston Height in Equilibrium for an Ideal Gas System

[winhog]

I'm having some trouble with a question involving a piston pressing down on an enclosed gas. I'm given the height of the cylinder, the number of moles of gas, the mass of the piston, and told that initially the gas is at STP. I need to find the height of the piston when the system is in equilibrium.

I thought I knew how to do it, using PV = nRT. I calculated the area of the piston at (about) 1 square meter, so I decided the pressure pushing up should be about the piston's mass * acceleration due to gravity, correct?

Since the mass of the piston is only 1.4 kg, I get about 14 Pa as the necessary pressure. But this is only an extremely small fraction of the original pressure of 1 atm, so the height difference seems negligible to me. Am I missing something, or am I going to be giving my answer in millimeters?

 

[Andrew Mason]

I'm having some trouble with a question involving a piston pressing down on an enclosed gas. I'm given the height of the cylinder, the number of moles of gas, the mass of the piston, and told that initially the gas is at STP. I need to find the height of the piston when the system is in equilibrium.

I thought I knew how to do it, using PV = nRT. I calculated the area of the piston at (about) 1 square meter, so I decided the pressure pushing up should be about the piston's mass * acceleration due to gravity, correct?

Since the mass of the piston is only 1.4 kg, I get about 14 Pa as the necessary pressure. But this is only an extremely small fraction of the original pressure of 1 atm, so the height difference seems negligible to me. Am I missing something, or am I going to be giving my answer in millimeters?

The system will be in equilibrium when the pressure of the gas multiplied by the area of the piston is equal to the weight of the piston. So:

PA = mg = nRTA/V = nRT/h where h = height of piston from end of cylinder
h=nRT/mg

AM

 

[winhog]

Thanks very much!

Though, it seems odd that the initial height of the cylinder they gave me was useless...wasn't it?

 

[winhog]

Sorry, I'm still having trouble. Using your formula, with m = 1.4 kg, T = 273 K, n = 0.1 mol

h = (.1)(8.314)(273) / ((1.4)(9.8)) = 16.5 meters.

The units all work out and everything, but the given height of the cylinder was only 2.4 meters. I know the piston would not shoot up to 16.5 meters, or go down to the floor...what am I doing wrong? Thanks in advance for any help.

 

[Andrew Mason]

Sorry, I'm still having trouble. Using your formula, with m = 1.4 kg, T = 273 K, n = 0.1 mol

h = (.1)(8.314)(273) / ((1.4)(9.8)) = 16.5 meters.

The units all work out and everything, but the given height of the cylinder was only 2.4 meters. I know the piston would not shoot up to 16.5 meters, or go down to the floor...what am I doing wrong? Thanks in advance for any help.

In my expression I omitted the atmospheric pressure P_a. It should be:

PA = mg + P_aA = nRTA/V = nRT/h where h = height of piston from end of cylinder
h=nRT/(mg + P_aA)

Try that.

AM

 

[winhog]

Hmmm, I'm back to my initial problem then, where my answer comes out to something like 0.2 millimeters. I guess it makes some sense since .1 moles of gas can't hold much up unless very compressed, but I've never been asked a problem with such a ridiculous answer from this book before.

 

[Andrew Mason]

Hmmm, I'm back to my initial problem then, where my answer comes out to something like 0.2 millimeters. I guess it makes some sense since .1 moles of gas can't hold much up unless very compressed, but I've never been asked a problem with such a ridiculous answer from this book before.

You will have to give me all the data. I am missing area A, height of cylinder, h.

AM

 

[winhog]

They give height = 2.4 meters, n = 0.1 mol, mass = 1.4 kg, and that the gas is initially at STP. I calculated the area at 0.93 m^2.

 

[Andrew Mason]

They give height = 2.4 meters, n = 0.1 mol, mass = 1.4 kg, and that the gas is initially at STP. I calculated the area at 0.93 m^2.

That seems like a huge piston. What is the diameter of the piston?