How to Calculate Work and Energy Changes in Expanding Helium?

[eprparadox]

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Homework Statement

Imagine some helium in a cylinder with an initial volume of 1 liter and an initial pressure of 1 atm. Somehow the helium is made to expand to a final volume of 3 liters, in such a way that its pressure rises in direct proportion to its volume.

(b) calculate the work done on the gas during the process, assuming no other types of work being done.

(c) calculate the change in the helium's energy content during this process.

(d) calculate the amount of head added to or removed from the helium during this process.

Homework Equations

W = - integral PdV

delta U = Q + W

The Attempt at a Solution

I know that the final pressure will be 3 atm since the pressure is directly proportional to the volume.

So I found the work done in going from the initial state (1 atm, 1 L) to the final state (3 atm, 3 L) by doing the integral above. This works out to be -4 J.

But I have no idea how to find the change in the internal energy. I figure you need to know the heat added or removed from the system and I don't know how to do that either. I guess once I can understand part c, then I can get part d, or vice versa.

Any ideas?

Thanks a lot for any help.

 

[naresh]

Hint: For an ideal gas, the change in internal energy is only a function of the change in temperature.

 

[eprparadox]

Hi,

Thanks for the response.

A couple of questions:

1) How do you know helium can be treated as ideal if it's not explicitly stated in the problem?
2) So if the internal energy is a function of temperature only for ideal gases, then I understand that if U(T) remains constant , then the change in U will also be zero. However, we know the temperature is changing because the pressure and volume are also changing. and I'm assuming the amount of helium is constant so T = PV/nR.

So the U is a function of T, but I do not know what relationship exists between U and T. All I know is that helium should have 3 degrees of freedom so that each atom will have 3/2kT energy. But I don't know how much helium there is total so I can't figure out the internal energy at any given temperature (because I believe U = N * 3/2kT). This is where I'm stuck.

Thanks again for your help.

 

[naresh]

1) How do you know helium can be treated as ideal if it's not explicitly stated in the problem?

Helium is as close as you will ever get to an ideal gas. I think the intermolecular forces are lesser in Helium than any other gas (most other gases, at least).

2) So if the internal energy is a function of temperature only for ideal gases, then I understand that if U(T) remains constant , then the change in U will also be zero. However, we know the temperature is changing because the pressure and volume are also changing. and I'm assuming the amount of helium is constant so T = PV/nR.

So the U is a function of T, but I do not know what relationship exists between U and T. All I know is that helium should have 3 degrees of freedom so that each atom will have 3/2kT energy. But I don't know how much helium there is total so I can't figure out the internal energy at any given temperature (because I believe U = N * 3/2kT). This is where I'm stuck.

Your relation between U and T is correct. You are right, we don't know what N is, and we don't know what T is. But we know that Helium is ideal, so we should know what N*T is! If you need a further hint, how are k and R related? What are N and n?

 

[eprparadox]

Wow, I'm such an idiot. This must be a sign that physics is not for me!

Thank you so much for your help.