# Ideal gas law, applied to non-rigid container

1. Nov 12, 2008

### ARQuattr

I'm trying to determine the internal volume of a container using the ideal gas law principle.

1. A container of known volume is pressurized with air to some known value relative to ambient.
2. The container of unknown volume is vented to ambient.
3. Temperature is constant for all air masses and throughout the experiment.
4. The vent on the container under test is sealed and the containers are allowed to communicate air mass.
5. The final equilibrium pressure of the combined air mass is measured.
6. Determine the unknown volume, given the control volume, initial pressure, final pressure.

Using PV = nRT, I have

PcVc = ncRTc, PuVu = nuRTu, and PfVf = nfRTf, or

nc = PcVc/RTc, nu = PuVu/RTu, and nf = PfVf/RTf

for subscripts c (control container), u (unknown container), and f (final), where

Vf = Vc + Vu, and nf = nc + nu

Then,

PfVf/RTf = PcVc/RTc + PuVu/RTu

and assuming T constant,

PfVf = PcVc + PuVu

Since the unknown container starts at ambient pressure, Pu = 0, so

PfVf = PcVc, or Pf(Vc+Vu) = PcVc

(I've tried this using absolute pressures and it works out the same.)

So the unknown quantity is

Vu = Vc(Pc - Pf)/Pf

Experimentally, this works fine for rigid unknown containers, but when I use a non-rigid container it doesn't. Why not?

By non-rigid, I'm referring to something like a plastic jug with a fairly well-defined volume, but if pressurized, even to only a fraction of a psig, the sides bulge out slightly. This drastically affects the final pressure, but I don't see how. I understand that the volume will increase slightly because of the measurement process, (and I'm able to accept the error in volume measurement due to the expansion of the container,) but the calculation yields a result that is way off.

I can also imagine how the container walls act like springs to increase resistance as it grows, but I don't see how that changes things. The pressure introduced by the control vessel causes the container to expand which allows the pressure to drop until it all reaches equilibrium and my end result should be the final container volume, right?

What am I missing?

2. Nov 13, 2008

### tankFan86

At first glance it sounds like you have a nasty semi-free-boundary problem that I don't even begin to know how to pose mathematically. Certainly the final pressure depends on the rigidity of the containing vessel.

3. Nov 13, 2008

### ARQuattr

Thanks tankFan86 for your reply. Experimentally and intuitively, you're right, but I can't see how the theory supports it. Do you know what I might look for to research this further?

Thanks again.

4. Nov 15, 2008

### tankFan86

I think you have posed an insanely hard question. If immediate results are required, I think collecting some experimental data might tell you lot about this problem. If you really want to explore the problem theoretically, you are dealing with the dynamics of a compressible fluid in a crazy geometry, something that I do not think has many good theoretical results. The theory must involve many coupled PDEs!