# Ideal Gas in a Cylinder with Piston

## Main Question or Discussion Point

Hey,

I'm trying to come up with a question, be it a simple one, where a mass of ideal gas (Oxygen) is bound in a cylindrical tube and has a circular piston applied to it - so that the gas becomes pressurised. Once the gas has been pressurised it will gain a certain temperature, which is the objective of the question I'm trying to compose.

If we imagine some mass of oxygen inside a cylindrical tube that is compressed by a circular piston then the volume of the enclosed space and thus the volume of the gas is given by:

$$V_{compressed}=\pi r^2h$$

Where 'r' is the radius of the circular piston and 'h' is the height of the compressed volume.

Applying some force to this piston causes the compression and pressure, the pressure from this force is given by:

$$F_{compressed}=\frac{P}{A}=\frac{P}{\pi r^2}$$

Where F is the force applied by the piston and A is the cross-sectional area of the circular piston.

Using these equations and the ideal gas equation:

$$PV=nRT\: ,\: \frac{F}{\pi r^2}\times \pi r^2h=PV$$

We find the product of PV is independent of the radius of the piston and only dependent on the force F and the height h. So resolving the ideal gas equation for temperature we find

$$\frac{Fh}{nR}=T$$

Now, I believe and hope this is all right so far... Provided it is I want to determine some reasonable values for 'F', 'h' and the mass of the oxygen such that it satisfies the ideal gas conditions - those being high temperature and low pressure.

I'm not really sure what qualifies as high temperature and low pressure, I'm not sure what a 'high' temperature is? I think it's just when the kinetic energy of the particles is much greater than the interaction energy between the particles - but once again I'm not sure for what temperature this would be.

Thanks for any help,
SK

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Rap
Oxygen is never a perfectly ideal gas, so you need to know beforehand how much deviation you are willing to settle for before its "ideal enough" for you. In other words, you have to figure out what is close enough to ideal for your application, then you can figure out what qualifies as high temperature and low pressure. Check out www.nist.gov/data/PDFfiles/jpcrd423.pdf [Broken] for thermodynamic properties of oxygen, that should help you figure out when the gas is ideal enough for your application.

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Ideal Gas

Ok I think I understand what you're saying - different properties reproduce results with varying deviations from the ideal gas assumption - in general though, is room temperature high enough or is this dependent on the molecular composition of the gas?

And when I say 'high enough' I mean so that the result the ideal equation law gives us is within about 10% of the deviation from the true result. After scanning through the document I came across two equations which have some association with the ideal gas laws and these equations reproduce good results for temperatures between 70K -1100K.

I do apologise if I'm being an idiot and completely missed your point. Accuracy to the exact true result isn't too much of an issue, I'm only looking for about within 10% deviation from the true result.

I'll continue reading through this document and probably come back with my issue resolved, thanks for getting back to me and finding this pdf!

Thanks,
SK

Rap
When you say 10% deviation, what thermodynamic parameter do you have in mind? Deviations from ideal are often measured in fugacity ratio (f), P/nkT where P is pressure, n is particle density, k is Boltzmann constant, T is temperature, so nkT is the pressure you would get if the gas were ideal, at the same n and T. So if you have f=1, the gas is ideal. Are you saying f=0.9 and above is ok? (I think for most gasses f is always less than 1, approaching 1 as density goes to zero and/or T goes to infinity.)

Ha this is what I mean about 'being an idiot' or just assuming the question is more simple than it is in principle. I'm not quite sure what I mean with reference to the fugacity ratio, I'm not sure if a fugacity ratio of 0.9 would mean that the ideal gas result is within a 10% deviation of the true real gas result.

I would need to look into this much more, I'll keep looking and get back to you when I understand my issue better.

Chestermiller
Mentor
Here is a reference which contains a plot of the compressibility factor z as a function of the reduced temperature (absolute temperature divided by critical temperature) and reduced pressure (absolute pressure divided by critical pressure).
http://www.thermopedia.com/content/806/
You will notice from the figure that ideal gas behavior is very closely approached if the reducted temperature is higher than 2.0 and the reduced pressure is less than 0.5.

Rap
Here is a reference which contains a plot of the compressibility factor z as a function of the reduced temperature (absolute temperature divided by critical temperature) and reduced pressure (absolute pressure divided by critical pressure).
http://www.thermopedia.com/content/806/
You will notice from the figure that ideal gas behavior is very closely approached if the reducted temperature is higher than 2.0 and the reduced pressure is less than 0.5.
Ah, it seems that compressibility factor is what I was calling fugacity ratio. That's a good link, thanks. I think it will ultimately answer the OP.

Thanks chestermiller & Rap

This is what Rap was getting at - but I was having trouble finding/interpreting the info. from the previous document. Though this information has now resolved my current issue! So thanks chestermiller & Rap - but I may be back with a further query.

Thanks again!