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>From: Pekka R{ty <praty@cc*.hu*.fi*>
I for one asked Juha to post actual data on the compressibility
factor for mixtures involving He. Thanks for following up.
>Because the purpose is to reach specific n number by measuring
>parameter P, equation is
>n= PV / (ZRT)
>Z is scheaduled factor, but also function of P. Also in a fill
>T is fucntion of P (when pressure icreases, also temperature
>increases).
>Space equation for ideal gas is PV = nRT
>Because real gases are close to ideal, but not ideal
>more correct equation is
> PV = nZRT
This equation is exact for all possible values of P,V and T
*if* you measure Z for all P, V and T. That's a lot of
different Z values to keep on hand.
For the purpose of understanding why real gases deviate
from the ideal gas law, the Van Der Waals model seem more
suitable (I believe a derivation was posted a while back
on this list).
The VDW's equation of state for a "real" gas reads (N is the
number of molecules and k Boltzmanns constant):
[p+ a(N/V)(N/V)] x [V - bN] = NkT.
Parameter "a" represents an attractive inter-molecular
force and "b" represents the finite volume of the gas
molecules themselves. "a" acts to lower the pressure.
"b" acts to increase the pressure.
Define v to be the "per mole volume" and we find for Z:
Z = v/(v-b) -a/(RTv).
"a" here is in units [liter-liter-atm/(mole-mole)]
and "b" [liter/mole].
For He: a = 0.03412 b = 0.02370
("Air": a = 1.33 b = 0.0366
This explains why Helium, an inert gas and the smallest
gas molecule, deviates so much from the ideal gas law
at low concentrations {pressures}: there is hardly any
intermolecular attraction to counter the small finite
size effect. For very high concentrations the 1/(v-b)
term in Z will make gases with large molecules deviate
way more from "ideal" than He.
While interesting in it's own right I'm not going to
check the Z values you gave against this formula. What I do
want to do is to hint at why there is no straightforward way
to use the Z values for gas mixtures (but I'm not saying
the error will be large if you do. A set of actual measurement
on real life fill will determine that).
For a mixture of say He and O2 the "a" term must incorporate
O2-O2 interactions, He-He interactions and the crossterms
He-O2 and O2-He (with the latter two being identical).
The difficulty is in the "b" factor. Imagine a tank being
filled with ball bearings. At some point the tank is full.
But there is still plenty of room left inbetween the giant
ball bearing molecules. Helium for example can be put into
the tank without affecting the overall Z (or pressure).
As you state the error encountered with diving gases and
typical pressures is rather small and I therefore think the
above formula for Z will turn out to give ok accuracy for
just about every dive imaginable (this must be verified
however. I have some data for binary fills involving
He so one day I can put a value on a(O2-He) and a(N2-He).
(Can't remember right now if you posted binary system
data or predictions).
I had actually hoped to both solve the problem in "b"
and present some results from this more compact approach
but it's unlikely I will get around to it soon so I decided to
post this approch without bringing much numeric data to
the table or pinning down the error when doing a mixed fill
without binary system data.
Bringing the temperature explicitly into the formula for Z
allows one to account for the temperature effects when filling.
The difficulty is now shifted to that of estimating the
temperature inside the tank. I had this idea for a 2D graph
of the fill pressure vs. temperatures to pin the temperature
in the tank down but that's another idea waiting.
In light of this I'm surprised to see you write:
>When mixing, one problem is when you stop the gas flow, the
>gas and cylinder starts to cool. During first minits the change
>of pressure is fastest due to stabilisation of the system - gas
>shares energy with the cylinder. After this short period the
>temperature measured from surface of cylinder, and the pressure
>inside cylinder start having linear correlation. (PV=nRT)
I.e., what happened to the temp dependence of Z or are
you assuming ideal behavior here ?
I too pondered weighting the tank but I never got it to work
in real life. How would you do it without having to disconnect
the whip ?
>In case of fast mixing, that is done in precense of time / heat
>factor, the factor can be conrolled by making several test fills.
>During these fills temperature is measured and deltaT factors defined.
>To avoid error from stabilization, it may be well argued to
>implement 10 min pause between different gases.
Delta P and Delta T (if you can measure them) would pin you
down to a specific "iso-n" in a 2D P,T plot.
>This text is purely information about scientific base only.
Ditto :-).
John
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