Who was Van der Waal anyway and what has he to do wiith my Nitrox fill?

JOHANNES DIDERIK VAN DER WAALS 1837-1923 Amsterdam University
1910 Nobel Prize for Physics for his work on the equation of state for gases and liquids.

This tends to indicate that he knew his stuff.

OK. The simple rules about pressure, volume and temperature that you learnt in Scuba for beginners are only an approximation. Good enough for some poor lad or lass who has to worry about doing a mask clear without drowning but now you're grown up and want to breath fancy stuff it just won't do anymore.

You haven't forgotton but we'll recap anyway.
Do you remember PV=kT?
The ideal gas law? Oh yes. The one that summerised all the others ones that we can't remember the names of. Well let us get rid of that nasty k for an arbitary constant and turn it into the real one PV=nRT. This is much better because we can calculate things with it rather than just do ratios. Let's name the parts in useful units:

PPressure in bar
VVolume in Litres
nQuantity of gas in mols
RThe universal gas constant 0.0831451 if we want to do bar and litres
TThe temperature in Kelvins (virtually Centigrade+273)
mols? Well it's a chemists trick to have a measure of something rather than work in boring units like grams.
A mol is a gram molecule. It is the amount of a substance that weighs X grams where X is the molecular weight of the molecule in question.
So Oxygen has an atomic weight of 16 (well 15.994 if you want to get all isotopic about it) so O2 has a molecular weight of 32 so 32 grammes of O2 is one mol and 64 grammes is 2 mols etc. Plutonium Oxide has a molecular weight of 536 so it take over half a kilo of that stuff to make a mol but chemists don't care. The trick is that 1 mol contains 6.02x1023 molecules. A mol is not so much a quantity as a head count and is great when you are working out how things react.

Great. Real numbers. Let's do an example. The 100% oxygen deco bottle is 3L, it's January at Stoney cove so it's 4C, that's 277K and it's 300bar.
So n=PV/RT so n= (fumbles with calculator) n=39 mols so at 32 grams to the mol we have 1.248 Kilos of Oxygen.

Then along comes Van der Waal in spoil sport mode and says "well not really".

The trouble is that this thing is the Ideal gas law. That is it only really would work for an ideal gas and all we have are real ones. An ideal gas is about as common as an ideal husband/boy friend or an ideal wife/girl friend. Yup. They don't exist. An ideal gas would be composed of infinitly small molecules that did not attract one another but real gases have real sized molecules taking up the space and they tend to attract and repel one another. Van der Waal was the man who set about solving the problem by working out how to allow for real gases.

What he came up with was not the exact answer but a much better gas equation.

If you look at it and remember PV=nRT you can see the old ideal gas equation in here but with two extra terms, one applying a fiddle factor to the pressure to allow for the attraction between molecules drawing them inwards and reducing the pressure on the outside world and the other fiddle factor is on the volume where it is effectivly reduced to allow for the fact that all these molecules are taking up space.
We get two constants a and b which depend on the gas we are considering.

Back to the example for our tank of O2 and use the Oxygen a value of 1.382 and b of 0.03186 and we get a different situation.
Putting our 1.248Kgs ie. our 39 mols of Oxygen into our 3L tank gives 780bar. Yikes!

OK lets graph that with mols along the bottom and pressure in bar up the left.
What's this mean? The nice straight purple line is gas pressure against mols of gas for the Ideal law. The blue line sweeping upwards is Van der Waal's calculation.
Down at 0 to 40 bar, where you did you school physics and your diving it is very good. Up at 100 bar it takes only 10 mols of Oxygen to get 100 bar rather than the 12 mols you hoped for (17% short), at 200 bar 26 expected but 17 delivered (35% short) and at 300 bar you expected 39 mols and you only got 22 (44% short).
Where's that blender! I want my money back!

It gets worse.

We assume for ideal gases that we can work out the partial pressures independently and just add them up (Dalton's Law) and moreover we assume that the ratios don't change with pressure. Now for nitrox it happens that the a and b values for Nitrogen are similar to Oxygen but if (simplistically) we put 100 bar of Oxygen in a tank and then topped it off to 300 bar with pure Nitrogen we do not have 33% Nitrox. It is a lot higher. More like 45%.

We can calculate it but Van der Waal's equation as stated above only applies to the simple monatomic gases and as the molecules of one gas see the others nothing is simple. However by using modified a and b constants for the mixed gas formed using the values for each gas and combined.


What on Earth? Yes. I did university physics and I winced at that one. What it is saying that for a gas made of n gases (1 to n) whose in ratios (of mols) are x1, x2, x3... xn and whose a and b values are a1, b1 etc. then you get the global a and b values by taking the formula to the right of the two sigma signs and adding up all the bits. Ie if you have three gases (Oxygen, Nitrogen and Helium for example) then
a = √(a1*a1)*x1*x1 + √(a1*a2)*x1*x2 + √(a1*a3)*x1*x3
+ √(a2*a1)*x2*x1 + √(a2*a2)*x2*x2 + √(a2*a3)*x2*x3
+ √(a3*a1)*x3*x1 + √(a3*a2)*x3*x2 + √(a3*a3)*x3*x3
and naturally b looks much the same. This was probably grief to poor old Van der Waal but we have spread-sheets on our home computers...
Once you have done this you can work out the partial pressures and add them up Dalton style because your partial pressure is allowing for the mixed gas situation. However don't try to work back from pressure to ratio. It's horrible.

So I suppose I ought to do a trimix calculation but its doesn't prove anything new. What I will do is give you the a and b values for the three gases we tend to worry about so you don't have to buy the great big book I did.

abmw
Oxygen1.3820.0318631.9988
Nitrogen1.3700.038728.01348
Helium0.03460.02384.0020602

One last thing. It is the partial pressure mix at 1 to 10 bar that we breath and so what we measure is what we get. You do not have to worry that the stuff in the tank is not going to be what you measure. The ratio of the number of gas molecules is what you care about so if it measures 38% it will stay 38%. In fact, let us admit that even on trimix people tend to be very careful on the mix, using tables that allow for Van der Waal's work but if the Oxygen measures right and it makes you sound like Donald Duck it's OK.