Partial-pressure blending — ideal gas, van der Waals and the real-world model

When I blend nitrox or trimix by the partial-pressure method, I do something that looks trivial: I let oxygen into the cylinder up to a certain pressure, then top it up with air. The gauge shows a pressure, and quietly I assume that pressure is the same thing as quantity of gas. With a cylinder filled to 200 or 300 bar that assumption starts to fall apart — and that is what this piece is about. You do not need it to dive nitrox safely. But if you like knowing what is really going on inside the cylinder, settle in.

The partial-pressure method in one paragraph

You want EAN32 — a mix with 32% oxygen — in a cylinder filled to 200 bar. First you let pure oxygen in to about 28 bar (I count in absolute pressure — on the gauge you will read roughly a bar less), then top up with air to 200 bar. The rest is arithmetic: the air adds its ~21% oxygen, your earlier pure oxygen makes up the rest, and the analyser should read 32%. The whole argument starts with one question: how many oxygen molecules did you really let in when you let in "28 bar"?

Model 1 — the ideal gas

The simplest model says: PV = nRT. The number of molecules (moles) is directly proportional to pressure, and the pressures of the components simply add up — that is Dalton’s law. "28 bar" of oxygen is exactly as many moles as it "should" be, and 32% worked out on the gauge is exactly 32% on the analyser. Clean, convenient and — at low pressures — perfectly accurate enough. This is the model behind every fill table and every blending calculator you use day to day.

Model 2 — van der Waals

The ideal gas pretends that molecules have no volume and do not attract one another. In 1873 van der Waals added two corrections: the constant b — molecules do take up some space, so it really is "tighter" in there — and the constant a — molecules attract one another, so the real pressure drops. We gather the whole effect into one number, the compressibility factor Z = PV/nRT. For an ideal gas Z = 1; van der Waals gives a Z different from one, depending on the gas and the pressure. It is an important historical step, and its more modern descendants (such as the Peng-Robinson equation) are genuinely decent — but the van der Waals equation itself, as you are about to see, can get even the direction wrong on this particular task.

Model 3 — the real-world model (the reference equation)

The most accurate thing we use today is multi-parameter equations of state fitted to thousands of laboratory measurements for each gas separately (the Helmholtz energy formalism — the same one that drives NIST’s REFPROP). They do not guess; they reproduce measured reality. For our gases at 15°C and 200 bar they give this picture:

• Oxygen: Z ≈ 0.93 — at 200 bar it is more compressible than the ideal.
• Nitrogen: Z ≈ 1.05 — a touch less compressible, working against the oxygen.
• Helium: Z ≈ 1.10 — clearly less compressible. A "bar of helium" is fewer moles than intuition suggests.

Worked example — EAN32 to 200 bar

Let us do it step by step. First the ideal recipe — how much pure oxygen to let in so that, after topping up with air, we land on 32%? Here Fₐ is the oxygen fraction in air (≈ 0.209) and Pₖ the final pressure:

You top up with air to 200 bar. In the ideal model the oxygen fraction is simply the summed partial pressures:

Exactly 32% — the ideal hits the target. Now the real-gas correction: we take the number of moles of each gas not from pressure alone, but from pressure divided by its compressibility factor:

The diluent — nitrogen in practice — has Z ≈ 1.05 at 200 bar, so it packs about 5% fewer moles than the ideal counts. Oxygen at 28 bar is almost ideal (Z ≈ 0.98), so its amount barely changes. The result: the oxygen fraction rises from 32.0% to 32.34%. To get back to a flat 32.0%, you let in about 0.8 bar less oxygen than the ideal recipe tells you to.

What comes out of it — EAN32 filled to 200 bar

We take the same recipe (oxygen to ~28 bar, air to 200) and ask each model: how much oxygen will the analyser show? Worked out honestly, at 15°C:

• Ideal gas: 32.00% — by definition it hits the target.
• Van der Waals: 31.92% — and here is the surprise. It not only gets it wrong, it claims the mix is leaner, when in truth it is richer. The wrong direction.
• Peng-Robinson (the more modern cousin): 32.13% — the direction is right now, though the deviation is underestimated.
• The real-world model: 32.34% — the mix comes out about 0.34 of a point richer in oxygen than the ideal recipe promises.

So where does this extra oxygen come from? Curiously, not from the oxygen itself — at 28 bar it is almost ideal (Z ≈ 0.98). It comes mainly from topping up with air to 200 bar: nitrogen there has Z ≈ 1.05, so each "bar" of air holds slightly fewer diluent moles than the ideal model assumes — and the share of the oxygen you let in earlier rises. That is why at 300 bar, where nitrogen’s Z climbs further still, the effect grows.

Three tenths of a percent is nothing? At 200 bar, indeed — it falls within the tolerance of a typical analyser. But this error grows with pressure. The same EAN32 filled to 300 bar is already ~33.3% oxygen by the real-world model, that is over a full percentage point above target. And that feeds straight into the MOD — the mix’s maximum operating depth: you work it out for 32%, and you breathe nearly 33%. If you wanted to hit a flat 32.0% at 200 bar, the real-world model advises: let in just under a bar less oxygen than the ideal recipe tells you to.

On the gauge you see pressure. Your body settles up in moles. The ideal gas pretends they are one and the same — the real one knows they are not.

Trimix — here the error gets bigger

Helium has Z ≈ 1.10, so it departs from the ideal more strongly than oxygen or nitrogen — and there is a lot of it in trimix. Take 18/45 (18% oxygen, 45% helium) filled to 200 bar by the same partial-pressure recipe. The real-world model says it really comes out at about 18.3% oxygen and 45.3% helium — both components drift upwards at once. Van der Waals gives 18.7% and 44.1% here, so it diverges even further and again in the uncertain direction. The more helium and the higher the pressure, the louder the ideal recipe lies.

Which model to choose

In short: for everyday planning and for the gauge itself the ideal gas is entirely good enough — simple, and at moderate pressures it stays within the analyser’s tolerance. Leave van der Waals to history and the textbooks: it was the first to show where the deviation comes from at all, but for real blending it can be worse than no correction, because it can point you the wrong way. When the numbers really start to matter — high pressures, trimix, an accurate reckoning of oxygen exposure — you reach for the reference equations. Those are the ones sitting inside any decent gas-blending software.

And the analyser still has the last word

All of this is physics worth understanding, but it does not replace two things. The first: real gas blending is a supervised procedure — it calls for training (a blender course), kit prepared for oxygen contact and plain humility before fire. The second, more important for you as a diver: you analyse every mix yourself, with your own analyser, just before the dive, and you sign the cylinder yourself. Even the most accurate model only explains why the number on the cylinder may not match the recipe. It is the analyser that decides what you are really about to breathe.