the molecolar interactions in mixtures
Looking at the real gas formula PV = ZnRT, we can see that by keeping “p”, “v” and “t” stable , increasing the value of Z, there is a decrease in the amount of gas “n” .
The difference between the experimental coefficient and those predicted by partial pressures can be attributed to the interactions between various combinations of the dissimilar molecules of the mixture.
The value calculated with Kay's rule deviates of 4.64% compared to the correct value, and the actual Z factor is outside the two values of the single gas !
A weighted average for two values is by definition between the two values, proportionally to each single value.
As you can see from the picture below, the behavior is the same, even with different values, for the whole pressure range.
The value of the mixture Z factor always remains external to the lines of the respective Z factor of the two individual gases, and always higher (and therefore expressing lower amounts of mixture) with respect to the area where we would expect the value calculated according to the principle of weighted average (between the blue and yellow lines):
The mixture blending method by partial pressures is affected not only by the individual gas compressibility but also by the molecular interaction between different gases.
On forming a mixture of two or more gases, new forces are brought into action which are not present in either of the pure components and the properties of mixtures may be very different from those of the pure constituting components, due to unlike forces.
It will be pointed out that energy, and not a geometrical effect, is the driving force for compound formation.
Some blending software calculate the density of a mixture with inadequate equations (usually van der Waals) applied to each single gas, not considering their molecular interactions.
They then use a common sense approach , calculating the total density of the mixture by the weighted average of the single gas values weighed by their mixture fractions.
This method is known in thermodynamics as "Kay's rule".
Unfortunately, at high pressures and especially with helium, it does not work.
This will be demonstrated using the data of Helium and Nitrogen at 200 bar gauge and 20 °C.
Their Z value calculated as single gas by the most advanced equations, and hence coincident with the experimental data are:
Oxygen – Helium mixtures
Up to revision 5 of the U.S. Navy Diving Manual, section "Breathing Gas Mixing Procedures", when explaining how to compensate the compressibility effects when mixing Heliox, there was reference to a specific manual, the U.S. Navy Diving-Gas Manual (NAVSEA 0994-LP-003-7010), nowadays virtually unobtainable and frequently confused (I'd have to say always, including bookshops and libraries ) with the U.S. Navy Diving Manual itself.
At the time when this study was produced, very few experimental data were found for high pressure Oxygen-Helium mixtures, so the comparisons of the algorithm with experimental information were difficult.
However, the source was very authoritative (we are talking about the U.S. NAVY DIVING SUPERVISOR OF NAVAL SHIPS SYSTEMS COMMAND), and the contents of these documents were of very high quality; the issue about the molecular interactions in Oxygen-Helium mixtures was discussed in depth and was conceptually similar (of course with different parameters) to the Helium- Nitrogen issue.
In one of my personal versions of Tech Gas Blender, I have implemented the algorithm code explained in this text. The results are pretty close to the more recent works, which fully confirms what has been said.
Tech Gas Blender, however, is based on different equations developed in more recent studies, with Oxygen-Helium-Nitrogen binary interaction parameters considered more accurate.
Z Factor He = 1.09513
Z Factor N = 1.05253
The expected Z Factor of the mixture , estimated by the Z weighted factors should therefore be :
(0.5 * 1.09513) + (0.5 * 1.05253) = 1.07383
The experimental value Z is 1.12609, higher than the Z average value.
It is well known that the thermodynamic quantities of even binary mixtures composed of simple spherical molecules cannot be obtained by simply adding the values for the pure components, weighted by the mole fractions.
Some reliable books in thermodynamic point out "… the procedures are not recommended for helium , hydrogen or neon unless special, modified critical constants are used ... "
The equations of state for gas mixtures He-N
In the extraction of helium from natural gases, the helium content of the mixture depends on process operations at the helium extraction; other than helium, nitrogen is the principal component in these gas mixtures.
The physical properties of Helium-Nitrogen mixtures over a wide range of pressures and temperatures are therefore of great technical importance to the industry. One of the objectives of the Helium research center is to develop a more effective process for the purification of Helium-Nitrogen mixtures.
The compressibility data of Helium and Nitrogen have been reported by a great number of investigators and a large part of these data have been represented by various equations of state for computing these mixture properties over a considerable temperature range with pressures over 300 atmospheres.
Due to their complexity, wide-ranging Equations of State can have a large number of adjustable coefficients, e.g., usually more than 30.
Compressibility Z factors have been then computed and compared with experimental data in the literature.
For example, in the report of investigation 6896 “EQUATION OF STATE FOR HELIUM-NITROGEN MIXTURES FROM 133.15° TO 748.15° K WITH PRESSURES TO 300 ATMOSPHERES” a comparison of the equation of state with 2.508 experimental compressibility factors within the pressure, gas composition and temperature range of the compressibility factors has been reported, with the result of an absolute deviation in Z of 0.0004