The Ideal Gas Law is an equality of pressure, volume, temperature and number of molecules. It is a combination of the gas laws of Boyle, Charles, and Gay-Lussac, and it is still taught in high school classes of chemistry and physics.

PV = nRT

Where R is the universal gas constant and n is the number of moles of gas present. A mole of gas is that amount of gas that has the same number of molecules as 12 grams of carbon 12. So n is not a measure of mass, but of number of molecules present.

That pressure should be dependent upon number of molecules rather than mass has always been a curious fact, a fact that has never been explained. In general, the pressure would be expected to be a function of the summed momentum of particles, but in a gas this may not be the case*.

The variable for temperature is a measurement of the velocity of the molecules. If we increase the temperature, we increase the average velocity of the gas. Therefore we would expect either the momentum or the kinetic energy of each molecule to be expressed by its mass and its velocity. In the macro-world, this would certainly be the case. Why is it not true at the molecular or atomic level?

The Ideal Gas Law is known to be inaccurate under extreme laboratory conditions. It was therefore corrected by van der Waals, giving us a state equation that is much more accurate. But van der Waals did not explain the mystery of the mole. He made two major corrections to the Ideal Gas Law, taking into account unavailable volume lost to the real size of the gas, and taking into account electrical forces between molecules—which caused variations near the surface of the gas. But he did not address the molecular reason for the unimportance of mass.

The reason that this unimportance of mass is curious is that molecules are not all the same. For example, a molecule of neon gas weighs five times more than a molecule of hydrogen gas. And yet in these equations they act the same. If both molecules have the same temperature, and therefore the same speed*, you would expect the molecules of neon to have five times the momentum. But they don’t. Or, if they do, this momentum somehow does not translate into pressure.

Van Der Waals state equation differentiates the two molecules by volume, since a larger molecule will cause more unavailable volume, which will tend to increase both the temperature and the pressure. But, as we have seen, this correction will only make a difference at extremes. At more normal temperatures and pressures, the ideal gas law will work. This fact means that van der Waals correction cannot explain the full mechanics of the situation. The fact that there is any situation in which increased mass does not cause increased pressure is a sign of the limits of our knowledge. Other signs of this limit are the two new constants in the van der Waals equation of state, which are different for different gases and which can be determined only from experiment. That is, they are heuristic corrections, with no theoretical underpinning.

The Physical Explanation

Expansion theory explains the mechanical reason for all this. Once mass is seen to be a real acceleration outward of each atom or molecule, the gas laws start to make sense. Pressure is caused by the collision of the molecules with the wall of the container. The time interval of this collision is not zero, but it is very small. It has always been treated as negligible. Mathematically it may be nearly negligible in most instances, but conceptually it is never negligible. In order to understand the mechanics of pressure, you must understand the mechanics of the collision.

As long as mass was assumed to be a measurement of substance, it could not be any theoretical help in explaining the situation above. That is to say, mass is not thought to vary over any interval, large or small, so studying mass closely during the collision of a molecule with the wall of the container could not possibly tell you anything. Now that I have shown that mass is internal acceleration, we can squeeze the problem for more information.

What we find is this: If mass is a measure of substance, and if a molecule of neon is five times more massive than a molecule of hydrogen, then we would expect the collision of a neon molecule with the wall of a container to create five times the force. Summed over all molecules, this would give five times the pressure. And yet we don’t see this. We see neon molecules that often impart the same force as a hydrogen molecule.*

On the other hand, if mass is a measure of internal acceleration, then, as the time interval of collision gets smaller, the time interval of internal acceleration also gets smaller. As the interval of internal acceleration gets smaller, the difference between different masses gets smaller. What we have in the collision of a molecule and the wall of the container is then not the expression of a mass and a velocity over a dt, but the expression of an acceleration and a velocity over a dt. In the former case, the mass would not vary; in the latter, the expression of the acceleration obviously would. Over a small enough dt, all objects with the same velocity would act (more nearly) the same.

To say it another way, according to expansion theory, two molecules with different “masses” would have different internal accelerations. But in a collision with the wall of the container, we are not measuring the difference in the acceleration a, we are measuring the difference in change in acceleration Δa over the interval of collision. As the interval gets smaller, the two Δa’s of the two different types of molecules approach equality. This explains why the Ideal Gas Law can ignore the masses of the molecules.

It also explains why the Ideal Gas Law works at the temperatures and pressures that it does. If my theory is correct, then we would expect the Ideal Gas Law to work best when the interval of collision was shortest. If the temperature or pressure is too low, then the velocity of the molecules is low, and the interval of collision increases. This will cause a deviation from the Ideal Gas Law using my theory, since the molecule hangs around near the wall for too long, and the acceleration force begins to be felt. This brings the “mass” into play. On the other hand, if the temperature is too high, the molecule stops acting like an elastic point particle—it squashes up against the wall and deforms. This causes the interval of collision to increase for a different reason. But it implies the same variation from the Ideal Gas Law.

Mass will least come into play, according to my theory, when the velocity is high enough to cause a very quick rebound from the wall of the container, but not so high that the molecule loses its elasticity.

The Argument against my Theory

Some will say my explanation is unnecessary, since temperature is a measurement of kinetic energy, not of velocity. If kinetic energy already takes into account the mass of each molecule, then my theory is not needed. All gases at the same temperature and pressure will have the same kinetic energies, but the average velocity of the molecules is not thought to be equal. It is not equal precisely because the masses are not equal. If you increase the mass of the molecule, you decrease the velocity, so that the kinetic energy stays the same.

My answer is twofold, 1) This historical application of temperature to kinetic energy is just an assumption. It has never been proved. To prove it would require comparing the average velocities of two different gases. This would require a direct measurement of the velocity of many individual molecules, not a calculation of velocity from kinetic energy. Any calculation down from an energy equation would be begging the question. Some kind of speed trap that could measure individual molecules would be required, and this has never been done, to my knowledge. 2) It probably will be done at some point in the near future, and my prediction is that the variation in velocities of different gases will not be nearly as great as has been assumed from historical theory. If there are the same number of molecules in a mole of gas, no matter the gas, and if the kinetic energies must be equal to explain the equal pressure, then the average velocities of different gases should be very different. By the equation mv²/2, a molecule of neon that has five times the mass of a molecule of hydrogen should be going 1/√5 times as fast. That is, more than twice as slow as the hydrogen molecule. I predict that the data will not bear this out. A difference in velocities between lighter gases and heavier gases will no doubt be found, but this difference will be much less than current theory can explain.

By using lasers, researchers can now measure molecular velocity up to about 500 ft/s. Micro PIV (Particle Image Velocimetry) is the latest technology that I am aware of. Unfortunately, the tests that have so far been run have been for industry and have not tested the kinetic energy theory of gases. I am not sure that the technology is yet sufficient to do the job, anyway. As an example, using the root mean square velocity equation, one can calculate that the velocity of an oxygen molecule should be around 500 m/s. This is already more than three times the limit of PIV, and oxygen is a heavy gas compared to hydrogen, being eight times as massive. Besides, according to my theory, the oxygen molecule should be expected to be going much faster than 500 m/s, since as regards temperature and pressure, it will act more like a hydrogen molecule at ideal gas conditions. That is, in an otherwise empty container with near-perfect elastic walls, the effects of the mass of a gas should be suppressed. All molecular velocities should be pushed toward equivalence by the experimental situation.

*See the last section of the paper if you disagree here.

Link to Miles Mathis's Website