Quantum Pressure! Or How The Pauli Exclusion Principle Converts Pressure from the Ideal Gas Law into Angular Velocity

I’ve spent months trying to figure out how to rewrite my Quantum Gravity Manuscript in a more physically rigorous way. My last post on the topic was over five weeks ago and in the intervening time I’ve tried to find a satisfying way to explain how steric effects can cause the curved supramolecular motion at the heart of the theory.

I always knew that I hadn’t really described the physical mechanism for quantum gravity in the paper. I had merely said that the sum-effect of fundamental interactions in combination with steric effect causes the curved motion of molecules. However, I left it intentionally vague how the fundamental interactions explicitly cause this curvature.

Some time ago I realized that the only interaction I need to explain this curved motion is Pauli exclusion principle. To simplify somewhat, the principle just means that molecules cannot be in the same place at the same time. This means that moving molecules are repulsed by their neighboring molecules, which causes them to reflected from surrounding molecules. I thought this would be so obvious that I wouldn’t need to explicitly state this. But based on reviewer comments, apparently, I do.

So, if the Pauli exclusion principle causes the curved motion of molecules, how is this seen in current physical laws describing nature? Here we get to one of the most important laws of physics to a chemist: the ideal gas law. The law is written in the empirical form:

pV = nRT,

Where p = pressure of gas, V = volume, n= number of molecules (in moles), R = gas constant and T = temperature. This means that the Pauli exclusion principle should explain the law. Here we get to an interesting realization that while basically all of the other terms in the equation is familiar to all (if n is seen as just a number), except the gas constant, the real understanding of them isn’t as straightforward as one would think.

First of all, for the uninitiated, the equation might look odd, because there are more than one term on both sides of the equation, so what does one learn from it? Basically in the equation there are four variables, of which three can be measured. This means that by measuring them, the fourth variable, number, can be determined. Or as an equation:

n = pV/RT.

So, if we have a chamber with a volume of 1 cubic meter, that has a thermometer measuring a temperature of 20 C (which is 293.15 K for the equation, because the equation only works on Kelvins) and a pressure of 1 atmosphere (which is 101325 Pascals, because the equation usually works them, although other units can be accommodated by the pick of the gas constant), the equation says that there are 41.6 moles of gas molecules in the chamber. And mole is just a very big number, which helps us using smaller number in our calculations. But If you want to know, 41.6 moles are roughly 25 000 000 000 000 000 000 000 000 molecules.

But let’s look at the equation more closely. V, or volume is really easy to understand, because to define it you just need a ruler and multiplications with a pen and paper. And the end result n is quite clear: a specific number of molecules. But what about temperature and pressure? These two are extremely familiar to all of us, but what do they mean physically? Well, temperature is directly related to the speed of the molecules with the equation:

where kB is the Boltzmann constant and m the mass of the molecule. The physics why temperature and velocity of molecules is related in this manner is a bit too complicated for this post, but let’s just assume (at least for now) that temperature is just another way to describe the speed (velocity) of molecules.

But what about pressure? According to Wikipedia:

Pressure (symbol: p or P) is the force applied perpendicular to the surface of an object per unit area over which that force is distributed.

Accompanying this description there is a picture of random collisions:

A bit further down there is a description:

In a static gas, the gas as a whole does not appear to move. The individual molecules of the gas, however, are in constant random motion. Because there are an extremely large number of molecules and because the motion of the individual molecules is random in every direction, no motion is detected.

This exact picture and accompanying paragraph are the fallacy behind the reason why it has taken so long to figure out the theory of everything. Glued on top of a very well thought theory there is an added layer of randomness that is not derived from any principle or any firm mathematics. Randomness is just assumed.

And this assumption of randomness would be okay if the mathematics and experimental results backed it up. However, when you try to combine this type of randomness with quantum effects, things don’t seem to add up.

But I’m getting ahead of myself. What is pressure on the scale of people? It is a specific force exerted onto a specific area. And a person exerts pressure on the ground because of gravity. If a large man weighs 100 kg, he pushes against the earth at a force of 981 N just by standing. According to Wikipedia, with a shoe size of 43 (European, or abut US 10), with a surface area of the pair being about 360 cm². So, dividing the force of gravity by area of the ground the person pushes the ground with a force of by about 27 kPa just by standing. However, as soon as any kind of motion is included, the calculations become more complicated.

But this calculation doesn’t make much sense for a single molecule, especially when it’s in motion. How is pressure defined in a molecular scale?

I asked ChatGPT about this:

If pressure is force divided by area, how is this linked to the pressure exerted by a single molecule of gas to its surroundings?

ChatGPT answered:

Pressure is indeed defined as

but for a single gas molecule, things become subtle because a single molecule does not exert a continuous force on a wall. Instead, it exerts force only during brief collisions.

A useful way to think about it is:

• A molecule moves freely between collisions.

• When it strikes a wall, it changes momentum.

• The wall experiences an equal and opposite impulse.

• The average force is the total momentum transferred per unit time.

For a molecule of mass moving perpendicular to a wall with velocity :

• Momentum before collision: mvx

• Momentum after collision: -mvx

• Change in momentum:

If the container has length L , the molecule returns to the same wall every

seconds.

The average force exerted by that molecule on the wall is therefore

If the wall area is A, then the molecule's time-averaged contribution to the pressure is

Since LA=V (the volume of the container),

This is the pressure contribution of one molecule averaged over time.

For N molecules, the contributions add:

Using the fact that molecular motion is equally distributed among the three spatial directions,

which gives the familiar kinetic-theory result

So the connection between "force divided by area" and molecular motion is that pressure arises from the rate at which molecules transfer momentum to surfaces through collisions. A single molecule does not produce a steady pressure in the everyday sense; it produces a sequence of impulses whose time average corresponds to a tiny pressure contribution.

As an illustration, in your earlier example of a 1 m³ container at 1 atm and 20°C, there are about 2.5 x 10^25 molecules. The average pressure of 101,325 Pa is the cumulative effect of an enormous number of molecular collisions occurring every second on every square centimeter of the container walls.

End of ChatGPT’s answer.

Because this answer is a bit too complicated to the layperson, I asked ChatGPT to simplify it to a single sentence. This is what it said:

On the molecular scale, pressure is the transfer of momentum between molecules (or between molecules and a surface) per unit area and per unit time.

But what is the unit area on a molecular scale? When I asked this from ChatGPT I got this answer:

This is an excellent question, because it gets to the point where the continuum concept of pressure begins to break down.

On a molecular scale, there is no unique "unit area" associated with a molecule. Instead, pressure is defined with respect to an imaginary surface that you choose.

For example, imagine an imaginary square of area floating inside a gas. During a time interval , molecules cross this surface and transfer momentum from one side to the other. The pressure is then defined as

The area belongs to the surface, not to any individual molecule.

At macroscopic scales, this works beautifully because:

• billions upon billions of molecules cross every tiny area,

• the fluctuations average out,

• pressure becomes a smooth quantity.

At molecular scales, however, things become noisy.

Suppose your imaginary surface is only in area. During a picosecond:

• perhaps no molecule crosses it,

• perhaps one crosses,

• perhaps several cross.

The "instantaneous pressure" fluctuates wildly.

Therefore, when statistical mechanics talks about local pressure, it usually means an average over some finite area and finite time interval.

There is another subtle point. If you zoom in far enough—say to a region containing only a few molecules—the concept of pressure itself becomes fuzzy.

For a single molecule, asking

"What pressure does this molecule exert?"

is not really well-defined.

What is well-defined is:

• its momentum,

• the impulses it delivers during collisions,

• the average momentum transfer it contributes to a chosen surface.

Pressure emerges only after averaging many such events.

In modern statistical mechanics, one often says:

Pressure is a coarse-grained quantity.

Just as temperature is not meaningfully assigned to a single molecule, pressure is not fundamentally attached to a single molecule or a single intermolecular collision. Rather, pressure is a statistical property of many momentum-transfer events occurring across a chosen surface.

So the answer to your question is:

On the molecular scale, the "unit area" is not associated with a molecule. It is an arbitrarily chosen real or imaginary surface across which momentum transfer is measured. Pressure becomes well-defined only after averaging the momentum flux through that surface over sufficiently many molecular events.

End of ChatGPT’s answer.

So, when pressed hard enough, the definition of pressure on a molecular level is super-undefined. Or some people might say that pressure is perfectly defined and the definition shows the randomness of reality. But I don’t buy this.

I’m saying: the emperor has no clothes! The molecular definition of pressure makes no sense, at least if randomness is introduced.

Instead of trying to tie my definition of molecular pressure to the existing definition, I’m going to do something a bit daring. I’ll try to see if I can get a definition of pressure from combining the theory of supramolecular motion to the ideal gas law. Going back to the equation we see that pressure is:

p =nRT/V

Looking at the equation, we see that all of the variables on the right can be explained by tangible phenomena: numbers, speed of molecules and volume. This indicates that pressure can be derived from these values. But to do this, we need to introduce the theory of supramolecular motion to the equation.

So, what does the theory of supramolecular motion state? It states that molecules in the gas (and partially also in the liquid) state move in toroidal (donut-like) curved orbitals. This means that for a given pressure and temperature, these orbitals occupy a specific fraction of the volume in which they reside. Or p/RT =n/V, where n/V is linearly correlated with the volume ratio of the molecules in the toroidal crust to the volume of the whole torus. The equation for the volume of a torus is

Where r = minor radius of the torus and R = major radius of the torus. However, if we also introduce the radius of the molecule and reserve r for it, we’ll change it to R1 = minor radius of the torus and R2 = major radius of the torus. To simplify the equation, we’ll assume that the thickness of the torus is 2r both on the outside and the inside, even though we know that in reality we know that the thickness on the inside is 4r. We do this to simplify the math.

This means that the total volume of the torus is:

And the volume of the hole inside the torus is

This means that the ratio of the crust of the torus to its total volume is:

This seems quite simple, but what does it mean in practice? Well, we’ve established that r is the radius of the molecule and it doesn’t change as conditions change. However, R1 does. Next, we need to ask what R1 means? It is the radius of the orbit of a string of molecules orbiting around a center of gravity.

I was also about to talk about the subject of angular momentum at length, but I realized that don’t know enough about it yet to do so. Rather, I’ll try to keep the idea of quantum pressure simple. The unit of pressure is Pascal, or Pa for short, but it is also defined as kg·m⁻¹·s⁻². From here we see that if a molecule has a specific mass (in kilograms) and it is whirled around at a specific angular velocity, the pressure is inversely proportional to the radius of the rotation. Or simply put, the smaller the radius of the rotation, the larger the pressure.

So, is the title of my post correct? Does the Pauli exclusion principle convert pressure from the ideal gas law into angular velocity? I think so, but I’ll have to go through the mathematics and physics to check for errors to be absolutely sure.

I would so like to state something more profound than the vague link of pressure to the radius of rotation of the string of molecules but I think if I want to be honest, I don’t know any more at the moment.

Unless I’m mistaken, this is the most profound physical revelation that I’ve had, right up there with the realization that there must exist an elementary particle of energy. But because the revelation is so brand new, I don’t yet know where it’ll take me.

All I know is that the scheme depicting pressure must be corrected based on this new revelation. The molecules still collide with surrounding molecules, exchanging momentum, but no longer in a random fashion: