My thinking... (2) On pressure

This is another part of the series. Just like the first one it was originally written in Polish and I have translated it for sharing here. And it similarly aims to strain the imagination by illustrating concepts with nothing more than words.

Let there be no misunderstanding: making tea is not the only thing I do in the kitchen. When I'm there, various instruments and dishes run through my hands, usually without provoking any deeper thoughts. But one time, when I was holding an ordinary empty jar in my hand, I surprised myself with another childish question: if I put it on a scale, would I weigh it together with the air inside?

When the lid is screwed on tight, I think so. And when it is off, or at least not sealed? If there is a difference, what causes it?

I quickly forgot what I was supposed to do with the jar as my thoughts started a great expedition in search of the answer to this puzzle.

The key to the solution is the air pressure - I realized when I started to imagine molecules hitting the glass walls, the scale and everything around.

And I immediately started having doubts whether my vague understanding of pressure would be enough, remembering the lesson I received with temperature. I decided to start from scratch and find out what pressure is and where it comes from.

According to the definition I remembered from school, pressure is a measure of the force with which gas pushes against a unit of surface area. It seemed simple, but I got used to poking holes, so I started to dismantle it into pieces. What does it mean that one thing acts on another with a force?

I remembered Newton's laws of dynamics, which say quite a lot about forces. First, that an object not affected by any forces moves with a constant velocity (and since velocity is a vector, in particular the direction of movement, which is one of the components of a vector, is constant). Second, if we multiply the acceleration (i.e. the measure of how much speed changes in a unit of time) and the weight of the object, we get the force that acts on it. And finally third that sounds similar to a common saying that every action meets with a reaction - that the force with which one object acts on another causes the latter to return the favor with the same strength (but in the opposite direction).

These laws describe forces through the prism of their effects. However, I felt something is missing in this approach. A catch phrase came to my mind: I understand how, but I do not understand why. And I went on to discover that it is possible to describe the same laws in a different way. For anyone familiar with physics this shouldn't be at all surprising, but it hit me.

One of the basic rules in physics is the law of conservation of momentum. It is similar to the conservation of energy, but refers to a different physical property. What is a momentum? For any given thing you can determine it by multiplying the mass and the velocity. Velocity is a vector and thus the same is true about momentum - it has a direction, it can be broken down into spatial components and so on.

Most importantly, however, the momentum never appears out of nowhere, nor does it disappear without a trace. Just like energy, it can at most be transferred from one thing to another, in whole or in part. How much of it arrived in some place, the same amount of it had to be lost somewhere. This behaviour suggests that momentum is an important element of the description of the world.

According to the second law of dynamics, one can calculate the force acting on a thing by multiplying the mass of the object and its acceleration, i.e. the change of its velocity per unit of time. But the change of velocity multiplied by the mass is nothing else but a change of momentum. Thus, the second law of dynamics turns out to be describing a force as a change of momentum per unit of time (or more mathematically: as a derivative of momentum with respect to time).

In particular, if the momentum of an object remains unchanged, it means exactly that no force is applied to it. If we also consider the law of the conservation of mass (which is in fact a camouflaged law of energy conservation, but this is a part of an entirely different story), then the invariability of the momentum results in the constant velocity, which leads to a conclusion identical to the first law of dynamics.

And what about the third law? When a given object is affected by force, it means that it is receiving some momentum (or losing it - these are two sides of the same coin, as removing momentum is exactly the same as adding it in the opposite direction). But the principle of conservation states that this momentum cannot come from nowhere. There must therefore be some other thing that gives the momentum to the first one. Then in both objects it changes by the same amount, but in opposite directions. They could be said to act on each other with opposing forces, but I liked a simple way of seeing it as a flow of momentum from one thing to another.

Everything started to look nice and clear when I suddenly realized that I had overlooked an important detail. The laws of dynamics only work if I observe the events from a right place - one that does not change its own speed relative to, roughly speaking, the rest of the world. When I look through the window of an accelerating car, I see the outside world speeding up and everything around seems to receive a lot of momentum from nowhere. It's a bit like if everything outside the car suddenly started to experience an additional gravity pulling backwards (it is also felt by passengers squeezed into seats), but it has no visible source and it's not obvious where all this momentum comes from. This association is not accidental; I know that it may in fact lead to the general theory of relativity - but it is a direction in which I definitely would not want to go at the moment, because I would undoubtedly be lost. So I decided to circumvent the problem and maintain that my observation point is always good enough to avoid additional reservations to the law of conservation of momentum.

Having recognized that I was able to cope with the forces, I came back to the pressure.

At this moment the matter seemed simple. The air consists of particles and each of them has its own momentum. When such speck hits something like a wall of my jar, it will most probably bounce back, i. e. it is going to get a momentum in the opposite direction. But then, however much momentum the particle received, the jar must have acquired just as much in reverse, total balance must be zero. This means that the particles of which the jar is made constantly absorb the momentum conveyed to them by the air molecules, and this is the force associated with pressure.

Something tempted me to try to capture it quantitatively and precisely. Perhaps a bit prematurely, because in the kitchen I had nothing to write, so I was doomed to handle mathematical formulas in memory. It can be very satisfying when it works out - but usually it is a source of repeated frustration, when upon trying to add another block to an intricate mental structure, everything suddenly falls apart and I no longer know what belongs where, and have to start from the beginning.

And yet I tried to keep it all as simple as possible. I imagined point particles of gas flying inside a closed box and bouncing perfectly elastically off its walls. The intuition told me that after such a bounce the speed of a gas molecule should remain the same - however fast it was approaching the wall, it would just as fast fly away from it. This can be explained by the fact that there is no energy flow between a particle and a box - and since the kinetic energy of a particle remains unchanged, the speed cannot change, because the energy of the particle is equal to the half of its mass multiplied by the square of its speed. This time I used the formula straight from the textbook, I was not as eager as to try poking holes in it.

Looking at these collisions from a statistical point of view (as I have done it before), the flow of energy between gas and walls has a chance of actually happening only when there is a temperature difference between them. And this would sooner or later even out, leaving afterwards a mostly unchanging balance of energy. The assumption that particles bounce elastically off walls did not therefore seem unreasonable, even if only as a statistical average.

However, although after a bounce the particle flies as fast as before, it does it in another direction. This means that the vector of velocity (and thus also the vector of momentum) has changed. So the speck was robbed of the momentum with which it rushed towards the wall and additionally it received just as large momentum in the opposite direction - in other words, its loss of momentum was double. And since the momentum is subject to strict accounting rules, this twice the momentum had to be added to the bill of the box wall.

In order to calculate the force exerted by these hits, it is necessary to estimate how much momentum is transferred in a unit of time. To start with something easy, I imagined a situation in which all particles fly with the same speed straight to one of the walls. Each of them then has the same momentum, so it is enough to count how many of them are going to hit the target within, say, a microsecond. In the world of molecules everything happens quite quickly, so it is better to choose not too large unit of time.

To say what speed something has is the same as to say what distance it would cover in a specific time unit. In my simplified variant all particles travel the same distance within a microsecond - so only those that are no further than within this distance from the wall are going to hit it. It is a thin layer of gas right next to the wall, and the box would receive twice the entire momentum of this slice.

This doubled momentum divided by a unit of time is a force acting on the box wall. When the magnitude of this force is divided by the surface area of the wall, it becomes pressure. However, I had a greedy plan to somehow associate pressure with the volume of gas, and to do that I had to play a little algebraic game.

Instead of a unit of time in my calculation, I could insert the thickness of the layer divided by speed of molecules - it actually is the same thing. After such a machination, the pressure became doubled magnitude of the momentum multiplied by the speed and then divided by the thickness of the layer and the area of the wall. The measure of momentum is the same as mass multiplied by speed - multiplying this once again by speed gave the mass multiplied by the square of speed (that was easy), and this was simply twice the kinetic energy. But because the momentum was doubled, it finally became four times the kinetic energy divided by... here my cunning plan paid out, because the thickness of the layer multiplied by the area of the surface is nothing other than the volume of the gas whose momentum I considered.

The pressure calculated in this way turned out to be four times the kinetic energy per unit of gas volume and this is something that could be called a quadruple energy density. I was convinced that I was heading in the right direction - but how could I refer to the real situation, where particles fly with various speeds and in all different directions?

It was no coincidence that I imagined a box instead of a jar, because six walls can be related to the basic three axes of the space and the momentum of any particle flying inside can be disassembled into components, each of which would be directed straight towards one of the walls. Statistically, each side of the box would receive roughly the same amount of momentum - at any given moment one sixth of the whole momentum contained in the gas should be directed towards a specific wall. But individual molecules would rush in that direction at different speeds - I cannot deny that this seemed troublesome.

I decided to divide the particles into groups corresponding to each possible speed. One can then calculate pressure exerted by each such group individually. Their number is probably unimaginably large, but many a mickle makes a muckle and collecting them all would give the total pressure acting on a wall. However, this would work only if the groups were completely independent - and this is not the case if the particles of gas collide also with each other.

Not seeing the solution to this problem, I have decided to downplay it. If the unit of time was short enough, the corresponding gas layers acting on the sides of the box would be really thin - and therefore the probability of collisions between the gas molecules would be rather low.

This way my calculations could come to an end, to my great relief. Fragmentary pressure exerted by one group of particles was four times the energy per unit of volume (which corresponds to how many particles with that velocity are in such volume inside the box, statistically). Taking the same unit for every group, I received a total of four times the entire kinetic energy per this volume. More precisely: the energy corresponding to the momentum facing a single wall of the box. Because this was only one sixth of the total energy of movement of the gas particles, the pressure was ultimately equivalent to the total energy multiplied by four and divided by six, and this is two thirds of the intriguing "energy density".

This was exactly what I was looking for. The energy density of the molecule movement is something one can imagine (and maybe calculate) in any region of space, regardless of whether the particles hit something or not. This helped me to understand how to talk about air pressure in any place where there is nothing beside it.

And I was delighted with all the details that were in line with my earlier intuitive understanding of pressure. If we put twice as many identical particles of gas into the box as it contained before, the total energy contained in its volume would double, and so would the pressure. This is how pumping more air into a container increases the pressure inside. A similar effect would be to place the same portion of gas in a box of half the size - this would be compression. Finally, the pressure would also increase if the average energy of particles increased, which is connected with higher temperature.

I confess, this computation has led me a bit astray. Although it was somewhat instructive, it did not by itself bring me closer to the solution of my initial problem. So with regret I abandoned further reflections on the density of energy and brought my thoughts back to the jar. Only to immediately replace it with an air-filled box, for simplicity.

The particles hitting the walls from the inside give them a lot of momentum - and yet the sides of the box do not fall off. The walls are held together as they are parts of a solid structure and even though each is pushed in a different direction, they are all in balance. Indeed, total exchange of momentum between the air and the container is zero, the opposite walls cancel out each other. However, forces acting on individual walls could still tear the box apart if it was not strong enough. That this usually does not happen is because the air is not only inside the box, but also outside. If the pressure is the same on both sides of the wall, the doses of momentum it receives are perfectly compensated (in other words, the forces are balanced). The box would only be threatened if it was placed in a vacuum or if it was inflated to a pressure much higher than outside.

Here I got to the key issue: is the pressure everywhere around me the same? The first thought that came to my mind was that if there is a place where there is less pressure, the air starts moving in that direction. Is this really the case?

I imagined that I had an additional wall in the box dividing it into two chambers, each of them with a different pressure inside. When I remove the barrier, the particles that would otherwise bounce off it will flow freely into the second cell (for my comfort and mental balance I once more decided to ignore the fact that they may also clash with each other). If the air in both chambers was the same in every aspect, then for any particle leaving the chamber the identical one would arrive from the other side - the effect would be the same as if they were simply bouncing off the now non-existent wall. So the exchange of momentum would then be exactly the same as in the presence of a barrier and the energy on both sides would remain the same - everything would be in balance.

What is different is when the second chamber has a higher pressure. Then different, more accelerated particles may arrive to replace the ones that have left the first cell. Or they may have similar speeds, but there may be many more of them - the details may vary, but in any variant more energy is going to flow into the first chamber. And it is going to continue until the density of that energy on both sides is equal. And it does not need to be associated with air flowing to the place with the lower pressure - it can even be the other way round. If there are many slowly moving particles in the first chamber (i. e. dense, but cool air) and in the second one relatively few very vigorous ones (i. e. a very hot and diluted gas), it might turn out that energy will rise in the first chamber, although more particles will flow into the second one.

This thought experiment has shed new light on what the pressure difference may look like, but there was an important detail I had not yet taken into account: gravity. It keeps pulling down all the things around me, by continuously giving each of them doses of momentum appropriate to its mass - as every object, no matter how light or heavy, is accelerated with the same speed by the Earth's attraction. They say this has something to do with the geometry of space, but this is yet another story for a different time.

Gas molecules are no exception: those that fly upwards are slowed down, while those flying downwards are accelerated even further. Total energy then remains unchanged - what some particles gain, others lose. But when looking at the momentum, it is no longer distributed the same way. If I put my hypothetical box on the kitchen table, gravity acting on the air inside would make more momentum be directed towards the bottom of the box than the top. And interestingly this difference of forces would correspond exactly to how much the air inside the box weighs.

So it seems that the bottom of the box should be affected by a greater force. And the density of energy next to it would also be larger - when it is taken away from particles flying upwards and instead given to those going down, it is natural that more energy is going to accumulate below. To state the matter simply, the air pressure would be higher at the bottom than at the top.

This is how it is everywhere. In my kitchen the air pressure is greater near the floor than under the ceiling. And a tiny little bit bigger under the table top than just above it. Just go a few millimetres higher and the pressure changes according to how much air in this thin layer weighs.

This also explains the origin of the high pressure that we face every day. It reflects the weight of all the air that fills the space above our heads - to the clouds and even higher. And when we climb a high mountain, or move a few kilometres up in any other way available to us, there will be less air above us and the pressure will drop significantly.

But what was more interesting to me at the moment were the differences in the small scale, like between the levels of my waist and my breast. This is seemingly unnoticeable, but sometimes its effects can be seen - it is enough to take a balloon and fill it with helium. The air pressure from above will be slightly lower than the pressure from below - there will be a difference in forces corresponding to the weight of the section of air at that height. But the balloon itself weighs less than that, since helium is much lighter than the air we breathe - so this difference of forces is going to outweigh the Earth's attraction, and the balloon is going to rise instead of falling.

This gradient of pressure does not shy away from my jar either. When I hold it in my hand, the pressure on the lid is lower than that on the bottom (unless I turn it upside down). If I put it on a scale, I thought, the air does not push it from below, but instead it presses on the weighing pan from underneath. Either way, the force from below is a bit bigger and the jar seems to be slightly lighter than it really is. And if I lay it on the side for a variety, the difference in pressure is going to decrease and the weight measurement could change if it were accurate enough. I even started wondering whether I had something very high and narrow at home, which I could weigh first standing and then lying on the side, and perhaps notice a difference even on a simple kitchen scale. Not seeing anything like this around me, I left the idea for later.

The measurement of the weight may then depend even on the weather - on some days the air is more dense, so the difference in pressure corresponding to these several centimetres of height varies. I started to suspect that there are days that are better than others for weighing things.

My thoughts had finally nowhere else to go but the air inside my jar. As everywhere else, its pressure changes with height - it pushes more on the bottom of the jar than on the lid from below. And since the difference in these forces corresponds to the weight of the air, when I put the jar on a scale I weigh it together with the air enclosed within. However this is to some extent compensated by the fact that the difference in pressure outside the jar lowers the measurement. If the air inside has exactly the same pressure as outside (e. g. when the lid is unsealed - but tightening it would not change anything at this point any more), these two effects might nearly perfectly cancel out. But if I moved a jar to a place with a different conditions, the results might vary.

It seemed to me earlier that there should have been a simple answer to my original question - but apparently there is not. It could be said that the answer is 'neither yes nor no' but without additional explanation such reply usually sounds like mocking the one asking. Thus it is better to tell the complete story. And maybe sum it up by saying: well, that is the way it is.