Physics question: pressure in a proton-filled chamber

[Winston Blake]

I'm curious about a little physics scenario that came into my head recently.

For an ordinary gas, the ideal gas law can tell you how much pressure would be required to hold a known mass of a known gas, inside a chamber of known volume, at a known temperature.

However, what if it was not a neutral gas, or even a plasma, but instead ionised hydrogen in an inert chamber with no electrons. I.e. just a 'proton gas', with all protons mutually repelling each other.

Given mass/volume/temperature, is there a simple relation to find the pressure required?

[Surlethe]

Off the top of my head, calculate the total electric repulsive force (i.e., force per particle * number of particles) and divide it by the surface area of the chamber, then add that to the ideal gas pressure? My guess is that the electric force will dominate.

[Simon_Jester]

Oh yeah. The space charge* forces are a lot more impressive than the purely kinetic pressure of the proton gas. This is basically a plasma physics problem; I wish you'd asked me four months from now because I'll hopefully be able to derive the equation quickly then...

Off the top of my head, temperature is going to be practically irrelevant, because for any substantial density of protons, the electrostatic potential (the energy you spent to cram that many positive charges into such a small space) will be much greater than the kinetic energy of the gas... unless the entire collection of protons is moving in a beam, in which case things are a little different.

I may be able to dig something out of the notes from the beam physics course I took last semester. Space charge figured prominently.

What are you making the container out of? Assuming it's made out of atoms, the proton gas should start pulling electrons off the walls in short order, which will have major effects on the nature of the gas.

*(technical term for electrostatic forces in a cloud of charged particles- I'm studying beam physics)

[Winston Blake]

My guess is that the electric force will dominate.

Off the top of my head, temperature is going to be practically irrelevant, because for any substantial density of protons, the electrostatic potential (the energy you spent to cram that many positive charges into such a small space) will be much greater than the kinetic energy of the gas.

So there's no way to contain the stuff at reasonable pressure? I was hoping the kinetic pressure would dominate so that the containment pressure could be minimised by reducing temperature.

What are you making the container out of? Assuming it's made out of atoms, the proton gas should start pulling electrons off the walls in short order, which will have major effects on the nature of the gas.

Something inert, I don't know what. At the moment I'm assuming some kind of box-like potential well, except the wall has a controllable 'temperature'.

I am open to ideas for containing protons densely without forming hydrogen. Perhaps a highly negatively charged rod with some means of preventing the protons from touching it? If it was current-carrying, could the B-field keep the protons mostly away from the rod, in a cylindrical cloud?

[Surlethe]

For a short period, but E-field drift will ultimately pull the protons toward the negative charge.

[Simon_Jester]

Off the top of my head, temperature is going to be practically irrelevant, because for any substantial density of protons, the electrostatic potential (the energy you spent to cram that many positive charges into such a small space) will be much greater than the kinetic energy of the gas.

So there's no way to contain the stuff at reasonable pressure? I was hoping the kinetic pressure would dominate so that the containment pressure could be minimised by reducing temperature.

I can't prove that the electrostatic pressure dominates without digging in my notes, but let's do a thought experiment:

At room temperature, the kinetic energy of a particle in an ideal gas is 1.5kBT, which works out to about 6*10-21 J.

What is the electrostatic repulsion energy between one proton in this gas and any one of its closest neighbors? Imagine that the density of protons in the gas is about the same as air at sea level: 1 kg/m3. That gives us ~6*1026 protons in the box, which means that the average spacing between protons is about 1.2 nm.

Now, what's the potential energy of a single proton brought within 1.2 nm of another single proton? This is given by U = kq2/R, where k is a physical constant, q is the proton charge, and R is the 1.2 nm average spacing. So: U = 1.9*10-19 J, about thirty times greater. And that's for one proton with one of its immediate neighbors. More realistically, it has four or six or so of those neighbors (150 times greater!), plus dozens and dozens of less immediate neighbors (!), plus... you get the idea.

If you somehow brought all these protons together to the density of one gram per liter by magic, then released them, all that potential would go kinetic... at which point the average kinetic energy of the protons would be equivalent to a temperature in the tens or hundreds of thousands of degrees. And they'd all be flying away from each other.

This is not going to be easy to contain. There's a good reason why all large conglomerations of matter we find in nature have a neutral net electric charge: the electrostatic repulsion between like charges is so great that it becomes overwhelming unless you introduce opposite charges to cancel it out.
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Something inert, I don't know what. At the moment I'm assuming some kind of box-like potential well, except the wall has a controllable 'temperature'.

If the box has magic force field walls, then this is at least possible, but containing a proton gas inside a force field is called a "tokamak," and you can look up how many troubles they've had trying to make those work. Of course, in a tokamak they're deliberately trying to force the temperature and pressure up to induce nuclear fusion, but the basic issues of instability and power consumption remain even at lower densities, I'd imagine.

I am open to ideas for containing protons densely without forming hydrogen. Perhaps a highly negatively charged rod with some means of preventing the protons from touching it? If it was current-carrying, could the B-field keep the protons mostly away from the rod, in a cylindrical cloud?

It's never going to make them easy to store.

What do you need the protons for, anyway?

[Winston Blake]

What do you need the protons for, anyway?

I'm using protons as a stand-in for exotic charged particles required by the tech of a reasonably-hard science fiction setting.

[Simon_Jester]

If they have a net electric charge, why can't you just have electrons orbiting them and store them as a neutral gas?

[Hawkwings]

If you want, you can store them as a neutral gas, then just shave off the electrons when you want to use them. Of course, whether this works or not for your exotic particles who knows?

[Winston Blake]

If they have a net electric charge, why can't you just have electrons orbiting them and store them as a neutral gas?

If you want, you can store them as a neutral gas, then just shave off the electrons when you want to use them. Of course, whether this works or not for your exotic particles who knows?

The basic idea is that this family of exotic particles (+, -, neutral) react with any nearby particles to release energy. The idea is that they are very heavy and unstable, but a quirk of future physics means they are metastable. This is the single element of unrealism, and all exotic technology (and the plot) flows from that, e.g. weapons, power sources and materials. The difficulty of storage and handling limits the powerful technology to a few competing, advanced factions, which are in the middle of an arms race, with vigorous espionage and deceptive scheming. Hence, no easy 'neutral gas' storage.

I had hoped that it would not be limited to magically advanced factions.

[Surlethe]

Why not tweak the physics so the particles can exist in neutral bound states, but they can only be made so in the presence of such-and-such a vacuum field, so to store them you have to be capable of manipulating vacua? And to liberate them, you have to be able to dump enough energy into the system so that they leave the bound state and the interact with each other/nearby particles.

[Simon_Jester]

Surlethe, could you define "vacuum field" for the benefit of the experimentalist(s) in the audience? I assume that it does not simply mean "a chamber containing no air or other matter."

[Surlethe]

I meant a field in a region absent a source charge. Was that incorrect terminology? I was thinking of something different then changed my mind halfway through the post.

[Simon_Jester]

I meant a field in a region absent a source charge. Was that incorrect terminology? I was thinking of something different then changed my mind halfway through the post.

I have no idea what the correct terminology is, because I am badly confused. I'm not clear on your intent, or on what sort of container you visualize as being possible for these neutral bound states.

Bear in mind that I do not know quantum electrodynamics, which may not be helping.

[Surlethe]

I'm just bad at communicating. I mean, instead of having the particles (+, -, n) stored alone, why not let the imaginary physics permit the +/- particles to be stored in bound states like electrons. To get them into the bound state, you have to boost them over some (imaginary) potential barrier so that the Coulomb potential takes over, then at very close distances the (imaginary) potential rises again. When they're bound, they're non-interacting unless a random collision happens to knock one out and it annihilates with a nearby particle or whatnot (you want to store them in a near-vacuum just to be safe). When they're unbound (by the imposition of a sufficiently strong electric field, maybe, or just a dose of high-frequency radiation?) they wander off to interact with other particles and blow up.

It's basically analogous to fission, except instead of the strong force and Coulomb repulsion you have the (imaginary) force, Coulomb repulsion, and then additional energy released when the unbound particles interact with normal matter in some (imaginary) way.

This could work politically in the story because (a) the energies required to generate and store these particles are very high, so only industrial giants (maybe ones building factories around neutron stars or some shit like that) can create them; (b) the energies required to liberate them are very high; and (c) it might introduce an interesting dynamic that if they're in their neutral bound state they might be available to lesser states, terrorists, and smugglers, sort of like radioactive material is today.

[Simon_Jester]

Yup. That works like a charm as an explanation. Seems good to me; thank you.

[Winston Blake]

It's basically analogous to fission, except instead of the strong force and Coulomb repulsion you have the (imaginary) force, Coulomb repulsion, and then additional energy released when the unbound particles interact with normal matter in some (imaginary) way.

Sounds like a good way forward. I shall call this bound state 'dalethonium' (future physicists switched to the Hebrew alphabet). However storing the neutral bound states may itself be a problem, since I imagine they would act like extremely heavy neutrons. Is it possible for such a state to have a strong magnetic moment, allowing it be confined by B-field gradients?

This could work politically in the story because (a) the energies required to generate and store these particles are very high, so only industrial giants (maybe ones building factories around neutron stars or some shit like that) can create them; (b) the energies required to liberate them are very high; and (c) it might introduce an interesting dynamic that if they're in their neutral bound state they might be available to lesser states, terrorists, and smugglers, sort of like radioactive material is today.

Dalethonium also conveniently eliminates the need for the neutral dalethon, which was previously temporarily created 'on demand' in the chambers of neutral particle beam weapons. Dalethonium can be instead be directly created by combining near-parallel positive and negative dalethon beams. The instability of all real-life 'onia' also gives a nice, natural reason for its short half-life. Neutral dalethons had to lack the metastability of charged dalethons so I could arbitrarily tune offensive & defensive space combat ranges.