I have $N$ ${\rm H}^+$ gas molecules in a sealed, electrically insulated container. What equation can I use to accurately calculate the pressure of the gas?
It seems to me that $PV = nRT$ will not apply to this scenario as the electrical forces between the molecules are significant, especially as the density of the gas increases.
Additional Thoughts
It is a complicated problem because there are a lot of interactions between energies and forces, lots of questions that need to be considered, for example:
- Is the pressure at every point in the container the same, or does the shape of the container affect the pressure as the molecules attempt to be distant from each other?
- Does the repulsive force of the gas cause the molecules to stay more or less stationary?
- How much energy is stored in the electric fields between the molecules?
Why this is interesting
I am trying to determine how effective this device would be as a capacitor/battery. How does this device compare to Li-Ion batteries in storage capacity? Could this be practically used as a piezoelectric or linear actuator?
See: http://imgur.com/Yb5fZ2x, http://imgur.com/UZbJjKB
Any insight would be appreciated!
Best Answer
Pressure inside the container: zero
Effective pressure on the container: proportional to$\frac{N^2}{r^4}$
(where $r$ indicates the size of the container)
The situation: There are $N$ protons inside an otherwise empty container.
Assumptions: The container does not interact with the protons except for completely stopping them from moving outside (essentially, a potential wall). The container is perfectly insulated, not allowing outside influence.
The charges in vacuum will act as in a conducting material, since there is nothing bonding them to their immediate vicinity. Thus, the protons will have migrated to the edge of the container. (See Gaussian surfaces.) This image is quite different from normal gas molecules. This is because gaseous groups of atoms simply move about and bump into each other. A group of charged particles, however, will "want" to be as far away from each other as possible; minimising the force between them.
What, then, is left over of our concept of "pressure"? When wanting to know the pressure of something in a container, it is usually to know the pressure on the container walls, or wanting to know what pressure a gas feels when added into the container. However, these protons do not interact with matter as gas usually does in the macroscopic world. I don't know my quantum mechanics (or chemistry?) well enough to consider these kinds of interactions, which is why we will only consider a "potential wall", and not a material one. (Perhaps Steve B's answer brings some light on the topic?)
Pressure of a gas within the container
As with all Gaussian surfaces, the electric field created by these photons is zero on every point inside the container. This means that aside from the chemical effects of the gas molecules directly bumping into them (which may or may not be significant, I don't know), the protons have no effect on any gas molecules inside of the container. This may be interpreted as the protons having no contribution to the pressure inside the container.
Pressure on the container itself
First of all, the pressure experienced by the container is equal to the pressure experienced by the protons on the edge of the container. This is true simply because of Newton's third law, and the fact that the protons are not moving perpendicularly to the surface of the container. We denote this value as $p$.
Second of all, the pressure on the protons is equal to the energy density caused by the electrical field produced by the protons themselves. This is as true for charged particles as it is for gaseous ones[1].
Luckily, the energy density $U_E$ of an electrical field $E$ in vacuum is easily found[2]: $$p = U_E = \frac{1}{2}\varepsilon_0 E^2$$
Thus, if the electric field on the surface of the container can be found (which depends only on the number of protons and the shape/size of the container), the pressure can be calculated for any area on the surface of the container.
Example: Spherical container
The electric field on the surface of a conducting sphere is also well known[3]: $$E = \frac{Ne}{4\pi\varepsilon_0r^2}$$
(where $e$ is the elementary charge; the charge of a proton, and $r$ is the radius of the sphere)
This results in a pressure across the surface of the sphere of: $$p_{sphere} = \frac{N^2e^2}{32\pi^2\varepsilon_0r^4}$$
The pressure experienced by differently shaped containers will not be so easily found; the pressure would not even be constant across the surface, as opposed to regular gaseous pressure. However, pressure on differently shaped containers will still follow the same proportionality to the number of protons $N$ and the size of the container $r$.
Edit:
As per rmhleo's comment, it is true that temperature will likely cause a distribution of protons with more at the edges and less in the centre. However, this effect should be minimal. Gas can extend space because the particles are neutral. Once a ranged force is present between these particles, however, the particles will clump up. An example of this would be the Earth's atmosphere (see that thin blue line?). Since the electromagnetic force is much, much stronger than gravity, this effect is observable even at scales much smaller than planetary.
To get a rough idea, the interior of the container is to the Gaussian surface as outer space is to Earth's atmosphere.