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Question is inspired by a recent burst of perpetuum mobile-type questions. It would be nice if one could simply discard them all (or at least the huge class that assumes some kind of perfect vacuum to eliminate friction) by an argument that shows it's impossible to create a perfect vacuum. Intuitively, I have some hope that there will be a thermodynamics/statistical mechanics argument that we can never even eliminate air friction completely, thereby ruling out all these elaborate constructions requiring specific arguments from the get-go. My question is therefore twofold:

  1. Does it take infinite energy to create a perfect vacuum (in a macroscopic box)?
  2. If yes, can you include a derivation? If no, can you give an explicit construction with a finite amount of work being done?
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The answer is no, or at least it is in the classical vacuum sense. I also don't see a rationale for why creating a vacuum would require infinite energy.

An explicit construction is to use a solid-phase reactive chemical "getter" to eliminate (nearly) all gas molecules present; in experimental practice, virtually all man-made materials still outgas slightly, which means that for practical purposes a true vacuum is difficult to achieve. As an example, one of the highest vacuums made on Earth was at CERN, with a density of 1 molecule per $\text{cm}^3$. However, this inability to create perfect vacuum is a problem of material science, rather than a side effect of theoretical impossibility.

In interstellar space, vacuums can approach 1 molecule per liter, which for all practical intents and purposes is perfect vacuum.

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Given an ideal piston/cylinder, starting with the piston completely inserted and zero volume, the work to make a perfect vacuum is simply:

(distance the piston moves) X (force) = (distance) X (area of the piston) X (exterior pressure).

So the work to make a vacuum of volume V, is V X P, where P is the exterior pressure, such as atmospheric pressure.

However, all real materials have a vapor pressure unless they are at a temperature of absolute zero. Molecules or atoms of the material of which the box is constructed will enter the vapor phase at all non-zero tempertures. A perfect vacuum in a macrosopic box cannot be made because reaching absolute zero in a finite number of steps is contrary to the third law of thermodynamics.

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In quantum scale, particles are appearing and disappearing out of random everywhere all the time, meaning that if you are actually able to create a perfect vacuum at a macro scale, it would be instantly denied by the quantum scale. Also, in the quantum world, there's a small chance of random "teleportation" of any particle or atom from your container to any place in the Universe, creating a whole that would allow other particles to come in the system. Or the other way around, randomly bringing particles or atoms into the system.

Even using infinite energy, a Three-dimensional container can not be able to create a perfect vacuum in our Universe.

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Perfect vacuum can't be created. Even if you somehow get rid of all material particles, there still will be blackbody photons from the container, not to mention virtual gravitons.

Generally, you can't be 100% sure that some part of space is perfect vacuum - to know that you should measure precisely the energy of that region, but it's forbidden by Heisenberg's uncertainity principle: $\Delta E \Delta t \geq \hbar$ where $\Delta E$ is uncertainity of energy measurement. I'd say that making sure that something is perfect vacuum would take infinite time.

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Perpetual motion machines

Take a UV-grade fused silica vessel with 1 cm walls, annealing point 1140 C. Heat it to 900 C under hard vacuum both sides (turbopump plus titanium sublimation pump) with electrodeless low pressure 184.9 nm Hg vapor UV flood to dissociate everything not silica. Let that proceed for a few days, then flame off the connection. Cool. Cool in superfluid liquid helium. There's your "perfect" vacuum. It still has zero-point energy (Casimir effect) plus thermal emission.

Here's a "perfect" perpetual motion machine of the Second Kind. Why is it crap?

A hermetically isolated hard vacuum envelope contains two closely spaced but not touching, in-register and parallel, electrically conductive plates having micro-spiked inner surfaces. They are connected with a wire, perhaps containing a dissipative load (small motor). One plate has a large vacuum work function material inner surface (e.g., osmium at 5.93 eV). The other plate has a small vacuum work function material inner surface (e.g., n-doped diamond "carbon nitride" at 0.1 eV). Above 0 kelvin, spontaneous cold cathode emission runs the closed isolated system. Emitted electrons continuously fall down the 5.8 volt potential gradient. Evaporation from carbon nitride cools that plate. Accelerated collision onto osmium warms that plate. Round and round. The plates never come into thermal equilibrium when electrically shorted. The motor runs forever.

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