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A gas is initially confined to a small portion of an empty container.

At time $t=0$ gas is allowed to expand to homogeneously occupy the container after some time $t$.

What is the time $t$?

This example is used to introduce entropy in undergraduate courses, because the homogenously distributed gas is entropically favored. But the expansion is driven my mechanical forces due to pressure imbalance at the gas-vacuum interface, so expansion rate should be convective and much faster than diffusive transport.

An example eliminating mechanical forces could be gas where particles in the two halves of the container are differently labeled (but have same physical properties), and they would homogenize at the diffusive rate.

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  • $\begingroup$ The expansion is not homogeneous. $\endgroup$
    – Chet Miller
    Commented May 18 at 10:37
  • $\begingroup$ @ChetMiller I read the OP as saying after time $t$ the gas is homogeneously in the container, not that the gas expands homogeneously at equilibrium $\endgroup$
    – Bob D
    Commented May 18 at 12:22
  • $\begingroup$ The initial expansion is "mechanical", some internal energy changes into macroscopic kinetic energy and entropy does not increase substantially; it's similar to a spring freely expanding after constraint keeping it squeezed is released. It is later in the decay of the flow and mechanical pressure waves that kinetic energy gets back into internal energy, and entropy increases substantially (the spring will stop vibrating and get hotter). $\endgroup$
    – Ján Lalinský
    Commented May 18 at 15:22
  • $\begingroup$ Calculating $t$ is not easy, it should depend on the initial and final volume, and be proportional to gas viscosity at the end $\eta$ (which is proportional to $\sqrt{k_B Tm}/d^2$, where $d$ is diameter of molecules), but the first constant of proportionality depends on details of the flow that are hard to calculate. Maybe some dimensional analysis could at least find how $t$ depends on $m,d,L,T$. $\endgroup$
    – Ján Lalinský
    Commented May 18 at 15:26
  • $\begingroup$ What you've proposed is essentially the setup for a shock tube experiment (if one side is totally evacuated). You should be able to find some approximate time scales and maybe formulas for it by researching this. $\endgroup$
    – RC_23
    Commented May 19 at 3:04

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