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if there's a large closed container, about a km in height, which is filled with a gas will there be a pressure differential created at the bottom of the container with respect to the top? like it does when the container is filled with a liquid

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  • $\begingroup$ Yes if there is gravity, like on a planet. $\endgroup$ Nov 17, 2018 at 18:51

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Yes. (See Earth's atmosphere for an example). The density distribution settles to one where pressure differential between adjacent heights is just enough to balance weight of gas between those heights.

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There is quiete a simple model that incorporates your thoughts: the scale height. Say, you are in this gas. When you travel the scale height, the pressure increases by a factor of $e$. This can be motivated as follows:

We want the exponential relation of height and pressure, such that $P=P_0\exp(-\frac{z}{H}$). This is also the solution to a differential equation of the form $\frac{\mathrm{d}P}{\mathrm{d}z}=-\frac{P}{H}$. But on the other hand it should also be true, that the change in pressure with height is proportional to the density times the gravitational acceleration -- higher density means higher pressure, but stronger gravitational pull should also imply higher pressure, so we have something like $\frac{\mathrm{d}P}{\mathrm{d}z}=-g\rho$. Bringing both equation together we find:

\begin{equation} \frac{P}{H}=g\rho \end{equation}

From an ideal gas and thermodynamics we know: $PV=nRT$ and by definition we have $\rho=\frac{M}{V}$, which leads to

\begin{align} \frac{nRT}{VH}&=g\frac{M}{V}\\ \frac{nRT}{H}&=gM\\ \frac{nRT}{gM}&=H \end{align}

So the pressure increases with $\exp(-\frac{z}{\frac{nRT}{gM}})=\exp(-\frac{zgM}{nRT})$.

So yes, there is always a pressure difference in a gas container, and the difference even increases exponentially!

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  • $\begingroup$ M/n is the molecular weight. $\endgroup$ Nov 18, 2018 at 5:00
  • $\begingroup$ @kalle: what you have shown is that if the pressure varies in an exponential way the decaying length is $nRT/gM$. However, this does not anwer the question if there should be or not a difference of pressure between top and bottom of the container. $\endgroup$ Nov 18, 2018 at 8:59

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