Will gravity create high pressure at the bottom of a container filled with a gas?

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

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

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.

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:

$$$$\frac{P}{H}=g\rho$$$$

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!

• M/n is the molecular weight. Nov 18, 2018 at 5:00
• @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. Nov 18, 2018 at 8:59