A confusion regarding gases

Question

A gas molecule is not affected by the gravity and can move anywhere in the confined atmosphere; yet Earth's atmosphere as a whole is retained by the same gravity. To make a simili i can move my hand at my will, but cannot make a hair on my hand fall off by my will. I have a control over the body as a whole, but does not have a control over minute part. Keeping this simili aside, what could be a theoretical explanation on the basis of physical forces, i wonder; help is requested.

Answer 1

The correct resolution of your question is that gravity does affect gas molecules but not so much that it matters to a measurable extent when doing laboratory-scale experiments that don't involve weighing the gas.

On a planetary scale the attraction matters as it is what provides an atmosphere. Gas molecules are attracted to the earth and there is a gradient of concentration (which we see as atmospheric pressure) when we travel a large distance vertically. The air at the top of Everest is about 1/3 as dense as it is at sea level. But Everest is nearly 9,000m above sea level. You don't normally have to worry about that distance in a laboratory as few have ceilings >10m above the floor.

It is also easy to see that gravity affects gases by weighing a vessel containing a gas compared to the same vessel in a vacuum. The gas adds to the weight and that is how early chemists worked out how heavy different gases were, an important part of the early understanding of chemical reactions.

If you are doing calculations about gas properties you worry about the pressure and temperature. But, often, pressure is atmospheric which doesn't vary enough across a lab for gravity to be taken into account.

Answer 2

We do have an effect of gravity. But, as we will see, it is very small unless you are dealing with heights much larger than you are likely to find in a laboratory.

Say you have an isothermal column of nitrogen gas. This is not a good model for the actual profiles in our atmosphere, of course, but it sets the scale of heights over which we can expect gravity to have an effect.

The equation for static pressure in the gas is the same as it would be for a liquid:

$\dfrac{dP}{dz}=-\rho g$

$P=\text{ pressure}$

$z=\text{ height}$

$\rho=\text{ density}$

$g=\text{ acceleration of gravity}$

We put in the Ideal Gas Law:

$P=\dfrac{\rho RT}{M}$

$R=\text{ gas constant}$

$T=\text{ absolute temperature}$

$M=\text{ molecular mass}$

Combining these two equations gives a differential equation for the debsity distribution which we can solve:

$\dfrac{d\rho}{dz}=-\dfrac{\rho g M}{RT}$

$\color{blue}{\rho=\rho_0\exp(-\dfrac{gMz}{RT})}$

Now put in numbers. Remember that everything is to be is SI units, so your molecular weight may seem a little weird:

$g=9.81\text{ m/s}^2$

$M=0.0280 \text{ kg/mol}$

$R=8.31\text{ J/(mol K)}$

$T=298\text{ K}$

Then

$\rho=\rho_0\exp(-1.11×10^{-4}\text{ m}^{-1}z)$

To get the exponential argument equal to 0.05, a 5% decrease in density, you need a height of $z=450\text{ m}$, compared with $416\text{ m}$ for the height of the Willis Tower observation deck in Chicago. So unless your laboratory workspace is as tall as the Willis Tower, you are not likely to see the effect of gravity versus the forces that drive molecular motion in gases.

To get large gravity effects, your gas column must be so tall that you would see the temperature change in the real atmosphere. This is no surprise because gravity ultimately drives temperature as well as pressure changes in our atmosphere (at least the troposphere). Meteorologists have to couple both effects in their models, precisely because you need that Willis Tower level of height, or more, before either temperature or pressure changes noticeably.