Gases in steady state

A steady state means that an equilibrium has been reached. When an equilibrium has been reached, there is no change in the overall system. For example if you have two bottles (one very large and one small) connected via a pipe fitted with a tap. Initially the tap is closed and the larger bottle is filled. Then the tap is opened to allow water to pass into the smaller bottle.

As this is happening, you will notice that the water is flowing. That means:

1. The water molecule's speed near the tap is larger than the one at the top of the bottle. That means there's a preference for water molecules to flow to the smaller bottle. The distribution of water molecules in the system is changing.

2. The distribution of water molecules also depend on direction. The distribution is increasing in the direction of the smaller bottle and decreasing in the larger bottle.

3. Also, the distribution is changing with time. With time the smaller bottle will start gaining more molecules and the larger bottle will lose molecules.

Eventually a point will be reached when the larger bottle is full to its ceiling and cannot take anymore water. This is the steady state. In the pipe the water molecules are still moving into the smaller bottle (due to random motion of fluid particles) but at the same time the same number of molecules are moving from the smaller bottle to the larger bottle. That means the net flow becomes zero and the distribution of molecules is no longer dependent on the above three properties.

For your gas system when a steady state is reached you can imagine that the box is made up of smaller imaginary compartments. The rate of filling those compartments is equal to the rate of emptying those compartments because of the random motion of the particles. At steady state there is no preference for the particles to be present at a particular place.

I see this is an old post, but would be interesting to know what text preceded the quote given. The text leads one to believe that a definition of "equilibrium" is being given, but the author uses the term "steady state". Thermal physics defines these two terms differently.

From Reif, Fundamentals of Statistical and Thermal Physics, Chapter 12:

Finally it is worth adding a very general comment about the distinction between equilibrium and steady-state situations. An isolated system is said to be in equilibrium when none of its parameters depends on the time. It is, however, also possible to have a non-equilibrium situation where a system A, which is not isolated, is maintained in such a way that all of its parameter are time-independent. The system A is then said to be in a "steady state", but the situation is not one of equilibrium, since the combined isolated system A0 consisting of A and its surrounding [A'] is not in equilibrium, i.e., since the parameters of A' vary in time.

He then gives an example of a copper rod, whose two ends are held at different temperatures by immersion in heat reservoirs, and enough time has passed so that the temperature distribution along the length of the rod is constant in time - i.e. it is time independent. That is not true, though of the heat reservoirs that are needed to maintain this state of affairs. Reif points out that if one waits long enough the temperatures of the reservoirs will become equal. "Steady state" is different from "equilibrium". Thus the question here is, is the gas described isolated or not?