Interpretation of Entropy on the Microscopic Scale - The Connection between
Randomness and Entropy
1.D.1 Entropy Change in Mixing of Two Ideal Gases
Consider an insulated rigid container of gas separated into two halves by a heat conducting
partition so the temperature of the gas in each part is the same.
One side contains air, the other
side another gas, say argon, both regarded as ideal gases.
The mass of gas in each side is such
that the pressure is also the same.
The entropy of this system is the sum of the entropies of the two parts:
Suppose the partition is taken away so the gases are free to diffuse throughout the volume.
an ideal gas, the energy is not a function of volume, and, for each gas, there is no change in
(The energy of the overall system is unchanged, the two gases were at the same
temperature initially, so the final temperature is the same as the initial temperature.)
entropy change of each gas is thus the same as that for a reversible isothermal expansion from
the initial specific volume
to the final specific volume,
For a mass
of ideal gas, the
entropy change is
The entropy change of the system is
Equation (D.1.1) states that there is an entropy increase due to the increased volume that each
gas is able to access.
Examining the mixing process on a molecular level gives additional insight.
Suppose we were
able to see the gas molecules in different colors, say the air molecules as white and the argon
molecules as red.
After we took the partition away, we would see white molecules start to move
into the red region and, similarly, red molecules start to come into the white volume.
watched, as the gases mixed, there would be more and more of the different color molecules in
the regions that were initially all white and all red.
If we moved further away so we could no
longer pick out individual molecules, we would see the growth of pink regions spreading into the
initially red and white areas.
In the final state, we would expect a uniform pink gas to exist
throughout the volume.
There might be occasional small regions which were slightly more red
or slightly more white, but these fluctuations would only last for a time on the order of several
In terms of the overall spatial distribution of the molecules, we would say this final state was
more random, more mixed, than the initial state in which the red and white molecules were
confined to specific regions.
Another way to say this is in terms of
“disorder”; there is more
disorder in the final state than in the initial state.
One view of entropy is thus that increases in
entropy are connected with increases in randomness or disorder.
This link can be made rigorous
and is extremely useful in describing systems on a microscopic basis.
While we do not have
scope to examine this topic in depth, the purpose of Section 1.D is to make plausible the link