7. Entropy in Isolated Systems
An isolated system exchanges neither heat nor work with its surroundings:
\(Q=0\) and \(W=0\).
The second law imposes a strict constraint on the entropy change of the isolated system:
Entropy production and the “arrow of time”
In isolated systems there is no entropy flow across the boundary, so any entropy increase must come from
entropy production due to irreversibility (unrestrained expansion, mixing, friction, finite gradients, etc.).
This is one practical expression of the “arrow of time”: isolated systems naturally evolve toward more probable macrostates.
Free expansion of an ideal gas (classic irreversible example)
In free expansion, a gas expands into a vacuum without doing boundary work.
For the actual process, \(Q=0\) and \(W=0\).
For an ideal gas, internal energy depends only on temperature, so the temperature stays constant:
\( \Delta U=0 \Rightarrow \Delta T=0 \).
Even though no heat is transferred in the actual expansion, entropy still increases because the process is irreversible.
Since entropy is a state function, compute it using a reversible reference path between the same endpoints:
Mixing of ideal gases in an isolated container
Two different ideal gases initially occupy separate compartments \(V_A\) and \(V_B\) in an insulated container.
When the partition is removed, each gas effectively expands into the total volume
\(V_{\text{tot}}=V_A+V_B\) (again, compute \(\Delta S\) via reversible reference expansions):
If the gases are identical, the “entropy of mixing” is taken as zero in the standard resolution of the Gibbs paradox
(the final macrostate is not meaningfully new compared to the initial labeling).
Reversible limit in an isolated system
The equality case \(\Delta S_{\text{isolated}}=0\) corresponds to a reversible process with no entropy production.
In practice, any real process has some irreversibility, so isolated systems typically satisfy \(\Delta S_{\text{isolated}}>0\).
Why the calculator includes a graph and animation
- Particle-box visual: illustrates “ordered → disordered” behavior for free expansion and mixing.
- Entropy vs progress plot: shows monotone \(S(\tau)\) rising to \(\Delta S\) for irreversible processes, and flat for the reversible limit.
