6.4. The Second Law[[Student version, January 17, 2003]] 183
Figure 6.3:(Schematic.) Expansion of gas into vacuum.thermodynamics:^2
Whenever we release an internal constraint on an isolated macroscopic system
in equilibrium, it eventually comes to a new equilibrium whose entropy is at
least as great as before.(6.11)
After enough time has passed to reestablish equilibrium, the system will be spending as much
time in the newly available states as in the old ones: Disorder will have increased. Entropy isnot
conserved.
Notice that “isolated” means in particular that the system’s surroundings don’t do any mechan-
ical work on it, nor does it do any work on them. Here’s an example.
Example Suppose we have an insulated tank of gas with a partition down the middle,N
molecules on the left side, and none on the right (Figure 6.3). Each side has volume
V.Atsome time a clockwork mechanism suddenly opens the partition and the gas
rearranges. What happens to the entropy?
Solution:Because the gas doesn’t push on any moving part, the gas does no work;
since the tank is insulated, no thermal energy enters or leaves either. Hence the
gas molecules lose no kinetic energy. So in Equation 6.6 nothing changesexceptthe
factorVN,and the change of entropy is∆S=kB[
ln(2V)N−ln(V)N]
=NkBln 2, (6.12)which is always positive.The corresponding increase in disorder after the gas expands, ∆I,is(K/kB)∆S,whereK=1/(ln 2).
Substitution gives ∆I=Nbits. That makes sense: Before the change we knew which side each
molecule was on, whereas afterward we have lost that knowledge. To specify the state to the
previous degree of accuracy, we’d need to specify an additionalNbinary digits. Chapter 1 already
made a similar argument, in the discussion leading up to the maximum osmotic work (Equation 1.7
on page 12).
Would this change ever spontaneously happen in reverse? Would we ever look again and find
allNmolecules on the left side? Well, in principle yes, but in practice no: We would have to wait
an impossibly long time for such an unlikely accident.^3 Entropy increased spontaneously when we
(^2) The “FirstLaw” was just the conservation of energy, including the thermal part (Section 1.1.2).
(^3) How unlikely is it? A mole occupies 22Lat atmospheric pressure and room temperature. IfV=1L,then the