6.5. Open systems[[Student version, January 17, 2003]] 187
Reviewing the algebra in the Example on page 184, we then see that the change in entropy for
system “a” is just ∆Sa=−NkLBδ. Requiring that this expression be positive would imply that
the piston always moves to the right—but that’s absurd. If the spring exerts more force per area
than the gas pressure, the piston will surely moveleft,reducing the entropy of subsystem “A.”
Something seems to be wrong.
Actually, we already met a similar problem in Section 1.2.2 on page 10, in the context of
reverse osmosis. The point is that we have so far looked only at thesubsystem’sentropy, whereas
the quantity that must increase is thewhole world’sentropy. We can get the entropy change
of system “B” from its temperature and Equation 6.9: T(∆SB)=∆EB =−∆Ea.Thusthe
quantity that must be positive in any spontaneous change of state is notT(∆Sa)but instead
T(∆Stot)=−∆Ea+T(∆Sa). Rephrasing this result, we find that the Second Law has a simple
generalization to deal with systems that are not isolated:
If we bring a small system “a”intothermal contact with a big system “B”in
equilibrium at temperatureT,then “B”will stay in equilibrium at the same
temperature (“a”istoosmall to affect it), but “a”will come to a new equilib-
rium, which minimizes the quantityFa=Ea−TSa.(6.14)
Thus the piston finds its equilibrium position, and so is no longer in a position to do mechanical
work for us, when its free energy is minimum. The minimum is the point whereFais stationary
under small changes ofL,orinother words when ∆Fa=0.
The quantityFaappearing in Idea 6.14 is called theHelmholtz free energyof subsystem “a.”
Idea 6.14 explains the name “free” energy: WhenFais minimum, then “a” is in equilibrium and
won’t change any more. Even though the mean energy〈Ea〉isn’t zero, nevertheless “a” won’t do
any more useful work for us. At this point the system isn’t driving to any state of lowerFa,and
can’t be harnessed to do anything useful for us along the way.
So a system whose free energy isnotat its minimum is poised to do mechanical or other useful
work. This compact principle is just a precise form of what was anticipated in Chapter 1, which
argued that the useful energy is the total energy reduced by some measure of disorder. Indeed,
Idea 6.14 establishes Equation 1.4 and Idea 1.5 on page 7.
Your Turn 6c
Apply Idea 6.14 to the system in Figure 6.4b, find the equilibrium location of the piston, and
explain why that’s the right answer.The virtue of the free energy is that it focuses all our attention on the subsystem of interest to us.
The surrounding system “B” enters only in a generic, anonymous way, throughone number,its
temperatureT.
Fixed-pressure case Another way in which “a” can interact with its surroundings is by expand-
ing its volume at the expense of “B.” We can incorporate this possibility while still formulating the
Second Law solely in terms of “a.”
Imagine, then, that the two subsystems have volumesVaandVB,constrained by a fixed total
volume:Va+VB=Vtot.First we again define temperature by Equation 6.9, specifying now that
the derivative is to be taken at fixed volume. Next we define pressure in analogy to Equation 6.9:
Aclosed system has
p=T
dS
dV∣∣
∣∣
E