8.2. Chemical reactions[[Student version, January 17, 2003]] 263
Let statejof the small subsystem have energyEj and particle numberNj.Wewantthe
probabilityPjfor “a” to be in statejregardless of what “B” is doing.
Your Turn 8b
Show that in equilibrium,
Pj=Z−^1 e(−Ej+μNj)/kBT, (8.5)
where now thegrand partition functionZis defined asZ=∑
je
(−Ej+μNj)/kBT.[Hint: Adapt
the discussion in Section 6.6.1.]The probability distribution you just found is sometimes called theGibbs,orgrand canonical,
distribution. It’s a generalization of the Boltzmann distribution (Equation 6.23 on page 193). Once
again, we see that most of the details about system “B” don’t matter; all that enters are two
numbers, the values of its temperature and chemical potential.
Thus indeed largeμmeans system “a” is more likely to contain many particles, justifying the
interpretation ofμas the availability of particles from “B.” It’s now straightforward to work out
results analogous to Your Turn 6g and the Example on page 198 (see Problem 8.8), but we won’t
need these later. (It’s also straightforward to include changes in volume as molecules migrate; again
see Problem 8.8.)
8.2 Chemical reactions
8.2.1 Chemical equilibrium occurs when chemical forces balance
Atlast we are ready to think about chemical reactions. Let’s begin with a very simple situation, in
which a molecule has two states (or isomers)α=1,2 differing only in internal energy: 2 > 1 .We
also suppose that spontaneous transitions between the two states are rare, so that we can think of
the states as two different molecular species. Thus we can prepare a beaker (system “B”) with any
numbersN 1 andN 2 welike, and these numbers won’t change.
But now imagine that in addition our system has a “phone booth” (called subsystem “a”) where,
like Superman and his alter ego, molecules of one type can duck in and convert (or isomerize) to
the other type. (We can think of this subsystem as a molecular machine, like an enzyme, though
wewill later argue that the same analysis applies more generally to any chemical reaction.)
Suppose type 2 walks in to the phone booth and type 1 walks out. After this transaction
subsystem “a” is in the same state as it was to begin with. Since energy is conserved, the big
system “B,” too, has the same total energy as it had to begin with. But now “B” has one fewer
type-2 and one more type-1 molecule. The difference of internal energies, 2 − 1 ,gets delivered to
the large system, “B,” as thermal energy.
No physical law prevents the same reaction from happening in reverse. Type-1 can walk into the
phone booth and spontaneously convert to type-2,drawingthe necessary energy from the thermal
surroundings.
Gilbert says: Of course this would never happen in real life. Energy doesn’t spontaneously organize
itself from thermal motion to any sort of potential energy. Rocks don’t fly out of the mud.
Sullivan: But transformations of individual molecules can go in either direction. If a reaction can
go forward, it can also go backward, at least once in a while. Don’t forget our buffalo (Figure 6.8
on page 195).
Gilbert: Yes, of course. I meant thenetnumber converting to the low-energy state per second