260 Chapter 8. Chemical forces and self-assembly[[Student version, January 17, 2003]]
In short, the machines of interest to us exchange both energyand moleculeswith the outside
world. To get started understanding chemical forces, then, we first examine how a small subsystem
in contact with a large one chooses to share each kind of molecule, temporarily neglecting the
possibility of interconversions among the molecule species.
8.1.1 μmeasures the availability of a particle species
We must generalize our formulas from Chapter 6 to handle the case where two systems, “A”
and “B,” exchange particles as well as energy. As usual, we will begin by thinking about ideal
gases. So we imagine an isolated system, for example, an insulated box of fixed volume withN
noninteracting gas molecules inside. LetS(E, N)bethe entropy of this system. Later, when we
wantto consider several species of molecules, we’ll call their populationsN 1 ,N 2 ,...,orgenerically
Nα,whereα=1, 2 ,....
The temperature of our system at equilibrium is again defined by Equation 6.9,T^1 =ddSE.Now,
however, we add the clarification that the derivative is taken holding theNα’s fixed:T−^1 =ddSE
∣∣
Nα.
(Take a moment to review the visual interpretation of this statement in Section 4.5.2 on page
120.) Since we want to consider systems that gain or lose molecules, we’ll also need to look at the
derivatives with respect to theNα’s: Let
μα=−T dS
dNα∣∣
∣∣
E,Nβ,β=α. (8.1)
Theμα’s are calledchemical potentials. This time, the notation means that we are to take the
derivative with respect to one of theNα’s, holding fixed both the otherN’s and the total energy
of the system. Notice that the number of molecules is dimensionless, soμhas the same dimensions
as energy.
Youshould now be able to show, exactly as in Section 6.3.2, that when two macroscopic sub-
systems can exchangebothparticles and energy, eventually each is overwhelmingly likely to have
energy and particle numbers such thatTA=TBand the chemical potentials match for each species
α:
μA,α=μB,α. matching rule for macroscopic systems in equilibrium (8.2)When Equation 8.2 is satisfied we say the system is inchemical equilibrium.Just asTA−TBgives
the entropic force driving energy transfer, soμA,α−μB,αgives another entropic force driving the
net transfer of particles of typeα.For instance, this rule is the right tool to study the coexistence
of water and ice at 0◦C;water molecules must have the sameμin each phase.
There is a subtle point hiding in Equation 8.1. Up to now, we have been ignoring the fact that
each individual molecule has some internal energy,for example the energy stored in chemical
bonds (see Section 1.5.3 on page 22). Thus the total energy is the sumEtot=Ekin+N 1 1 +···of
kinetic plus internal energies. In an ideal gas, the particles never change, and so the internal energy
is locked up: It just gives a constant contribution to the total energyE,which we can ignore. In this
chapter, however, we will need to account for the internal energies, which change during a chemical
reaction. Thus it’s important to note that the derivative in the definition of chemical potential
(Equation 8.1) is to be taken holding fixed thetotalenergy, including its internal component.
Toappreciate this point, let’s work out a formula for the chemical potential in the ideal-gas
case, and see howcomes in. Our derivation of the entropy of an ideal gas is a useful starting