262 Chapter 8. Chemical forces and self-assembly[[Student version, January 17, 2003]]
for the chemical species in question at some standard concentrationc 0. Usually we don’t need
to worry about the details of the solvent interactions; we’ll regardμ^0 as just a phenomenological
quantity to be looked up in tables.
Forgases the standard concentration is taken to be the one obtained at atmospheric pressure
and temperature, roughly one mole per 22L.Inthis book, however, we will nearly always be
concerned withaqueous solutions(solutions in water), not with gases. For this situation the
standard concentrations are all taken to be^1 c 0 =1M≡ 1 mole/L,and we introduce the shorthand
notation [X]=cX/(1M)for the concentration of any molecular species X in molar units. A solution
with [X]=1 is called a “onemolarsolution.”
Youcan generalize Equation 8.3 to situations where in addition to,each molecule also has
an extra potential energyU(z)depending on its position. For example a gravitational field gives
U(z)=mgz.Amore important case is that of an electrically charged species, whereU(z)=qV(z).
In either case we simply replaceμ^0 byμ^0 +U(z)inEquation 8.3. (In the electric case some authors
call this generalizedμtheelectrochemical potential.) Making this change to Equation 8.3 and
applying the matching rule (Equation 8.2) shows that in equilibrium every part of an electrolyte
solution has the same value ofc(z)eqV(z)/kBT.This result is already familiar to us—it’s equivalent
to the Nernst relation (Equation 4.25 on page 126).
Setting aside these refinements, the key result of this subsection is that we have found a quantity
μdescribing theavailability of particlesjust asT describes the availability of energy; for dilute
systems it separates into a part with a simple dependence on the concentration, plus a concentration-
independent partμ^0 (T)involving the internal energy of the molecule. More generally, we have
afundamental definition of this availability (Equation 8.1), and a result about equilibrium (the
matching rule, Equation 8.2), applicable to any system, dilute or not. This degree of generality is
important, since we know that the interior of cells is not at all dilute—it’s crowded (see Figure 7.2
on page 222).
The chemical potential goes up when the concentration increases (more molecules are available),
but it’s also greater for molecules with more internal energy (they’re more eager to dump that energy
into the world as heat, increasing the world’s disorder). In short:
Amolecular species will be highly available for chemical reactions if its concen-
trationcis big, or its internal energyis big.The chemical potential, Equation 8.3, describes the overall availability.
T 2 Section 8.1.1′on page 294 makes some connection to more advanced treatments, and to quan-
tum mechanics.
8.1.2 The Boltzmann distribution has a simple generalization accounting for particle exchange
From here it’s straightforward to redo the analysis of Section 6.6.1. We temporarily continue to
suppose that particles cannot interconvert, and a smaller system “a” is in equilibrium with a much
larger system “B.” Then the relative fluctuations ofNin “a” can be big, since “a” may not be
macroscopic; it could even be a single molecule. So we cannot just computeNfrom Equation 8.2;
the best we can do is to give theprobability distributionPjof its various possible statesj.System
“B,” on the other hand,ismacroscopic, and so has some equilibrium characterized byT andμ,
whose relative fluctuations of particle number are small.
(^1) With some exceptions—see Section 8.2.2 below.