264 Chapter 8. Chemical forces and self-assembly[[Student version, January 17, 2003]]
must be positive.
Sullivan: But wait! We’ve seen before how even that isn’t necessarily true, as long as somebody
pays the disorder bill. Remember our osmotic machine; it can draw thermal energy out of the
environment to lift a weight (Figure 1.3a on page 11).
Sullivan has a good point. The discussion above, along with the defining Equation 8.1, implies
that when the reaction takes one step converting type-1 to type-2, the world’s entropy changes^2 by
(−μ 2 +μ 1 )/T.The Second Law says that a net, macroscopic flow in this direction will happen if
μ 1 >μ 2 .Soitmakes sense to refer to the difference of chemical potentials as a “chemical force”
driving isomerization. In the situation sketched above, 1 < 2 ,butis only a part of the chemical
potential (Equation 8.3). If theconcentrationof the low-energy isomer is high (or that of the
high-energy isomer is low), then we can haveμ 1 >μ 2 ,and hence a net flow 1→2! And indeed
some spontaneous chemical reactions are “endothermic” (heat-absorbing): Think of the chemical
icepacks used to treat sprains. The ingredients inside the ice pack spontaneously put themselves
into a higher-energy state, drawing the necessary thermal energy from their surroundings.
What does Sullivan mean by “pays the disorder bill?” Suppose we prepare a system where
initially species #1 far outnumber #2. This is a state with some order. Allowing conversions
between the isomers is like connecting two tanks of equal volume but with different numbers of gas
molecules. Gas whooshes through the connection to equalize those numbers, erasing that order. It
can whoosh through even if it has to turn a turbine along the way, doing mechanical work. The
energy to do that work came from the thermal energy of the environment, butthe conversion from
thermal to mechanical energy was paid for by the increase of disorder as the system equilibrated.
Similarly in our example, if #1 outnumbers #2 there will be an entropic force pushing the conversion
reaction in the direction 1→2, even if this means going “uphill,” that is, raising the stored chemical
energy. As the reaction proceeds, the supply of #1 gets depleted (andμ 1 decreases) while that of
#2 gets enriched (μ 2 increases), untilμ 1 =μ 2 .Then the reaction stalls. In other words,
Chemical equilibrium is the point where the chemical forces balance. (8.6)More generally, if mechanical or electrical forces act on a system, we should expect equilibrium
when the net ofallthe driving forces, including chemical ones, is zero.
The discussion just given should sound familiar. Section 6.6.2 on page 194 argued that two
species with a fixed energy difference ∆Ewould come to an equilibrium with concentrations related
byc 2 /c 1 =e−∆E/kBT(Equation 6.24 on page 194). Taking the logarithm of this formula shows
that for dilute solutions it’s nothing but the condition thatμ 2 =μ 1 .Ifthe two “species” have many
internal substates, the discussion in Section 6.6.4 on page 198 applies; we just replace ∆Ebythe
internalfreeenergy difference of the two species. The chemical potentials include both the internal
entropy and the concentration-dependent part, so the criterion for equilibrium is stillμ 2 =μ 1.
There is another useful interpretation of chemical forces. So far we have been considering
an isolated system, and discussing the change in its entropy when a reaction takes one step. We
imagined dividing the system into subsystems “a” (the molecule undergoing isomerization) and “B”
(the surrounding test tube), and required that the entropy of the isolated system a+B increase.
But more commonly a+B is in thermal contact with an even larger world, as for example when a
reaction takes place in a test tube, sitting in our lab. In this case the entropy change of a+B will
notbe (−μ 2 +μ 1 )/T,because some thermal energy will be exchanged with the world in order to
(^2) It’s crucial that we definedμas a derivative holding total energy fixed. Otherwise (−μ 2 +μ 1 )/Twould describe
an impossible, energy-nonconserving process.