204 Chapter 6. Entropy, temperature, and free energy[[Student version, January 17, 2003]]
- Free energy: Consider a small system “a” of fixed volume, sealed so that matter can’t go in
or out. If we bring system “a” into thermal contact with a big system “B” in equilibrium atT,
then “B” will stay in equilibrium at the same temperature (“a” is too small to affect it), but
“a” will come to a new equilibrium, which minimizes its Helmholtz free energyFa=Ea−TSa
(Idea 6.14). If “a” is not macroscopic, we replaceEaby〈Ea〉and use Shannon’s formula for
Sa(Equation 6.32).
Suppose instead that small system “a” can also exchange volume (push on) the larger system,
and the larger system has pressurep.Then “a” will instead minimize its Gibbs free energy
Ga=Ea−TSa+pVa(Equation 6.16). When chemical reactions occur in water,Vais essentially
constant, so the difference betweenFandGis also essentially a constant. - Equipartition: When the potential energy of a subsystem is a sum of terms of the form
kx^2 (that is, an ideal spring), then each of the displacement variables shares average thermal
energy^12 kBTin equilibrium (Your Turn 6f). - Partition function: The partition function for system “a” in contact with a thermal reser-
voir “B” of temperatureT isZ=
∑
je
−Ej/kBT (Equation 6.33). The free energy can be
reexpressed asFa=−kBTlnZ.Further reading
Semipopular:
The basic ideas of statistical physics: Feynman, 1965, chapter 5; Ruelle, 1991, chapters 17–21
Intermediate:
Many books take a view of this material similar to the one here, for example: Schroeder, 2000,
and Feynman et al., 1996, chapter 5.
Heat engines: Feynman et al., 1963a,§44.
Technical:
Statistical physics: Callen, 1985, chapters 15–17; Chandler, 1987
Maxwell’s demon: Feynman et al., 1996; von Baeyer, 1999; Leff & Rex, 1990
More on optical tweezers: Bustamante et al., 2000; Mehta et al., 1999; Svoboda & Block, 1994
Another nearly two-state system: Finzi & Gelles, 1995