206 Chapter 6. Entropy, temperature, and free energy[[Student version, January 17, 2003]]
Statistical Postulate for this purpose, and see if it gives experimentally testable results. For more
on this viewpoint, see Callen, 1985,§15–1; see also Sklar, 1993.
- The Statistical Postulate is certainly not graven in stone the way Newton’s Laws are. Point
(1) above has already pointed out that the dividing line between “statistical” and “deterministic”
systems is fuzzy. Moreover, even macroscopic systems may not actually explore all of their allowed
states in any reasonable amount of time, a situation called “non-ergodic behavior,” even though
they do make rapid transitions between some subset of their allowed states. For example, an iso-
lated lump of magnetized iron won’t spontaneously change the direction of its magnetization. We
will ignore the possibility of nonergodic behavior in our discussion, but misfolded proteins, such
as the “prions” thought to be responsible for neurological diseases like scrapie, may provide exam-
ples. In addition, single enzyme molecules have been found to enter into long-lived substates with
significantly different catalytic activity from those of identical neighboring molecules. Even though
the enzymes are constantly bombarded by the thermal motion of the surrounding water molecules,
such bombardment seems unable to shake them out of these states of different “personality,” or to
do so extremely slowly.
6.2.2′
- The Sakur–Tetrode formula (Equation 6.6) is derived in a more careful way in Callen, 1985,
§16–10. - Equation 6.6 has another key feature:Sisextensive.This means that the entropy doubles when
weconsider a box with twice as many molecules, twice the volume, and twice the total energy.
Your Turn 6j
a. Verify this claim (as usual supposeNis large). [Remark: You’ll notice that the final factor
ofN!iscrucial to get the desired result; before people knew about this factor they were puzzled
bythe apparent failure of the entropy to be extensive.]
b. Also show that the entropydensityof an ideal gas isS/V=−ckB[ln(c/c∗)]. (6.36)Herec=N/Vas usual, andc∗is a constant depending only on the energy per molecule, not the
volume.- Those who question authority can find the area of a higher-dimensional sphere as follows. First
let’s deliver on another deferred promise (from the Example on page 68) to compute
∫+∞
−∞dxe
−x^2.We’ll call this unknown numberY. The trick is to evaluate the expression
∫
dx 1 dx 2 e−(x^1(^2) +x 22 )
in two different ways. On one hand, it’s just
∫
dx 1 e−x^12 ·∫
dx 2 e−x^22 ,orY^2 .Onthe other hand,
since the integrand depends only on the length ofx,this integral is very easy to do in polar
coordinates. We simply replace
∫
dx 1 dx 2 by∫
rdrdθ.Wecan do theθintegral right away, since
nothing depends onθ,toget
∫
rdr· 2 π.Comparing our two expressions for the same thing then
givesY^2 =2π
∫
rdre−r
2
.But this new integral is easy. Simply change variables toz=r^2 to show
that it equals 1/2, and soY=
√
πas claimed in Chapter 3.
Tosee what that’s got to do with spheres, notice that the factor of 2πarising above is the
circumference of a unit circle, or the “surface area of a sphere in 2-dimensional space.” In the same