6.3. Temperature[[Student version, January 17, 2003]] 179
ABFigure 6.1:(Schematic.) Two systems thermally insulated from the world, but only partially insulated from each
other. The hatching denotes thermal insulation, with one small break on the common wall. The boxes don’t exchange
particles, only energy. The two sides may contain different kinds of molecules.
Other factors like (m/ 2 π^2 ^2 )^3 N/^2 just add a constant to the entropy per molecule, and won’t affect
derivatives like dS/dE.(Later on however, when studying the chemical potential in Chapter 8, we
will need to look at these factors again.)
T 2 Section 6.2.2′on page 206 makes several more comments about the Sakur–Tetrode formula,
and derives the formula for the area of a higher-dimensional sphere.
6.3 Temperature
6.3.1 Heat flows to maximize disorder
Having constructed the rather abstract notion of entropy, it’s time to see some concrete conse-
quences of the Statistical Postulate. We begin with the humblest of everyday phenomena, the flow
of thermal energy from a hot object to a cool one.
Thus, instead of studying one isolated box of gas, we now imagineattachingtwosuch boxes,
called “A” and “B,” in such a way that they’re still isolated from the world but can slowly exchange
energy with each other (Figure 6.1). We could put two insulated boxes in contact and make a small
hole in the insulation between them, leaving a wall that transmits energy but does not let particles
cross. (You can imagine this wall as a drumhead, which can vibrate when a molecule hits it.) The
twosides containNAandNBmolecules, respectively, and the total energyEtotis fixed. Let’s
explore how the boxes share this energy.
Tospecify the total state of the combined system, choose any state of “A” and any state of
“B,” with energies obeyingEtot=EA+EB.The interaction between the systems is assumed to
beso weak that the presence of “B” doesn’t significantly affect the states of “A,” and vice versa.
NowEAcan go up as long asEBgoes down to compensate. SoEBisn’t free. After the boxes have
been in contact a long time, we then shut the thermal door between them, isolating them, and let
each come separately to equilibrium. According to Equation 6.5, the total entropy of the combined
system is then the sum of the two subsystems’ entropies:Stot(EA)=SA(EA)+SB(Etot−EA). We
can make this formula more explicit, since we have a formula for the entropy of an ideal gas, and
weknow that energy is conserved. The Example on page 178 gives
Stot(EA)=kB[
NA
( 3
2 lnEA+lnVA)
+NB
( 3
2 ln(Etot−EA)+lnVB)]
+const. (6.7)