6.3. Temperature[[Student version, January 17, 2003]] 181
0.2 0.4 0.6 0.8 1.00.4 0.5 0.65.10.15.20.25.30.35.40.ab
EA/EtotEA/EtotS/kB
P
(EStot )ASB SAN=10N=300N=7000Figure 6.2: (Mathematical functions.) The disorder (entropy) of the joint system is maximal when the two
subsystems share the total energy according to Equation 6.8. (a)Entropy of subsystem “A” (rising curve) and “B”
(descending curve), as a function ofEA/Etot. Each chamber hasN=1 0 molecules. A constant has been added
and the entropy is expressed in units ofkB;thusthe actual functions plotted are a constant plus ln(EA)^3 N/^2 and
ln(Etot−EA)^3 N/^2 respectively. The dashed line shows the sum of these curves (total system entropy plus a constant);
it’s maximal when the subsystems share the energy equally. (b)Probability distribution corresponding to the dashed
curve in (a) (low, wide curve), and similar distributions withN=300 and 7000 molecules on each side. Compare
the related behavior seen in Figure 4.3.
this idea, let the quantityTfor any system be defined by
T=
(
dS
dE)− 1
. fundamental definition of temperature (6.9)
Your Turn 6a
a. Verify that the dimensions work in this formula.
b. For the special case of an ideal gas, use the Sakur–Tetrode formula to verify that the temper-
ature really is (3kB/2) times the average kinetic energy, as required by Idea 3.21.
c. Show that quite generally the condition for maximum entropy in the situation sketched in
Figure 6.1 is
TA=TB. (6.10)Your Turn 6b
Show that if we duplicate a system (consider two disconnected, isolated boxes, each withN
molecules and each with total energyE), then the entropy doubles butT,defined by applying
Equation 6.9 to the combined system, stays the same. That is, find the change inStotwhen we
add a small amount dEof additional energy to the combined system. It may seem that one needs
to know how dEgot divided between the two boxes; show that on the contrary it doesn’t matter.
[Hint: Use a Taylor series expansion to expressS(E+dE)interms ofS(E)plus a correction.]Wesummarize what you just found by saying thatSis anextensivequantity, whileTisintensive.