Statistical thermodynamics 1: the concepts - W.H. Freeman

PC8eC16 1/26/06 14:34 Page 585
DISCUSSION QUESTIONS 585
6
1
Partition function, q
4
2
Entropy, SNk /
0.5
10
5
0
/kT
5 10
0 0.5 1
Internal energy, UN /
Internal energy, UN /
Entropy., SNk /
1
0.5
10
1
0.5
10
5
5
0
/kT
0
/kT
Fig. 16.20 The partition function, internal energy, and entropy of
a two-level system extended to negative temperatures but plotted
against β = 1/kT (modified here to the dimensionless quantity ε/kT).
5
5
10
10
Fig. 16.21 The variation of the entropy with internal energy for a
two-level system extended to negative temperatures.
to the shift towards population of the upper state alone as more
energy is packed into the system.
The phenomenological laws of thermodynamics survive largely
intact at negative temperatures. The First Law (in essence, the
conservation of energy) is robust, and independent of how
populations are distributed over states. The Second Law survives
because the definition of entropy survives (as we have seen above).
The efficiency of heat engines (Section 3.2), which is a direct
consequence of the Second Law, is still given by 1 − T cold /T hot .
However, if the temperature of the cold reservoir is negative, then
the efficiency of the engine may be greater than 1. This condition
corresponds to the amplification of signals achieved in lasers.
Alternatively, an efficiency greater than 1 implies that heat can be
converted completely into work provided the heat is withdrawn from
a reservoir at T < 0. If both reservoirs are at negative temperatures,
then the efficiency is less than 1, as in the thermal equilibrium case
treated in Chapter 3. The Third Law requires a slight amendment
on account of the discontinuity of the populations across T = 0: it is
impossible in a finite number of steps to cool any system down to
+0 or to heat any system above −0.
Discussion questions
16.1 Describe the physical significance of the partition function.
16.2 Explain how the internal energy and entropy of a system composed of
two levels vary with temperature.
16.3 Enumerate the ways by which the parameter β may be identified with
1/kT.
16.4 Distinguish between the zipper and Zimm–Bragg models of the
helix–coil transition.
16.5 Explain what is meant by an ensemble and why it is useful in statistical
thermodynamics.
16.6 Under what circumstances may identical particles be regarded as
distinguishable