208 Chapter 6. Entropy, temperature, and free energy[[Student version, January 17, 2003]]
not fundamental. Equation 6.9isfundamental, and we’ve seen that when we define temperature
this wayanytwobig systems (not just ideal gases) come to equilibrium at the same value ofT.
- In Your Turn 6b you showed that duplicating an isolated system doubles its entropy. Actually
the extensive property of the entropy is more general than this. Consider two weakly interacting
subsystems, for example our favorite system of two insulated boxes touching each other on small
uninsulated patches. The total energy of the state is dominated by the internal degrees of freedom
deep in each box, so the counting of states is practically the same as if the boxes were independent,
and the entropy is thus additive, givingStot ≈SA+SB.Evenifwedraw a purelyimaginary
wall down the middle of a big system, the two halves can be regarded as only weakly interacting.
That’s because the two halves exchange energy (and particles, and momentum... ) only across
their boundary, at least if all the forces among molecules are short-ranged. If each subsystem is big
enough, the surface-to-volume ratio gets small, again the total energy is dominated by the degrees of
freedom interior to each subsystem, and the subsystems are statistically nearly independent except
for the constraints of fixed total energy (and volume). Then the total entropy will again be the
sum of two independent terms. More generally still, in a macroscopic sample the entropy will be
an entropydensitytimes the total volume. In the extreme case of a noninteracting (ideal) gas, you
already showed in Your Turn 6j thatSis extensive in this sense.
More precisely, we define a macroscopic system as one that can be subdivided into a large
number of subsystems, each of which still contains many internal degrees of freedom and interacts
weakly with the others. The previous paragraph sketched an argument that the entropy of such a
system will be extensive. See Landau & Lifshitz, 1980,§§2, 7 for more on this important point.
6.4.2′One can askwhythe Universe started out in such a highly ordered state, that is, so far from
equilibrium. Unfortunately, it’s notoriously tricky to apply thermodynamics to the whole Universe.
Forone thing, the Universe can never come to equilibrium: Att→∞it either collapses or at least
forms black holes. But a black hole has negative specific heat, and so can never be in equilibrium
with matter!
Forour purposes, it’s enough to note that the Sun is a hot spot in the sky, and most other
directions in the sky are cold. This unevenness of temperature is a form of order. It’s what (most)
life on Earth ultimately feeds on.
6.6.1′
- Our derivation of Idea 6.23 implicitly assumed that only the probabilities of occupying the
various allowed states of “a” depend on temperature; the list of possible states themselves, and
their energies, was assumed to be temperature-independent. As mentioned in Section 1.5.3 on page
22, the states and their energy levels come (in principle) from quantum mechanics, and so are
outside the scope of this book. All we need to know is that there is some list of allowed states. - Skeptics may ask why we were allowed to drop the higher-order terms in Equation 6.22. The
justification goes back to a remark in Section 6.2.2′on page 206: The disorder of a macroscopic
system isextensive.Ifyou double the system size, the first expansion coefficientIBin Equation 6.22
doubles, the second one ddEIBB stays the same, and the next one^12 d
(^2) IB
dEB^2 drops to half. So each
successive term of Equation 6.22 is smaller by an extra factor of a small energyEadivided by a big
energyEB.
Doesn’t this throw out baby with the bathwater? Shouldn’t we truncate Equation 6.22 after
thefirstterm? No: when we exponentiate Equation 6.22 the first term is a constant, which got