182 Chapter 6. Entropy, temperature, and free energy[[Student version, January 17, 2003]]
More precisely, the temperature of an isolated macroscopic system can be defined once it has
come to equilibrium; it is then a function of how much energy the system has, namely Equation 6.9.
When two isolated macroscopic systems are brought into thermal contact, heat will flow until a
new equilibrium is reached. In the new equilibrium there is no net flux of thermal energy, and each
subsystem has the same value ofT,atleast up to small fluctuations. (Theywon’thave the same
energy—a coin held up against the Eiffel Tower has a lot less thermal energy than the tower, but
the same temperature.) As mentioned at the end of Section 6.3.1, the fluctuations will be negligible
for macroscopic systems. Our result bears a striking resemblance to something we learned long
ago: We saw in Section 4.4.2 on page 115 how a difference inparticle densitiescan drive a flux of
particles,via Fick’s law (Equation 4.18).
Subdividing the freezing and boiling points of water into 100 steps, and agreeing to call freezing
“zero,” gives theCelsius scale.Using the same step size, but starting at absolute zero, gives the
Kelvin(or “absolute”) scale. The freezing point of water lies 273 degrees above absolute zero, which
wewrite as 273K.Wewill often evaluate our results at the illustrative valueTr= 295 K,which we
will call “room temperature.”
Temperature is a subtle, new idea, not directly derived from anything you learned in classical
mechanics. In fact a sufficiently simple system, like the Moon orbiting Earth, has no useful concept
ofT;its motion is predictable, so we don’t need a statistical description. In a complex system, in
contrast, the entropyS,and henceT,involveall possiblemicrostates that are allowed, based on
the veryincompleteinformation we have. Temperature is a simple, qualitatively new property of a
complex system not obviously contained in the microscopic laws of collisions. Such properties are
called “emergent” (see Section 3.3.3).
T 2 Section 6.3.2′on page 207 gives some more details about temperature and entropy.
6.4 The Second Law
6.4.1 Entropy increases spontaneously when a constraint is removed
Wehave seen how the Zeroth Law is rooted in the fact that when we remove an internal constraint
on a statistical physical system, the disorder, or entropy, of the final equilibrium will be larger than
that of the initial one. A system with extra order initially (less entropy; energy separated in such a
waythatTA=TB)willlosethat order (increase in entropy) until the temperatures match (entropy
is maximal).
Actually, even before people knew how big molecules were, before people were even quite sure
that molecules were real at all, they could still measure temperature and energy, and they had
already concluded that a system in thermal equilibrium had a fundamental propertySimplicitly
defined by Equation 6.9. It all sounds very mysterious when you present it from the historical point
of view; people were confused for a long time about the meaning of this quantity, until Ludwig
Boltzmann explained that it reflects thedisorderof a macroscopic system in equilibrium, when all
wehave is limited, aggregate, knowledge of the state.
In particular, by the mid-nineteenth century a lot of experience with steam engines had led
Clausius and Kelvin to a conclusion so remarkable that it came to be known as theSecond Lawof