186 Chapter 6. Entropy, temperature, and free energy[[Student version, January 17, 2003]]
external energy supply (and a heat sink) after all.
- The formula for the entropy of an ideal gas, Equation 6.6, applies equally to a dilutesolutionof
Nmolecules of solute in some other solvent. Thus for instance Equation 6.12 gives theentropy of
mixingwhen equal volumes of pure water and dilute sugar solution mix. Chapter 7 will pick this
theme up again and apply it to osmotic flow. - The Statistical Postulate, claims that the entropy of anisolated, macroscopicsystem must not
decrease. Nothing of the sort can be said about individual molecules, which are neither isolated
(they exchange energy with their neighbors), nor macroscopic. Indeed, individual moleculescan
certainly fluctuate into special states. For example, we already know that any given air molecule
in the room will often have energy three times as great as its mean value, since the exponential
factor in the Maxwell velocity distribution (Equation 3.25 on page 78) is not very small when
E=3·^32 kBTr.
T 2 Section 6.4.2′on page 208 touches on the question of why entropy should increase.
6.5 Open systems
Point (3) in Section 6.4.2 will prove very important for us. At first it may seem like a discouraging
remark: If individual molecules don’t necessarily tend always toward greater disorder, and we want
to study individual molecules, then what was the point of formulating the Second Law? This section
will begin to answer this question by finding a form of the Second Law that is useful when dealing
with a small system, which we’ll call “a,” in thermal contact with a big one, called “B.” We’ll call
such a systemopento emphasize the distinction fromclosed,orisolated systems. For the moment
wecontinue to suppose that the small system is macroscopic. Section 6.6 will then generalize
our result to handle the case of microscopic, even single-molecule, subsystems. (Chapter 8 will
generalize still further, to consider systems free to exchange molecules, as well as energy, with each
other.)
6.5.1 The free energy of a subsystem reflects the competition between entropy and energy
Fixed-volume case Let’s return to our gas+piston system (see Figure 6.4b). We’ll refer to the
subsystem including the gas and the spring as “a.” As shown in the figure, “a” can undergo interal
motions, but its total volume as seen from outside does not change. In contrast to the Example
on page 184, this time we’ll assume that “a” is not thermally isolated, but rather rests on a huge
block of steel at temperatureT (Figure 6.4b). The block of steel (system “B”) is so big that
its temperature is practically unaffected by whatever happens in our small system: We say it’s a
thermal reservoir.The combined system, a + B, is still isolated from the rest of the world.
Thus, after we release the piston and let the system come back to equilibrium, the temperature
of the gas in the cylinder will not rise, as it did in the Example on page 184, but will instead stay
fixed atT,bythe Zeroth Law. Even though all the potential energy lost by the spring went to raise
the kinetic energy of the gas molecules temporarily, in the end this energy was lost to the reservoir.
ThusEkinremains fixed at^32 NkBT,whereas the total energyEa=Ekin+Espringgoesdownwhen
the spring expands.