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Chapter 3 gravitational attractions. There was a possibility that gravitational attractions might
The Second Law of Thermodynamics eventually overcome the expansion, thereby causing the universe to begin to contract,
ultimately bringing all matter together again. Perhaps a new Big Bang would then ini-
tiate a new cycle of expansion and contraction. An alternative possibility was that
there was not enough matter to prevent the expansion from continuing forever.
If the cyclic expansion–contraction cosmological model is correct, what will hap-
pen in the contraction phase of the universe? If the universe returns to a state essentially
the same as the initial state that preceded the Big Bang, then the entropy of the universe
would decrease during the contraction phase. This expectation is further supported by
the arguments for a direct connection between the thermodynamic and cosmological
arrows of time. But what would a universe with decreasing entropy be like? Would time
run backward in a contracting universe? What is the meaning of the statement that
“time runs backward”?
Astronomical observations made in 1998 and subsequent years have shown the
startling fact that the rate of expansion of the universe is increasing with time, rather
than slowing down as formerly believed. The accelerated expansion is driven by a
mysterious entity called dark energy, hypothesized to fill all of space. Observations
indicate that ordinary matter constitutes only about 4% of the mass–energy of the
universe. Another 22% is dark matter, whose nature is unknown (but might be as yet
undiscovered uncharged elementary particles). The existence of dark matter is in-
ferred from its observed gravitational effects. The remaining 74% of the universe is
dark energy, whose nature is unknown. The ultimate fate of the universe depends on
the nature of dark energy, and what is now known about it seems to indicate that the
expansion will likely continue forever, but this is not certain. For discussion of the
possibilities for the ultimate fate of the universe and how these possibilities depend
on the properties of dark energy, see R. Vaas, “Dark Energy and Life’s Ultimate
Future,” arxiv.org/abs/physics/0703183.

3.9 SUMMARY

We assumed the truth of the Kelvin–Planck statement of the second law of ther-
modynamics, which asserts the impossibility of the complete conversion of heat to
work in a cyclic process. From the second law, we proved that dq /T is the differ-
rev
ential of a state function, which we called the entropy S. The entropy change in a
2
process from state 1 to state 2 is S dq /T, where the integral must be eval-
1
rev
uated using a reversible path from 1 to 2. Methods for calculating S were dis-
cussed in Sec. 3.4.
We used the second law to prove that the entropy of an isolated system must
increase in an irreversible process. It follows that thermodynamic equilibrium in an
isolated system is reached when the system’s entropy is maximized. Since isolated
systems spontaneously change to more probable states, increasing entropy corre-
sponds to increasing probability p. We found that S k ln p a, where the Boltzmann
constant k is k R/N and a is a constant.
A
Important kinds of calculations dealt with in this chapter include:
• Calculation of S for a reversible process using dS dq /T.
rev
• Calculation of S for an irreversible process by finding a reversible path between
the initial and final states (Sec. 3.4, paragraphs 5, 7, and 9).
• Calculation of S for a reversible phase change using S H/T.
• Calculation of S for constant-pressure heating using dS dq /T (C /T) dT.
rev
P
• Calculation of S for a change of state of a perfect gas using Eq. (3.30).
• Calculation of S for mixing perfect gases at constant T and P using Eq. (3.33).