138 3 The Second and Third Laws of Thermodynamics: Entropyrandomness generally corresponds to greater lack of information about the mechanical
state of the system, and the statement has some validity. The fact that a liquid has
greater entropy than a solid corresponds not to the fact that a liquid has a more irregular
structure, but to the fact that we do not know which disordered state it is in.
The connection between the statistical entropy and lack of information about the
microstate can be illustrated by an example. Assume that a deck of 52 playing cards can
be shuffled randomly in some way so that the order of the cards is completely unknown.
(Note that a perfect shuffle, in which the deck is divided exactly in half and the cards
are shuffled one card at a time from each half, results in an order that is known if the
order was known prior to the shuffle. It is said that a sequence of a certain number of
perfect shuffles restores the original order of the cards.) Lay the cards out face down
on a table in four rows of 13 cards without looking at them. The statistical entropy of
the cards is nowSstkBln(52!)(1. 3807 × 10 −^23 JK−^1 )ln(8. 0658 × 1067 )
2. 1588 × 10 −^21 JK−^1Now turn each card over without changing its position. The cards are still in an arrange-
ment without any regular pattern, but we can now see which card is in each position.Ω
is now equal to 1 and the statistical entropy is 0, even though the cards are still in the
same disordered arrangement as when they were lying face down. It is not the lack of
an ordered arrangement (the “randomness”) that corresponds to an increase in entropy.
It is the lack of information.PROBLEMS
Section 3.4: Statistical Entropy
3.27 Assume that you have 1.00 mol of normal six-sided dice.
a.If nothing is known about the orientation of the dice,
what is the statistical entropy?
b.If it is somehow known that every die is oriented with
six spots on the top face, what is the statistical entropy?3.28 Assume that a card game uses a deck of 96 unique cards.
Calculate the change in statistical entropy of a deck of 96
cards if it is randomly shuffled. That is, it goes from a state
in which the order of the cards is known to a completely
unknown ordering.
3.29 Assume that a card game uses a deck of 96 cards,
consisting of 24 sets of cards such that each set of four
cards consists of four identical cards. Calculate the change
in statistical entropy of a deck of 96 cards if it is randomly
shuffled. That is, it goes from a state in which the order of
the cards is known to a completely unknown ordering.
3.30 Assume that you have 100 coins, each of which can lie on
a table with “heads” up or “tails” up.
a.How many different arrangements are there for the
coins?
b. If you can turn one coin over per second, how long
would it take you to go through all of the possible
conformations of the coins?
3.31Assume that an element exists that has two isotopes, and
that each isotope has an abundance of 50.00%. Using
the formula for the entropy of mixing for an ideal gas,
calculate∆Smixfor a sample of 1.000 mol of this element.
3.32 Tell whether the thermodynamic entropy of the system
increases or decreases in each of the following processes,
and tell why it behaves as it does. Do the same for the
statistical entropy in each process.
a.A sample of a gas is heated at constant volume.
b.A sample of gas is expanded at constant temperature.
c.A sample of liquid water is heated.
d.A sample of liquid water is frozen.
e.A sample of liquid water is vaporized.