3.4 Statistical Entropy 137
SSstkBln(Ω)+S 0 (general relation) (3.4-19)whereSis the thermodynamic entropy and whereS 0 is a constant that can be taken
equal to zero if that is convenient.EXAMPLE3.15
a.Find the coordinate contribution to the statistical entropy corresponding to the value of
ln(Ωcoord) in Example 3.13.
b.Find the coordinate contribution to the statistical entropy corresponding to the value of
ln(Ωcoord) corrected for indistinguishability in Example 3.14.Solution
a. ScoorkBln(Ωcoord)(
1. 3807 × 10 −^23 JK−^1)
(3. 64 × 1025 )503JK−^1b. ScoorkBln(Ωcoord)(1.^3807 ×^10 −^23 JK−^1 )(4.^03 ×^1024 )
55 .6JK−^1The value in part a is much larger than the total entropy of a monatomic gas near room
temperature. For exampleS◦m 146 .327JK−^1 mol−^1 at 298.15 K. The correction for indis-
tinguishability in part b is obviously needed.Exercise 3.15
Find the value ofΩfor a system if its entropy is equal to 210 J K−^1.The Interpretation of Entropy
Entropy has both macroscopic and molecular aspects, as we have seen. The thermo-
dynamic entropy is defined in terms of heat transferred in a reversible process. When
heat is transferred from a higher to a lower temperature, the entropy of the universe
increases. Heat at a lower temperature is less efficient in driving a Carnot engine, so
entropy has some connection with the efficiency with which heat can be turned into
work. In a later chapter we will discuss free energy, which is closely associated with
the thermodynamic entropy.
On the microscopic level, the statistical entropy is a measure of lack of information
about the mechanical state of a system. It is commonly said that entropy is a measure of
“randomness” with larger values corresponding to greater randomness. This statement
is an imprecise and possibly misleading way of stating the connection between entropy
and lack of information about the mechanical state. A room in a state of disarray is
sometimes said to have a higher entropy than a room that has been organized. This is
correct only if we have no information about the disorder in the room. If the disordered
room can be viewed so that it is in a known state of disorder, its statistical entropy is
no greater than that of the room if it is in a known state of order. However, disorder or