3.4 Statistical Entropy
135
is defined as the product of all of the integers starting with N and ranging down to unity:
N! N(N − 1)(N − 2)(N − 3) … (2)(1) (definition)
(3.4-4)
If the particles are actually indistinguishable, we have overcounted the value of Ω by
this factor. We should replace the expression in Eq. (3.4-3) by
Ω coor
M N
N!
(3.4-5)
so that
ln(Ω coor ) N ln(M) − ln(N!)
(3.4-6)
For very large values of N the use of the exact expression for N! is inconvenient.
For fairly large values of N we can apply Stirling’s approximation:
N! ≈ (2πN)
1/2 N
N e
−N
(3.4-7)
ln(N!) ≈
1
2
ln(2πN) + N ln(N) − N
(3.4-8)
For very large values of N we can neglect the first term on the right-hand side of this
equation:
ln(N!) ≈ N ln(N) − N
(3.4-9)
With this approximation
ln(Ω) ≈ N ln(M) − N ln(N) + N
(3.4-10)
E X A M P L E 3.14
Find the value of ln(Ω coord ) in Example 3.13 using Eq. (3.4-8).
Solution
ln(Ω coord ) 3.64 × 10 25 − (6.022 × 10 23 ) ln(6.022 × 10 23 ) + 6.022 × 10 23
4.03 × 10 24
Ω coord e 4.03×10 24 10 1.75×10 24
Exercise 3.14
a. List the 36 possible states of two dice and give the probability for each sum of the two numbers
showing in the upper faces of the dice.
b. Determine how many possible states occur for four dice.
c. Determine how many possible states occur for two “indistinguishable” dice, which means
that there is no difference between a four on the first die and a five on the second die, or
between a five on the first die and a four on the second, etc. Explain why the correct answer
is not equal to 18.
135
is defined as the product of all of the integers starting with N and ranging down to unity:
N! N(N − 1)(N − 2)(N − 3) … (2)(1) (definition)
(3.4-4)
If the particles are actually indistinguishable, we have overcounted the value of Ω by
this factor. We should replace the expression in Eq. (3.4-3) by
Ω coor
M N
N!
(3.4-5)
so that
ln(Ω coor ) N ln(M) − ln(N!)
(3.4-6)
For very large values of N the use of the exact expression for N! is inconvenient.
For fairly large values of N we can apply Stirling’s approximation:
N! ≈ (2πN)
1/2 N
N e
−N
(3.4-7)
ln(N!) ≈
1
2
ln(2πN) + N ln(N) − N
(3.4-8)
For very large values of N we can neglect the first term on the right-hand side of this
equation:
ln(N!) ≈ N ln(N) − N
(3.4-9)
With this approximation
ln(Ω) ≈ N ln(M) − N ln(N) + N
(3.4-10)
E X A M P L E 3.14
Find the value of ln(Ω coord ) in Example 3.13 using Eq. (3.4-8).
Solution
ln(Ω coord ) 3.64 × 10 25 − (6.022 × 10 23 ) ln(6.022 × 10 23 ) + 6.022 × 10 23
4.03 × 10 24
Ω coord e 4.03×10 24 10 1.75×10 24
Exercise 3.14
a. List the 36 possible states of two dice and give the probability for each sum of the two numbers
showing in the upper faces of the dice.
b. Determine how many possible states occur for four dice.
c. Determine how many possible states occur for two “indistinguishable” dice, which means
that there is no difference between a four on the first die and a five on the second die, or
between a five on the first die and a four on the second, etc. Explain why the correct answer
is not equal to 18.