Physical Chemistry Third Edition

3.4 Statistical Entropy 135

is defined as the product of all of the integers starting withNand 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 ofΩby

this factor. We should replace the expression in Eq. (3.4-3) by

Ωcoor

MN

N!

(3.4-5)

so that

ln(Ωcoor)Nln(M)−ln(N!) (3.4-6)

For very large values ofNthe use of the exact expression forN! is inconvenient.

For fairly large values ofNwe can applyStirling’s approximation:

N!≈(2πN)^1 /^2 NNe−N (3.4-7)

ln(N!)≈

1

2

ln(2πN)+Nln(N)−N (3.4-8)

For very large values ofNwe can neglect the first term on the right-hand side of this

equation:

ln(N!)≈Nln(N)−N (3.4-9)

With this approximation

ln(Ω)≈Nln(M)−Nln(N)+N (3.4-10)

EXAMPLE3.14

Find the value of ln(Ωcoord) in Example 3.13 using Eq. (3.4-8).

Solution

ln(Ωcoord) 3. 64 × 1025 −(6. 022 × 1023 )ln(6. 022 × 1023 )+ 6. 022 × 1023

 4. 03 × 1024

Ωcoorde^4.^03 ×^10

24

 101.^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.