Eight Molecules In Box Example

This problem is equivalent to tossing 8 coins and looking at the number of heads and tails that come up.

The Microstate of the system is equal to the total number of ways the molecules could be arranged in the two halves of the container.

Ω_total = 2^N = 256 ways

The Macrostates are the number of ways that m molecules could be in the right half of the container. (Equivalently, the number of ways that m heads could tossed)

Arrangement

State

m

Ω_m

Prob.

S

S (bits)

0

1

.39%

k ln(1)=0

0

100 %

1

8

3.13%

k ln(8)

3.00

51.05%

2

28

10.94%

k ln(28)

4.81

21.66%

3

56

21.88%

k ln(56)

5.81

5.25%

4

70

27.34%

k ln(70)

6.13

0 %

5

56

21.88%

k ln(56)

5.81

5.25 %

6

28

10.94%

k ln(28)

4.81

21.66%

7

8

3.31%

k ln(8)

3.00

51.05%

8

1

.39%

k ln(1)=0

0

100 %

Arrangement	State m	Ω_m	Prob.	S	S (bits)
	0	1	.39%	k ln(1)=0	0	100 %
	1	8	3.13%	k ln(8)	3.00	51.05%
	2	28	10.94%	k ln(28)	4.81	21.66%
	3	56	21.88%	k ln(56)	5.81	5.25%
	4	70	27.34%	k ln(70)	6.13	0 %
	5	56	21.88%	k ln(56)	5.81	5.25 %
	6	28	10.94%	k ln(28)	4.81	21.66%
	7	8	3.31%	k ln(8)	3.00	51.05%
	8	1	.39%	k ln(1)=0	0	100 %