A system consists of two special dice. They are cubes, like regular dice, but in

Question

A system consists of two special dice. They are cubes, like regular dice, but instead of the numbers 1 through 6, they have the numbers 3 through 8. A macrostate is the sum of what is rolled on the die.

a. List the possible macrostates and the corresponding microstates of the system.   For example, the macrostate 8 can be achieved when you roll 4 and 4, 5 and 3, 3 and 5, for a total number of 3 microstates for this one macrostate.

b. Which macrostate is most probable?

c. Which macrostate has the most entropy? Which macrostates have the lowest entropy?

d. Calculate the entropy of the most disordered macrostate, and the most ordered.

Explanation / Answer

(a) This the table for all microstates

probablity=

no.of microstate/total microstates

(8,7) , (7,8)

2/34

(8,6), (6,8)

(7,7)

(8,5), (5,8)

(7,6), (6,7)

(8,4) (4,8)

(7,5) (5,7)

(6,6)

(8,3) (3,8)

(7,4), (4,7)

(6,5), (5,6)

(6,4), (4,6)

(5,5)

(6,3), (3,6)

(5,4), (4,5)

(5,3) (3,5)

(4,4)

7

(4,3) (3,4)

(b) From table, above we can see that MACROSTATE =11, for which probability is 6/34 = 0.1764 is the maximum, is the most probable.

(c)That macrostate has most entropy, which has the highest number of microstates. From table it is evident that MACROSTATE=11 for which total microstate=6 has the most entropy. And macrostates 8 & 6 have the lowest entropy(only 1 microstate for both).

(d)

Most disordered macrostate has maximum no. of microstates, hence macrostate=11 is the most disordered macrostate. Hence

Entropy = k ln(6) .# here 6 is no. of microstates.

Macrostates possible combination total microstates

probablity=

no.of microstate/total microstates

16 (8,8) 1 1/34 15

(8,7) , (7,8)

2

2/34

14

(8,6), (6,8)

(7,7)

3 3/34 13

(8,5), (5,8)

(7,6), (6,7)

4 4/34 12

(8,4) (4,8)

(7,5) (5,7)

(6,6)

5 5/34 11

(8,3) (3,8)

(7,4), (4,7)

(6,5), (5,6)

6 6/34 10

(6,4), (4,6)

(5,5)

3 3/34 9

(6,3), (3,6)

(5,4), (4,5)

4 4/34 8

(5,3) (3,5)

(4,4)

3 3/34

7

(4,3) (3,4)

2 2/34 6 (3,3) 1 1/34 sum = 34

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