1. A sample consists of 4 molecules with a total energy E-58. Each molecule can
ID: 2304294 • Letter: 1
Question
1. A sample consists of 4 molecules with a total energy E-58. Each molecule can occupy any one of a set of evenly spaced energy levels: 0, E, 28, 38, 48, 58 a. Identify how many different ways the total energy E-5? can be distributed among these molecules: ie. identify all these different "distributions" by describing how many particles, Nj are in the jth energy level with energy je. (Hint: there should be a total of 6 distinct "distributions" - label them as Di, D2, ..D b. For each of the distributions, calculate the number of possible microstates W c. For the "distribution" that has 3 molecules each with an energy , draw diagrams to show all the possible ways (there are W of them) that distribution can be achieved.Explanation / Answer
a)
Energy levels
D1
D2
D3
D4
D5
D6
5?
O
4?
O
3?
O
O
2?
O
OO
O
?
O
OO
O
OOO
0
OOO
OO
OO
O
O
W
4
12
12
12
12
4
b)
General formula to calculate the number of microstates = N! / (n1! x n2!...)
Where N is the total number of particles. Here N = 4
n1, n2 ... are the number of particles in each state.
Total number of microstates for D1, W1 = 4!/(3! x 1!) = 4
For D2, W2 = 4!/(2! x 1! x 1!) = 4 x 3 = 12
For D3, W3 = 4!/(2! x 1! x 1!) = 4 x 3 = 12
For D4, W4 = 4!/(2! x 1! x 1!) = 4 x 3 = 12
For D5, W5 = 4!/(2! x 1! x 1!) = 4 x 3 = 12
For D6, W6 = 4!/(3! x 1!) = 4
c)
Consider that each molecules are a, b, c and d.
Energy levels
D6
(i)
D6
(ii)
D6
(iii)
D6
(iv)
5?
4?
3?
2?
d
c
b
a
?
abc
abd
acd
bcd
0
Energy levels
D1
D2
D3
D4
D5
D6
5?
O
4?
O
3?
O
O
2?
O
OO
O
?
O
OO
O
OOO
0
OOO
OO
OO
O
O
W
4
12
12
12
12
4
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