Question

Consider a 4-particle system. Each particle may be in one of three levels called "A", "B", and "C". Answer the questions below, using the process of finding micro- and macrostates described in the January 11th lecture. Assume the particles are distinguishable from one another.(a)  How many microstates does this system have?(b)  For this system, define a macrostate as a set of microstates with the same number of particles in each named state, e.g., (AAAB) and (AABA) are in the same macrostate, but in a different macrostate from (BBBA). How many macrostates are there?(c)  Determine the multiplicity of each macrostate using the multiplicity equation from class and confirm that you have accounted for all of the microstates for the system. (d)  What is the probability of getting the particles in the levels AABB in any order? (e)  What is the most likely macrostate?

Answer 1

(a) Macrostate possible for three energy level and 4 distinguishable particles are as follow:

(4,0,0), (3,1,0), (2,2,0), (1, 2, 1)

Now we will calculate number of microstates associated with every macrostate by using the formula given as follow:

w = N! / n!

N = Total number of particles

n! = number of ways

For (4,0,0) w = 4!/4!0!0! = 1

For (3, 1, 0) w = 4!/3!1!0! = 4

For (2, 2, 0) w =4!/ 2!2!0! = 6

For (1, 2, 1) w= 4!/ 1!2!1! = 12

Therefore total number of microstate = 1+4+6+12 = 23

(b) There are total four macrostate

(c) We have already calculated multiplicity of each macrostate in part (a)

(d) probability = w / wtotal

For the macrostate (2,2,0) P = 6 / 23 = 0.2608

(e) For most likely macrostate we will calculate probability of each macrostate by the formula used in part (d)

For (4,0,0) P = 1/23 = 0.0434

For (3,1,0) P = 4/23 = 0.1739

For (2,2,0) P = 6 / 23 = 0.2608

For (1,2,1) P = 12/23 = 0.5217

Macrostate with highest probability is the most likely macrostate

Therefore microstate (1,2,1) is most probable.