Stat. Mech. Macrostates for four particles?

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In summary, the problem at hand involves finding the thermodynamic probabilities for macrostates of 4 particles distributed among two energy levels, with one level being twofold degenerate. The correct solution for part b is 5, 4, 3, 2, 1, and for part c it is 16, 32, 24, 48, 24. The formula used for part b is W = N! (Pi)j gj^(Nj) / Nj!, which is for Bose-Einstein statistics, and for part c, the same formula can be used. The incorrect values for W1 and W0 may be due to using the incorrect formula or a miscalculation. The problem
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Lagraaaange
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Homework Statement


a. Find macro states of 4 particles among two energy level one of which is twofold degenerate. SOLVED
b. Find the thermodynamic probability of each macrostate for indistinguishable particles. SOLVED
c. ------------------distinguishabled problems (need help)
d. the assembly for b and c.
Answers: a. 5. b. 5,4,3,2,1, c. 16,32,24,48,24. d. 84,15

Homework Equations


Concept: MBE statistics applies to classical distinguishable particles not obeying exclusion principle. FD applies to indistinguishable obeying exclusion principle. Bose Einstein applies to indistinguishable without exclusion.

Thermo. Prob. = W = (gj+Nj -1)! / (gj-1)!Nj! this is for BE stats which I used for b and obtained correcting answer
W = N! (Pi)j gj^(Nj) / Nj! (FD statistics I thought to use for c.)

where (Pi) is big pi standing for product, gj is degeneracy (2 here), Nj is particles in Energy state, N is total particle (4) FD statistics I thought to use for c.

The Attempt at a Solution


W = N! (Pi)j gj^(Nj) / Nj!
applying for macro state of 4 where one can have different arrangements of 4 particles in each level.
W4 = 16 CORRECT
W3 = 32 CORRECT
W2 = 24 CORRECT
W1 = 48 INCORRECT
W0 = 24 INCORRECT
Am I using the wrong formula? I know for d you can just sum up each macro state so I mainly need help with c please.
 
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SOLVED.
 

Related to Stat. Mech. Macrostates for four particles?

1. What is the definition of a macrostate in statistical mechanics?

A macrostate is a set of possible configurations or arrangements of a system's microscopic particles that have the same macroscopic properties, such as energy, volume, and pressure. It is described by a set of macroscopic variables that are constant and do not change over time.

2. How many macrostates are possible for a system of four particles in statistical mechanics?

For a system of four particles, the number of possible macrostates is given by the formula N!/(n1!n2!n3!...), where N is the total number of particles and n1, n2, n3... are the number of particles in each energy level. In this case, there are four particles in total and each can be in one of the energy levels, so the number of macrostates is 4!/(1!1!1!1!) = 24.

3. How does the number of microstates relate to the number of macrostates in a system?

The number of microstates is the total number of possible arrangements of a system's particles, while the number of macrostates is the number of possible combinations of these microstates that have the same macroscopic properties. The number of macrostates is much smaller than the number of microstates, as there are usually many different microstates that can result in the same macrostate.

4. What is the significance of entropy in statistical mechanics?

Entropy is a measure of the disorder or randomness in a system. In statistical mechanics, it is related to the number of microstates that correspond to a particular macrostate. The higher the entropy, the more microstates are available, and the more disordered the system is. Entropy is also related to the probability of a system being in a particular macrostate.

5. How do macrostates and microstates help in understanding the behavior of a system?

Macrostates and microstates provide a way to analyze and predict the behavior of a system at the microscopic level. By understanding the number of microstates and how they are distributed among macrostates, we can determine the most probable macrostate and make predictions about the system's behavior, such as its temperature, pressure, and energy distribution.

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