A hypothetical system contains 16 particles which can occupy nondegenerate equally space energy levels of spacing e. The system is set up so that it has a total energy of 18e, and it is the most probable macrostate, which is shown below:
(see the attached file for the chart)
(i) How many microstates are there in this macrostate? What is the entropy of the system in units of kg?
(ii) Based on the distributions of the particles, make a rought estimate of the temperature of the systems in units of e/Kb.
You can solve this problem as follows. For each of the 16 particles you introduce a variable that indicates the energy divided by epsilon of its quantum state it is in. Let's call this variable E_i for particle nr. i. Then the E_i are integers. The total energy of the system of 16 particles is 18 epsilon, therefore:
E_1 + E_2 + E_3 +....+ E_16 = E (1)
where E = 18
Any solution of equation (1) with E_i integers larger than or equal to zero defines a possible microstate of the system with energy E. Also two different solutions define different microstates (assuming that the particles are not identical) So, the number of microstates is the number of solutions of equation (1) for E = 18. To count the number of solutions, let's consider a different problem described by the same equation. Suppose during ...
The following posting helps with problems involving macrostates, microstates and the temperature of a system.
Einstein solids: macrostates and microstates
Consider a system of two Einstein solids, A and B, each containing 10 oscillators, sharing a total of 20 units of energy. Assume that the solids are weakly coupled, and that the total energy is fixed.
(a) How many different macrostates are available to this system?
(b) How many different microstates are available to this system?
(c) Assuming that this system is in thermal equilibrium, what is the probability of
finding all the energy in solid A?
(d) What is the probability of finding exactly half of the energy in solid A?
(e) Under what circumstances would this system exhibit irreversible behaviorView Full Posting Details