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Macrostates, microstates and temperature of a system

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.


Solution Preview

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 ...

Solution Summary

The following posting helps with problems involving macrostates, microstates and the temperature of a system.