Homework Question about Statistical Mechanics

[Elizabeth Chick]

Homework Statement

Consider the system of two large, identical Einstein solids, each with oscillators, in thermal contact with each other. Suppose the total energy of the system is 2 units of the energy quanta, i.e., =2ℏ, (i) how many MACRO-states (e.g., one macro-state corresponds to one where all the energy =2ℏ is taken by the first solid, and another corresponds to the case where half of the is taken by the first solid and another half by the second. The latter is the most likely state); (ii) Find a general (approximate) expression for the total number of MICRO-states Ω for the combined system, (iii) An expression for Ω=Ωmax when the system is in its most likely macro-state. (iv) Express the entropy of the combined system when (a) all of the micro-states are allowed, and (b) when the system is in its most likely macro-state, using the Sterling approximation: ln!=ln−.

Full Question as below
https://ibb.co/eZrrTx

Homework Equations

The Attempt at a Solution

 

[Chandra Prayaga]

Start by writing down all relevant quantities: One solid has N₁ oscillators, etc. Total energy must ne 2ħω, not just 2ħ (units!) where ω is the frequency of the oscillators. Ask yourself questions. If you distribute 2 quanta in N₁ oscillators, how many microstates is that? That should start you off.

 

[Elizabeth Chick]

Start by writing down all relevant quantities: One solid has N₁ oscillators, etc. Total energy must ne 2ħω, not just 2ħ (units!) where ω is the frequency of the oscillators. Ask yourself questions. If you distribute 2 quanta in N₁ oscillators, how many microstates is that? That should start you off.

Got it, thanks so much!

 

[Elizabeth Chick]

Start by writing down all relevant quantities: One solid has N₁ oscillators, etc. Total energy must ne 2ħω, not just 2ħ (units!) where ω is the frequency of the oscillators. Ask yourself questions. If you distribute 2 quanta in N₁ oscillators, how many microstates is that? That should start you off.

I am trying to work on question (ii), the general expression of approximation of one solid would be Ω = (q+N/q)^q⋅(q+N/N)^N, and the total Ω = Ω1⋅Ω2 now there are two solids in the system with a total energy of 2N units, what should be the q? If I substitute 2N into q, it means that each solid has 2N units of energy?

 

[Chandra Prayaga]

I am confused. What is q? And, are you distributing 2 quanta of energy, or 2N quana of energy?