# two-state system of N particles in thermal reservoir. Entropy change of system and reservoir when one particle changes state

take N non interacting particles of which $n_1$ are in state 1 with energy $E_1$, $n_2$ in state 2 with energy $E_2$. This system sits in a thermal reservoir at constant temperature T . One particle flips from state 2 to 1 .
- Assume both $n_1$ and $n_2$ >>1
- Assume system + reservoir together as an isolated system and the energy of the reservoir a lot bigger than the system
- this system doesn't have to be a gas. I was thinking of f.e. a line of N non interacting electrons with 2 spin states: up and down

what is the entropy change of the system?
what is the change of entropy of the reservoir?
use these to find a ratio of $\frac{n_1}{n_2}$

I found the entropy of the system using the ways to select $n_1$ particles of N and using $S=k_b \ln\Omega$ and then do the same for $(n_1-1)$ particles. I get $\Delta S = ln \left(\frac{n_1}{n_2}\right)$ for the system. Here the entropy change depends only on the initial number of particles in each state.

Another way:
The system loses energy $\Delta E = E_2 - E_1$ and $\Delta S = \frac{\Delta E}{T}$ by assuming there are no changes in the temperature T, the volume and total number of particles N so that $dE = TdS$
Here the entropy change seems to depend only on the energy difference between the two states.

I feel one of the two is wrong.

I assume the following for the entropy of the reservoir:
The system loses energy $\Delta E = E_2 - E_1$
The reservoir wins the same amount of energy because (system + reservoir) is an isolated system. The entropy change of the reservoir is thus $\Delta S = \frac{\Delta E}{T}$

I feel this can't be right because the same argument can be used for the system then and the total entropy change of system+ reservoir would be zero then. This can't lead to an ideal ratio of $n_1$ and $n_2$.

I also calculated $Z= {(e^{-\beta*E_1}+e^{-\beta*E_2})}^N$ as the partition function of the canonical ensemble but this gives me an (unchanging) entropy for the system and doesn't help me on calculating entropy changes.

Can i possibly use the grand canonical? I see 2 non interacting systems: the collection of all the particles in state 1 and the collection of all the particles in state 2. Each can interact with the reservoir, exchanging particles and energy. Then $\mathscr Z=\mathscr Z_1*\mathscr Z_2$...

Can anyone nudge me in the right direction?

• why is my question downvoted? – Kash Nirukhi Jan 20 '17 at 21:38
• Thank you Countto10 for your upvote and your advice on MathJax. – Kash Nirukhi Jan 20 '17 at 21:45
• No problem, there are users far more experienced than I am, so I can't guess. But if I had a euro/dollar....for every down vote I got.....well, I would have about 4 dollars :) but I am just lucky, seriously don't worry about it. – user140606 Jan 20 '17 at 21:52