Statistical Mechanics - Change in Entropy

[NewtonApple]

Homework Statement

A system of N distinguishable particles is arranged such that each particle can exist in one of the two states: one has energy [itex]\epsilon_{1}[/itex], the other has energy [itex]\epsilon_{2}[/itex]. The populations of these states are [itex]n_{1}[/itex] and [itex]n_{2}[/itex] respectively, ([itex]N = n_{1}+n_{2}[/itex]). The system is placed in contact with a heat bath at temperature T. A simple quantum process occurs in which the populations change: [itex]n_{2}\rightarrow n_{2} - 1[/itex] and [itex]n_{1}\rightarrow n_{1} + 1[/itex] with the energy released going into the heat bath.

(a) Calculate the change in the entropy of the two level system.
(b) Calculate the change in the entropy of the heat bath.
(c) If the process is reversible, what is the ratio of [itex]n_{2}[/itex] to [itex]n_{1}[/itex]?

Homework Equations

Boltzmann's Hypothesis - Entropy (S) is [itex]S=k_{B}ln(W)[/itex]

Stirling's approximation for large factorials [itex] ln N! = N\,ln\,N - N [/itex]

The Attempt at a Solution

The number of ways for initial state

[itex]W_{i}=\frac{N!}{n_{1}!\, n_{2}!}[/itex]​

The number of ways for final state

[itex]W_{f}=\frac{N!}{({n_{2}-1)!\, (n_{1}+1)!}}[/itex]​

Using [itex]S=k_{B}ln(W)[/itex] Change in Entropy is

[itex]\Delta S_{2LS} = S_{f}-S_{i} = k_{B}\Big [ln W_{f} - ln W_{i}\Big ] = k_{B}\Big [\frac {ln W_{f}}{ln W_{i}}\Big ][/itex]​

Substituting values from above and simplifying

[itex]\Delta S_{2LS} = S_{f}-S_{i} =k_{B} ln \Big[\frac{n_{1}!\, n_{2}!} {(n_{2}-1)!\, (n_{1}+1)!} \Big ][/itex]​

Using Stirling's approximation [itex] ln N! = N\,ln\,N - N [/itex]

[itex]\Delta S_{2LS}=k_{B} \Big[ \frac{ \big(n_{1}\, ln(n_{1})-n_{1}))(n_{2}\, ln(n_{2})-n_{2})\big)}{ \big( (n_{2}-1)ln(n_{2}-1)-(n_{2}-1)\big ) \big((n_{1}+1)ln(n_{1}+1)-(n_{1}+1)\big) } \Big] [/itex]​

and simplifying

[itex]\Delta S_{2LS}=k_{B} [n_{1} \, ln(n_1) + n_{2} \, ln(n_2) - (n_{2}-1) \, ln(n_{2}-1) - (n_{1}+1) \, ln(n_{1} + 1)] [/itex]​

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Suppose to get following Solution

[itex]\Delta S_{2LS}=k_{B}\, ln(n_{1}/n_{2}) [/itex]​

No idea how to get it used all simplification techniques. Please give me some hints.

 

[TSny]

Hello, NewtonApple and welcome to PF!

[itex]\Delta S_{2LS} = S_{f}-S_{i} =k_{B} ln \Big[\frac{n_{1}!\, n_{2}!} {(n_{2}-1)!\, (n_{1}+1)!} \Big ][/itex]

You should be able to greatly simplify this without resorting to Stirling.

 

[NewtonApple]

Thanks a bunch TSny!