deletions | additions       diff --git a/Gibbs.tex b/Gibbs.tex  index d06416f..84ff4eb 100644  --- a/Gibbs.tex  +++ b/Gibbs.tex 
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Once two distinct gases, $A$ and $B$, mix, there is an increase in entropy and it will take some work to separate them again. We can also extract work from their mixing (maximum if quasistatic). Some simple questions to answer:  \begin{enumerate}  \item What is entropy change (and hence change?   \item Hence  what is the maximum work extractable from the mixing process?   \item How could one extract this work?   \end{enumerate}     Answers:   \begin{enumerate}   \item For {\it distinguishable} particles, mixing two volume-$\frac{V}{2}$ boxes of $\frac{N}{2}$ ideal-gas particles each incurs an entropy increase of $\Delta S = N\ln2$.   \item The maximum work extractable from {\it this} process is therefore $W_{\rm max} = NT\ln2$. (This is also the  minimum work needed required  to recover the  original state)? state.)  \item Not sure yet, but here's an idea. The total number of particles $N$ is fixed, but not necessarily the final volume. If we set $v^{\rm final} < V^{\rm init}_1 + V^{\rm init}_2$, then perhaps pressure is developed to do work?  \end{enumerate}  To do:   \begin{itemize}   \item Extension to non-ideal gas?   \item Extension to arbitrary initial particle numbers and volumes.   \item Check the case $v^{\rm final} < V^{\rm init}_1 + V^{\rm init}_2$: is this a viable way to extract work from the process? Is it maximal?   \end{itemize}  %%--------------------------------------------------------------------------------------  \subsection{Distinguishing bit-strings}