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\sect{Chemical equilibrium volume-polymer}


Consider a polymer composed with ${M}$ monomers. The polymer is in a
gas with temperature ${\beta}$ and chemical potential ${\mu}$. The
gas molecules can absorb the polymer's  monomers. The connection
energy of the gas molecule to the monomer is ${\varepsilon}$. The
natural length of a monomer is ${a}$, when a gas molecule is
absorbed to it, it's length is ${b}$.

\begin {itemize}
\item[(a)]

Calculate ${Z_{N}}$ for the polymer, and from that, calculate ${Z}$.

\item[(b)]
Calculate ${Z}$ by the factorization.

Guideline: in paragraph b' write the polymer's states in this form
${|n_{r}\left(r=1...M\right)>}$ when ${n_{r}=0,1}$. Accordingly, if
there is or there is no absorption. Write ${N_{\left(n_{r}\right)}
E_{\left(r\right)}}$, and show the sum you need to calculate for
${Z}$ is factorized.

\item[(c)]
Calculate the average length ${L}$ of the polymer.

Guideline: Express ${\hat{L}}$ through ${\hat{N}}$. Calculate
${N\equiv \langle \hat{N}\rangle}$ in two ways:

Way I - to derive from ${Z}$ (page

Way II - Express ${\hat{N}}$ through ${\hat{n}_{r}}$ and then use
the probability theory and the result for ${\langle
\hat{n}_{r}\rangle}$.

\end {itemize}




