= Solution
At fixed volume and temperature, equilibrium minimizes the <Helmholtz free energy> $F=U-TS$. Mixing increases translational <entropy>, whereas favourable contacts between <chromatin> segments can lower the interaction contribution to <internal energy>. At low enough temperature, favourable contacts can compensate the <entropy> lost on concentrating the arrays; at higher temperature, the mixing contribution can dominate. This gives the upper-critical dome assumed in (a).
A simple illustrative <Flory–Huggins solution theory> writes the dimensionless mixing free energy per site as
$$
g(\phi)=\frac{\phi}{N}\ln\phi+(1-\phi)\ln(1-\phi)+\chi(T)\phi(1-\phi),\qquad \chi(T)=A/T.
$$
Here $N$ is an effective polymer size, not automatically the number 12, and $A>0$ represents an effective energetic incompatibility of polymer and solvent. The <common-tangent construction> determines coexisting compositions. Solving $g''=g^{(3)}=0$ gives
$$
\boxed{\phi_c=\frac{1}{1+\sqrt N},\qquad \chi_c=\frac12(1+N^{-1/2})^2.}
$$
Since $\chi$ decreases on heating, the coexistence interval narrows and closes at $T_c=A/\chi_c$. For fixed pressure the corresponding potential is <Gibbs free energy>, with the enthalpic term in $G=H-TS$; enthalpy and <internal energy> should not be interchanged without stating the mechanical constraint. Hydration, ion release or temperature-dependent effective interactions can change this simple temperature trend.
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