= Solution
The rigid-link <freely jointed chain> stores extension entirely in orientation. It predicts $\langle z\rangle\to L=Nb$ and a divergent <force> as that limit is approached. Real chemical bonds and bond angles can deform, adding energetic compliance. This permits the measured chain to stretch farther at the same large <force> and removes the rigid-link vertical asymptote over the experimentally accessible range. Other limitations include segment correlations, solvent effects and eventual failure; a fit alone does not isolate their contributions.
Introduce one axial stretch modulus $S$ with units of <force>. A useful phenomenological <extensible freely jointed chain> law adds a linear strain contribution:
$$
\boxed{\langle z\rangle=L\left[\coth\left(\frac{fb}{k_BT}\right)-\frac{k_BT}{fb}\right]+\frac{L}{S}f}.
$$
Equivalently, if each aligned segment is a spring of stiffness $k_s$ in <force> per length, $S=bk_s$ and the additional extension is $Nf/k_s$. At high <force>, $\langle z\rangle\simeq L-Nk_BT/f+Lf/S$; the extra term supplies finite axial compliance. At small <force>, the compliances add: $d\langle z\rangle/df=Nb^2/(3k_BT)+L/S$.
This additive law is a simple requested one-term modification, rather than an exact angular average for harmonic links. Another common approximation is $L\mathcal L(fb/(k_BT))(1+f/S)$, which couples stretching to orientation. Neither should be extrapolated through bond rupture or claimed to predict a universal finite maximum for an extensible chain. A positive fitted $S$ could improve the high-force extension while leaving the entropic part largely intact.
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